Integral Equations and Operator Theory - Volume 64

Integr. equ. oper. theory 64 (2009), 1–20 c 2009 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-20, published o...

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Integr. equ. oper. theory 64 (2009), 1–20 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010001-20, published online April 24, 2009 DOI 10.1007/s00020-009-1675-0

Integral Equations and Operator Theory

Admissibility and Observability of Observation Operators for Semilinear Problems Mahmoud Baroun and Birgit Jacob Abstract. This paper deals with semilinear evolution equations with unbounded observation operators. Sufficient conditions are given guaranteeing that the output function of a semilinear system is in L2loc ([0, ∞); Y ). We prove that the Lebesgue extension of the observation operators are invariant under nonlinear globally Lipschitz continuous perturbations. Further, relations between the corresponding Λ-extensions are studied. We show that exact observability of linear autonomous system is conserved under small Lipschitz perturbations. The obtained results are illustrated by several examples. Mathematics Subject Classification (2000). Primary 47H20; Secondary 93C73, 93B07, 93C25. Keywords. Admissible observation operators, exact observability, Lebesgue extension, nonlinear semigroup, semilinear problems.

1. Introduction In this work we consider the following abstract semilinear evolution equation u0 (t) = Au(t) + F (u(t)),

u(0) = x,

t ≥ 0,

x ∈ X,

(1.1)

equipped with the output equation y(t) = Cu(t),

(1.2)

where A is assumed to be the infinitesimal generator of a C0 -semigroup (T (t))t≥0 in a Banach space X and F is a nonlinear continuous function on X. Further, it is assumed that C, the observation operator, is a linear bounded operator from D(A), the domain of A, to another Banach space Y . It is well-known that global Lipschitz continuity of the nonlinearity F implies that the problem (1.1) admits a unique mild solution given by the variation of

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parameters formula Z

t

T (t − σ)F (u(σ, x))dσ, t ≥ 0, x ∈ X.

u(t, x) = T (t)x + 0

We define a nonlinear semigroup (S(t))t≥0 associated to the solution of (1.1) by S(t)x = u(t, x). Hence the output function is formally given by y(t) = CS(t)x. The output function is only well-defined if C is bounded, i.e. if the operator C can be extended to a linear bounded operator from X to Y . However, in case of unbounded observation operators, even if x ∈ D(A), it might happen that u(t, x) is not in D(A), so that Cu(t, x) is not defined. We call the operator C admissible for the nonlinear semigroup (S(t))t≥0 if the output function y is well-defined as locally square integrable function with values in Y . The problem of admissibility has been studied by many authors, e.g., [3, 4, 8, 18], but in their works they are interested in linear systems only. In this article we extend the definition of admissibility of the observation operator C for semilinear systems and we develop conditions guaranteeing that the set of admissible observation operators for the semilinear problem coincides with the set of admissible observation operators for the linearized system. In applications, it is often required that the system is exactly observable, that is, the initial state x ∈ X can be recovered from the output function y by a bounded operator. This problem is well studied for linear systems, see e.g. [9, 10, 14, 17, 21]. In this paper, we generalize the concept of exact observability to semilinear problems and we develop conditions guaranteeing that the semilinear system is exactly observable if and only if the linearized system has this property. Our paper is organized as follows. In Section 2 we describe, as a preliminary, the solutions of (1.1) by a semigroup of nonlinear operators and we summarize some properties of this semigroup. In Section 3 we introduce the definition of admissible observation operators C for semilinear systems and we develop conditions on the nonlinearity guaranteeing that the set of admissible observation operators for the semilinear problem coincides with the set of admissible observation operators for the linearized system. Some examples are given to illustrate our theoretical results. In Section 4, we study the invariance of the Lebesgue extension under globally Lipschitz continuous perturbations of the generator. We give also some relations between the Λ-extensions of such observation operators with respect to the original generator and the perturbed generator. Finally, in Section 5, we study the concept of exact observability for semilinear systems and we prove that the exact observability is not changed under small Lipschitz perturbations. We conclude Section 5 by means of an example: a semilinear wave equation with Neumann boundary observation.

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2. Notations and preliminaries We begin by introducing some notions and preliminary tools. We denote by XA := (D(A), k · kA ) the Banach space D(A) endowed with the graph norm, i.e., kxkA := kxk + kAxk, for x ∈ D(A). The resolvent set of A is denoted by ρ(A) and its resolvent operator by R(λ, A) := (λ − A)−1 for λ ∈ ρ(A). Throughout this paper, we suppose that (H1) F : X → X is globally Lipschitz continuous, i.e, kF (x) − F (y)k ≤ Lkx − yk, for all x, y ∈ X, where L is a positive constant and F (0) = 0. Under the assumption (H1), equation (1.1) admits a unique mild solution u(·, x) given by the variation of parameters formula Z t T (t − σ)F (u(σ; x))dσ, t ≥ 0, (2.1) u(t; x) = T (t)x + 0

y(t) = Cu(t; x).

(2.2)

Let (S(t))t≥0 be the family of nonlinear operators defined in X by S(t)x = u(t; x), for t ≥ 0, x ∈ X.

(2.3)

The operators S(t) map X into itself and they satisfy the two properties below: (P1) S(0)x = x, S(t + s)x = S(t)S(s)x for s, t ≥ 0 and x ∈ X. (P2) For each x ∈ X, the X-valued function S(·)x is continuous over [0, +∞). The first property is obtained through the uniqueness of mild solutions, and the second property follows from the fact that the solution u(t; x) to (2.1) is continuous. By a nonlinear semigroup on X we mean a family (S(t))t≥0 of nonlinear operators on X with the above mentioned properties (P1) and (P2). If in particular a semigroup on X provides mild solutions of (1.1) in the sense of (2.3), we call it the nonlinear semigroup on X associated with the semilinear evolution equation (1.1) and we have Z t S(t)x = T (t)x + T (t − σ)F (S(σ)x)dσ, t ≥ 0, x ∈ X. (2.4) 0

Since (T (t))t≥0 is a C0 -semigroup, there exists the constants M ≥ 1, ω ∈ R, such that kT (t)k ≤ M eωt for all t ≥ 0. Moreover, we have the following property Proposition 2.1. For every x, y ∈ X and t ≥ 0, we have kS(t)xk ≤ M e(ω+M L)t kxk, (ω+M L)t

kS(t)x − S(t)yk ≤ M e

kx − yk.

(2.5) (2.6)

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Proof. Let x, y ∈ X. Since F is globally Lipschitz continuous, it follows that for t ≥ 0, Z t kS(t)x − S(t)yk ≤ kT (t)x − T (t)yk + kT (t − σ)[F (S(σ)x) − F (S(σ)y)]kdσ 0 Z t ≤ M eωt kx − yk + M Leω(t−σ) kS(σ)x − S(σ)ykdσ. 0

By Gronwall’s lemma, we obtain the assertion (2.6). Writing y = 0 in (2.6), we get the assertion (2.5). Corollary 2.2. If ω < −M L, then (T (t))t≥0 and (S(t))t≥0 are exponentially stable.

3. Admissibility We start this section with the definition of finite-time (resp. infinite-time) admissibility of output operators C for linear semigroups. Definition 3.1. Let C ∈ L(D(A), Y ). We say that C is a finite-time admissible observation operator for (T (t))t≥0 , if for every t0 > 0, there is some Kt0 > 0 such that Z t0 kCT (t)xk2Y dt ≤ Kt0 kxk2 , (3.1) 0

for any x ∈ D(A). Definition 3.2. Let C ∈ L(D(A), Y ). Then C is called an infinite-time admissible observation operator for (T (t))t≥0 , if there is some K > 0 such that Z ∞ kCT (t)xk2Y dt ≤ Kkxk2 , (3.2) 0

for any x ∈ D(A). Note that the admissibility of C guarantees that we can extend the mapping x 7→ CT (·)x to a bounded linear operator from X to L2 ([0, t0 ]; Y ) for every t0 > 0. Similarly, if C is an infinite-time admissible observation operator, we can extend this mapping to a bounded linear operator from X to L2 ([0, ∞); Y ). The reader is referred to see [8, 18, 19, 20] for more details on this concept of admissibility. Next, we introduce the concept of finite-time (resp. infinite-time) admissibility of output operators C for the nonlinear semigroup (S(t))t≥0 given by (2.4) as follows: Definition 3.3. Let C ∈ L(D(A), Y ) with S(t)D(A) ⊂ D(A) for every t ≥ 0. We say that C is a finite-time admissible observation operator for (S(t))t≥0 , if for every t0 > 0, there is some Kt0 > 0 such that Z t0 kCS(t)x − CS(t)yk2Y dt ≤ Kt0 kx − yk2 , (3.3) 0

for any x, y ∈ D(A).

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Definition 3.4. Let C ∈ L(D(A), Y ) with S(t)D(A) ⊂ D(A) for every t ≥ 0. Then C is called an infinite-time admissible observation operator for (S(t))t≥0 , if there is some K > 0 such that Z ∞ kCS(t)x − CS(t)yk2Y dt ≤ Kkx − yk2 , (3.4) 0

for any x, y ∈ D(A). Equation (3.3) (resp. (3.4)) implies that the mapping x 7→ CS(·)x has a continuous extension from X to L2 ([0, t0 ]; Y ) for every t0 > 0 (resp. L2 ([0, ∞); Y )). Remark 3.5. (i) It is immediately clear that for a linear semigroup equation (3.3) (resp. (3.4)) is equivalent to equation (3.1) (resp. (3.2)). (ii) It is not difficult to verify that C is a finite-time admissible observation operator for (T (t))t≥0 (resp. (S(t))t≥0 ) if (3.1) (resp. (3.3)) holds for one t0 > 0. (iii) If (T (t))t≥0 (resp. (S(t))t≥0 ) is exponentially stable, then the notion of finitetime admissibility and infinite-time admissibility are equivalent. The objective of this section is to find sufficient conditions guaranteeing that the output function y of the system (1.1) is in L2 ([0, t0 ]; Y ). To begin with, we introduce another Banach space that contains the range of F and has the following properties: Definition 3.6. (Desch, Schappacher [5, Definition 4]) Let A be the infinitesimal generator of a linear C0 -semigroup (T (t))t≥0 on X. A Banach space (Z, | · |Z ) is said to satisfy assumption (Z) with respect to A if and only if (Z1) Z is continuously embedded in X; (Z2) for all continuous functions ϕ : [0, ∞) → Z we have Z t T (t − s)ϕ(s)ds ∈ D(A) for all t > 0, 0

and there exists a continuous nondecreasing function γ : [0, ∞) → [0, ∞) such that γ(0) = 0 and

Z t

A T (t − s)ϕ(s)ds

≤ γ(t) sup |ϕ(s)|Z . 0

0≤s≤t

Important examples of Banach spaces that satisfy assumption (Z) with respect to A are provided by: (1) XA = (D(A), k · kA ) with k · kA the graph norm of A. (2) The Favard class of A, given by 1 Z = FA = x ∈ X| sup kT (t)x − xk < ∞ , 0
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This set is often also called the generalized domain of the semigroup (T (t))t≥0 . In practice, it can also be determined via the characterization

FA = x ∈ X| sup λA(λ − A)−1 x < ∞ . λ>0

Recall that in case of a reflexive space X we have FA = XA , see [6] for details. (3) If A generates an analytic semigroup we may take either Z = D((−A)α ), α ∈ (0, 1), the fractional power space of −A, or Z = DA (α, p), α ∈ (0, 1), p ∈ [1, ∞] the interpolation space. For more details and definitions of these spaces, we refer the reader to [6, 13, 15]. One main result concerning admissibility is Theorem 3.7. Let (Z, | · |Z ) satisfy assumption (Z) with respect to A and C ∈ L(D(A), Y ). We assume additionally that F maps X to Z and that F : X → Z is globally Lipschitz continuous. Then the following assertions are equivalent: (i) C is finite-time admissible for (T (t))t≥0 . (ii) C is finite-time admissible for (S(t))t≥0 . Proof. To begin with, we show that (i) implies (ii). Let x, y ∈ D(A) and t0 ≥ 0. We have, for 0 ≤ t ≤ t0 ,

Z t

T (t − s)[F (S(s)x − F (S(s)y)]ds

0

X

A

Z t

A = T (t − s)[F (S(s)x − F (S(s)y)]ds

0

Z t

T (t − s)[F (S(s)x − F (S(s)y)]ds +

0

Z

t

≤ γ(t) sup |F (S(s)x − F (S(s)y)|Z + M R 0≤s≤t

eω(t−s) |F (S(s)x − F (S(s)y)|Z ds

0

Z

t

≤ γ(t)L sup kS(s)x − S(s)yk + M LR 0≤s≤t

eω(t−s) kS(s)x − S(s)ykds

0 (ω+M L)t0

Z

2

t

≤ γ(t)LM max{1, e }kx − yk + M LR eω(t−s) e(ω+M L)s kx − ykds 0 ≤ γ(t)LM max{1, e(ω+M L)t0 } + M Re(ω+M L)t kx − yk. Since γ is nondecreasing and positive, we obtain

Z t

2

T (t − s)[F (S(s)x − F (S(s)y)]ds

0

XA

2

2

2

2(ω+M L)t0

≤ 2 γ(t0 ) L M max{1, e

} + M 2 R2 e2(ω+M L)t kx − yk2

≤ 2M 2 max{γ(t0 )2 L2 , R2 } max{1, e2(ω+M L)t0 }kx − yk2 .

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On the other hand, we have Z t0 kCS(t)x − CS(t)yk2Y dt 0 Z t0 ≤2 kCT (t)x − CT (t)yk2Y dt 0

Z

t0

+2

0

0 t0

Z ≤2

2

Z t

C T (t − s)[F (S(s)x − F (S(s)y)]ds

dt

kCT (t)x −

Y

CT (t)yk2Y

dt

0

Z

t0

kCk2L(XA ,Y )

+2 0

2

Z t

T (t − s)[F (S(s)x − F (S(s)y)]ds

0

dt

XA

≤ 2Kt0 kx − yk2 + 4M 2 kCk2L(XA ,Y ) max{γ(t0 )2 L2 , R2 } max{1, e2(ω+M L)t0 } t0 kx − yk2 . Defining Kt00 := 2Kt0 +4M 2 kCk2L(XA ,Y ) max{γ(t0 )2 L2 , R2 } max{1, e2(ω+M L)t0 }t0 , this implies that Z t0 kCS(t)x − CS(t)yk2Y dt ≤ Kt00 kx − yk2 . 0

Conversely, suppose that (ii) holds. Using the formula, Z t CT (t)x = CS(t)x − C T (t − s)F (S(s)x)ds, x ∈ D(A), 0

and by similar calculations as above, we have Z t0 kCT (t)x − CT (t)yk2Y dt ≤ Kt0 kx − yk2 x, y ∈ D(A).

(3.5)

0

Therefore C is finite-time admissible for (T (t))t≥0 by Remark 3.5 (i).

Theorem 3.8. Suppose that the assumptions of Theorem 3.7 are satisfied. If (T (t))t≥0 and (S(t))t≥0 are exponentially stable, then the following statements are equivalent: (i) C is infinite-time admissible for (T (t))t≥0 . (ii) C is infinite-time admissible for (S(t))t≥0 . Proof. The proof follows immediately from Remark 3.5 (iii).

Note that by Corollary 2.2 the exponential stability of both semigroups is for example satisfied if ω < −M L. Now, we consider the situation where A generates an analytic semigroup (T (t))t≥0 . For 0 < α < 1, we call a Banach space Xα with norm k · kα an intermediate space between D(A) and X, or a space of class Jα , if D(A) ,→ Xα ,→ X

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and there is a constant c > 0 such that kxkα ≤ ckxk1−α kxkα A , x ∈ D(A), where k·kA is the graph norm associated to A. Examples of Xα are D((−A)α ), α ∈ (0, 1), the domains of the fractional powers of −A, the real interpolation spaces DA (α, ∞), α ∈ (0, 1), defined as follows ( DA (α, ∞) := x ∈ X : [x]α = sup0
. For more details about interand the abstract H¨ older spaces DA (α) := D(A) mediate spaces, see [6, Chap. II, Section 5.b] and [13]. Another main result of this section is the following theorem. Theorem 3.9. Assume that A generates an analytic semigroup (T (t))t≥0 and that C ∈ L(Xα , Y ). Then the following assertions are equivalent: (i) C is finite-time admissible for (T (t))t≥0 . (ii) C is finite-time admissible for (S(t))t≥0 . Proof. We may assume without loss of generality that 0 ∈ ρ(A). Then the graph norm on D(A) is equivalent to the norm x 7→ kAxk, which will be used here. Let t0 ≥ 0. For x ∈ Xα and 0 ≤ t ≤ t0 , we have T (t)x ∈ D(A) ,→ Xα and Z t v(t) = T (t − s)F (S(s)x)ds 0

is bounded with values in Xα because kT (t − s)kL(X,Xα ) ≤ cM 1−α max{1, eω(1−α)t0 } (t − s)−α . We set sup kS(t)k ≤ M1 := M max{1, e(ω+M L)t0 }, 0≤t≤t0

and c1 := cM 1−α max{1, eω(1−α)t0 }. Then kv(t)kXα

Z t

T (t − s)F (S(s)x)ds =

0

Z

t

≤

kT (t − s)F (S(s)x)kXα ds 0

Xα

Z

t

≤

kT (t − s)kL(X,Xα ) kF (S(s)x)k ds Z t ≤ c1 L (t − s)−α kS(s)xkds. 0

0

Therefore

Z t

kv(t)kXα = T (t − s)F (S(s)x)ds

0

Xα

≤ c1 LM1

t1−α 0 kxk. 1−α

(3.6)

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Then S(t)x ∈ Xα . Moreover we have for every x, y ∈ Xα Z t0 kCS(t)x − CS(t)yk2Y dt 0 Z t0 ≤2 kCT (t)x − CT (t)yk2Y dt 0

2

Z t

T (t − s)[F (S(s)x) − F (S(s)y)]ds +2

dt

0 0 Xα Z t0 1−α 2 t c1 LM1 0 ≤ 2Kt0 kx − yk2 + 2kCk2L(Xα ,Y ) dtkx − yk2 1−α 0 " 2 # t1−α 2 0 ≤ 2Kt0 + kCkL(Xα ,Y ) c1 LM1 t0 kx − yk2 . 1−α Z

t0

kCk2L(Xα ,Y )

The converse can be obtain by the same procedure as above and in the same way as in the second part of the proof of Theorem 3.7. Theorem 3.10. Suppose that the assumptions of Theorem 3.9 are satisfied. If (T (t))t≥0 and (S(t))t≥0 are exponentially stable, then the following statements are equivalent: (i) C is infinite-time admissible for (T (t))t≥0 . (ii) C is infinite-time admissible for (S(t))t≥0 . Proof. The proof follows immediately from Remark 3.5 (iii).

We conclude this section by two examples to illustrate our theory. Example. Let Ω be a bounded domain with smooth boundary ∂Ω in R2 and let Γ be an open subset of ∂Ω. Consider the following nonlinear initial and boundary value problem  ¨ t) = −∆2 w(x, t) + w3 (x, t), t ≥ 0, x ∈ Ω,   w(x, w(x, t) = ∆w(x, t) = 0, t ≥ 0, x ∈ ∂Ω, (3.7)   w(x, 0) = w0 (x), w(x, ˙ 0) = w1 (x), x ∈ Ω, with the output function ∂ w(x, ˙ t) |Γ . (3.8) ∂ν We take H = L2 (Ω) and A : D(A) ⊂ H → H the linear unbounded operator defined by Aϕ = ∆2 ϕ, where D(A) = {φ ∈ H 4 (Ω) ∩ H01 (Ω)| ∆φ = 0 on ∂Ω}, y(t) =

1

D(A 2 ) = H 2 (Ω) ∩ H01 (Ω). Setting W := (w, w), ˙ the problem (3.7) can be rewritten as an abstract semilinear 1 equation in the Hilbert space X = D(A 2 ) × H of the form ˙ (t) = AW (t) + F (W (t)), W

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1 0 IX is defined on a domain D(A) = D(A) × D(A 2 ). Then −A 0 A is the generator of a C0 -group on X. The nonlinear mapping F (W ) = (0, w3 ) maps X into D(A). Next we show that for W = (w, w), ˙ V = (v, v) ˙ ∈ X we have where A :=

kF (W )kD(A) ≤ Kkwk2L∞ (Ω) kW kX kF (W ) − F (V )kD(A) ≤

K(kwk2L∞ (Ω)

+

(3.9)

kvk2L∞ (Ω) )kW

− V kX .

(3.10)

1

Indeed, since D(A 2 ) ⊂ H 2 (Ω) it follows from Sobolev’s Imbedding Theorem [1] that H 2 (Ω) ,→ L∞ (Ω) and that there is a constant K such that 1

kwkL∞ (Ω) ≤ KkwkH 2 (Ω) for w ∈ D(A 2 ).

(3.11) 1

Denoting by D any first order differential operator we have for every w ∈ D(A 2 ) |D2 w3 | ≤ K(6|w| |Dw| + 3|w|2 |D2 w|), and therefore kw3 kH 2 (Ω) ≤ 6K(kwkL∞ (Ω) kwkH 2 (Ω) + kwk2L∞ (Ω) kwkH 2 (Ω) ) kw3 kH 2 (Ω) ≤ 6KkwkL∞ (Ω) (kwkL∞ (Ω) + 1)kwkH 2 (Ω) . Then kF (W )kD(A)

0

=

w3

D(A)

= kw3 kH 2 (Ω) ≤ 6KkwkL∞ (Ω) (kwkL∞ (Ω) + 1)kwkH 2 (Ω)

w

≤ 6KkwkL∞ (Ω) (kwkL∞ (Ω) + 1)

w˙ . X

The inequality (3.10) is proved similarly using Leibniz’s formula for the derivatives of products and the estimate (3.11). It follows from the inequalities (3.9) and (3.10) that F : X → D(A) = Z, and it is Lipschitz continuous in D(A). Next, we define the output space Y = L2 (Γ) and we can rewrite (3.8) as y(t) = Cx(t), where C = (0 C0 ), C0 w =

1 ∂w |Γ ∀w ∈ D(A 2 ). ∂ν 1

In [11, p. 287], the author proved that C0 ∈ L(D(A 2 ), Y ) is an admissible observation operator for the linear problem, i.e. for all T ≥ 0 there exist a constant KT > 0 such that Z TZ ky(t)k2 dΓdt ≤ KT2 (kw0 k2H 2 (Ω) + kw1 k2L2 (Ω) ), 0

Γ

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1

for all (w0 , w1 ) ∈ D(A) × D(A 2 ). Moreover, one deduces from Theorem 3.7 that C ∈ L(D(A), Y ) is an admissible observation operator for the problem (3.7). Example. Let Ω be a bounded domain with smooth boundary ∂Ω in Rn . We consider the following nonlinear initial value problem  ˙ t) = ∆w(x, t) + sin(w(x, t)), x ∈ Ω, t ≥ 0,   w(x, w(x, t) = 0, x ∈ ∂Ω, t ≥ 0, (3.12)   w(x, 0) = w0 (x), x ∈ Ω, with the output function ∂w(x, t) |∂Ω . (3.13) ∂ν Let X = L2 (Ω), Y = L2 (∂Ω). Consider the operator A : D(A) → X, Aϕ = ∆ϕ, with D(A) = H 2 (Ω) ∩ H01 (Ω). Recall that A generates an analytic semigroup ε+ 43 3 (T (t))t≥0 . Consider Cϕ = ∂ϕ ) → Y, for ∂ν |∂Ω ∈ Y. Since C : Xε+ 4 := D((−A) every ε > 0, is bounded, see [12, Section 3.1], and by Theorem 2.6.13 of [15] we have that y(t) =

7

3

7

kC(−A)1−γ T (t)k2 = kC(−A)−ε− 4 (−A)−γ+ε+ 4 T (t)k ≤ ct2γ−2ε− 2 is integrable near 0 for every γ > 54 . This means that C ∈ L(X1−γ , Y ) is admissible for γ > 45 . Considering the function F : X −→ X, F (x) = sin(x), it is easy to see that F is globally Lipschitz. Now Theorem 3.9 guarantees that C is an admissible observation operator for the problem (3.12)-(3.13).

4. Invariance under perturbations In this section we show that the Lebesgue extension of C is invariant under Lipschitz perturbations and we give relations between the Λ-extension of admissible operators with respect to the semigroup (T (t))t≥0 and the nonlinear semigroup (S(t))t≥0 . Definition 4.1. Let X, Y be Banach spaces, (T (t))t≥0 a C0 -semigroup on X with generator A and C ∈ L(D(A), Y ). We define the operator CL : D(CL ) → Y, the Lebesgue extension of C with respect to (T (t))t≥0 by Z 1 τ CL x = lim C T (t)xdt, (4.1) τ ↓0 τ 0 where D(CL ) = {x ∈ X| the limit in (4.1) exists} . On the domain D(CL ) we define the norm

Z

1 kxkD(CL ) = kxk + sup

0<τ ≤1 τ

0

τ

T (t)xdt

.

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Then (D(CL ), k · kD(CL ) ) is a Banach space. We then have D(A) ⊂ D(CL ) ⊂ X with continuous injections, and CL ∈ L(D(CL ), Y ). For this definition and further properties we refer the reader to [18, Section 4]. In a similar manner we define the Lebesgue extension of C with respect to a nonlinear semigroup (S(t))t≥0 : Definition 4.2. Let X, Y be Banach spaces, (S(t))t≥0 a nonlinear semigroup on X given by (2.4) and C ∈ L(D(A), Y ). We define the operator CL0 : D(CL0 ) → Y, the Lebesgue extension of C with respect to (S(t))t≥0 by Z 1 τ S(t)xdt, (4.2) CL0 x = lim C τ ↓0 τ 0 where D(CL0 ) = {x ∈ X| the limit in (4.2) exists} . Theorem 4.3. Let (T (t))t≥0 be a C0 -semigroup with generator A, let (S(t))t≥0 be the nonlinear semigroup given by (2.4) and C ∈ L(D(A), Y ). Then the Lebesgue extension of C with respect to (S(t))t≥0 is the same as with respect to (T (t))t≥0 . Proof. Let x ∈ X, τ > 0, and let CL0 be the Lebesgue extension of C with respect to (S(t))t≥0 , given by Z Z Z 1 τ 1 τ t CL0 x = lim C T (t)xdt + C T (t − σ)F (S(σ)x)dσdt , (4.3) τ ↓0 τ 0 τ 0 0 if this limit exists. If we can prove that

Z τZ t

1

T (t − σ)F (S(σ)x)dσdt lim

= 0, τ ↓0 τ 0

0

(4.4)

A

then the second term on the right-hand side of (4.3) tends to 0. Therefore, the limit in (4.3) exists if and only if the limit in (4.1) exists, and the two limits are equal. Now, we have to show that limit (4.4) exists. By Fubini’s theorem we have Z τZ t Z τZ τ T (t − σ)F (S(σ)x)dσdt = T (t − σ)F (S(σ)x)dtdσ. 0

0

0

Rτ

R τ −σ

σ

The integral σ T (t − σ)F (S(σ)x)dt = 0 T (t)F (S(σ)x)dt belongs to D(A) and Rτ A σ T (t − σ)F (S(σ)x)dt = (T (τ − σ) − I)F (S(σ)x). It follows that Z τZ t T (t − σ)F (S(σ)x)dσdt ∈ D(A), 0

0

and Z

τ

Z

t

Z

τ

T (t − σ)F (S(σ)x)dσdt =

A 0

0

Z A

Z0 τ

τ −σ

T (t)F (S(σ)x)dtdσ 0

(T (τ − σ) − I)F (S(σ)x)dσ.

= 0

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Hence, 1 A τ

Z

τ

Z

t

T (t − σ)F (S(σ)x)dσdt = 0

0

1 τ

Z

τ

(T (τ − σ) − I)F (S(σ)x)dσ. 0

We decompose T (τ − σ)F (S(σ)x) − F (S(σ)x) = T (τ − σ) [F (S(σ)x) − F (x)] + (T (τ − σ) − I)F (x) − [F (S(σ)x) − F (x)] , and we denote M := max kT (t)k. t∈[0,1]

Fix x ∈ X and let ε > 0. Then there exists δε ∈ (0, 1], such that for t ∈ [0, δε ] ε ε ε kS(t)x − xk ≤ , k(T (t) − I)F (x)k ≤ and kF (S(t)x) − F (x)k ≤ . 3M L 3 3 Then for τ ∈ (0, δε ] and σ ∈ [0, τ ] we obtain kT (τ − σ)F (S(σ)x) − F (S(σ)x)k ≤ ε, which implies

Z τZ t

1

A T (t − σ)F (S(σ)x)dσdt

τ

≤ ε. 0 0 On the other hand, it is not difficult to verify that Z Z 1 τ t T (t − σ)F (S(σ)x)dσdt = 0. lim τ ↓0 τ 0 0 Consequently

Z τZ t

1

lim T (t − σ)F (S(σ)x)dσdt

= 0. τ ↓0 τ 0 0 A

Remark 4.4. This result coincides with Weiss’ result (see [18, Theorem 5.2]), if one considers F ∈ L(X). In [19, 20], Weiss introduced another extension of C, the Λ-extension. Definition 4.5. Let (T (t))t≥0 be a C0 -semigroup with generator A and let C ∈ L(D(A), Y ). We define the Λ-extension CΛ of C by D(CΛ ) := x ∈ X| lim CλR(λ, A)x exists , λ→+∞ (4.5) CΛ x := lim CλR(λ, A)x, x ∈ D(CΛ ). λ→+∞

D(CΛ ) endowed with the norm kxkD(CΛ ) = kxk + sup kCλR(λ, A)xkY , λ≥λ0

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for λ0 ∈ C such that [λ0 , +∞) ⊂ ρ(A), is a Banach space satisfying the continuous embedding D(A) ,→ D(CΛ ) ,→ X, and CΛ ∈ L(D(CΛ ), Y ). The following result is due to Weiss. The proof was given for Lebesgue extension CL of C, see [18, Theorem 4.5]. Since D(CΛ ) contains D(CL ) (see [20, Remark 5.7]), we obtain the following. Theorem 4.6. Let x ∈ X. Assume that C is an admissible observation operator for (T (t)t≥0 ). Then T (t)x ∈ D(CΛ ) for all t ≥ 0, and CΛ T (·)x ∈ L2 ([0, τ ], Y ) for all τ > 0. The following proposition is proved in [7, Proposition 3.3] Proposition 4.7. Let f ∈ L2loc (R+ , X). Suppose that C is an admissible observation Rt operator for (T (t))t≥0 . Then (T ∗ f )(t) := 0 T (t − s)f (s)ds ∈ D(CΛ ) for all t ≥ 0 and kCΛ (T ∗ f )kL2 ([0,τ ],Y ) ≤ c(τ )kf kL2 ([0,τ ],Y ) , for all τ > 0 with c(τ ) > 0 is independent of f. Moreover, limτ ↓0 c(τ ) = 0. Theorem 4.8. Let C be an admissible observation operator for (T (t)t≥0 ) and let (S(t))t≥0 be the nonlinear semigroup given by (2.4). Then S(t)x, S(t)y ∈ D(CΛ ) for all x, y ∈ X and kCΛ S(t)x − CΛ S(t)ykL2 ([0,τ ],Y ) ≤ Kτ kx − yk, for τ, Kτ > 0. Proof. Let x, y ∈ X. From Theorem 4.6 and Proposition 4.7, we deduce that S(t)x, S(t)y ∈ D(CΛ ) and kCΛ (T ∗ F (S(·)x)) − CΛ (T ∗ F (S(·)y))kL2 ([0,τ ],Y ) ≤ c(τ )kF (S(·)x) − F (S(·)y)kL2 ([0,τ ],X) , for τ > 0. On the other hand, we have kF (S(·)x) − F (S(·)y)k2L2 ([0,τ ],X) Z τ = kF (S(t)x) − F (S(y)x)k2 dt 0 Z τ ≤ L2 kS(t)x − S(t)yk2 dt 0 Z τ 2 2 ≤L M e2(ω+M L)t dtkx − yk2 =: η(τ )kx − yk2 . 0

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15

Using formula (2.4), we can write kCΛ S(t)x − CΛ S(t)ykL2 ([0,τ ],Y ) ≤ kCΛ T (t)x − CΛ T (t)ykL2 ([0,τ ],Y ) + kCΛ (T ∗ F (S(·)x)) − CΛ (T ∗ F (S(·)y))kL2 ([0,τ ],Y ) ≤ c0 (τ )kx − yk + c(τ )η(τ )kx − yk ≤ Kτ kx − yk.

5. Exact observability The object of this section is to prove that exact observability is not changed under small Lipschitz perturbations. We start by giving the definition of exact observability of linear system described by the equations u0 (t) = Au(t),

u(0) = x,

y(t) = Cu(t), t ≥ 0,

(5.1)

and of the semilinear system (1.1), respectively. Definition 5.1. Let C ∈ L(D(A), Y ) be an admissible observation operator for the linear C0 -semigroup (T (t))t≥0 and let τ > 0. Then, the system (5.1) is exactly observable if there is some K > 0 such that kCT (·)xkL2 ([0, ∞);Y ) ≥ Kkxk,

x ∈ D(A),

(5.2)

and (5.1) is τ -exactly observable if there is some Kτ > 0 such that kCT (·)xkL2 ([0, τ ];Y ) ≥ Kτ kxk,

x ∈ D(A).

(5.3)

Definition 5.2. Let C ∈ L(D(A), Y ) be an admissible observation operator for the nonlinear semigroup (S(t))t≥0 given by (2.4) and let τ > 0. Then, the system (1.1) is exactly observable if there is some K 0 > 0 such that kCS(·)x − CS(·)ykL2 ([0, ∞);Y ) ≥ K 0 kx − yk, and (1.1) is τ -exactly observable if there is some kCS(·)x − CS(·)ykL2 ([0, τ ];Y ) ≥

Kτ0 kx

Kτ0

x, y ∈ D(A),

(5.4)

> 0 such that

− yk,

x, y ∈ D(A).

(5.5)

Remark 5.3. (i) It is well-known that the notion of τ -exactly observable may depend on τ , see [14, section 5]. (ii) If (T (t))t≥0 (resp. (S(t))t≥0 ) is exponentially stable then for the system (5.1) (resp. (1.1)-(1.2)) there is an equivalence between exact observability and τ -exact observability for some τ > 0, see [17]. Throughout this section, we suppose that we have the following condition (H2): (i) For all τ > 0 and ϕ ∈ C([0, τ ]; X) Z τ T (τ − s)ϕ(s)ds ∈ D(A). (5.6) 0

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(ii) There exists τ0 > 0 and a constant α > 0 such that

Z τ0

A T (τ0 − s)ϕ(s)ds

≤ α sup kϕ(s)k,

(5.7)

s∈[0, τ0 ]

0

for all ϕ ∈ C([0, τ ]; X). Remark 5.4. (a) It is easy to see that (H2) holds if X satisfies assumption (Z). (b) If there exists τ > 0 and p ∈ [1, ∞) such that (5.6) holds for all ϕ ∈ Lp ([0, τ ]; X), then (H2) is satisfied. Indeed, by [5, Proposition 8], one can see that X satisfies the assumption (Z) and as a consequence of (a), we obtain our result. Here we give a useful exponential estimate of the inequality (5.7). Lemma 5.5. Let ϕ satisfy (H2), let τ0 > 0 and M ≥ 1, ω ∈ R such that kT (t)k ≤ M eωt for all t ≥ 0. Then, for all τ > 0, we have

Z τ

A

≤ N (ω, τ, τ0 )α sup kϕ(s)k, T (τ − s)ϕ(s)ds (5.8)

s∈[0, t]

0

where  e|ω|τ0   M eωτ , ω > 0,  ωτ0 − 1|    |e τ , ω = 0, N (ω, τ, τ0 ) := M 1 + (5.9) τ0     M   , ω < 0. ωτ |e 0 − 1| Rt Proof. We set (V A ϕ)(t) := A 0 T (t − s)ϕ(s)ds for all ϕ ∈ C([0, ∞); X) and let t ≥ τ ≥ 0. One first has to verify the following equality, (V A ϕ)(t) = T (t − τ )(V A ϕ)(τ ) + (V A ϕτ )(t − τ ),

(5.10)

where ϕτ := ϕ(· + τ ). Indeed, using integration by parts of (V A ϕ)(t) and (5.6) we obtain Z τ Z t A (V ϕ)(t) = A T (t − s)ϕ(s)ds + A T (t − s)ϕ(s)ds 0 τ Z τ Z t−τ = AT (t − τ ) T (τ − s)ϕ(s)ds + A T (t − τ − s)ϕ(τ + s)ds 0 0 Z τ Z t−τ = T (t − τ )A T (τ − s)ϕ(s)ds + A T (t − τ − s)ϕ(τ + s)ds 0

0

= T (t − τ )(V A ϕ)(τ ) + (V A ϕτ )(t − τ ). The remaining of the proof follows the proof of Boulite et al. [2, Proposition 4]. Now, we can state the main result of this section as follows. Theorem 5.6. Let L be the Lipschitz constant of F and τ > 0. Then we have:

Vol. 64 (2009)

Admissibility and Observability

(a) There exists a constant L0 > 0 If L < L0 and the system (5.1) is τ -exactly observable. (b) There exists a constant L1 > 0 If L < L1 and the system (1.1) is τ -exactly observable.

17

such that: is τ -exactly observable, then the system (1.1) such that: is τ -exactly observable, then the system (5.1)

Proof. (a) We assume that the system (5.1) is exactly observable on [0, τ ] for τ > 0. Let x, y ∈ D(A). We have CT (τ )x − CT (τ )y τ

Z = CS(τ )x − CS(τ )y − C

T (τ − σ)[F (S(σ)x) − F (S(σ)y)]dσ. 0

Using the hypotheses (H2), we obtain kCT (τ )x − CT (τ )yk2Y CS(τ )yk2Y

≤ 2kCS(τ )x −

Z

+ 2

C

0

τ

2

T (τ − σ)[F (S(σ)x) − F (S(σ)y)]dσ

Y

≤ 2kCS(τ )x − CS(τ )yk2Y

Z τ

2

+ 2kCk2L(D(A),Y ) T (τ − σ)[F (S(σ)x) − F (S(σ)y)]dσ

0

A

≤ 2kCS(τ )x − CS(τ )yk2Y

Z τ

2

T (τ − σ)[F (S(σ)x) − F (S(σ)y)]dσ + 2kCkL(D(A),Y ) A

0

2

Z τ

T (τ − σ)[F (S(σ)x) − F (S(σ)y)]dσ +

0

≤ 2kCS(τ )x − CS(τ )yk2Y + 2kCk2L(D(A),Y ) α N (ω, τ, τ0 ) sup kF (S(σ)x) − F (S(σ)y)k 0≤σ≤τ

+ M 2 Leωτ

Z

τ

eM Ls kx − ykds

2

0

≤ 2kCS(τ )x − CS(τ )yk2Y 2 + 4kCk2L(D(A),Y ) α N (ω, τ, τ0 )M L max{1, e(ω+M L)τ } kx − yk2 + 4kCk2L(D(A),Y ) M 4 L2 τ 2 e2(ω+M L)τ kx − yk2 ≤ 2kCS(τ )x − CS(τ )yk2Y + 8kCk2L(D(A),Y ) α N (ω, τ, τ0 )M 2 Lτ

2

max{1, e2(ω+M L)τ }kx − yk2 .

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Set M2 := kCk2L(D(A),Y ) M 4 τ 2 . Then Z τ kCS(r)x − CS(r)xk2Y dr 0 Z 1 τ kCT (r)x − CT (r)xk2Y dr ≥ 2 0 Z τ N 2 (ω, r, τ0 ) max{1, e2(ω+M L)r }kx − yk2 dr − 4α2 M2 L2 0 Z 1 τ kCT (r)x − CT (r)yk2Y dr ≥ 2 0 Z τ 2 2 2(ω+M L)τ N 2 (ω, r, τ0 )drkx − yk2 . − 4α M2 L max{1, e } 0

Consequently, Z τ 1 2 2 2 kCS(r)x − CS(r)ykY dr ≥ Kτ − 4η(L)α L kx − yk2 , 2 0 Rτ where η(L) := M2 max{1, e2(ω+M L)τ } 0 N 2 (ω, r, τ0 )dr. Set f (L) = 21 Kτ − 4η(L)α2 L2 . The function f is continuous from [0, +∞) to (−∞, 21 Kτ ] and strictly decreasing, hence it is bijective. Then there exists a unique L0 > 0 such that f (L0 ) = 0. The parenthesis above becomes positive for L < L0 , which implies that the system (1.1) is τ -exact observable. The proof of (b) is easy since we use the same procedure as above. Corollary 5.7. Let L be the Lipschitz constant of F . If the semigroups (T (t))t≥0 and (S(t))t≥0 are exponentially stable then we have: (a) There exists a constant L0 > 0 such that: If L < L0 and the system (5.1) is exactly observable, then the system (1.1) is exactly observable. (b) There exists a constant L1 > 0 such that: If L < L1 and the system (1.1) is exactly observable, then the system (5.1) is exactly observable. The statements of the Theorem 5.6 and the Corollary 5.7 still hold if we drop the assumption (H2) and instead it is just assumed that A generates an analytic semigroup (T (t))t≥0 , F : X → X is globally Lipschitz and C ∈ L(Xα , Y ). The proof is similar to the Theorem 5.6 using (3.6). Example. Let Ω = (0, π) × (0, π) and let Γ = ([0, π] × 0) ∪ (0 × [0, π]) be a subset of ∂Ω. We consider the following semilinear problem for the wave equation with Neumann boundary observation:  ¨ t) = ∆w(x, t) + λw3 (x, t), x ∈ Ω, t ≥ 0,   w(x, w(x, t) = 0, x ∈ ∂Ω, t ≥ 0, (5.11)   w(x, 0) = w0 (x), w(x, ˙ 0) = w1 (x), x ∈ Ω,

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19

with the output function ∂w(x, t) |Γ , (5.12) ∂ν where λ > 0. Let X = L2 (Ω), Y = L2 (Γ). We set x := (w, w). ˙ The system (5.11)1/2 (5.12) can be written in the form (1.1)-(1.2) in the Hilbert space H = D(A0 )×X, where 1 0 IX A := , D(A) = D(A0 ) × D(A02 ), −A0 0 y(t) =

A0 φ = −∆φ ∀φ ∈ D(A0 ), D(A0 ) = H 2 (Ω) ∩ H01 (Ω), ∂φ |Γ ∀φ ∈ D(A0 ). ∂ν It is known that the operator A generates a C0 -group on H and the nonlinear mapping F (x) = (0, λw3 ) is globally Lipschitz continuous from H to D(A) as in Example 3. From [11, p. 44], it follows that C ∈ L(D(A), Y ) is an admissible observation operator for the linearized problem (5.11)-(5.12) and in [16, Theorem 6.2] it is shown that the linearized system of (5.11)-(5.12) is exactly observable in some time τ . Now for a small constant λ, all assumptions of Theorem 5.6 are satisfied and hence the semilinear problem (5.11)-(5.12) is exactly observable in some time τ . 1

D(A02 ) = H01 (Ω), C = (C0 , 0), C0 φ =

Acknowledgement The authors would like to thank the reviewer for his/her comments that helped to improve the paper.

References [1] R. A. Adams, Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [2] S. Boulite, A. Idrissi, L. Maniar, Robustness of controllability under some unbounded perturbations. J. Math. Anal. Appl. 304 (2005), no. 1, 409–421. [3] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Theory, Lecture Notes in Control and Information Sciences, Vol. 8, Springer-Verlag, 1978. [4] R. F. Curtain and G. Weiss, Well posedness of triples of operators (in the sense of linear systems theory), International Series of Numerical Mathematics 91 Birkh¨ auserVerlag, (1989), 41–58. [5] W. Desch and W. Schappacher, Some generation results for pertubed semigroup, Semigroup Theory and Applications (Cl´emnet, Invernizzi, Mitidieri, and Vrabie, eds.) Lect. Notes Pure Appl. Math. 116, (1989), 125–152. [6] K. J. Engel, and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., Springer-Verlag, 1999.

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[7] S. Hadd, Unbounded perturbations of C0 -semigroups on Banach spaces and applications, Semigroup Forum 70 (2005), no. 3, 451–465. [8] B. Jacob, J.R. Partington, Admissibility of control and observation operators for semigroups: a survey, in: J.A. Ball, J.W. Helton, M. Klaus, L. Rodman (Eds.), Current Trends in Operator Theory and its Applications, Proceedings of IWOTA 2002, Operator Theory: Advances and Applications, vol. 149, Birkh¨ auser, Basel, 199–221. [9] B. Jacob, H. Zwart, Exact observability of diagonal systems with a one-dimensional output operator. Infinite-dimensional systems theory and operator theory (Perpignan, 2000). Int. J. Appl. Math. Comput. Sci. 11 (2001), no. 6, 1277–1283. [10] B. Jacob, H. Zwart, Exact observability of diagonal systems with a finite-dimensional output operator. Systems Control Lett. 43 (2001), no. 2, 101–109. [11] J.-L. Lions, Contrˆ olabilit´e exacte, perturbations et stabilisation de syst`emes distribu´es. Tome 1, Recherches en Math´ematiques Appliqu´ees, Vol. 8, Masson, Paris, 1988. [12] I. Lasiecka, R. Triggiani, Control theory for partial differential equations: continuous and approximation theories, Volume I, Cambridge University Press, Cambridge, 2000. [13] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh¨ auser, Basel, 1995. [14] J.R. Partington, S. Pott, Admissibility and exact observability of observation operators for semigroups. Irish Math. Soc. Bull. 55 (2005), 19–39. [15] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. [16] K. Ramdani, T. Takahashi, G. Tenenbaum, M. Tucsnak, A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator. J. Funct. Anal. 226 (2005), no. 1, 193–229. [17] D. L. Russell, G. Weiss, A general necessary condition for exact observability. SIAM J. Control Optimization 32 (1994), 1–23. [18] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math. 65 (1989) 17–43. [19] G. Weiss, Regular linear systems with feedback, Math. Control Signals Systems 7 (1994) 23–57. [20] G. Weiss, Transfer functions of regular linear systems. Part I: Characterization of regularity, Trans. Amer. Math. Soc. 342 (1994) 827–854. [21] G. Q. Xu, C. Liu, S. P. Yung, Necessary conditions for the exact observability of systems on Hilbert spaces. Systems Control Lett. 57 (2008), no. 3, 222–227. Mahmoud Baroun and Birgit Jacob Institute for Mathematics, University of Paderborn Warburger Str. 100, 33098 Paderborn, Germany e-mail: [email protected] [email protected] Submitted: August 6, 2008. Revised: March 5, 2009.

Integr. equ. oper. theory 64 (2009), 21–33 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010021-13, published online March 25, 2009 DOI 10.1007/s00020-009-1670-5

Integral Equations and Operator Theory

On the Structure of L1 of a Vector Measure via its Integration Operator J.M. Calabuig, J. Rodr´ıguez and E.A. S´anchez-P´erez Abstract. Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L1 (m). In this paper we study the cases of the integration operator being: (i) p-concave on Lp (m), or (ii) positive p-summing on L1 (m) (where 1 ≤ p < ∞). We prove that (i) is equivalent to saying that L1 (m) contains continuously the Lp space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L1 (m) is order isomorphic to the L1 space of a non-negative scalar measure. Mathematics Subject Classification (2000). 46E30, 46G10. Keywords. Banach function space, integration operator, p-concave operator, positive p-summing operator, vector measure.

1. Introduction Let E be an order continuous Banach function space (over a non-negative scalar measure) having weak order unit. It is known that there is a vector measure m such that E is order isomorphic to L1 (m), see [3, Theorem 8] or [12, Proposition 3.30]. In this case, we say that m represents E. This representation is not unique. However, the properties of the integration operator associated to some/every vector measure representing E determine some features of E. From this point of view, properties of the integration operator like compactness or weak compactness have already been studied (see [12, Section 3.3] and the references therein). J.M. Calabuig was supported by MEC and FEDER (MTM2005-08350-C03-03) and Generalitat Valenciana (GV/2007/191). J. Rodr´ıguez was supported by MEC and FEDER (MTM200508379) and Generalitat Valenciana (GVPRE/2008/312). E.A. S´ anchez-P´ erez was supported by MEC and FEDER (MTM2006-11690-C02-01).

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In this paper we analyze the continuous injection in E of usual Lebesgue spaces Lr (λ) (λ being a non-negative scalar measure and 1 ≤ r < ∞). Some results in this direction can be found in [12, Chapter 6]. Our arguments are closely related to the ones presented there. In our main result, Theorem 2.3, we characterize the vector-valued norm inequalities that the integration operator associated to certain vector measure m representing E must satisfy in order to have the following property: There is a (non-negative scalar) control measure λ for m such that Lr (m) ,→ Lr (λ) ,→ L1 (m) ' E. Moreover, in Theorem 2.3 we also prove that the property above is equivalent to saying that the integration operator on Lr (m) is r-concave. Here, the notation E1 ,→ E2 (where E1 and E2 are Banach function spaces over non-negative scalar measures defined on the same measurable space) means that ‘identity’ mapping is a well-defined one-to-one operator (i.e. linear continuous mapping) from E1 to E2 . The last part of the paper is devoted to the ‘extreme case’: when is E order isomorphic to L1 (λ) for some non-negative scalar measure λ? In Theorem 2.7 we show that the positive p-summability (1 ≤ p < ∞) of the integration operator associated to some/every vector measure representing E provides a complete answer to the previous question. Terminology and preliminaries. All unexplained terminology can be found in our standard references [6], [9] and [12]. All our vector spaces are real. Given a Banach space Y , the symbol Y 0 stands for the topological dual of Y and the duality is denoted by h·, ·i. We write BY to denote the closed unit ball of Y . If in addition Y is a Banach lattice, we write Y + and BY+ for the positive cone of Y and its intersection with BY , respectively. A relevant class of Banach lattices is that of Banach function spaces. Given a finite measure space (Ω, Σ, µ), a linear subspace E of L0 (µ) equipped with a complete norm k · kE is called a Banach function space over µ if the following conditions are satisfied: (i) if f ∈ L0 (µ) and g ∈ E are such that |f | ≤ |g| (for the µ-a.e. order), then f ∈ E and kf kE ≤ kgkE ; (ii) every simple function belongs to E; and (iii) the ‘identity’ defines a one-to-one operator from E to L1 (µ). Throughout this paper X is a Banach space, (Ω, Σ) is a measurable space and m : Σ → X is a (countably additive) vector measure. By a control measure for m we mean a non-negative scalar measure λ on (Ω, Σ) such that λ(A) = 0 if and only if kmk(A) = 0, where kmk denotes the semivariation of m. For each x0 ∈ X 0 we write hm, x0 i to denote the scalar measure defined by hm, x0 i(A) := hm(A), x0 i, for all A ∈ Σ. From now on we fix a Rybakov control measure for m, that is, a control measure of the form µ = |hm, x00 i| with x00 ∈ BX 0 , cf. [6, p. 268]. In this way, a property holds µ-a.e. if and only if it holds kmk-a.e. A Σ-measurable function f : Ω → R is m-integrable if f is integrable with 0 0 0 respect for R to hm, x i for every xR ∈ X and, R each A ∈ Σ, there exists a vector A f dm ∈ X such that h A f dm, x0 i = A f dhm, x0 i for all x0 ∈ X 0 . Given

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On the Structure of L1 of a Vector Measure

23

1 ≤ p < ∞, the space Lp (m) is the Banach function space over µ made up of all equivalence classes of functions f such that |f |p is m-integrable, endowed with the norm Z p1 kf kLp (m) := sup |f |p d|hm, x0 i| . x0 ∈BX 0

Ω

p

The space L (m) is p-convex, order continuous and has weak unit. Observe that kf kpLp (m) = k|f |p kL1 (m) for all f ∈ Lp (m). For the basic properties of this space, we refer the reader to [7] and [12, Chapter 3]. R 1 1 The operator Im : L1 (m) → X defined by Im (f ) := Ω f dm is called the integration operator associated to m. Since Lp (m) ,→ L1 (m) (cf. [12, p. 122]), we can also consider the operator on Lp (m) defined by Z p p Im : Lp (m) → X, Im (f ) := f dm. Ω 1

The fact that L (m) is order continuous ensures that its topological dual L1 (m)0 coincides with its K¨ othe dual L1 (m)× (cf. [11, Corollary 2.6.5]) and we identify each functional ϕ ∈ L1 (m)0 with the (unique) function u ∈ L1 (µ) such R that hf, ϕi = Ω f u dµ for all f ∈ L1 (m). AsR usual, we write u dµ to denote the real-valued measure on (Ω, Σ) given by A u dµ. A R R For simplicity, from now on we just write the symbol instead of Ω to denote any ‘integral’ over Ω. Remark 1.1. Let λ be a control measure for m. The following statements are equivalent: (1) L1 (m) ,→ L1 (λ). (2) λ = u dµ for some u ∈ L1 (m)0 , u ≥ 0. 1 Proof. (1)⇒(2). We can write λ = uRdµ for some R u ∈ L (µ), u ≥ 0. Since the linear 1 functional on L (m) given by f f dλ = f u dµ is continuous, it follows that u belongs to L1 (m)0 . (2)⇒(1). For each f ∈ L1 (m) we have f ∈ L1 (λ) and Z kf kL1 (λ) = |f |u dµ ≤ kf kL1 (m) kukL1 (m)0 ,

hence L1 (m) ,→ L1 (λ).

2. Results Let r ≥ 1 and p, q > 1 be real numbers such that 1/r = 1/p + 1/q. Then the product f g belongs to Lr (m) whenever f ∈ Lp (m) and g ∈ Lq (m), with kf gkLr (m) ≤ kf kLp (m) kgkLq (m) , cf. [12, (3.88)]. Therefore, we can consider the bilinear continuous mapping Lp (m) × Lq (m) → X defined by Z r (f, g) Im (f g) = f g dm.

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Calabuig, Rodr´ıguez and S´ anchez-P´erez

IEOT

In [8] this approach has been used to obtain factorization theorems for operators defined between Banach lattices satisfying adequate convexity/concavity properties. In our main result, Theorem 2.3 below, we discuss the r-concavity of the r integration operator Im : Lr (m) → X in terms of the bilinear mapping described r above. Moreover, we show that the r-concavity of Im is equivalent to the fact that r r 1 L (m) ,→ L (λ) ,→ L (m) for some control measure λ for m. We stress that the 1 1-concavity of Im is analyzed in [12] (see Section 3.4 and Chapter 6), where some relevant examples can also be found. To state our Theorem 2.3 we need the following: Definition 2.1. A set S ⊂ BL+1 (m)0 is positively norming for L1 (m) if kf kL1 (m) = sup h|f |, ϕi

for all f ∈ L1 (m).

ϕ∈S

Remark 2.2. Some examples of positively norming sets: • BL+1 (m)0 is positively norming for L1 (m). • If λ is a non-negative scalar measure on (Ω, Σ), then the singleton {χΩ } ⊂ BL+∞ (λ) is positively norming for L1 (λ). R • Given x0 ∈ BX 0 , define ϕx0 ∈ BL+1 (m)0 by ϕx0 (f ) := f d|hm, x0 i|. The RadonNikod´ ym derivative of |hm, x0 i| with respect to µ d|hm, x0 i| dhm, x0 i = dµ dµ is the function associated to ϕx0 via the identification L1 (m)0 ' L1 (m)× . Clearly, the set {ϕx0 : x0 ∈ BX 0 } is positively norming for L1 (m). Condition (3) in Theorem 2.3 involves some spaces of multiplication operators recently studied in [2]. Recall that if E1 and E2 are two Banach function spaces over non-negative scalar measures defined on (Ω, Σ), then an operator T : E1 → E2 is called a multiplication operator if there is (a unique) h ∈ E2 such that T (f ) = f h for all f ∈ E1 ; in this case we write T = Mh . The space M(E1 , E2 ) of all multiplication operators from E1 to E2 becomes a Banach space when endowed with the operator norm, cf. [10]. Theorem 2.3. Let r ≥ 1 and p, q > 1 be such that 1/r = 1/p+1/q. Let S ⊂ BL+1 (m)0 be a weak∗ compact convex set which is positively norming for L1 (m). The following statements are equivalent: (1) There is a constant K > 0 such that the inequality n Z n n

r r1

X

X X p1 q1

≤ K |fi |p |gi |q

fi gi dm

p q i=1

i=1 p

L (m)

i=1 q

L (m)

holds for every f1 , . . . , fn ∈ L (m) and g1 , . . . , gn ∈ L (m), n ∈ N. (2) There exist a constant K > 0 and u0 , v0 ∈ S such that the inequality

Z

Z p1 Z q1

p |f | u0 dµ |g|q v0 dµ

f g dm ≤ K

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On the Structure of L1 of a Vector Measure

25

holds for every f ∈ Lp (m) and g ∈ Lq (m). (3) There exist u0 , v0 ∈ S such that: – u0 dµ and v0 dµ are control measures for m. – Each f ∈ Lp (u0 dµ) induces a multiplication operator Mf ∈ M(Lq (v0 dµ), L1 (m)). – The mapping f Mf is a one-to-one operator from Lp (u0 dµ) to q 1 M(L (v0 dµ), L (m)). r/p r/q (4) There exist u0 , v0 ∈ S such that for h0 = u0 v0 ∈ BL+1 (m)0 we have Lr (m) ,→ Lr (h0 dµ) ,→ L1 (m). (5) There is a control measure ν for m such that Lr (m) ,→ Lr (ν) ,→ L1 (m). r (6) The integration operator Im : Lr (m) → X is r-concave, that is, there is a constant K > 0 such that the inequality n n Z

X

r r1 r1 X

|fi |r r ≤K

fi dm i=1

i=1

L (m)

holds for every f1 , . . . , fn ∈ Lr (m), n ∈ N. Proof. (1)⇒(2). Observe first that S × S is a convex compact subset of the linear space L1 (m)0 × L1 (m)0 endowed with the (locally convex) product topology T obtained from (L1 (m)0 , weak∗ ). We now divide the proof of the implication (1)⇒(2) in several steps. Step 1. Fix f1 , . . . , fn ∈ Lp (m) and g1 , . . . , gn ∈ Lq (m). Using (1), the fact that S is positively norming and Young’s inequality we obtain n Z

r X

fi gi dm i=1

≤ K r sup

n Z X

h∈S

Kr r sup ≤ p h∈S

i=1 n Z X i=1

n Z pr X rq |fi |p h dµ sup |gi |q h dµ h∈S

(2.1)

i=1

n Z Kr r X |fi | h dµ + sup |gi |q h dµ . q h∈S i=1 p

Define the function φ : S × S → R (depending on the fi ’s and gi ’s) by Z n Z n Z

r X X 1 1

r p φ(u, v) := |fi | u dµ + |gi |q v dµ

fi gi dm − K r p q i=1 i=1

(2.2)

for all (u, v) ∈ S × S. Clearly, φ is affine (hence convex) and T-continuous. Inequality (2.1) can be read as inf (u,v)∈S×S

φ(u, v) ≤ 0

26

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IEOT

and, since this infimum is attained (bear in mind that S × S is T-compact and φ is T-continuous), it follows that φ(uφ , vφ ) ≤ 0 for some (uφ , vφ ) ∈ S × S. Step 2. Let Φ be the family made up of all φ’s which can be constructed (as in Step 1 ) from different sets of functions in Lp (m) and Lq (m). We claim that Φ is a convex cone of RS×S . Indeed, take α1 , α2 ≥ 0 and, for each j ∈ {1, 2}, take nj ∈ N and choose functions f1,j , . . . , fnj ,j ∈ Lp (m) and g1,j , . . . , gnj ,j ∈ Lq (m) whose associated function belonging to Φ (via (2.2)) is denoted by φj . Let φ ∈ Φ be the function associated to the collection: 1/p

αj fi,j ∈ Lp (m),

1/q

αj gi,j ∈ Lq (m),

j ∈ {1, 2}, i ∈ {1, . . . , nj }.

A direct computation shows that α1 φ1 + α2 φ2 = φ. This proves the claim. Step 3. Ky Fan’s lemma (cf. [5, Lemma 9.10]) applied to the family Φ ensures the existence of u0 , v0 ∈ S such that φ(u0 , v0 ) ≤ 0 for all φ ∈ Φ. In particular, for each f1 ∈ Lp (m) and g1 ∈ Lq (m) we have Z

r

Z K r r Z Kr r

(2.3) |f1 |p u0 dµ + |g1 |q v0 dµ .

f1 g1 dm ≤ p q Take f ∈ Lp (m) and g ∈RLq (m). Suppose without loss of generality that a := R p ( |f | u0 dµ)1/p and b := ( |g|q v0 dµ)1/q are non-zero. Inequality (2.3) applied to f1 := (1/a)f ∈ Lp (m) and g1 := (1/b)g ∈ Lq (m) yields Z Z

r K r r Z 1 Kr r

p f g dm ≤ |f | u dµ + |g|q v0 dµ

0 r r p q a b pa qb Kr r Kr r + = Kr, = p q R R R hence k f g dmk ≤ K( |f |p u0 dµ)1/p ( |g|q v0 dµ)1/q . This completes the proof of the implication (1)⇒(2). (2)⇒(3). We first show that u0 dµ isRa control measure for m. To this end, take A ∈ Σ with (u0 dµ)(A) = 0, that is, χA u0 dµ = 0. Given any B ⊂ A with B ∈ Σ, condition (2) applied to f = χB and g = 1 implies that m(B) = 0, hence kmk(A) = 0. Similarly, v0 dµ is a control measure for m. Let Y1 ⊂ Lp (u0 dµ) and Y2 ⊂ Lq (v0 dµ) be the linear subspaces made up of all simple functions. Given f ∈ Y1 and g ∈ Y2 , their product f g is again a 1 simple function, R so it belongs to L (m). Its norm can be computed as kf gkL1 (m) = supz∈BL∞ (µ) k f gz dmk (cf. [12, Lemma 3.11]) and condition (2) yields

Z

kf gkL1 (m) = sup f gz dm ≤ K kf kLp (u0 dµ) kgkLq (v0 dµ) . z∈BL∞ (µ)

Thus we can define a bilinear continuous mapping P : Y1 × Y2 → L1 (m) by P(f, g) := f g. Since Y1 and Y2 are dense in Lp (u0 dµ) and Lq (v0 dµ), respectively, a standard argument ensures the existence of a bilinear continuous mapping P : Lp (u0 dµ) × Lq (v0 dµ) → L1 (m) extending P. We claim that f g ∈ L1 (m) and f g = P(f, g) whenever f ∈ Lp (u0 dµ) and g ∈ Lq (v0 dµ). Indeed, choose sequences (fn ) in Y1 and (gn ) in Y2 such that

Vol. 64 (2009)

On the Structure of L1 of a Vector Measure

27

kfn − f kLp (u0 dµ) → 0 and kgn − gkLq (v0 dµ) → 0. We can assume without loss of generality that fn → f u0 dµ-a.e. and gn → g v0 dµ-a.e. Then fn gn → f g µ-a.e. On the other hand, the continuity of P ensures that P(fn , gn ) = fn gn → P(f, g) in L1 (m), and so we have fn gn → P(f, g) in L1 (µ) as well (because µ = |hm, x00 i| for some x00 ∈ BX 0 ). It follows that f g ∈ L1 (m) and f g = P(f, g). Therefore, for each f ∈ Lp (u0 dµ) we can define a multiplication operator Mf : Lq (v0 dµ) → L1 (m),

Mf (g) := f g,

Mf is a one-to-one with norm kMf k ≤ kPk kf kLp (u0 dµ) . The natural mapping f operator from Lp (u0 dµ) to M(Lq (v0 dµ), L1 (m)), as required. r/p r/q (3)⇒(4). Set h0 := u0 v0 . Since 0 ≤ h0 ≤ pr u0 + rq v0 (by Young’s inequality) and pr u0 + rq v0 ∈ BL+1 (m)0 , we also have h0 ∈ BL+1 (m)0 . Moreover, since u0 dµ and v0 dµ are control measures for m, we can assume without loss of generality that u0 > 0 and v0 > 0 pointwise. Fix h ∈ Lr (h0 dµ). Set r r v pq u pq r r 0 0 ∈ Lp (u0 dµ) and g := |h| q ∈ Lq (v0 dµ). f := sign(h)|h| p u0 v0 According to (3), h = f g ∈ L1 (m) and khkL1 (m) ≤ kMf k kgkLq (v0 dµ) ≤ K kf kLp (u0 dµ) kgkLq (v0 dµ) for some constant K > 0 independent of h. But p1 Z Z q1 r |h| h0 dµ kf kLp (u0 dµ) kgkLq (v0 dµ) = |h|r h0 dµ = khkLr (h0 dµ) , hence khkL1 (m) ≤ KkhkLr (h0 dµ) . This shows that the ‘identity’ mapping from Lr (h0 dµ) to L1 (m) is a welldefined one-to-one operator. In particular, h0 dµ is a control measure for m and so L1 (m) ,→ L1 (h0 dµ), hence Lr (m) ,→ Lr (h0 dµ). (4)⇒(5). Just bear in mind that the condition Lr (m) ,→ Lr (h0 dµ) implies that h0 dµ is a control measure for m. (5)⇒(6). By Remark 1.1 we can write λ = h dµ for some h ∈ L1 (m)0 with h ≥ 0. We can assume further that h ∈ BL+1 (m)0 . Let K > 0 be a constant such that kf kL1 (m) ≤ Kkf kLr (λ) for all f ∈ Lr (λ). Given simple functions f1 , . . . , fn ∈ Lr (m), we have n Z n

r r1 X X r1

≤ kfi krL1 (m)

fi dm i=1

i=1

≤K

n X

kfi krLr (λ)

r1

i=1

=K

=K

n Z X

r1 |fi |r dλ

i=1

n Z X i=1

|fi |r

n

X r1 r1

h dµ ≤K |fi |r i=1

Lr (m)

.

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IEOT

Since simple functions are dense in Lr (m) (because this space is order continuous, r cf. [12, Proposition 3.28]), the r-concavity of Im can be deduced easily from the previous chain of inequalities. To this end, it suffices to bear in mind that the mapping from Lr (m) to L1 (m) given by f |f |r is continuous because, as in the case of scalar measures (cf. [13, Chapter 3, Exercise 24]), the inequality

r r−1

|f | − |g|r 1 r ≤ r kf kr−1 Lr (m) + kgkLr (m) kf − gkL (m) L (m) holds for all f, g ∈ Lr (m). (6)⇒(1). Given f1 , . . . , fn ∈ Lp (m) and g1 , . . . , gn ∈ Lq (m), each product r fi gi belongs to Lr (m) and the r-concavity of Im yields n n Z

X

r r1 r1 X

(2.4) ≤ K |fi gi |r r .

fi gi dm L (m)

i=1

i=1

By H¨ older’s inequality (for real numbers!) we have n n n X pr X rq X |fi gi |r ≤ |fi |p |gi |q , i=1

i=1

i=1

0

hence for each x ∈ BX 0 the inequality Z X Z X n n n pr X rq |fi gi |r d|hm, x0 i| ≤ |fi |p |gi |q d|hm, x0 i| i=1

i=1

i=1

holds and again H¨ older’s inequality (now for integrals!) applied to the right hand side of the previous inequality allows us to conclude that Z X n |fi gi |r d|hm, x0 i| i=1

≤

n Z X

n pr Z X rq |fi |p d|hm, x0 i| |gi |q d|hm, x0 i|

i=1

i=1

n

X p1 r

≤ |fi |p p

L (m)

i=1

n

X 1 r

q q |g |

q i

As x0 ∈ BX 0 is arbitrary, it follows that n n

X

X r1 p1

|fi gi |r ≤ |fi |p

r i=1

L (m)

i=1

.

L (m)

i=1

Lp (m)

n

X q1

|gi |q

i=1

Lq (m)

,

which combined with (2.4) yields the inequality in (1). The proof of the theorem is over. Remark 2.4. In the previous theorem, the equivalence (1)⇔(2) can also be obtained as a particular case of a result of Defant [4, Theorem 1]. The equivalence (4)⇔(6) can be found essentially in [12, Section 6.4], see in particular Lemma 6.39, Proposition 6.40 and Theorem 6.41.

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On the Structure of L1 of a Vector Measure

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Given 1 ≤ p < ∞, a Banach function space E is called p-concave (resp. pconvex) if the identity operator on E is p-concave (resp. p-convex), that is, there is a constant K > 0 such that the inequality n n

X X p1 p1

(resp. the reverse one) kzi kpE ≤K |zi |p i=1

i=1

E

holds for every z1 , . . . , zn ∈ E, n ∈ N. It is known that E is order isomorphic to the Lp space of a non-negative scalar measure whenever it is simultaneously p-concave and p-convex, cf. [9, p. 59]. As Lp (m) is always p-convex, the following result (cf. [12, Proposition 3.74]) can be seen as a specialized version of the previous statement. Corollary 2.5. The following statements are equivalent: (1) Lp (m) is p-concave for some 1 ≤ p < ∞. (2) L1 (m) is 1-concave. (3) Lp (m) is p-concave for every 1 ≤ p < ∞. 1 (4) The integration operator Im : L1 (m) → X is 1-concave. + (5) There is h0 ∈ BL1 (m)0 such that the ‘identity’ map from L1 (m) to L1 (h0 dµ) is an isomorphism. (6) There is a control measure λ for m such that L1 (m) is order isomorphic to L1 (λ). Proof. The equivalence (1)⇔(2)⇔(3) follows from a simple computation. (2)⇒(4) is straightforward. (4)⇒(5) follows from the implication (6)⇒(4) in Theorem 2.3 (taking there r = 1). For (5)⇒(6) just observe that h0 dµ is a control measure for m. Finally, the implication (6)⇒(2) is a consequence of the 1-concavity of L1 (λ) and the general fact that p-concavity is preserved by order isomorphisms (cf. [9, Proposition 1.d.9]). Following [1], we say that an operator T from a Banach function space E to X is positive p-summing (where 1 ≤ p < ∞) if there is a constant K > 0 such that the inequality n n X p1 X p1 kT zi kp ≤ K sup |hzi , z 0 i|p z 0 ∈BE 0

i=1

i=1

+

holds for every z1 , . . . , zn ∈ E . This property lies strictly between being absolutely p-summing and being p-concave, see [1]. For more information about this subject, we refer the reader to [5]. We will need the following folk characterization of positive p-summing operators. Remark 2.6. Let T be an operator from a Banach function space E to X and let 1 ≤ p < ∞. Then T is positive p-summing if and only if there is a constant K > 0 such that the inequality n n X p1 X p1 kT zi kp |h|zi |, z 0 i|p ≤ K sup i=1

z 0 ∈BE 0

i=1

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holds for every z1 , . . . , zn ∈ E. We arrive at the last result of the paper. Theorem 2.7. Let E be an order continuous Banach function space having weak order unit. The following statements are equivalent: (1) For every vector measure ν representing E and every 1 ≤ p < ∞, the integration operator Iν1 is positive p-summing. (2) There exist a vector measure ν representing E and 1 ≤ p < ∞ such that Iν1 is positive p-summing. (3) There exist a vector measure ν representing E and 1 ≤ p < ∞ such that Iν1 is absolutely p-summing. (4) E is order isomorphic to the L1 space of a non-negative scalar measure. Proof. (1)⇒(2) and (3)⇒(2) are obvious. For the implication (4)⇒(3), just bear in mind that the integration operator of the L1 space of a non-negative scalar measure has rank 1 and, therefore, it is absolutely p-summing for any 1 ≤ p < ∞. (2)⇒(4). The case p = 1 follows from Corollary 2.5 since Iν1 is p-concave. Assume now that p > 1 and let q > 1 such that 1/p + 1/q = 1. By Corollary 2.5, we only have to check that Lp (ν) is p-concave. To this end, fix f1 , . . . , fn ∈ Lp (ν). Take arbitrary g1 , . . . , gn ∈ BLq (ν) and denote by µ0 a fixed Rybakov control measure for ν. Since Iν1 is positive p-summing, Remark 2.6 ensures that n n Z

p X X

|h|fi gi |, hi|p

fi gi dν ≤ K p sup i=1

= Kp = Kp

h∈BL1 (ν)0 i=1 n X

sup

h|fi gi |, |h|ip

h∈BL1 (ν)0 i=1 n Z X

sup

(2.5)

|fi gi ||h| dµ0

p

h∈BL1 (ν)0 i=1

for some constant K > 0 which depends only on Iν1 . For each 1 ≤ i ≤ n and each older’s inequality implies h ∈ BL1 (ν)0 , H¨ Z Z p1 Z q1 p |fi gi ||h| dµ0 ≤ |fi | |h| dµ0 |gi |q |h| dµ0 Z p1 Z p1 p ≤ |fi | |h| dµ0 kgi kLq (ν) ≤ |fi |p |h| dµ0 , which combined with (2.5) yields n Z n Z

p X X

|fi |p |h| dµ0

fi gi dν ≤ K p sup h∈BL1 (ν)0 i=1

i=1

=K

p

sup h∈BL1 (ν)0

Z X n i=1

n

X p1 p

|fi |p |h| dµ0 ≤ K p |fi |p p i=1

L (ν)

.

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On the Structure of L1 of a Vector Measure

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R Since each kfi kLp (ν) can be computed as the supremum of k fi g dνk where g runs over BLq (ν) (cf. [12, (3.64)]), we conclude that n X

kfi kpLp (ν)

p1

n

X p1

≤ K |fi |p i=1

i=1

Lp (ν)

.

It follows that Lp (m) is p-concave, as required. (4)⇒(1). Let T : L1 (λ) → L1 (ν) be an order isomorphism, where λ is a nonnegative scalar measure. We can assume without loss of generality thatRkT −1 k = 1. Then the functional h ∈ BL+1 (ν)0 defined by the formula hf, hi := T −1 (f ) dλ satisfies Z Z kf kL1 (ν) −1 h|f |, hi = T (|f |) dλ = |T −1 (f )| dλ = kT −1 (f )kL1 (λ) ≥ kT k for all f ∈ L1 (ν). Given f1 , . . . , fn ∈ L1 (ν)+ , we have n n n Z

X X X

hfi , hi ≤ kT k kfi kL1 (ν) ≤ kT k

fi dν ≤ i=1

i=1

i=1

sup

n X

z 0 ∈BL1 (ν)0 i=1

|hfi , z 0 i|.

Therefore, the operator Iν1 is positive 1-summing. By [1, Proposition 2], Iν1 is also positive p-summing for all 1 ≤ p < ∞. The proof is over. Remark 2.8. For an order continuous Banach function space E having weak order unit, in general the statements of Theorem 2.7 are not equivalent to the following one: For every vector measure ν representing E and every 1 ≤ p < ∞, the integration operator Iν1 is absolutely p-summing. Indeed, observe that the E-valued measure A χA (the characteristic function of A) represents E and its corresponding integration operator is just the identity mapping on E, which is not absolutely p-summing (for any 1 ≤ p < ∞) whenever E is infinite-dimensional. We finish the paper with two questions: 1. We have shown that concavity type properties for the integration operator characterize the continuous injection of Lebesgue spaces in L1 (m). Is it possible to generalize these ideas to characterize the continuous injection of other classical spaces (Lorentz, Orlicz, etc.) in L1 (m)? 2. Lozanovskii lattice interpolation spaces obtained from Lebesgue spaces and L1 (m) are a well described class of Banach function spaces. Which are the vector-valued norm inequalities for the integration operator that characterize the continuous injection of such spaces in L1 (m)?

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References [1] O. Blasco, Positive p-summing operators on Lp -spaces, Proc. Amer. Math. Soc. 100 (1987), no. 2, 275–280. MR 884466 (88c:47033) [2] J.M. Calabuig, F. Galaz-Fontes, E. Jim´enez-Fern´ andez and E.A. S´ anchez-P´erez, Strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series, Math. Z. 257 (2007), no. 2, 381–402. MR 2324807 (2008f:46058) [3] G.P. Curbera, Operators into L1 of a vector measure and applications to Banach lattices, Math. Ann. 293 (1992), no. 2, 317–330. MR 1166123 (93b:46083) [4] A. Defant, Variants of the Maurey-Rosenthal theorem for quasi K¨ othe function spaces, Positivity 5 (2001), no. 2, 153–175. MR 1825653 (2002a:46031) [5] J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297 (96i:46001) [6] J. Diestel and J. J. Uhl, Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977, With a foreword by B. J. Pettis, Mathematical Surveys, No. 15. MR 0453964 (56 #12216) [7] A. Fern´ andez, F. Mayoral, F. Naranjo, C. S´ aez and E.A. S´ anchez-P´erez, Spaces of p-integrable functions with respect to a vector measure, Positivity 10 (2006), no. 1, 1–16. MR 2223581 (2006m:46053) [8] A. Fern´ andez, F. Mayoral, F. Naranjo, C. S´ aez and E.A. S´ anchez-P´erez, Spaces of integrable functions with respect to a vector measure and factorizations through Lp and Hilbert spaces, J. Math. Anal. Appl. 330 (2007), no. 2, 1249–1263. MR 2308439 (2008f:46034) [9] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II Function spaces, Results in Mathematics and Related Areas, vol. 97, Springer-Verlag, Berlin, 1979. MR 540367 (81c:46001) [10] L. Maligranda and L.E. Persson, Generalized duality of some Banach function spaces, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 3, 323–338. MR 1020026 (91b:46028) [11] P. Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MR 1128093 (93f:46025) [12] S. Okada, W.J. Ricker and E.A. S´ anchez-P´erez, Optimal domain and integral extension of operators. Acting in function spaces, Operator Theory: Advances and Applications, vol. 180, Birkh¨ auser Verlag, Basel, 2008. MR 2418751 [13] W. Rudin, Real and complex analysis, Third Edition, McGraw-Hill Book Co., New York, 1987. MR 924157 (88k:00002) J.M. Calabuig Instituto Universitario de Matem´ atica Pura y Aplicada (IUMPA-UPV) Universidad Polit´ecnica de Valencia Camino de Vera, s/n, 46022 Valencia Spain e-mail: [email protected]

Vol. 64 (2009)

On the Structure of L1 of a Vector Measure

J. Rodr´ıguez Departamento de Matem´ atica Aplicada Facultad de Inform´ atica Universidad de Murcia 30100 Espinardo (Murcia) Spain e-mail: [email protected] E.A. S´ anchez-P´erez Instituto Universitario de Matem´ atica Pura y Aplicada (IUMPA-UPV) Universidad Polit´ecnica de Valencia Camino de Vera, s/n, 46022 Valencia Spain e-mail: [email protected] Submitted: December 16, 2008.

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Integr. equ. oper. theory 64 (2009), 35–59 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010035-25, published online April 24, 2009 DOI 10.1007/s00020-009-1679-9

Integral Equations and Operator Theory

State Space Formulas for a Solution of the Suboptimal Nehari Problem on the Unit Disc Ruth F. Curtain and Mark R. Opmeer

Abstract. We give state space formulas for a (“central”) solution of the suboptimal Nehari problem for functions defined on the unit disc and taking values in the space of bounded operators in separable Hilbert spaces. Instead of assuming exponential stability, we assume a weaker stability concept (the combination of input-, output- and input-output stability), which allows us to solve the problem for general H-infinity functions. Mathematics Subject Classification (2000). Primary 47B35; Secondary 41A30, 47N70, 93B28. Keywords. Infinite-dimensional linear systems, J-spectral factorizations, Hankel operators, Lyapunov equations, Nehari problem.

1. Introduction The Nehari problem on the unit disc can be formulated as follows. Given (separable) Hilbert spaces U and Y and the coefficients Gn ∈ L(U, Y ) of the power series ∞ X

Gn z n ,

n=1

find coefficients Gn with n ≤ 0 such that the Laurent series ∞ X n=−∞

Gn z n ,

(1)

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defines a L∞ (T; L(U, Y )) function on the unit circle. It is well-known that the Hankel operator H : l2 (N; U ) → l2 (N; Y ) formed from the problem data as   G1 G 2 G3 . . .  G2 G 3     G3    .. . plays a crucial role: the problem is solvable if and only if H is a bounded operator and the norm of H equals the infimum of the L∞ norms of all the Laurent series solutions. In the scalar case this is due to Nehari [13] and in the Hilbert space case it is due to Page [17] (see also Nikolskii [14] and Peller [18] for treatments in book form and further references). The suboptimal Nehari problem is: for a given σ > kHk, parameterize all solutions with L∞ norm smaller than or equal to σ. This problem was solved in the scalar case by Adamjan, Arov and Krein [1, 2] and in the Hilbert space case by Kheifets [12] (see also Peller [18]). The suboptimal Nehari problem has many applications in control theory, e.g. the problem of designing robustly stabilizing controllers (see e.g. Curtain and Zwart [8]). For such applications the above mentioned abstract existence and parametrization results are not enough. More specific information about (at least) one of the solutions is needed in the form of so-called state space formulas, which we explain below. In the applications to control theory that we have in mind the power series (1) defines a function in the Hardy space H∞ (D; L(U, Y )) of the unit disc. This is often called input-output stability. Moreover, there is a realization in terms of operators A ∈ L(X), B ∈ L(U, X) and C ∈ L(X, Y ), where the Hilbert space X is called the state space, and Gn = CAn+1 B, n > 0. Whenever the Hankel operator is bounded, there is always a ‘trivial’ state space realization of (1) called the shift realization with X = l2 (N; Y ) and (Ax)n = xn+1 ,

Bu = (Gn u)n∈N ,

Cx = x1 .

In fact, there are infinitely many state space realizations. However, in control applications the state space parameters A, B and C are given and they have physical significance. The to be determined operators Gn with n ≤ 0 should also ˜ B ˜ and C˜ be given in state space form, since these state space parameters A, are needed for implementation of the controller. Consequently, in this paper we ˜ ∈ L(U, X) and seek expressions for the to be determined operators A˜ ∈ L(X), B −n−1 ˜ ˜ ˜ ˜ C ∈ L(X, Y ) in terms of A, B, C such that Gn = C A B for n < 0. This is what we mean by a state space solution to the suboptimal Nehari problem. Since the state space solutions given in Section 7 involve the expressions LB =

∞ X k=0

A∗k B ∗ BAk ,

LC =

∞ X k=0

Ak CC ∗ A∗k ,

(2)

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The Suboptimal Nehari Problem

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we require that both of these expressions be in L(X). This is satisfied if and only if the realization is input and output stable (see Section 2). A stronger sufficient condition is that A is exponentially stable, i.e., its spectral radius is strictly less than one. Our three basic assumptions of input stability, output stability and input-output stability are much weaker than the exponential stability assumption. It is interesting to note that our state space formulas applied to the specific case of the shift realization give an explicit formula for Gn with n ≤ 0 in terms of the Hankel operator H, the shift τ and the first element e1 of the standard basis of l2 (N): −n+1 XHe1 u , Gn u = −HH ∗ τ ∗ Tτ X −1 1

with X = (σ 2 − HH ∗ )−1 , the inverse of the square of the defect operator of H and Tτ = (σ 2 − τ HH ∗ τ ∗ )−1 , the inverse of the square of the defect operator of H ∗ τ ∗ . Since the Hankel operator is bounded, the condition LB , LC ∈ L(X) is automatically satisfied for the shift realization. So in the light of the known results on the Nehari problem the condition LB , LC ∈ L(X) is a natural one. Note also that the above formulas in terms of the Hankel operator make perfect sense even if the power series is not in the Hardy space H∞ . Although our proof breaks down in this case, we conjecture that the formulas remain valid. We now compare our results with existing results in the literature on state space formulas for solutions of the suboptimal Nehari problem. There are many results in the half-plane case under the assumption of exponential stability: e.g. Curtain and Zwart [8], Glover et al. [10], Ran [19], Curtain and Ran [7], Curtain and Zwart [3], Curtain and Ichikawa [4]. We generalized these results for systems with realizations that are input stable, output stable and input-output stable in [5], [6, Section 6]. As already mentioned, not every H∞ function has an exponentially stable realization, but it does always possess an input stable, output stable and input-output stable realization. The best existing result for the disc case is by Foias et al in [9, Section VI.8] where they assume exponential stability of the state space realization. Their approach uses commutant lifting results which is very different from our approach and unfortunately the formulas are given in a different form. (Their central solution does agree with ours). In contrast, we use the J-spectral factorization approach that we used in the half-plane case [5], [6, Section 6] adapted to the case of the disc. In obtaining the formulas for the J-spectral factorization we were aided by the exposition for the case of rational functions in Ionescu et al ˜ should read R). ˆ [11, Section 8] (in spite of the typo in (8.94) there: R Our paper is arranged as follows. We begin in Section 2 with some background on infinite-dimensional discrete-time state space systems and in Section 3 we introduce the special Riccati equations that lie behind the formulas for the solution to the suboptimal Nehari problem. In Section 4 we introduce and solve the

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associated J-spectral factorization problem. For notational simplicity in the proofs we assume that σ = 1. The inverse of the solution to the J-spectral factorization problem found in Section 4 is analyzed in Section 5. These results are then used in Section 6 to obtain an explicit state space solution to the suboptimal Nehari problem for the case σ = 1. Finally, in Section 7 the solution to the general case (where σ may not equal one) is derived. In the sequel we shall use the following notation G(z) =

∞ X n=1

Gn z n ,

Z(z) =

0 X n=−∞

Gn

1 , zn

K(z) =

0 X

Gn z n .

n=−∞

2. Transfer functions, characteristic functions and stability We recall some basic facts on infinite-dimensional discrete-time state space systems (a more comprehensive treatment can be found in Opmeer [16]). For the state space system Σ(A, B, C, D) defined by a bounded operator A B X X ∈L , U Y C D with U , X, Y separable Hilbert spaces, we define the transfer function G(z) = D +

∞ X

CAi Bz i+1 .

i=0

The system is called input-output stable if G is analytic and uniformly bounded on the unit disc, i.e. G ∈ H∞ (D, L(U, Y )). The Hankel operator of the system has the infinite matrix representation   CB CAB CA2 B . . .  CAB CA2 B CA3 B . . .     CA2 B CA3 B CA4 B . . .  .   .. .. .. .. . . . . Note that all of this is consistent with what was mentioned in the introduction (where D = 0). The characteristic function of the state space system is G(z) := Cz(I − −1 zA)−1 B + D, which for z 6= 0 may also be written as C z1 − A B + D. The characteristic function and the transfer function of a state space system are equal for |z| < 1/r(A), where r(A) is the spectral radius of the operator A, but they might have different values at other points (see [8, Example 4.3.8] or [21]). If one assumes exponential stability (also called power stability), i.e. r(A) < 1, then G = G on the closed unit disc and the difference is insignificant for the Nehari problem. In this article we do not assume exponential stability and therefore we do have to be careful about this difference. The main advantage of the characteristic

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The Suboptimal Nehari Problem

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function is that it lends itself better for algebraic computations, but it is the transfer function of the system that we are in fact interested in. At several points in this article we will need to find a state space realization of the inverse of a characteristic function given a realization of that function itself. The following simple lemma will be used for that purpose. Lemma 2.1. The characteristic function of the state space system Σ(A, B, C, D) is invertible if and only if D is invertible. In this case it is the characteristic function of the state space system Σ(A − BD−1 C, BD−1 , −D−1 C, D−1 ). Proof. By multiplying out we obtain # " # " −1 −1 1 1 −1 −1 −1 −1 I −A B + D × −D C I − A + BD C BD + D C z z −1 1 1 1 = I+C I −A × I − A + BD−1 C − I − A − BD−1 C z z z −1 1 I − A + BD−1 C BD−1 × z = I. That the product in the reverse order is also the identity follows similarly.

The following stability concept plays an important role in this article. Here H2 (D, Y ) is the usual Hardy space. Definition 2.2. A state space system is called output stable if, with ∞ X C(z) := CAi z i , i=0

for every x ∈ X we have C(·)x ∈ H2 (D, Y ). Note that the transfer function of an output stable system, being equal to D + zC(z)B, is analytic in the open unit disc. Output stability is equivalent to the observation Lyapunov equation A∗ LA − L + C ∗ C = 0

(3)

having a nonnegative, selfadjoint solution. The observability gramian LC is the smallest nonnegative, selfadjoint solution of the observation Lyapunov equation. An explicit formula for the observability gramian is given in (2). A state space system is input stable if the dual system Σ(A∗ , C ∗ , B ∗ , D∗ ) is output stable. The controllability gramian LB of the system is the observability gramian of this dual system. It is is the smallest nonnegative, selfadjoint solution of the control Lyapunov equation ALA∗ − L + BB ∗ = 0. The transfer function of an input stable system is analytic in the open unit disc, just as the transfer function of an output stable system is, as we saw above.

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The following key result was first established, in a different context, in Weiss and Weiss [20] for the continuous-time case. See also Oostveen [15, Lemma 4.2.6]. The dashes indicate unimportant entries. Lemma 2.3. Let LC be the observability gramian of the output stable and inputoutput stable system Σ(A, B, C, −). Then Σ(A, −, B ∗ LC A, −) is output stable. Proof. Since the system is input-output stable we have G ∈ H∞ (D, L(U, Y )) and since it is output stable we have Cx ∈ H2 (D, Y ) for all x ∈ X. It follows that G∗ Cx ∈ L2 (T, U ) for all x ∈ X with T the unit circle. From this it follows that the analytic part of G∗ Cx is in H2 (D, U ). We calculate for z ∈ T (so that z¯ = 1/z): ! ∞ ! ∞ X X ∗ ∗ ∗k ∗ −k−1 m m CA z G(z) C(z) = B A C z . m=0

k=0

It follows that the analytic part is: ∞ ∞ X X B ∗ A∗k C ∗ CAm z m−k−1 . k=0 m=k+1

Substituting j = m − k − 1 we obtain for this analytic part ∞ X ∞ X B ∗ A∗k C ∗ CAk Aj+1 z j , j=0 k=0

and using that LC =

∞ X

A∗k C ∗ CAk ,

k=0

we can rewrite this as

∞ X

B ∗ LC AAj z j .

j=0

It follows that the analytic part of G∗ C equals the ‘C’ function of the system Σ(A, −, B ∗ LC A, −) defined in Definition 2.2. It follows from this definition and the above that this system is output stable. Our stability assumption on the state space realization Σ(A, B, C, D) of G is that it is input stable, output stable and input-output stable. This assumption is implied by and is strictly weaker than exponential stability. Any function in H∞ (D, L(U, Y )) has a realization that satisfies our stability assumption (for example the shift realization mentioned in the introduction).

3. Algebraic Riccati equations In this section we introduce two algebraic Riccati equations that play a key role in the solution of the suboptimal Nehari problem. We first recall some general facts about algebraic Riccati equations.

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Definition 3.1. The control algebraic Riccati equation of a system Σ(A, B, C, D) is the equation A∗ QA − Q + C ∗ C = F ∗ SF, where S := I + D∗ D + B ∗ QB, F := −S −1 (B ∗ QA + D∗ C). The closed-loop system corresponding to a nonnegative selfadjoint solution Q is   A + BF BS −1/2  (4) F S −1/2  . C + DF DS −1/2 The filter algebraic Riccati equation of a system Σ(A, B, C, D) is the equation AP A∗ − P + BB ∗ = LRL∗ , where R := I + DD∗ + CP C ∗ , L := −(AP C ∗ + BD∗ )R−1 . The following theorem can be proven as was done for the corresponding continuous-time result in [6] (see [16, Proposition 6.34, Corollary 6.40]). Lemma 3.2. If the control Riccati equation of Σ(A, B, C, D) has a nonnegative, selfadjoint solution Q, then the closed-loop system (4) is output stable and inputoutput stable. Moreover, the H∞ norm of its transfer function is bounded from above by one. If the filter algebraic Riccati equation of Σ(A, B, C, D) also has a nonnegative, selfadjoint solution, then the afore-mentioned closed-loop system is also input stable. Standing hypothesis. From here on, until section 7, we assume that G ∈ H∞ (D, L(U, Y )) is such that its Hankel operator has norm strictly smaller than one and G(0) = 0. We take σ = 1 and fix an input stable, output stable and input-output stable realization Σ(A, B, C, 0) of G. The spectral radius of the product LB LC of the controllabilty and observability gramians equals the square of the norm of the Hankel operator HG (see [16, Lemma 3.1.8]). Since we take σ = 1 > kHG k it follows that I − LB LC has a bounded inverse and that, with

∗

N : = (I − LB LC )−1 ,

(5)

W : = N LB ,

(6)

X : = LC N,

(7)

∗

we have W = W ≥ 0, X = X ≥ 0. Some nontrivial algebraic manipulations show that W and X satisfy the algebraic Riccati equations given below. We first define the following operators TX := (I + B ∗ XB)1/2 , TW := (I + CW C ∗ )1/2 .

(8)

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Lemma 3.3. The nonnegative selfadjoint operator X defined by (7) is a solution of the control algebraic Riccati equation of the system A B , (9) −1 TW CN 0 −2 −2 ∗ A∗ XA − X + N ∗ CTW CN = A∗ XBTX B XA.

The nonnegative selfadjoint operator W defined by (6) is a solution of the filter algebraic Riccati equation of the system −1 A N BTX , (10) C 0 −2 ∗ ∗ −2 AW A∗ − W + N BTX B N = AW C ∗ TW CW A∗ .

(11)

Proof. We only give a proof the case of the system (10) since that for the system (9) is very similar. We first note that the following identities hold: I + BB ∗ X = (I − ALB A∗ LC ) (I − LB LC )−1 , ∗

∗

−1

I + C CW = (I − A LC ALB ) (I − LC LB )

.

(12) (13)

We will prove only (13) since the proof for (12) is similar. Using the definition (6) of W and the observation Lyapunov equation (3) we have I + C ∗ CW = I + (LC − A∗ LC A) LB (I − LC LB )−1 . This can be rewritten as (I − LC LB + LC LB − A∗ LC ALB ) (I − LC LB )−1 , which simplifies to the right-hand side of (13). Our goal is to show that (11) holds. We first note that by the definition (5) of N we have N B(I + B ∗ XB)−1 B ∗ N ∗ = (I − LB LC )−1 (I + BB ∗ X)−1 BB ∗ (I − LC LB )−1 . Next we use the control Lyapunov equation to rewrite this as (I − LB LC )−1 (I + BB ∗ X)−1 (LB − ALB A∗ ) (I − LC LB )−1 , which by (12) in turn equals −1

(I − ALB A∗ LC )

(LB − ALB A∗ ) (I − LC LB )−1 .

Rewriting this first as −1

(I − ALB A∗ LC )

([I − ALB A∗ LC ]LB − ALB A∗ [I − LC LB ]) (I − LC LB )−1

and then simplifying this via −1

LB (I − LC LB )−1 − (I − ALB A∗ LC )

ALB A∗

to −1

LB (I − LC LB )−1 − ALB (I − A∗ LC ALB )

A∗

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and using the definition (6) of W gives W − AW (I − LC LB )(I − A∗ LC ALB )−1 A∗ . We now use (13) and obtain for the above W − AW (I + C ∗ CW )−1 A∗ , which we first rewrite as W − AW (I + C ∗ CW )−1 (−C ∗ CW + I + C ∗ CW ) A∗ and subsequently simplify to W + AW (I + C ∗ CW )−1 C ∗ CW A∗ − AW A∗ . So we finally obtain N B(I + B ∗ XB)−1 B ∗ N ∗ = W + AW (I + C ∗ CW )−1 C ∗ CW A∗ − AW A∗ , which is easily seen to be equivalent to the desired (11).

From Lemmas 3.2 and 3.3 we obtain the following. Lemma 3.4. The system  B AX  −T −2 B ∗ XA 0  , X −1 TW CN 0

(14)

AX := (I + BB ∗ X)−1 A,

(15)



where is output stable and input-output stable. The system −2 AW −AW C ∗ TW C 0

−1 N BTX 0

,

(16)

where AW := A(I + W C ∗ C)−1 ,

(17)

is input stable and input-output stable. Proof. The system (14) is the closed-loop system (up to an insignificant similarity transformation in the input space) of the control Riccati equation for the system (9). So by Lemma 3.2 it is output stable and input-output stable. The argument for the system (16) is similar, but based on the filter Riccati equation for (10). The following important result shows that N intertwines AX and AW . Lemma 3.5. The following identity holds: N AX = AW N.

(18)

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Proof. Using the definitions of AX , AW and N we see that (18) is equivalent to A(I − LB LC )(I + W C ∗ C) = (I + BB ∗ X)(I − LB LC )A, and using X = LC N and W = N LB this in turn is equivalent to A(I − LB LC + LB C ∗ C) = (I + BB ∗ LC − LB LC )A, which is easily seen to be true by using C ∗ C = LC − A∗ LC A as well as BB ∗ = LB − ALB A∗ . The following formula that relates N A to AN will also be useful. Lemma 3.6. The following identity holds: N A = AN − N ALB C ∗ CN + N BB ∗ LC AN. ∗

(19)

∗

Proof. This follows easily after substituting C C = LC − A LC A and BB ∗ = LB − ALB A∗ .

4. J-spectral factorization The following J-spectral factorization plays a crucial role. I G∗ −I 0 I 0 −I 0 −X X∗ ≤ 0. 0 I 0 I G I 0 I The objective is to find a analytic function on the open unit disc X such that the above inequality holds on the open unit disc. This function X is subsequently used to define a function K and the above inequality is used to show that this K is a solution of the suboptimal Nehari problem. It was first realized in [6] that it is this inequality that is important and not equality on the unit circle, which was used in all previous approaches via J-spectral factorization to the Nehari problem. Our candidate solution is the following. Definition 4.1. Denote by X the transfer function and by X the characteristic function of the system   −1 −1 A −N BTX AW C ∗ TW −1 −1   B ∗ LC A . (20) TX B ∗ LC AW C ∗ TW C 0 TW By an application of Lemma 2.3 we obtain the following stability result. Lemma 4.2. The system given by (20) is output stable. Proof. By our standing hypothesis Σ(A, −, C, −) is input-output and output stable. It then immediately follows from Lemma 2.3 that Σ(A, −, [B ∗ LC A; C], −) is output stable and then the same conclusion can be made about the system (20). It will prove useful to have the following alternative formula for X.

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Lemma 4.3. We have X = X1 T with X1 the transfer function of the system   A −N B N ALB C ∗  B ∗ LC A  (21) I 0 C 0 I and

−1 −1 TX B ∗ LC AW C ∗ TW T := . 0 TW The corresponding result for characteristic functions also holds.

(22)

Proof. The only thing to prove is that −1 −1 Bh := [−N B, N ALB C ∗ ] T = −N BTX , AW C ∗ TW ,

(23)

for the first component this is trivial and for the second component this amounts to proving that −1 −1 −N BB ∗ LC AW C ∗ TW + N ALB C ∗ TW = AW C ∗ TW .

(24)

∗

Multiplying (19) from the right with LB C and noting that W = N LB gives N ALB C ∗ = AW C ∗ − N ALB C ∗ CW C ∗ + N BB ∗ LC AW C ∗ , −1 which after rearranging and multiplication to the right with TW gives (24).

We have the following useful identities involving the operator T . Lemma 4.4. With the notation J :=

−1 0

0 1

,

the following identities hold: −I + B ∗ LC B + B ∗ LC AW A∗ LC B B ∗ LC AW C ∗ ∗ T JT = , 2 CW A∗ LC B TW ∗ −B ∗ N ∗ B + B ∗ LC AW A∗ ∗ T JT = CLB A∗ N ∗ CW A∗ and −B ∗ N ∗ ∗ ∗ [−N B, N ALB C ] T JT = AW A∗ − W. CLB A∗ N ∗

(25) (26)

(27)

Proof. To prove (25) we only need to show that −2 −2 −TX + B ∗ LC AW C ∗ TW CW A∗ LC B = −I + B ∗ LC B + B ∗ LC AW A∗ LC B. (28)

By the algebraic Riccati equation for W we have for the left-hand side of (28) −2 −2 ∗ ∗ −TX + B ∗ LC AW A∗ − W + N BTX B N LC B. Thus (28) is equivalent to −2 −2 ∗ ∗ −TX − B ∗ LC W LC B + B ∗ LC N BTX B N LC B = −I + B ∗ LC B,

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i.e. that −2 ∗ ∗ 2 ∗ 2 ∗ 2 2 ∗ −TX B LC W LC B + TX B LC N BTX B N LC B + TX − TX B LC B = I.

Using that LC N = X and LC W = XLB this is equivalent to −2 ∗ 2 ∗ 2 ∗ 2 2 ∗ −TX B XLB LC B + TX B XBTX B XB + TX − TX B LC B = I. −2 −2 ∗ Furthermore, we have B ∗ XBTX = TX B XB so that the above is equivalent to 2 ∗ 2 2 ∗ −TX B XLB LC B + B ∗ XBB ∗ XB + TX − TX B LC B = I.

Using the definition of TX from (8) this is equivalent to − B ∗ XLB LC B − B ∗ XBB ∗ XLB LC B + B ∗ XBB ∗ XB + B ∗ XB − B ∗ LC B − B ∗ XBB ∗ LC B = 0.

(29)

Combining the first and fourth term on the left-hand side gives −B ∗ XLB LC B + B ∗ XB = B ∗ X(I − LB LC )B = B ∗ LC B, so that these terms cancel against the fifth term on the left-hand side of (29). The second and third term on the left-hand side of (29) are − B ∗ XBB ∗ XLB LC B + B ∗ XBB ∗ XB = B ∗ XBB ∗ X (I − LB LC ) B = B ∗ XBB ∗ LC B, so that they cancel against the sixth term on the left-hand side of (29). So (29) holds and consequently (28) and (25) hold. Using (25) we see that the left-hand side of (26) equals ∗ B − B ∗ LC BB ∗ − B ∗ LC AW A∗ LC BB ∗ + B ∗ LC AW C ∗ CLB A∗ N ∗ . (30) −CW A∗ LC BB ∗ + CLB A∗ + CW C ∗ CLB A∗ Now using the Lyapunov equations we see that the first component of (30) equals B ∗ (I − LC LB + LC ALB A∗ − LC AW A∗ LC LB + LC AW A∗ LC ALB A∗ +LC AW LC LB A∗ − LC AW A∗ LC ALB A∗ ) N ∗ . We see that the fifth and seventh terms cancel against each other. Using that W = LB N ∗ and the definition of N we obtain B ∗ (I − LC LB + LC AW [(I − LC LB )A∗ − A∗ LC LB + LC LB A∗ )]) N ∗ . Cancelling terms we obtain B ∗ (I − LC LB + LC AW A∗ [I − LC LB ]) N ∗ , which equals B ∗ + B ∗ LC AW A∗ , as desired. Using that W = LB N ∗ and the definition of N we see that the second component of (30) equals CW (−A∗ LC BB ∗ + (I − LC LB )A∗ + C ∗ CLB A∗ ) N ∗ .

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Now using the Lyapunov equations we see that this equals CW HN ∗ , with H := −A∗ LC LB + A∗ LC ALB A∗ + (I − LC LB )A∗ + LC LB A∗ − A∗ LC ALB A∗ . After cancellations we see that this equals CW (−A∗ LC LB + A∗ ) N ∗ = CW A∗ , as desired. The equality (27) follows by recognizing that the left-hand side equals Bh JBh∗ with −1 −1 Bh = [−N BTX , AW C ∗ TW ]

(compare (23)), so that (27) is nothing else than the Riccati equation (11).

The following lemma is proven by more algebraic manipulations. Lemma 4.5. On 1/ρ(A) we have

G∗ I

−I 0 I 0 −I 0 −X X∗ 0 I G I 0 I ∗ |z|2 − 1 L0 L ∗ ∗ L M L 0 + , = 0 M 0 |z|2 I 0

(31)

where L is the characteristic function of the system

W 1/2 A ∗ B LC A 0

,

(32)

M is the characteristic function of the system

A C

W 1/2 0

,

(33)

and L0 is the characteristic function of the system

A 1/2 LC

B 0

.

(34)

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Proof. We have XJX∗ = X1 T JT ∗ X1 ∗ −1 1 B LC A [−N B, N ALB C ∗ ] T ∗ JT = T JT ∗ + −A C z −∗ 1 −B ∗ N ∗ ∗ [A∗ LC B, C ∗ ] + T JT −A CLB A∗ N ∗ z ∗ −1 1 B LC A [−N B, N ALB C ∗ ] T ∗ JT + −A C z −∗ 1 −B ∗ N ∗ × −A [A∗ LC B, C ∗ ] . CLB A∗ N ∗ z Using (25), (26) and (27) we see that the right-hand side equals

−I + B ∗ LC B + B ∗ LC AW A∗ LC B B ∗ LC AW C ∗ 2 CW A∗ LC B TW −1 ∗ 1 B LC A −A [B + AW A∗ LC B, AW C ∗ ] (35) + C z ∗ −∗ 1 B + B ∗ LC AW A∗ + − A [A∗ LC B, C ∗ ] CW A∗ z ∗ −1 −∗ 1 1 B LC A ∗ + −A (AW A − W ) −A [A∗ LC B, C ∗ ] . C z z

We proceed by considering the separate components of this 2 by 2 matrix. The (1,1) component of (35) equals −1 −∗ 1 1 ∗ − I + B LC B + B LC A −A B+B −A ALC B z z −1 −∗ 1 1 1 + − 1 B ∗ LC A −A W −A A∗ LC B, 2 |z| z z ∗

∗

where we have used the identity

1 − 1 W |z|2 ∗ ∗ 1 1 1 1 = −A W − A + AW −A + − A W A∗ + AW A∗ − W. z z z z

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Using a similar calculation we obtain −∗ −1 1 1 G∗ G = B ∗ −A C ∗C −A B z z −∗ ∗ ∗ 1 1 1 1 ∗ × = B −A − A LC A + − A LC −A z z z z 1 1 ∗ + A LC − A + 1 − 2 LC z |z| −1 1 × −A B z −1 −∗ 1 1 ∗ ∗ = B LC A −A B + B LC B + −A A∗ LC B z z −∗ −1 1 1 1 −A LC −A B. + 1 − 2 B∗ |z| z z where we have used the observation Lyapunov equation. It follows from the formulas obtained for XJX∗ and G∗ G that the (1,1) component of the left-hand side of (31) equals 1 1 − 2 (H1 + H2 ) , |z| with −∗ −1 1 1 −A LC −A B, H1 = B z z −1 −∗ 1 1 H2 = B ∗ LC A −A W −A A∗ LC B, z z ∗

as desired. The proof for the other components is obtained by similar tedious calculations and hence is omitted. The following lemma gives the corresponding result for transfer functions. Lemma 4.6. On the open unit disc we have I G∗ −I 0 I 0 −I 0 −X X∗ 0 I 0 I G I 0 I ∗ |z|2 − 1 L ∗ L0 ∗ L M L0 0 + , = M 0 |z|2 where L is the transfer function of the system (32), M is the transfer function of the system (33) and L0 is the transfer function of the system (34). Proof. From Lemma 4.5 we obtain equality of the transfer functions in a neighborhood of zero. The transfer functions involved are all analytic on the open unit disc: for G, M and L0 this follows directly from our stability assumptions, while for

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X this follows from Lemma 4.2 and for L this follows using Lemma 2.3. The leftand right-hand sides of the equation are not analytic on the unit disc, but they are real-analytic there. This last assertion follows from the fact that both analytic functions and their adjoints are real-analytic and that the product of real-analytic functions is again real-analytic. By the identity theorem for real-analytic functions the equality holds on the whole unit disc (see [5, Appendix]). Remark 4.7. We note that the right-hand sides of the equations obtained in Lemmas 4.5 and 4.6 are strictly speaking not defined for z = 0, since we divide by |z|2 . Noting that the feedthrough terms of the systems (32), (33) and (34) are zero this singularity is seen to be removable, i.e. the functions can be continuously (and even analytically) extended to z = 0. It is in this sense that the equalities hold in z = 0. From Lemma 4.6 we deduce the following. Lemma 4.8. On the open unit disc we have I G∗ −I 0 I 0 −I ≤X 0 I 0 I G I 0

0 I

X∗ .

Proof. The term between brackets on the right-hand side of the equation obtained in Lemma 4.6 is of the form T1∗ T1 + T2∗ T2 and so it is nonnegative. The fraction in 2 front, |z||z|−1 2 , is negative since by assumption |z| < 1. It follows that the right-hand side of the equation obtained in Lemma 4.6 is, for every z ∈ D, nonpositive. It follows that the left-hand side is, which gives the desired inequality. The following lemma gives important formulas for the inverse of one of the components of X. Lemma 4.9. The inverse of the (2, 2) component of X on the open unit disc is the transfer function of the input stable and input-output stable system −2 AW C ∗ TW AW . (36) −1 −1 −TW C TW Proof. That the characteristic functions of (36) is the inverse of that of the (2,2) component of (20) follows by an application of Lemma 2.1 together with the identity −2 A − AW C ∗ TW C = AW , which is easily proven using only the definition of TW . That the system (36) is input and input-output stable follows from Lemma 3.4. Since the characteristic functions of the two systems are inverses it follows that the product of the two transfer functions equals the identity for z in a neighborhood of zero. This equality extends to the whole of the open unit disc since these transfer functions are analytic on the open unit disc by the stability properties of the given realizations.

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5. The inverse of the spectral factor Definition 5.1. Define V as the transfer function and V as the characteristic function of the system   B −ALB C ∗ A  T −1 B ∗ XA TX −T −1 B ∗ XALB C ∗  . (37) X X −1 −1 0 TW TW CN Lemma 5.2. The system given by (37) is input stable. Proof. This follows similarly as Lemma 4.2.

Lemma 5.3. We have V = T −1 V1 with V1 the transfer function of the system   A B −ALB C ∗  B ∗ LC AN I  (38) 0 CN 0 I and

−1 ∗ TX −TX B XALB C ∗ T = (39) −1 0 TW the inverse of the operator T defined by (22). The corresponding result for characteristic functions also holds. −1

Proof. We first show that the right-hand side of (39) is indeed T −1 . It is easily seen that the inverse of T defined by (22) is given by −2 TX −TX B ∗ LC AW C ∗ TW . (40) −1 0 TW To show that this equals (39) we need to show that −2 −1 ∗ TX B ∗ LC AW C ∗ TW = TX B XALB C ∗ ,

or equivalently that −2 ∗ 2 B ∗ LC AW C ∗ = TX B XALB C ∗ TW .

Since W = N LB , with (19) we see that the left-hand side equals B ∗ LC N [ALB + ALB C ∗ CN LB − BB ∗ LC AN LB ] C ∗ , which by using X = LC N and W = N LB is seen to equal B ∗ X AN −1 − BB ∗ XN −1 A(I + W C ∗ C)−1 (I + W C ∗ C)W C ∗ . Using (15) and (17) this can be rewritten as 2 B ∗ X(I + BB ∗ X) AX N −1 − (I + BB ∗ X)−1 BB ∗ XN −1 AW W TW C ∗. From (18) we have AX N −1 = N −1 AW so that the above equals 2 B ∗ X(I + BB ∗ X) I − (I + BB ∗ X)−1 BB ∗ X AX N −1 W TW C ∗.

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Simplifying shows that this equals 2 B ∗ XAX LB TW C ∗,

and using (15) shows that this in turn equals −2 ∗ 2 TX B XALB TW C ∗,

as desired. To show that V = T V1 it only remains to show that −1 ∗ −1 ∗ TX B XA = TX B ∗ LC AN − TX B XALB C ∗ CN.

After multiplying (19) from the left by B ∗ LC we obtain B ∗ LC N A = B ∗ LC AN − B ∗ LC N ALB C ∗ CN + B ∗ LC N BB ∗ LC AN. But LC N = X and so rearranging we obtain B ∗ XA = (I + B ∗ XB)B ∗ LC AN − B ∗ XALB C ∗ CN. −1 Multplication from the left by TX gives the desired equality.

The following is the analogue of Lemma 4.9 and is proven similarly. Lemma 5.4. The inverse of the (1, 1) component of V on the open unit disc is the transfer function of the output stable and input-output stable system −1 −BTX AX . (41) −2 ∗ −1 TX B XA TX Proof. That the characteristic function of the system (41) is the inverse of that of the (1, 1) component of (37) follows by an application of Lemma 2.1 together with the identity −2 ∗ A − BTX B XA = AX ,

which is easily proven using only the definition of TX . That the system (41) is output and input-output stable follows from Lemma 3.4. Since the characteristic functions of the two systems are inverses it follows that the product of the two transfer functions equals the identity for z in a neighborhood of zero. This equality extends to the whole of the open unit disc since these transfer functions are analytic on the open unit disc by the stability properties of the given realizations.

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The following lemma can be proven by algebraic manipulation. Lemma 5.5. On 1/ρ(A) we have VX = I = XV, where X is the characteristic function of the system (21) and V is the characteristic function of the system (38). Proof. We first note that equivalently we may show that the characteristic functions of the systems (21) and (38) are each others inverses. Define ∗ B LC A Bl := [−N B, N ALB C ∗ ], Cl := . C We apply Lemma 2.1 to obtain the inverse of the characteristic function of the system (21). This is the characteristic function of the system A − Bl Cl Bl . (42) −Cl I The system (38) may be written in terms of Bl and Cl as −N −1 Bl A , Cl N I

(43)

provided that A − Bl Cl = N AN −1 . But after multiplication from the right by N and substituting the definitions of Bl and Cl this simplifies to (19) The case of transfer functions follows immediately. Lemma 5.6. On the open unit disc we have VX = I = XV, where X is the transfer function of the system (21) and V is the transfer function of the system (38). Proof. Using Lemma 5.5 for the corresponding result for characteristic functions, the output stability of the system (21) and the input stability of the system (38) (which follows from the dual version of Lemma 2.3) we obtain the desired result. From the previous lemma and the J-spectral factorization inequality obtain in Lemma 4.8 we obtain the following. Lemma 5.7. On the open unit disc we have I G∗ −I 0 I V 0 I 0 I G

0 I

V∗ ≤

−I 0

0 I

.

Proof. This follows from multiplying both sides of the inequality obtained in Lemma 4.8 from the left with V, from the right with V∗ and using that V is the inverse of X by Lemma 5.6.

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6. The Nehari problem In this section we give an explicit solution to our Nehari problem using the inverse J-spectral factor V found in Section 5. Definition 6.1. Define Z as the transfer function of the system A∗X −A∗X XB . CLB A∗X −CLB A∗X XB

(44)

Lemma 6.2. The system (44) is input stable. Proof. Since AX = (I+BB ∗ X)−1 A, we see that the adjoint of the control operator, B ∗ XAX , equals (I + B ∗ XB)−1 B ∗ XA. Using Lemma 3.4, specifically that the system (14) is output stable, we obtain the desired result. The following can be proven by algebraic manipulation. Lemma 6.3. The characteristic function of the system (44) equals the characteristic function of the system −A∗W LC B A∗W . (45) CW A∗W −CW A∗W LC B Proof. From (18) it follows by multiplication from the left with LC and from the right with LB that XAX LB = LC AW W , which in turn implies that the rightbottom terms in (44) and (45) are equal. Using (18) it follows that the top righthand corner of (44) equals that of (45) multiplied from the left by N ∗ , the bottom left-hand corner of (44) equals that of (45) multiplied from the right by N −∗ and the top left-hand corner of (44) equals that of (45) multiplied from the right by N −∗ and from the left by N ∗ . In the calculation of the characteristic functions this causes cancellation, showing that the characteristic functions are equal. The following can be proven in a similar manner as Lemma 6.2. Lemma 6.4. The system (45) is output stable. Lemma 6.5. The transfer functions of the systems (44) and (45) are equal on the open unit disc. Proof. Since the system (44) is input stable, its transfer function is analytic on the open unit disc. Since the system (45) is output stable, the same holds for this system. By Lemma 6.3 the characteristic functions of these systems are equal. So the transfer functions restricted to the open unit disc are analytic extensions of the same function. By the identity theorem for analytic functions they must be equal. The following lemmas relate Z to the J-spectral factor X and its inverse V. The first of these lemmas deals with the corresponding characteristic functions and can by proven by algebraic manipulation.

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Lemma 6.6. On 1/ρ(A∗ ) ∩ 1/ρ(A∗X ) we have Z(z) = V12 (¯ z )∗ V11 (¯ z )−∗ .

(46)

On 1/ρ(A∗ ) ∩ 1/ρ(A∗W ) we have Z(z) = −X22 (¯ z )−∗ X21 (¯ z )∗ .

(47)

−2 ∗ Proof. We first note that TX B XA = B ∗ XAX , and that if follows from this −2 ∗ equality using Lemma 3.5 that TX B XA = B ∗ LC AW N . −2 ∗ Using Definition 5.1, Lemma 5.4 and the identity TX B XA = B ∗ LC AW N , we have

V11 (z)−1 V12 (z) −1 1 BB ∗ LC AW W C ∗ I − AX = −B LC AW W C + z −1 1 1 −2 ∗ −2 ∗ + TX B XA I − AX I − AX + BTX B XA − z z −1 1 I −A ALB C ∗ . × z The term in square brackets is easily seen to equal − z1 I − A , so that the above simplifies to −1 1 −2 ∗ −B ∗ LC AW W C ∗ + TX B XA I − AX [BB ∗ LC AW W C ∗ − ALB C ∗ ] . z ∗

∗

−2 ∗ TX B XA

The term in square brackets simplifies to −AX LB C ∗ . Using this, the identity −2 ∗ TX B XA = B ∗ XAX and comparing to the formula for Z in (44) (and (45) for the constant term) we obtain (46). The equality (47) is proven similarly. Lemma 6.7. On the open unit disc we have Z(z) = V12 (¯ z )∗ V11 (¯ z )−∗ = −X22 (¯ z )−∗ X21 (¯ z )∗ .

(48)

Proof. This follows from Lemma 6.6 and the stability properties of the given realizations of the involved transfer functions. The following is the main result of this article. Theorem 6.8. Define K(z) := Z(1/z), where Z is defined as in Definition 6.1. Then K ∈ H∞ (D+ , L(U, Y )) and kG + KkL∞ (T,L(U,Y )) ≤ 1. Proof. Noting that all the transfer functions G, V, and Z are holomorphic on D, we perform some elementary calculations on D. We first verify that −∗ I I 0 ∗ V11 (z) = V(z) G(z) + Z(¯ z) G(z) I 0

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by expanding the right hand-side: I 0 V11 (z)∗ V21 (z)∗ V11 (z)−∗ G(z) I V12 (z)∗ V22 (z)∗ 0 I 0 I = G(z) I V12 (z)∗ V11 (z)−∗ I , = G(z) + Z(¯ z) where by Lemma 6.7 Z(¯ z ) = V12 (z)∗ V11 (z)−∗ holds on the open unit disc. Using the just established identity we have ∗

(G(z) + Z(¯ z )) (G(z) + Z(¯ z )) − I ∗ I −I 0 I = G(z) + Z(¯ z) 0 I G(z) + Z(¯ z) ∗ ∗ −∗ V11 (z) I 0 −I 0 = V(z) × 0 G(z) I 0 I −∗ I 0 V11 (z) × V(z)∗ . G(z) I 0

(49)

Applying Lemma 5.7 we obtain ∗

(G(z) + Z(¯ z )) (G(z) + Z(¯ z )) − I ∗ V11 (z)−∗ −I 0 V11 (z)−∗ ≤ 0 0 I 0

(50)

= −V11 (z)−1 V11 (z)−∗ ≤ 0. This shows that the analytic function Z is bounded in norm on D and so it is in H∞ (D, L(U, Y )). We also obtain the estimate kG(z) + Z(¯ z )k ≤ 1 for z ∈ D. By taking nontangential limits we obtain the same estimate almost everywhere on the unit circle. It follows from Z ∈ H∞ (D, L(U, Y )) that K(z) = Z(1/z) is in ¯ H∞ (D+ ; L(U, Y )). The boundary functions of Z and K are related by K(ζ) = Z(ζ) with ζ on the unit circle. So K satisfies the following for almost all ζ on the unit circle kG(ζ) + K(ζ)k ≤ 1. Remark 6.9. Theorem 6.8 gives one solution of the sub-optimal Nehari problem. With little extra effort one can actually obtain infinitely many. To show this we −1 first note that V21 V11 equals the second component of the transfer function of the closed-loop system (as defined by (4)) of the system (9). The proof is very similar −1 to that of Lemma 6.6. It follows from Lemma 3.2 that kV21 V11 k∞ ≤ 1. So if kQk∞ < 1, then V11 +QV21 has a well-defined inverse in H∞ . Now define Z(¯ z) = (V12 (z)∗ +V22 (z)∗ Q(z)∗ )(V11 (z)∗ +V21 (z)∗ Q(z)∗ )−1 . Entirely analogously to the proof of Theorem 6.8 it follows that K defined by K(z) = Z(1/z) is a solution of

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the suboptimal Nehari problem (Theorem 6.8 being the special case Q = 0). In the analogue of inequality (50) we used that kQk∞ ≤ 1 (strict inequality is not needed for this). So for any Q ∈ H∞ (D, L(U, Y )) with kQk∞ < 1 we obtain a solution. From the analogy with similar formulas known to give a parametrization of all solutions of the suboptimal Nehari problem, one would expect that the above given linear fractional formula (possibly rewritten in Redheffer form) with parameter Q ranging over the closed unit ball kQk∞ ≤ 1 should give rise to all solutions. However, justification of such a statement would require further work beyond the scope of the present article.

7. The general case In this section we make some elementary observations that allow us to obtain the general case of the suboptimal Nehari problem from the special with D = 0 and σ = 1 considered above. Remark 7.1. If K is a solution of the suboptimal Nehari problem for data G−G(0), then K + G(0) is a solution of the suboptimal Nehari problem for data G. Let a, b > 0. If K is a solution of the suboptimal Nehari problem for data G/a and parameter b, then aK is a solution of the suboptimal Nehari problem for data G and parameter ab. If S := Σ(A, B, C, D) is a realization of G, then Sa := Σ(A, B, aC, aD) is a realization of aG. If LC is the observability gramian of S, then a2 LC is the observability gramian of Sa . From the above remark and Theorem 6.8 it follows that if G is the transfer function of the state space system Σ(A, B, C, D) and σ > kHG k, then the function K defined for |z| > r(AXσ ) by K(z) = −D − CLB A∗Xσ Xσ B − CLB A∗Xσ (z − A∗Xσ )−1 A∗Xσ Xσ B extends to K ∈ H∞ (D+ , L(U, Y )) which satisfies kG + KkL∞ (T,L(U,Y )) ≤ σ. Here Xσ = (σ 2 I − LC LB )−1 LC , AXσ = (I + BB ∗ Xσ )−1 A.

References [1] V. M. Adamjan, D. Z. Arov, and M. G. Kre˘ın. Infinite Hankel matrices and generalized Carath´eodory-Fej´er and I. Schur problems. Funkcional. Anal. i Priloˇzen., 2(4):1–17, 1968. [2] V. M. Adamjan, D. Z. Arov, and M. G. Kre˘ın. Infinite Hankel block matrices and related problems of extension. Izv. Akad. Nauk Armjan. SSR Ser. Mat., 6(2-3):87– 112, 1971.

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[3] Ruth Curtain and Hans Zwart. The Nehari problem for the Pritchard-Salamon class of infinite-dimensional linear systems: a direct approach. Integral Equations Operator Theory, 18(2):130–153, 1994. [4] Ruth F. Curtain and Akira Ichikawa. The Nehari problem for infinite-dimensional linear systems of parabolic type. Integral Equations Operator Theory, 26(1):29–45, 1996. [5] Ruth F. Curtain and Mark R. Opmeer. The suboptimal Nehari problem for wellposed linear systems. SIAM J. Control Optim., 44(3):991–1018, 2005. [6] Ruth F. Curtain and Mark R. Opmeer. Normalized doubly coprime factorizations for infinite-dimensional linear systems. Math. Control Signals Systems, 18(1):1–31, 2006. [7] Ruth F. Curtain and A. C. M. Ran. Explicit formulas for Hankel norm approximations of infinite-dimensional systems. Integral Equations Operator Theory, 12(4):455– 469, 1989. [8] Ruth F. Curtain and Hans Zwart. An introduction to infinite-dimensional linear systems theory, volume 21 of Texts in Applied Mathematics. Springer-Verlag, New York, 1995. [9] C. Foias, A. E. Frazho, I. Gohberg, and M. A. Kaashoek. Metric constrained interpolation, commutant lifting and systems, volume 100 of Operator Theory: Advances and Applications. Birkh¨ auser Verlag, Basel, 1998. [10] Keith Glover, Ruth F. Curtain, and Jonathan R. Partington. Realisation and approximation of linear infinite-dimensional systems with error bounds. SIAM J. Control Optim., 26(4):863–898, 1988. [11] Vlad Ionescu, Cristian Oar˘ a, and Martin Weiss. Generalized Riccati theory and robust control: A Popov function approach. John Wiley & Sons Ltd., Chichester, 1999. [12] A. Kheifets. Parametrization of solutions of the Nehari problem and nonorthogonal dynamics. In Operator theory and interpolation (Bloomington, IN, 1996), volume 115 of Oper. Theory Adv. Appl., pages 213–233. Birkh¨ auser, Basel, 2000. [13] Zeev Nehari. On bounded bilinear forms. Ann. of Math. (2), 65:153–162, 1957. [14] Nikolai K. Nikol0 ski˘ı. Treatise on the shift operator, volume 273 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1986. [15] Job Oostveen. Strongly stabilizable distributed parameter systems, volume 20 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. [16] Mark R. Opmeer. Model reduction for controller design for infinite-dimensional systems. PhD thesis, University of Groningen, 2006. [17] Lavon B. Page. Bounded and compact vectorial Hankel operators. Trans. Amer. Math. Soc., 150:529–539, 1970. [18] Vladimir V. Peller. Hankel operators and their applications. Springer Monographs in Mathematics. Springer-Verlag, New York, 2003. [19] A. C. M. Ran. Hankel norm approximation for infinite-dimensional systems and Wiener-Hopf factorization. In Modelling, robustness and sensitivity reduction in control systems (Groningen, 1986), volume 34 of NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., pages 57–69. Springer, Berlin, 1987.

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[20] Martin Weiss and George Weiss. Optimal control of stable weakly regular linear systems. Math. Control Signals Systems, 10(4):287–330, 1997. [21] Hans Zwart. Transfer functions for infinite-dimensional systems. Systems Control Lett., 52(3-4):247–255, 2004. Ruth F. Curtain Department of Mathematics University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands e-mail: [email protected] Mark R. Opmeer Department of Mathematical Sciences University of Bath Claverton Down Bath BA2 7AY United Kingdom e-mail: [email protected] Submitted: July 17, 2008. Revised: November 9, 2008.

Integr. equ. oper. theory 64 (2009), 61–81 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010061-21, published online April 24, 2009 DOI 10.1007/s00020-009-1680-3

Integral Equations and Operator Theory

Eigenvalues of Integral Operators Defined by Smooth Positive Definite Kernels J. C. Ferreira and V. A. Menegatto Abstract. We consider integral operators defined by positive definite kernels K : X × X → C, where X is a metric space endowed with a strictly-positive measure. We update upon connections between two concepts of positive definiteness and upgrade on results related to Mercer like kernels. Under smoothness assumptions on K, we present decay rates for the eigenvalues of the integral operator, employing adapted to our purposes multidimensional versions of known techniques used to analyze similar problems in the case where X is an interval. The results cover the case when X is a subset of Rm endowed with the induced Lebesgue measure and the case when X is a subset of the sphere S m endowed with the induced surface Lebesgue measure. Mathematics Subject Classification (2000). Primary 45P05; Secondary 42A82, 45C05, 43A35, 41A99. Keywords. Integral operators, eigenvalue estimates, positive definiteness, Mercer’s theorem, trace, trace norm.

1. Introduction This paper is concerned with the analysis of decay rates for eigenvalues of integral operators construct from positive definite kernels on subsets of metric spaces. Such operators appear quite naturally in approximation theory, integral equations and operator theory, playing an important role in many problems. We will consider two different notions of positive definiteness as explained below. If X is a nonempty set, a kernel K : X × X → C is positive definite when the inequality n X ci cj K(xi , xj ) ≥ 0, i,j=1

The first author was partially supported by FAPESP, grants # 2005/56694−4, # 2007/58086−7.

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holds for all n ≥ 1, x1 , x2 , . . . , xn ∈ X and scalars c1 , c2 , . . . , cn . If X is endowed with a measure ν and K belongs to L2 (X × X, ν × ν), we say that K is L2 -positive definite when the corresponding integral operator Z K(f )(x) := K(x, y)f (y) dν(y), f ∈ L2 (X, ν), x ∈ X, (1.1) X

is positive, that is, when the following condition holds Z Z K(x, y)f (y) dν(y) f (x) dν(x) ≥ 0, X

f ∈ L2 (X, ν).

X

We will write P D(X) and L2 P D(X, ν) to denote these two classes of kernels. Needless to say that the expression on the left-hand side of the above inequality is just hK(f ), f i2 , in which h·, ·i2 is the inner product of L2 (X, ν). Keeping the context as general as possible, we will investigate possible connections between these two concepts of positive definiteness. Assuming positive definiteness of the kernel and reasonable additional smoothness assumptions on K, we will update on the corresponding Mercer’s theory and analyze decay rates for the eigenvalues of the integral operator (1.1). A quite general formulation for the classical result of Mercer is as follows (see [11, 13]). Theorem 1.1 (Mercer’s Theorem). Let X be a topological Hausdorff space equipped with a finite Borel measure ν. Then for every continuous positive definite kernel K : X × X → C there exist a scalar sequence {λn } ∈ l1 , λ1 ≥ λ2 ≥ . . . ≥ 0 and an orthonormal system {φn } in L2 (X, ν) consisting of continuous functions only, such that the expansion K(x, y) =

∞ X

λn (K)φn (x)φn (y),

x, y ∈ supp(ν),

n=1

converges uniformly. If the integral operator has countably many eigenvalues λ1 (K) ≥ λ2 (K) ≥ · · · ≥ 0, the basic decay rate given by Mercer’s theory is λn (K) = o(n−1 ), as n → ∞. The following example ([16]) shows that this rate can not be improved, unless additional assumptions are added. Indeed, consider X = [−1, 1] endowed with the Lebesgue measure µ. The kernel K(x, y) =

∞ X n=1

1 np+1+

cos(nπx) cos(nπy),

x, y ∈ [−1, 1],

where p is a nonnegative integer and > 0, is an element of L2 P D([−1, 1], µ). If φn (x) := cos(nπx),

x ∈ [−1, 1],

then the sequence {φn } is L2 ([−1, 1], µ) -orthonormal and λn (K) = n−1−p− , n = 1, 2, . . .. As so, λn (K) = O(n−1−p− ) = o(n−1−p ), as n → ∞, but

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λn (K) 6= o(n−1−p− ), as n → ∞. The series ∞ X n=1

nq np+1+

,

0 ≤ q ≤ p,

being convergent, it is not hard to see that the partial derivatives of order at most p of K are continuous. As so, K is of class C p . Improvements on the basic estimate exhibit above can be found in many contexts under different sets of hypotheses. For instance, reference [15] considers the case when X is a compact interval while references [12, 13] analyze the case when X is a compact metric space or even a differentiable manifold, endowed with a finite measure. In [2], now dropping the compactness assumption on X, Buescu and Paix˜ ao investigated generalizations when X is a closed interval and K satisfies certain smoothness hypotheses. A similar analysis can be found in [3, 4]. In this paper these questions will be take up again, just assuming that X is a metric space endowed with a strictly-positive measure. In Section 2, we investigate possible connections between the notions of positive definiteness and recover Mercer’s Theorem in this new setting. In Section 3, we analyze important properties of the square root of an integral operator, under the light of the assumptions adopted on X and on the kernel. In particular we show that the square root is also an integral operator and deduce a recovery formula via the kernel defining the original operator. In Section 4, using the results in Section 3, we discuss basic finite approximations for the integral operator based upon special decompositions of X. The last result of the section describes an estimate for the sum of the eigenvalues of the operator deduced from such finite approximations. Section 5 begins with the concept of (q, t)-compactness, a very special decomposition for metric spaces endowed with a measure. Later in the section, we deduce the main results of the paper. They describe decay rates for the eigenvalues of the integral operator when X is (q, t)-compact and the generating kernel satisfies a smoothness condition of Lipschitz type. It is important to emphasize that the approach we take here is based upon arguments found in [2] and references therein.

2. Positive definiteness and Mercer’s theory revisited The results in this section indicate possible contexts in which the classes P D(X) and L2 P D(X, ν) coincide. Henceforth, if X is a metric space, we will write C(X) to denote the set of continuous functions on X and CB (X) to denote the subset of C(X) formed by bounded functions vanishing outside a bounded subset of X. The letter ν will be used to denote a measure over X. If X is a subset of Rm and the measure is the restriction of the usual Lebesgue measure of Rm to X, the letter µ will be used instead. The first result describes a setting where the inclusion P D(X) ⊂ L2 P D(X, ν) holds.

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Theorem 2.1. If X is a measurable subset of Rm endowed with the usual Lebesgue measure µ then P D(X) ∩ C(X × X) ∩ L2 (X × X, µ × µ) ⊂ L2 P D(X, µ). Proof. Let K be in P D(X) ∩ C(X × X) ∩ L2 (X × X, µ × µ). Since CB (X) is dense in L2 (X, µ) ([8, p.217]), to show that hK(f ), f i2 ≥ 0, f ∈ L2 (X, µ), it suffices to verify that hK(f ), f i2 ≥ 0, f ∈ CB (X). Let f ∈ CB (X) and write Xf to denote a bounded subset of X for which f (x) = 0, x ∈ X \Xf . There exists a sequence {An } of compact subsets of Xf such that An ⊂ An+1 , n = 1, 2, . . ., and limn→∞ µ(Xf \ An ) = 0. In particular, the kernel Kf given by Kf (x, y) = K(x, y)f (x)f (y), x, y ∈ X, is uniformly continuous in An ×An . The Monotone Convergence Theorem shows that {Kf χAn ×An } converges to Kf in L1 (X × X, µ × µ). Next, for each n, we can km find r = r(n) > 0 so that An ⊂ [−r/2, r/2]m . Writing [−r/2, r/2]m = ∪j=1 Cjk , k k k in which C1 , C2 , . . . , Ckm , are m-dimensional cubes having sides of length r/n, parallel to the coordinate axes, we may decompose An in the form m

An = ∪kj=1 Akj , Choosing

xkj

∈

Akj ,

Akj ⊂ Cjk ,

j = 1, 2, . . . , k

m

Akj ∩ Akl = ∅,

l 6= k.

and defining

m

gkn =

k X

K(xki , xkj )f (xki )f (xkj )χAki ×Akj ,

i,j=1

{gkn }

it is easily seen that converges uniformly to Kf χAn ×An in An × An , when k → ∞. Also, since K ∈ P D(X), it follows that gkn (x, y) ≥ 0, x, y ∈ An . Taking into account that Kf χAn ×An is bounded and the fact that µ(An ) < ∞, we can use the Dominated Convergence Theorem to deduce that Z Z Kf (x, y) dµ(x) dµ(y) = Kf (x, y) dµ(x) dµ(y) Xf ×Xf

X×X

Z Kf (x, y) dµ(x) dµ(y)

= lim

n→∞

An ×An

= lim

n→∞

Z lim

k→∞

gkn (x, y) dµ(x) dµ(y) ≥ 0.

An ×An

2

Thus, K ∈ L P D(X, µ).

In a similar manner, the following extension can be proved. Corollary 2.2. If X is a locally compact Hausdorff space endowed with a Radon measure ν that is finite on compact subsets then P D(X) ∩ C(X × X) ∩ L2 (X × X, ν × ν) ⊂ L2 P D(X, ν). The other inclusion can be guaranteed in a quite general context. If X is a topological space, we say that a measure ν on X is strictly-positive when it is a Borel measure fulfilling the following requirements: every open nonempty subset

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of X has positive measure and every x ∈ X belongs to an open subset of X having finite measure. Theorem 2.3. If a topological space X is endowed with a strictly-positive measure ν, then L2 P D(X, ν) ∩ C(X × X) ⊂ P D(X). Proof. Let K ∈ L2 P D(X) ∩ C(X × X), x1 , x2 , . . . , xn ∈ X and c1 , c2 , . . . , cn ∈ C. Due to the continuity of K, for each > 0 and j ∈ {1, 2, . . . , n} there exist open sets Xj so that xj ∈ Xj and x ∈ Xi ,

|K(x, y) − K(xi , xj )| < ,

y ∈ Xj ,

i, j = 1, 2, . . . , n.

Since ν is strictly-positive, we can assume that 0 < µ(Xj ) < ∞, j = 1, 2 . . . , n. As so, integration implies that Z Z 1 |K(x, y) − K(xi , xj )| dν(x) dν(y) < . ν(Xi )ν(Xj ) Xi Xj In particular, 1 1 lim →0+ ν(Xi ) ν(Xj )

Z

Z K(x, y) dν(x) dν(y) = K(xi , xj ).

Xi

Since the functions f :=

Xj

n X j=1

cj χX , µ(Xj ) j

> 0,

belong to L2 (X, ν), the inequality 0 ≤ hK(f ), f i2 =

n X

ci cj

i,j=1

leads to

1 ν(Xi )ν(Xj ) n X

0≤

Z

Z K(x, y) dν(x) dν(y)

Xi

Xj

ci cj K(xi , xj ),

i,j=1

that is, K ∈ P D(X).

Next, we introduce a general formulation for Mercer’s Theorem. We will need some additional notation attached to a measure space (X, ν). Precisely, we will write A(X, ν) to denote the subset of C(X × X) ∩ L2 P D(X, ν) formed by all kernels K : X × X → C for which x ∈ X → K(x, x) is an element of L1 (X, ν). Theorem 2.4 below includes in its statement a generalization of the classical Mercer’s Theorem. The proof we include here uses adaptations of methods introduced in [1, 6, 12, 14, 18]. Recall that an operator T on a Hilbert space H is trace-class ([9]), when its square root |T | := (T ∗ T )1/2 satisfies the condition X h|T |(f ), f iH < ∞, (2.1) f ∈B

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for every orthonormal basis B of (H, h·, ·iH ). The trace acts linearly over the vector space of all trace-class operators over H. If T is trace-class, the sum in (2.1) does not depend upon the basis and it is called the trace of T , here denoted by tr(T ). If T is a compact operator, then |T | is compact, positive and self-adjoint. As so, denoting by {sn (T )} the sequence of eigenvalues ofP|T |, each repeated as often as ∞ its multiplicity, then T is trace-class if and only if n=1 sn (T ) < ∞. As a matter of fact, the sum coincides with the trace of |T | when T is trace-class. If T is also self-adjoint, then sn (T ) = |λn (T )|, n = 1, 2, . . . , so that ∞ X tr (T ) = λn (T ). n=1

An important family of trace-class operators is that encompassing all finite rank operators on H. If H is a separable Hilbert space, the set of all trace-class operators on H is a vector space and the formula ∞ X kT ktr := sn (T ) n=1

defines a norm on it, the so-called trace norm. From now on, all general Hilbert spaces mentioned in the paper are assumed to be separable. Theorem 2.4. Let X be a metric space endowed with a strictly-positive measure ν. If K ∈ A(X, ν) then the following assertions hold: (i) The range of K is a subset of C(X) ∩ L2 (X, ν); (ii) The operator K is compact and selfadjoint, having an L2 (X, ν)-convergent series representation in the form ∞ X K(f ) = λn (K)hf, φn i2 φn , f ∈ L2 (X, ν). n=1

The series is absolutely and uniformly convergent on compact subsets of X; (iii) K has a L2 (X × X, ν × ν)-convergent series representation in the form ∞ X K(x, y) = λn (K)φn (x)φn (y), x, y ∈ X, n=1

{λn (K)} decreases to 0 and {φn } is L2 (X, ν)-orthonormal. The convergence of the series is absolute and uniform on compact subsets of X × X. If λn (K) > 0 then φn is an eigenfunction of K associated with the eigenvalue λn (K), taking into account multiplicities; (iv) The operator K is trace-class and Z tr(K) = K(x, x) dν(x). X

Proof. Assume K ∈ A(X, ν). Since ν is strictly-positive, Theorem 2.3 shows that K ∈ P D(X). In particular, |K(x, y)|2 ≤ K(x, x)K(y, y),

x, y ∈ X.

(2.2)

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If y ∈ X → K(y, y) is integrable, it is quite clear that every function y ∈ X → K(x, y), x ∈ X, belongs to L2 (X, ν). Next, we show that the mapping x ∈ X → K(x, ·) ∈ L2 (X, ν) is continuous. Let {xn } be a sequence in X converging to x0 ∈ X. Since K is continuous, the sequence {K(xn , y)} converges to K(x0 , y), for every y ∈ X fixed. Using (2.2), we deduce that |K(xn , y) − K(x0 , y)|2 ≤ |K(xn , y)|2 + 2|K(xn , y)||K(x0 , y)| + |K(x0 , y)|2 ≤ K(y, y) (K(xn , xn ) + K(x0 , x0 )) + 2K(y, y)K(xn , xn )1/2 K(x0 , x0 )1/2 ≤ 4 sup {K(xm , xm )} K(y, y),

y ∈ X.

m∈Z+

The Dominated Convergence Theorem leads to Z |K(xn , y) − K(x0 , y)|2 dν(x) = 0. lim n→∞

X

The continuity of x ∈ X 7→ K(x, ·) ∈ L2 (X, ν) follows. Since K(f )(x) = hf, K(x, ·)i2 ,

f ∈ L2 (X, ν),

x ∈ X,

assertion (i) follows. Since (X, ν) is a measure space and K ∈ L2 (X × X, ν × ν), the integral operator K : L2 (X, ν) → L2 (X, ν) is compact ([5, p.86]). Being K hermitian, K is selfadjoint. Applying the Spectral Theorem for compact selfadjoint operators ([5, p.93]), we can deduce that K is an L2 (X, ν)-convergent series of the form K(f ) =

∞ X

λn (K)hf, φn i2 φn ,

f ∈ L2 (X, ν),

(2.3)

n=1

where {λn (K)} decreases to 0 and {φn } is L2 (X, ν)-orthonormal. Next, we consider auxiliary kernels Kp , p ≥ 1, given by the formula Kp (x, y) = K(x, y) −

p X

λn (K)φn (x)φn (y),

x, y ∈ X.

n=1

Obviously, Kp ∈ L2 (X × X, ν × ν) ∩ C(X × X) while standard computations show that Kp ∈ A(X, ν). Lemma 2.2 in [6] reveals that Kp (x, x) ≥ 0, x ∈ X, that is, p X

λn (K)|φn (x)|2 ≤ K(x, x),

x ∈ X.

n=1

The inequality p+q 2 p+q X X λ (K)hf, φ i φ (x) ≤ λ (K) sup K(y, y) |hf, φn i2 |2 , n 2 n 1 n=p n y∈Y n=p

x ∈ Y,

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holds whenever Y is a compact subset of X and q, p ≥ 1. As so, the convergence of the series in (2.3) is uniform on compact subsets of X. This takes care of (ii). From the Cauchy-Schwarz inequality we now obtain p+q 2 p+q X X λn (K)φn (x)φn (y) ≤ (K(x, x) + 1) λn (K)|φn (y)|2 , x, y ∈ X, p, q ≥ 1. n=p n=p The Cauchy P∞ Criterion for convergence and the continuity of K imply that the series n=1 λn (K)φn (x)φn (y) is convergent to a function in C(X), when one of the variables is held fixed. However, due to (i) and (ii), ! Z ∞ ∞ X X λn (K)φn (x)φn (y) f (y) dν(y) = λn (K)hf, φn i2 φn (x) = K(f )(x), X

n=1

n=1 2

whenever f ∈ L (X, ν) and x ∈ X. Using this information with a convenient choice for f and recalling our assumption on X, we deduce that ∞ X

λn (K)φn (x)φn (y) = K(x, y),

x, y ∈ X.

n=1

Dini’s Theorem leads to ∞ X

λn (K)|φn (x)|2 = K(x, x),

x ∈ X,

(2.4)

n=1

with uniform and absolute convergence on compact subsets of X. Finally, the Cauchy Criterion for uniform convergence and the Cauchy-Schwarz inequality imply uniform and absolute convergence of the series on compact subsets of X × X. The Monotone Convergence Theorem along with (2.4) resolves (iv). The assumptions listed in the previous theorem are to be assumed from now on. Theorem 2.4 provides basic information on decay rates for the eigenvalues of the integral operator K, at least when K fits the description considered there. That we quote in a separated result. Corollary 2.5. Under the conditions stated in Theorem 2.4, it holds λn (K) = o(n−1 ), as n → ∞.

3. The square root of K In this section, we will list some of the properties the square root K1/2 of the integral operator K has, when K fits the assumptions in Theorem 2.4. Such properties will be used ahead in some key arguments. The existence of K1/2 of K is guaranteed by a well-known result from Hilbert space theory ([19, p.142]).

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Lemma 3.1. Let X and ν be as in Theorem 2.4 and K ∈ A(X, ν). Consider the representation for K provided by Theorem 2.4-(iii). Then K1/2 coincides with the integral operator S : L2 (X, ν) → L2 (X, ν), in which S ∈ L2 P D(X, ν) is the kernel S(x, y) :=

∞ X

λn (K)1/2 φn (x)φn (y),

x, y ∈ X.

(3.1)

n=1

Proof. Due to Theorem 2.4-(iv), it is easily seen that the series ∞ X

λn (K)1/2 φn ⊗ φn ,

n=1

in which φn ⊗ φn (x, y) := φn (x)φn (y), x, y ∈ X, converges in L2 (X × X, ν × ν) (see Theorem 4.11 in [19]. Hence, Formula (3.1) defines an element S in L2 (X×X, ν×ν). On the other hand, due to Theorem 2.4-(ii), Z ∞ X S(x, y)f (y) dν(y) = λn (K)1/2 hf, φn i2 φn (x) X

n=1 1/2

=K

(f )(x), x ∈ X,

f ∈ L2 (X, ν).

Thus, S = K1/2 . The L2 -positive definiteness of S is clear.

Lemma 3.2 below describes a crucial information regarding the range of K

1/2

.

Lemma 3.2. Under the conditions stated in Lemma 3.1, the range of K1/2 is a subset of C(X) ∩ L2 (X, ν). Proof. The proof uses the formula ∞ X K1/2 (f )(x) = λn (K)1/2 hf, φn i2 φn (x),

x ∈ X,

f ∈ L2 (X, ν).

(3.2)

n=1

If λn (K) > 0, Theorem 2.4-(iii) asserts that φn is an eigenfunction of K associated with the eigenvalue λn (K). Hence, Theorem 2.4-(i) implies that φn is continuous. Thus, to reach the continuity of K1/2 , it suffices to show that the series in (3.2) converges uniformly on compact subsets of X. But, that follows from the inequalities p+q 2 p+q p+q X X X λn (K)1/2 hf, φn i2 φn (x) ≤ |λn (K)1/2 φn (x)|2 |hf, φn i2 |2 n=p n=p n=p ≤ hf, f i2

p+q X

λn (K)|φn (x)|2 ,

x ∈ X,

p, q ≥ 1,

n=p

consequences of the Cauchy-Schwarz and Bessel inequalities.

Lemma 3.3 establishes an integral connection between the kernels K and S associated with K and S = K1/2 respectively.

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Lemma 3.3. Under the conditions stated in the previous lemmas, K can be recovered from S through the formula Z S(x, u)S(x, v) dν(x) = K(v, u), u, v ∈ X. X

Proof. Due to Lemma 3.1, S ∈ L2 (X × X, ν × ν). For y ∈ X, define Syj

j X

:=

λn (K)1/2 φn (y) φn ,

j = 1, 2, . . . .

n=1

Since Syj ∈ L2 (X, ν), j = 1, 2, . . . and S(·, y) ∈ L2 (X, ν), it is easily seen that hS(·, y) −

Syj , S(·, y)

−

Syj i2

=

∞ X

λn (K)|φn (y)|2 ,

y ∈ X,

j = 1, 2, . . . .

n=j+1

Due to the continuity of the inner product, it now follows that lim hSuj , Svj i2 = hS(·, u), S(·, v)i2 ,

j→∞

u, v ∈ X.

Meanwhile, the orthonormality of {φn } implies that lim hSuj , Svj i2 = lim

j→∞

j→∞

j X

λn (K)φn (v)φn (u) = K(v, u),

u, v ∈ X.

n=1

By uniqueness, hS(·, u), S(·, v)i2 = K(v, u), u, v ∈ X, and the result follows.

4. Finite rank kernels Let (X, d) be as in the statement of Theorem 2.4. In this section we will deal with the integral operator F generated by the kernel F given by the formula F (x, y) :=

Γ X

1 χC (x)χCn (y), ν(Cn ) n n=1

x, y ∈ X,

(4.1)

in which {Cn : n = 1, 2, . . . , Γ} is a family of subsets of X satisfying the following two requirements: 0 < ν(∪Γn=1 Cn ) < ∞ and ν(Cn ∩ Cl ) = 0, n 6= l. The inequality is needed in order to guarantee that F is an element of L2 (X × X, ν × ν) while the other condition enters in some orthonormality arguments (see the beginning of the proof of Lemma 4.1 below for example). The symbol χCn will stand for the usual characteristic function of Cn . Depending on the family {Cn : n = 1, 2, . . . , Γ}, F can be used to construct a convenient finite rank approximation to K, with respect to the trace norm, at least when K ∈ A(X, ν). That will become clear at the end of the section when we estimate the sum of all eigenvalues of K. The symbol I stands for the identity operator.

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Lemma 4.1. The following assertions hold: (i) The integral operator F : L2 (X, ν) → L2 (X, ν) is positive of rank at most Γ; (ii) The operator I − F is positive. Proof. The set {ν(Cn )−1/2 χCn : n = 1, 2, . . . , Γ} being L2 (X × X, ν × ν)-orthonormal, we can write F in the form F(f ) =

Γ D X

f, ν(Cn )−1/2 χCn

n=1

E 2

ν(Cn )−1/2 χCn ,

f ∈ L2 (X, ν),

(4.2)

and assertion (i) follows. The same representation shows that F is positive, having 0 and 1 as the only possible eigenvalues. As so, it is an operator of norm 1. It is now clear that, for all f ∈ L2 (X, ν), h(I − F)(f ), f i2 = hf, f i2 − hF(f ), f i2 ≥ hf, f i2 − kFkhf, f i2 = 0,

(4.3)

and the proof is complete.

In Lemma 4.2 below we will deal with the operator K1/2 FK1/2 . Lemma 4.2. Let K be an element of A(X, ν). The following assertions hold: (i) K1/2 FK1/2 is an integral operator whose kernel is an element of A(X, ν); (ii) The number Γ is an upper bound for the rank of K1/2 FK1/2 ; (iii) The operator K − K1/2 FK1/2 is positive. Proof. Let us write S := K1/2 . Recalling the proof of Lemma 3.1, we deduce that Z Z Z SFS(f )(x) = S(x, u) F (u, v) S(v, y)f (y) dν(y) dν(v) dν(u), X

X

X

2

whenever f ∈ L (X, ν) and x ∈ X. Due to Fubini’s Theorem, we conclude that SFS is an integral operator on L2 (X, ν), with kernel G ∈ L2 (X × X, ν × ν) given by the formula Z Z G(x, y) = S(x, u)F (u, v)S(v, y) dν(u) dν(v), x, y ∈ X, X

or, alternatively, Z G(x, y) = ∪Γ n=1 Cn

X

Z S(x, u)F (u, v)S(v, y) dν(u) dν(v),

x, y ∈ X.

(4.4)

∪Γ n=1 Cn

Returning to the definition of F , Z Γ Z X χCn (v) χCn (u) dν(v) dν(u) S(v, y) G(x, y) = S(x, u) 1/2 ν(Cn )1/2 ν(Cn ) X n=1 X Z Γ Z X χCn (u) χCn (v) = S(x, u) dν(u) S(y, v) dν(v), 1/2 ν(Cn ) ν(Cn )1/2 X n=1 X

x, y ∈ X.

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It follows that SFS has rank at most Γ. Lema 3.2 reveals that G is continuous while Lemma 4.1-(i) justifies hSFS(f ), f i2 = hFS(f ), S(f )i2 ≥ 0,

f ∈ L2 (X, ν).

In other words, G ∈ L2 P D(X) ∩ C(X × X). To finish the proof, first we use the Cauchy-Schwarz inequality to obtain 2 X Γ Z Γ Z X S(x, u) χCn (u) dν(u) ≤ |S(x, u)|2 dν(u), x ∈ X. 0 ≤ G(x, x) = 1/2 ν(C ) n X X n=1 n=1 Due to Lemma 3.3, it follows that 0 ≤ G(x, x) ≤ K(x, x)Γ,

x ∈ X,

and, therefore, the function x ∈ X → G(x, x) belongs to L1 (X, ν). This takes care of (i) and (ii). From Lemma 4.1-(ii) and (4.3), we can write hK(f ), f i2 = hS(f ), S(f )i2 ≥ hFS(f ), S(f )i2 = hSFS(f ), f i2 ,

f ∈ L2 (X, ν).

Assertion (iii) follows.

In Lemma 4.3 below, we will deduce a formula that allows one to compare the traces of K and K1/2 FK1/2 . Lemma 4.3. If K is an element of A(X, ν) then Z Z Γ X 1 [K(u, u) − K(v, u)] dν(u) dν(v) tr(K) − tr(K1/2 FK1/2 ) = ν(Cn ) Cn Cn n=1 Z + K(u, u) dν(u). X\(∪Γ n=1 Cn )

Proof. If K ∈ A(X, ν) then, due to Lemma 4.2-(i), Theorem 2.4-(iv) can be applied to both K and K1/2 FK1/2 . Hence, Z Z tr(K) − tr(K1/2 FK1/2 ) = K(x, x) dν(x) − G(x, x)dν(x). X

X

We compute these two integrals separately. Employing (4.4), Fubini’s Theorem and then Lemma 3.3, it is not hard to see that Z Z Z G(x, x) dν(x) = F (u, v)K(v, u) dν(u) dν(v) X ZX X Z = F (u, v)K(v, u) dν(u) dν(v). ∪Γ n=1 Cn

∪Γ n=1 Cn

The definition of F reveals that Z Γ X F (u, v) dν(u) = χCn (v) = 1, ∪Γ n=1 Cn

n=1

v ∈ ∪Γn=1 Cn , a.e.,

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so that Z

"Z

Z

#

K(u, u) dν(u) =

F (v, u) dν(v) K(u, u) dν(u)

∪Γ n=1 Cn

∪Γ n=1 Cn

∪Γ n=1 Cn

"Z

Z

#

=

F (u, v) dν(u) K(v, v) dν(v). ∪Γ n=1 Cn

Hence, Z

73

∪Γ n=1 Cn

Z

Z

K(u, u) dν(u) =

K(u, u) dν(u) + X\(∪Γ n=1 Cn )

X

K(u, u) dν(u) ∪Γ n=1 Cn

Z =

K(u, u) dν(u) X\(∪Γ n=1 Cn )

Z

Z

+

F (u, v)K(v, v) dν(u) dν(v). ∪Γ n=1 Cn

Thus, denoting S = K1/2 , Z tr(K − SFS) =

∪Γ n=1 Cn

K(u, u) dν(u)

X\(∪Γ n=1 Cn )

Z

Z F (u, v) [K(v, v) − K(v, u)] dν(u) dν(v).

+ ∪Γ n=1 Cn

∪Γ n=1 Cn

The formula in the statement of the lemma follows from (4.2).

Proposition 4.4 below is an extension of a result on best approximation by finite rank operators, originally found in [15]. Proposition 4.4. Let T be a compact self-adjoint operator on a Hilbert space H and consider its series representation T (f ) =

∞ X

λn (T )hf, φn iH φn ,

f ∈ H,

n=1

as given by the spectral theorem for such operators. If R ∈ L(H) has rank at most k then kT − Rktr ≥ kT − Tk ktr , where Tk ∈ L(H) is the truncated sum Tk (f ) =

k X

λn (T )hf, φn iH φn ,

f ∈ H.

n=1

Proof. Since T − R is compact and self-adjoint, we may consider its spectral representation ∞ X (T − R)(f ) = λn (T − R)hf, ψn iH ψn , f ∈ H. n=1

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Defining A0 = R and Ap (f ) = R(f ) +

p X

λn (T − R)hf, ψn iH ψn ,

f ∈ H,

p = 1, 2, . . . ,

n=1

it is easily seen that Ap has rank at most j + p and (T − Ap )(f ) =

∞ X

λn (T − R)hf, ψn iH ψn ,

f ∈ H.

n=p+1

Using Theorem 2.5 in [9] it is now seen that |λp+1 (T − R)| = kT − Ap k ≥ kT − Tj+p k ≥ |λj+p+1 (T )|,

p = 0, 1, . . . ,

and the proof follows.

Next, using Proposition 4.4, we describe a method to estimate the eigenvalues of K using the family {Cn : n = 1, 2, . . . , Γ} behind the definition of F . Theorem 4.5. Let K be an element of A(X, ν). If {Cn : n = 1, 2, . . . , Γ} is a family of subsets of X such that 0 < ν(∪Γn=1 Cn ) < ∞ and ν(Cn ∩ Cl ) = 0, n 6= l, then ∞ X

λn (K) ≤

n=Γ+1

Γ X

1 ν(Cn ) n=1 Z +

Z

Z [K(u, u) − K(v, u)] dν(u) dν(v)

Cn

Cn

K(u, u) dν(u).

X\(∪Γ n=1 Cn )

Proof. Consider the series representation for K as described in Theorem 2.4-(ii) and write T to denote the operator obtained from the series by truncating it at Γ: T (f )(x) =

Γ X

λn (K)hf, φn i2 φn ,

f ∈ L2 (X, ν).

n=1

Proposition 4.4 implies that ∞ X

λn (K) = kK − T ktr ≤ kK − K1/2 FK1/2 ktr ,

n=Γ+1

in which F is the kernel described in (4.1). Since kK − K1/2 FK1/2 ktr = tr (K) − tr (K1/2 FK1/2 ), the inequality in the statement of the theorem follows from Lemma 4.3. An alternative for the inequality in Theorem 4.5 is provided below.

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Theorem 4.6. Let K be an element of A(X, ν). If {Cn : n = 1, 2, . . . , Γ} is a family of subsets of X such that 0 < ν(∪Γn=1 Cn ) < ∞ and ν(Cn ∩ Cl ) = 0, n 6= l, then Z Z ∞ Γ X X K(u, u) + K(v, v) 1 − K(v, u) dν(u) dν(v) λn (K) ≤ ν(Cn ) Cn Cn 2 n=1 n=Γ+1 Z + K(u, u) dν(u). X\(∪Γ n=1 Cn )

Proof. It is analogous to the proof of Theorem 4.5, but using a version of Lemma 4.3 leading to an inequality involving the kernel 2−1 (K(u, u) + K(v, v)) − K(v, u). The details are left to the readers.

5. Decay rates for the eigenvalues under Lipschitz conditions Keeping the context described in Theorem 2.4, this section describes decay rates for the eigenvalues of the integral operator K, at least when the kernel K comes from A(X, ν) and satisfies a convenient Lipschitz condition. The rates hold when the metric space (X, d) fits in the description below. Let q be a positive integer and t a positive real. The space (X, d) is said to be (q, t)-compact when there exist x0 ∈ X and positive real numbers a, b, c and r0 for which the following condition holds: if N ∈ Z+ and r ≥ r0 there exist a family {Cnr : n = 1, 2, . . . , k(N )} of subsets of X, all having finite measure, such that (i) ν(Cnr ∩ Clr ) = ∅, n 6= l; (ii) d(x, y) ≤ art N −t , x, y ∈ Cnr , n = 1, 2, . . . , k(N ); (iii) k(N ) ≤ bN q ; k(N ) (iv) B[x0 , r c] := {x ∈ X : d(x, x0 ) ≤ r c} = ∪n=1 Cnr . Example.√Let X be a measurable subset of Rm having positive Lebesgue measure. Set a = m, b = 1, c = 1/2 and choose x0 ∈ X. Clearly, B[x0 , r/2] is a subset of the m-dimensional (closed) cube of edge r. Subdividing the cube in N m (not necessarily closed) m-dimensional (disjoint) cubes Qrn , n = 1, 2, . . . , N m := k(N ), then Cnr := Qrn ∩B[x0 , r/2] satisfy the conditions in the definition above with t = 1 and q = m. The number r0 can be any positive real. Example. Results in [7] and [17, p.219] reveal that a subset of the unit sphere S m−1 in Rm , endowed with its usual Lebesgue measure, is (m − 1, 1)-compact. The numbers in the definition are now a = π/2, c = 1, and r0 = 2. The constant b can be 10 while the point x0 can be any point in S m−1 . Example. A similar process can be applied to a subset X of a p-dimensional surface in Rm , endowed with its surface measure. It can be shown that (X, d), in which d is the metric induced by the usual norm of Rm , is (m1 , 1)-compact for some m1 ≤ m. If X is a subset of a p-dimensional C k -manifold M , endowed with some measure which is finite on balls, then using a Whitney-type Theorem ([10, p.54]) it can be shown that (X, d) is (m, 1)-compact whenever m is large enough and

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d(x, y) := d1 (f (x), f (y)), where f : M → Rm+p is an embedding and d1 is a metric in f (M ), induced by the usual norm of Rm+p . Next, we introduce the Lipschitz condition we will adopt. Let α > 0 and s ≥ 0 be constants. A kernel K : X × X → C belongs to the Lipschitz class Lipα,s (X, ν) when the following two conditions hold: (i) There exist δ > 0 and a locally integrable function A : X → [0, +∞] so that |K(x, x) − K(x, y)| ≤ A(x)d(x, y)α , (ii) There exists B ≥ 0 such that Z −s lim sup r r→∞

x, y ∈ X,

A(x) dν(x) ≤ B,

d(x, y) ≤ δ;

y ∈ X.

(5.1)

(5.2)

B[y,r]

The definition above is a weaker version of others in the literature (see [7, 12, 13]). For instance, the first inequality above is easily found in a nonlocal form such as |K(x, y) − K(x, y 0 )| ≤ A(x)d(y, y 0 )α ,

x, y, y 0 ∈ X.

Theorem 5.1. Let X be (q, t)-compact and K ∈ A(X, ν) ∩ Lipα,s (X, ν). Assume there exist β > 0 and C > 0 such that Z lim sup rβ K(x, x) dν(x) ≤ C, y ∈ X. (5.3) r→∞

X\B[y,r]

Define γ := tαβ(β + s + tα)−1 . If N is large enough then there exists a constant C1 > 0 such that ∞ X C1 λn (K) ≤ γ , N n=k(N )+1

for some k(N ) ∈ {0, 1, . . . , bN q }. Proof. Let x0 , a, b, c, and r0 be as in the definition of (q, t)-compactness and let δ, A and B as in the definition of the class Lipα,s (X, ν). Write S := K1/2 . Due to (5.2), there exists a rx0 > 0 such that Z A(x)dν(x) ≤ Brs , r ≥ rx0 . (5.4) B[x0 ,r c]

Without loss of generality we can assume that r0 > rx0 and r0 c > rx0 . For each N ∈ Z+ and r ≥ r0 consider families {Cnr : n = 1, 2, . . . , k(N )} as described in the definition of (q, t)-compactness. Theorem 4.5 implies that ∞ X n=k(N )+1

k(N )

λn (K) ≤

1 ν(Cnr ) n=1 Z + X

Z

Z [K(u, u) − K(v, u)] dν(u) dν(v)

r Cn

X\B[x0 ,r c]

r Cn

K(u, u) dν(u).

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By increasing r0 if necessary, we can use (5.3) to conclude that Z Cc−β . K(u, u) dν(u) ≤ rβ X\B[x0 ,r c] If art N −t < δ, which is always guaranteed when N is large enough, we can use (5.1) to write k(N ) ∞ X X 1 Z Z Cc−β α A(u)d(u, v) . λn (K) ≤ dν(u) dν(v) + ν(Cnr ) Cnr Cnr rβ n=1 n=k(N )+1

It is now clear that ∞ X

k(N )

λn (K) ≤

n=k(N )+1

X n=1

1 ν(Cnr )

Z

Z

r Cn

r Cn

A(u)aα

r tα Cc−β dν(u) dν(v) + N rβ

r tα k(N X) Z Cc−β A(u) dν(u) + ≤a N rβ r n=1 Cn Z r tα Cc−β ≤ aα . A(u) dν(u) + N rβ B[x0 ,r c] α

Recalling (5.4), we finally deduce that ∞ r tα αt+s X Cc−β Cc−β αr Brs + = Ba + , λn (K) ≤ aα N rβ N αt rβ

(5.5)

n=k(N )+1

as long as r ≥ r0 and N is large enough. To conclude the proof, we will apply the above estimate using a special choice of r. Precisely, we will put r = r(N ) := N αt/(β+αt+s) . Since limN →∞ r(N ) = ∞, r(N ) ≥ r0 when N is large enough. Since r(N ) = 0, N the inequality art N −t < δ can be equally captured. Since σ := αt/(β + αt + s) satisfies αt − σ(αt + s) = σβ, inequality (5.5) takes the form ∞ X Baα + Cc−β λn (K) ≤ αtβ/(αt+β+s) . N lim

N →∞

n=k(N )+1

A special case is as follows. Theorem 5.2. Let X be (q, t)-compact and K ∈ A(X, ν) ∩ Lipα,s (X, ν). If either X is bounded or K vanishes outside of a bounded set and N is large enough then there exists a constant C1 > 0 such that ∞ X C1 λn (K) ≤ tα , N n=k(N )+1

for some k(N ) ∈ {0, 1, . . . , bN q }.

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Proof. Under either condition mentioned in the statement of the theorem, there exists r0 > 0 such that Z K(u, u) dν(u) = 0, r > r0 , y ∈ X. X\B[y,r]

Repeating the arguments used in the proof of Lemma 5.1 and adjusting r0 , if necessary, inequality (5.5) reduces itself to ∞ X

λn (K) ≤ Baα

n=k(N )+1

rαt+s N αt

as long as r ≥ r0 and N is large enough. In particular, ∞ X

λn (K) ≤ Baα

n=k(N )+1

(r0 + 1)αt+s N αt

for N arbitrarily large.

In order to re-phrase the previous results in a language a little bit more familiar, we will need a lemma ([7]). Lemma 5.3. Let {an } be a non-increasing sequence of nonnegative real numbers. Let l, q and N0 be nonnegative integers, p a positive integer at least 1 and γ ∈ R. Suppose there exists a constant C > 0 satisfying the following property: if N ≥ N0 , there exists k(N ) ≤ pN q such that ∞ X

an ≤

n=k(N )+l+1

C . Nγ

Then, the set {n1+γ/q an : n = 1, 2, . . .} is bounded. In particular, an = O(n−1−γ/q ),

as n → ∞.

The main results of the paper are as follows. Theorem 5.4. Let X be (q, t)-compact and K ∈ A(X, ν) ∩ Lipα,s (X, ν). If there exist β > 0 and C ≥ 0 such that Z β lim sup r K(x, x) dν(x) ≤ C, y ∈ X, (5.6) r→∞

X\B[y,r]

then λn (K) = O(n−1−γ/q ), −1

where γ := tαβ(β + s + tα)

as n → ∞,

(5.7)

.

Proof. This follows from Theorem 5.1 and Lemma 5.3.

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Theorem 5.5. Let X be (q, t)-compact and K ∈ A(X, ν)∩Lipα,s (X, ν). If for every β > 0 there exists C = C(β) ≥ 0 such that Z lim sup rβ K(x, x) dν(x) ≤ C, y ∈ X, (5.8) r→∞

X\B[y,r]

then λn (K) = o(n−1−θ/q ),

as n → ∞,

β , β + s + tα

β ∈ [0, ∞),

whenever θ ∈ [0, tα). Proof. The function γ(β) := tα

is continuous with range [0, tα). As so, the previous theorem implies that λn (K) = O(n−1−θ/q ),

as n → ∞,

λn (K) 6= o(n−1−γ0 /q ),

as n → ∞,

whenever θ ∈ [0, tα). If

for some γ0 ∈ [0, tα), then there would exist C > 0 such that lim sup{n−1−γ0 /q λn (K)} ≥ C. n→∞

But this would imply in unbounded sequences {n1+θ/q λn (K)} when θ ∈ (γ0 , tα), a clear contradiction. Theorem 5.6. Let X be (q, t)-compact and K ∈ A(X, ν) ∩ Lipα,s (X, ν). If either X is bounded or K vanishes outside a bounded set then λn (K) = O(n−1−tα/q ),

as n → ∞.

Proof. This follows from Theorem 5.2 and Lemma 5.3.

The reader is advised that the results above generalize some of the results proved in [2] to the multi-dimensional case. There, the authors use a particular case of Theorem 4.6 to obtain decay rates for the eigenvalues of K under differentiability hypotheses. We intend to use Theorem 4.6 to investigate multi-dimensional versions of such context in a future work. Lets return to the context of Rm . If X is a finite union of convex subsets m of R and the assumption K ∈ Lipα,s (X, µ) is changed to the existence and boundedness of ∂K/∂x then the mean value inequality and the continuity of K show that K ∈ Lip1,m (Xl , µ), in which Xl is a convex component of X. In view of this, some of the proofs can be adapted to show that the estimate (5.7) holds with γ := β(β + m + 1)−1 . Finally, we would like to observe that the existence of a β > 0 so that lim sup |x|β+m K(x, x) < ∞ |x|→∞

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implies condition (5.6) in Theorem 5.4. Here, | · | stands for the usual norm in Rm . A similar remark applies to condition (5.8) in Theorem 5.5.

References [1] Buescu, J., Positive integral operators in unbounded domains. J. Math. Anal. Appl. 296 (2004), no. 1, 244–255. [2] Buescu, J.; Paix˜ ao, A. C., Eigenvalues of positive definite integral operators on unbounded intervals. Positivity 10 (2006), no. 4, 627–646. [3] Buescu, J.; Paix˜ ao, A. C., Eigenvalue distribution of positive definite kernels on unbounded domains. Integral Equations Operator Theory 57 (2007), no. 1, 19–41. [4] Buescu, J.; Paix˜ ao, A. C., Eigenvalue distribution of Mercer-like kernels. Math. Nachr. 280 (2007), no. 9-10, 984–995. [5] Cheney, E. W., Analysis for applied mathematics. Graduate Texts in Mathematics, 208. Springer-Verlag, New York, 2001. [6] Ferreira, J. C.; Menegatto, V. A.; Oliveira, C. P., On the nuclearity of integral operators, Positivity, to appear. [7] Ferreira, J. C.; Menegatto, V. A.; Peron, A. P., Integral operators on the sphere generated by positive definite smooth kernels, J. Complexity, 24 (2008), no. 5-6, 632– 647. [8] Folland, G. B., Real analysis. Modern techniques and their applications. Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. [9] Gohberg, I.; Goldberg, S.; Krupnik, N., Traces and determinants of linear operators. Operator Theory: Advances and Applications, 116. Birkh¨ auser Verlag, Basel, 2000. [10] Guillemin, V.; Pollack, A., Differential Topology. Prentice Hall, 1974. [11] K¨ onig, H., Eigenvalue distribution of compact operators. Operator Theory: Advances and Applications, 16. Birkh¨ auser Verlag, Basel, 1986. [12] K¨ uhn, T., Eigenvalues of integral operators with smooth positive definite kernels. Arch. Math. (Basel), 49 (1987), no. 6, 525–534. [13] K¨ uhn, T., Eigenvalues of integral operators generated by positive definite H¨ older continuous kernels on metric compacta. Indag. Math. 49 (1987), no. 1, 51–61. [14] Novitski˘ı, I. M., Representation of kernels of integral operators by bilinear series. (Russian) Sibirsk. Mat. Zh. 25 (1984), no. 5, 114–118. [15] Reade, J. B., Eigenvalues of positive definite kernels. SIAM J. Math. Anal. 14 (1983), no. 1, 152–157. [16] Reade, J. B., On the sharpness of Weyl’s estimate for eigenvalues of smooth kernels. SIAM J. Math. Anal., vol. 16 (1985), no. 3, 137–142. [17] Reimer, M., Multivariate polynomial approximation, Vol. 144, Birkh¨ auser, Berlin, 2003. [18] Sun, Hongwei, Mercer theorem for RKHS on noncompact sets. J. Complexity 21 (2005), no. 3, 337–349.

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[19] Young, N., An introduction to Hilbert space. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge, 1988. J. C. Ferreira and V. A. Menegatto Departamento de Matem´ atica ICMC-USP - S˜ ao Carlos Caixa Postal 668 13560-970 S˜ ao Carlos SP Brasil e-mail: [email protected] [email protected] Submitted: September 17, 2008. Revised: March 4, 2009.

Integr. equ. oper. theory 64 (2009), 83–113 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010083-31, published online April 24, 2009 DOI 10.1007/s00020-009-1678-x

Integral Equations and Operator Theory

Generalized Polar Decompositions for Closed Operators in Hilbert Spaces and Some Applications Fritz Gesztesy, Mark Malamud, Marius Mitrea and Serguei Naboko Abstract. We study generalized polar decompositions of densely defined closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators. Mathematics Subject Classification (2000). Primary: 47A05, 47A07; Secondary: 47A55. Keywords. Polar decomposition, relatively bounded and relatively form bounded perturbations, relatively compact and relatively form compact perturbations.

1. Introduction This paper had its origin in attempts of proving that certain operators of the type (A + IH )−1/2 B(A + IH )−1/2 ,

(1.1)

in a complex, separable Hilbert space H (where S denotes the closure of the operator S and IH is the identity operator in H), are bounded, respectively, compact, where A > 0 is self-adjoint in H, and B is a densely defined closed operator in H. To prove such a result, it became desirable to replace the standard polar decomposition of B (cf. [5, Sect. IV.3], [11, Sect. VI.2.7]), B = U |B| = |B ∗ |U on dom(B) = dom(|B|),

(1.2)

by some modified polar decomposition of the type B = |B ∗ |1/2 U |B|1/2 on dom(B) = dom(|B|),

(1.3)

Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 and FRG-0456306, and the Austrian Science Fund (FWF) under Grant No. Y330.

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and then reduce boundedness, respectively, compactness of the operator (1.1) to that of |B|1/2 (A + IH )−1/2 and |B ∗ |1/2 (A + IH )−1/2 . (1.4) With (1.3) in mind, it is natural to try to establish that, in fact, the following version of (1.3) holds B = |B ∗ |α U |B|1−α on dom(B) = dom(|B|)

(1.5)

for all α ∈ [0, 1]. In fact, after this was accomplished, it became clear that the following rather general polar-type decomposition can be established B = φ(|B ∗ |)U ψ(|B|) on dom(B) = dom(|B|),

(1.6)

where φ and ψ are Borel functions on R with the property that φ(λ)ψ(λ) = λ, λ ∈ R, and such that dom(|B|) ⊆ dom(ψ(|B|)). Finally, an even more general version of (1.6) is to show that an operator T introduced as T = V A1 = A2 V on dom(T ) = dom(A1 ), (1.7) also has the representation T = φ(A2 )V ψ(A1 ) on dom(T ) = dom(A1 )

(1.8)

for any pair of self-adjoint (in fact, also normal) operators Aj , j = 1, 2, and any bounded operator V satisfying V dom(A1 ) ⊆ dom(A2 ), assuming also dom(A1 ) ⊆ dom(ψ(A1 )) (cf. Theorems 2.1 and 2.3 for details). In Section 2 we provide proofs of (1.6) and (1.8), and in Section 3 we discuss some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint operators. In the final Section 4 we discuss some applications to m-sectorial operators.

2. Generalized polar decompositions To set the stage, let Hj , j = 1, 2, be two separable complex Hilbert spaces with scalar products and norms denoted by (·, ·)Hj and k · kHj , j = 1, 2, respectively. The identity operators in Hj are written as IHj , j = 1, 2. We denote by B(H1 , H2 ) (resp., B∞ (H1 , H2 )) the Banach space of linear bounded (resp., compact) operators from H1 into H2 . If H1 = H2 = H, these spaces are denoted by B(H) (resp., B∞ (H)). The domain, range, kernel (null space), resolvent set, and spectrum of a linear operator will be denoted by dom(·), ran(·), ker(·), ρ(·), and σ(·), respectively. Finally, we let S stand for the closure of an operator S. We assume that Aj are self-adjoint operators in Hj with domains dom(Aj ), j = 1, 2,

(2.1)

and that V ∈ B(H1 , H2 )

(2.2)

V dom(A1 ) ⊆ dom(A2 ).

(2.3)

satisfies

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In addition, suppose that V A1 = A2 V on dom(A1 ).

(2.4)

Next, given a self-adjoint operator A in a complex separable Hilbert space H, we denote by {EA (λ)}λ∈R the family of spectral projections associated with A, and we introduce the function ρf by ( R → [0, ∞), f ∈ H. (2.5) ρf : λ 7→ kEA (λ)f k2H , Clearly, ρf is bounded, non-decreasing, right-continuous, and lim ρf (λ) = 0,

λ↓−∞

lim ρf (λ) = kf k2H ,

λ↑∞

f ∈ H.

(2.6)

Hence, ρf generates a measure, denoted by dρf , in a canonical manner. A function φ : R → C is then called dEA -measurable if it is dρf -measurable for all f ∈ H. Standard examples of dEA -measurable functions are all continuous functions, all step functions, all pointwise limits of step functions, and all Borel measurable functions. Given a dEA -measurable function φ, the operator φ(A) is then defined in terms of the spectral representation of A as usual by Z Z dkEA (λ)f k2H |φ(λ)|2 < ∞ . φ(A) = dEA (λ) φ(λ), dom(φ(A)) = f ∈ H R

R

(2.7)

Our first result then reads as follows: Theorem 2.1. Suppose Aj , j = 1, 2, and V satisfy (2.1)–(2.4), and consider the operator T given by T = V A1 = A2 V on dom(T ) = dom(A1 ).

(2.8)

(i) If ψ is both a dEA1 - and dEA2 -measurable function on R, then V dom(ψ(A1 )) ⊆ dom(ψ(A2 ))

(2.9)

V ψ(A1 ) = ψ(A2 )V on dom(ψ(A1 )).

(2.10)

and (ii) Assume that φ and ψ are simultaneously dEA1 - and dEA2 -measurable functions on R such that φ(λ)ψ(λ) = λ,

λ ∈ R,

(2.11)

and dom(A1 ) ⊆ dom(ψ(A1 )).

(2.12)

T = φ(A2 )V ψ(A1 ) on dom(T ) = dom(A1 ).

(2.13)

Then

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Proof. Since V A1 ⊆ A2 V , one infers V (A1 − zIH1 ) ⊆ (A2 − zIH2 )V and hence V (A1 − zIH1 )−1 = (A2 − zIH2 )−1 V,

z ∈ C\R.

(2.14)

In the following we denote by {EAj (λ)}λ∈R the family of (strongly right-continuous) spectral projections of the self-adjoint operators Aj , j = 1, 2. Then, the representation (cf. [11, Sect. VI.5.2]) EAj (λ) = IHj − 21 Uj (λ) + Uj (λ)2 , λ ∈ R, j = 1, 2, (2.15) where 2 ε↓0, R↑∞ π

Z

Uj (λ) = s-lim

R

−1 dη (Aj −λIHj ) (Aj −λIHj )2 +η 2 IHj ,

λ ∈ R, j = 1, 2,

ε

(2.16) (here s-lim denotes the strong limit in Hj ) yields V EA1 (λ) = EA2 (λ)V,

λ ∈ R.

Next, choose f ∈ dom(ψ(A1 )). Then Z R Z R dkEA2 (λ)V f k2H2 |ψ(λ)|2 = dkV EA1 (λ)f k2H2 |ψ(λ)|2 −R −R Z dkV EA1 (λ)f k2H2 |ψ(λ)|2 ≤ kV k2B(H1 ,H2 ) kψ(A1 )f k2H1 . −→ R↑∞

(2.17)

(2.18)

R

Thus, f ∈ dom(ψ(A1 )) implies V f ∈ dom(ψ(A2 )), proving (2.9). Choosing f ∈ dom(ψ(A1 )) and g ∈ H2 then yields Z (V ψ(A1 )f, g)H2 = d(V EA1 (λ)f, g)H2 ψ(λ) ZR = d(EA2 (λ)V f, g)H2 ψ(λ) R

= (ψ(A2 )V f, g)H2 ,

(2.19)

and hence (2.10) is proven. Finally, (2.13) follows from (2.10)–(2.12) since φ(A2 )V ψ(A1 ) = φ(A2 )ψ(A2 )V = A2 V = T, concluding the proof.

(2.20)

Remark 2.2. (i) The crucial intertwining relation (2.17) also follows from (2.14) and the Stieltjes inversion formula for (finite) complex measures (cf., e.g., [23, App. B]). Indeed, Z Z d(V EA1 (λ)f, g)H2 (λ − z)−1 = d(EA2 (λ)V f, g)H2 (λ − z)−1 , (2.21) R R z ∈ C\R, f ∈ H1 , g ∈ H2 , implies d(V EA1 (·)f, g)H2 = d(EA2 (·)V f, g)H2 ,

f ∈ H1 , g ∈ H2 ,

(2.22)

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and hence (2.17). (ii) In the special case where in addition to (2.1)–(2.4), Aj are bounded, Aj ∈ B(Hj ), j = 1, 2, one can also derive (2.13) for functions φ and ψ continuous in an open neighborhood of the spectra of A1 and A2 using a Stone–Weierstrass approximation argument. Now we turn our attention to a pair of normal operators Aj , j = 1, 2, with the aim of proving the analog of Theorem 2.1 in this case. For an extensive treatment of normal operators and the spectral family and spectral theorem associated with them, we refer to [23, Sects. 5.6 and 7.5]. Thus, we assume that Aj are normal operators in Hj with domains dom(Aj ), j = 1, 2, (i.e.,

Aj A∗j

=

A∗j Aj

and dom(Aj ) =

dom(A∗j ),

(2.23)

j = 1, 2) such that

ρ(A1 ) ∩ ρ(A2 ) 6= ∅.

(2.24)

V ∈ B(H1 , H2 )

(2.25)

V dom(A1 ) ⊆ dom(A2 ),

(2.26)

In addition, suppose that satisfies and assume that V A1 = A2 V on dom(A1 ). (2.27) Given a normal operator A in a complex separable Hilbert space H we denote by {EA (ν)}ν∈C the family of spectral projections associated with A. We recall that (A + A∗ )/2 and (A − A∗ )/(2i) are self-adjoint, in addition, we denote by {E(A+A∗ )/2 (λ)}λ∈R and {E(A−A∗ )/(2i) (λ)}λ∈R the corresponding family of spectral projections. Then the family of spectral projections {EA (ν)}ν∈C for the normal operator A is given by (cf. [23, Theorem 7.32]) EA (ν) = E(A+A∗ )/2 (λ) E(A−A∗ )/(2i) (µ) = E(A−A∗ )/(2i) (µ) E(A+A∗ )/2 (λ), ν = λ + iµ ∈ C, λ, µ ∈ R.

(2.28)

In analogy to the self-adjoint case one then defines the function τf by ( C → [0, ∞), τf : f ∈ H. (2.29) ν 7→ kEA (ν)f k2H , As discussed in [23, Appendix A.1], introducing N = L × M = {z ∈ C | Re(z) ∈ L, Im(z) ∈ M }

(2.30)

for arbitrary intervals L, M ⊆ R, then τf (N ) = kEA (N )f k2H = kE(A+A∗ )/2 (L) E(A−A∗ )/(2i) (M )f k2H

(2.31)

defines a regular interval function and hence a measure dτf for each f ∈ H. A function φ : C → C is then called dEA -measurable if it is dτf -measurable for all f ∈ H.

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Theorem 2.3. Suppose Aj , j = 1, 2, and V satisfy (2.23)–(2.27), and consider the operator T given by T = V A1 = A2 V on dom(T ) = dom(A1 ).

(2.32)

(i) If ψ is both a dEA1 - and dEA2 -measurable function on C then V dom(ψ(A1 )) ⊆ dom(ψ(A2 ))

(2.33)

and V ψ(A1 ) = ψ(A2 )V on dom(ψ(A1 )). (2.34) (ii) Assume that φ and ψ are simultaneously dEA1 - and dEA2 -measurable functions on C such that φ(λ)ψ(λ) = λ, λ ∈ C, (2.35) and dom(A1 ) ⊆ dom(ψ(A1 )). (2.36) Then T = φ(A2 )V ψ(A1 ) on dom(T ) = dom(A1 ). (2.37) Proof. The idea of the proof is to try to reduce the case of normal operators to that of self-adjoint ones treated in Theorem 2.1. With this goal in mind, pick z ∈ C\(σ(A1 ) ∪ σ(A2 )) for the remainder of this proof. Then V A1 ⊆ A2 V implies again V (A1 − zIH1 )−1 = (A2 − zIH2 )−1 V, (2.38) and hence also V eiζ(A1 −zIH1 )

−1

−1

= eiζ(A2 −zIH2 ) V,

ζ ∈ C,

(2.39)

applying (2.38) repeatedly to all terms in the norm convergent Taylor expansion of both exponentials in (2.39). (Here ζ denotes the complex conjugate of ζ ∈ C.) In particular, −1

−1

V = eiζ(A2 −zIH2 ) V e−iζ(A1 −zIH1 ) ,

ζ ∈ C.

(2.40)

Thus, one obtains ∗

−1

∗

eiζ(A2 −z) V e−iζ(A1 −z) ∗

−1

−1

−1

−1

∗

= eiζ(A2 −z) eiζ(A2 −zIH2 ) V e−iζ(A1 −zIH1 ) e−iζ(A1 −z) = eiB2 (ζ) V e−iB1 (ζ) ,

−1

ζ ∈ C,

(2.41)

where we have set B1 (ζ) = ζ(A∗1 − z)−1 + ζ(A1 − zIH1 )−1 = B1 (ζ)∗ , B2 (ζ) = ζ(A∗2 − z)−1 + ζ(A2 − zIH2 )−1 = B2 (ζ)∗ . Consequently,

iζ(A∗ −z)−1 −iζ(A∗ −z)−1

e

2 1 Ve = kV kB(H1 ,H2 ) , B(H1 ,H2 )

ζ ∈ C.

(2.42)

(2.43)

Since the left-hand side of (2.41) is entire with respect to ζ ∈ C, the uniform boundedness in (2.43) and Liouville’s theorem yield that the left-hand side of

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(2.41) is actually constant with respect to ζ ∈ C. Thus, the left-hand side of (2.41) equals its value at ζ = 0, allowing one to conclude that ∗

−1

∗

eiζ(A2 −z) V e−iζ(A1 −z)

−1

= V,

ζ ∈ C.

(2.44)

Differentiating (2.44) with respect to ζ and subsequently taking ζ = 0, then yields V (A∗1 − z)−1 = (A∗2 − z)−1 V,

(2.45)

and consequently, V A∗1 = A∗2 V on dom(A∗1 ). Equations (2.32) and (2.46) together imply

(2.46)

V (A1 ± A∗1 ) = (A2 ± A∗2 )V on dom(A1 ) = dom(A∗1 ).

(2.47)

Next we will show that (2.47) extends to the closures of Aj ± A∗j , j = 1, 2, as follows: First, we note that Aj ± A∗j , j = 1, 2, are symmetric and hence closable. Next, pick arbitrary f± ∈ dom A1 ± A∗1 and let f±,n ∈ dom(A1 ) = dom(A∗1 ) be such that

lim kf±,n −f± kH1 = 0 and lim (A1 ±A∗1 )f±,n −(A1 ± A∗ )f± = 0. (2.48) n→∞

1

n→∞

H1

Given that V ∈ B(H1 , H2 ), one also has

lim kV f±,n − V f± kH = 0 and lim V (A1 ± A∗1 )f±,n − V (A1 ± A∗ )f± 2

n→∞

1

n→∞

H2

= 0.

(2.49) Since A2 ± A∗2 are closable and

lim (A2 ± A∗2 )V f±,n − V (A1 ± A∗1 )f± H2 n→∞

= lim V (A1 ± A∗1 )f±,n − V (A1 ± A∗1 )f± H2 = 0,

(2.50)

n→∞

one obtains V f±,n ∈ dom A2 ± A∗2

and

lim (A2 ± A∗2 )V f±,n − (A2 ± A∗2 )V f± H2 = 0,

n→∞

(2.51) and thus, (2.52) V (A1 ± A∗1 )f± = (A2 ± A∗2 )V f± . ∗ Upon recalling that f± ∈ dom A1 ± A1 were arbitrary, this finally implies that V (A1 ± A∗1 ) = (A2 ± A∗2 )V on dom A1 ± A∗1 . (2.53) Next, we recall that (Aj + A∗j )/2 and (Aj − A∗j )/(2i), j = 1, 2, are selfadjoint, and we denote by {E(Aj +A∗ )/2 (λ)}λ∈R and {E(Aj −A∗ )/(2i) (λ)}λ∈R , j = j j 1, 2, the corresponding family of spectral projections. Analogously to (2.28), the families of spectral projections {EAj (ν)}ν∈C for the normal operators Aj , j = 1, 2, are given by EAj (ν) = E(Aj +A∗ )/2 (λ) E(Aj −A∗ )/(2i) (µ) = E(Aj −A∗ )/(2i) (µ) E(Aj +A∗ )/2 (λ), j

j

j

j

ν = λ + iµ ∈ C, λ, µ ∈ R, j = 1, 2.

(2.54)

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As in the proof of (2.17), equations (2.53) then yield V E(A1 +A∗ )/2 (λ) = E(A2 +A∗ )/2 (λ)V, 1

2

V E(A1 −A∗ )/(2i) (µ) = E(A1 −A∗ )/(2i) (µ)V, 2

2

λ, µ ∈ R.

(2.55)

From (2.54) and (2.55) one then deduces that V EAj (ν) = EAj (ν)V,

ν ∈ C, j = 1, 2.

(2.56)

With this in hand, the proof is then completed by following the last part of the RR R proof of Theorem 2.1 step by step (replacing −R by |ν|6R , etc.). Remark 2.4. We note that the strategy just employed to prove that (2.32) implies (2.46) is essentially outlined in the special context of similarity and unitarity of normal operators (where A2 = A1 ) in [23, p. 219]. After completing this proof, we became aware of the detailed history of this type of results: Apparently, Fuglede [6] first proved that V A ⊆ AV , with V bounded and A normal, implies V A∗ ⊆ A∗ V . This was extended by Putnam [15] to the result at hand, viz., V A1 ⊆ A2 V , with V bounded and Aj normal, j = 1, 2, implies V A∗1 ⊆ A∗2 V . Finally, the proof of (2.46) we presented is basically due to Rosenblum [19]. For the convenience of the reader (and for some measure of completeness) we decided to keep the short proof of (2.46). For a detailed history of this circle of ideas we refer to [16, p. 9–11]. Remark 2.5. For Aj ∈ B(Hj ) (Aj not necessarily normal), j = 1, 2, and functions φ, ψ analytic in an open neighborhood of the spectra of A1 and A2 , one can also use the Dunford–Taylor functional calculus (see, e.g., [3, Sect. VII.3]) to prove (2.37). To make the connection with the polar decomposition of densely defined closed operators in Hilbert spaces, and some of its generalizations, which originally motivated the writing of this paper, we next recall a few facts: Given a densely defined closed linear operator S : dom(S) → H2 , dom(S) ⊆ H1 , the self-adjoint operator |S| is defined as usual by |S| = (S ∗ S)1/2 ≥ 0.

(2.57)

Moreover, we denote by PM the orthogonal projection onto the closed linear subspace M of a Hilbert space. The basic facts about the polar decomposition of closed linear operators then read as follows: Theorem 2.6. ([11, Sect. VI.2.7] (see also [5, Sect. IV.3])) Let T : dom(T ) ⊆ H1 → H2 be a densely defined closed linear operator. Then, T = U |T | = |T ∗ |U = U T ∗ U on dom(T ) = dom(|T |), ∗

∗

∗

∗

∗

T = U |T | = |T |U = U T U ∗

∗

∗

∗

∗

on dom(T ) = dom(|T |),

∗

|T | = U T = T U = U |T |U on dom(|T |), ∗

∗

∗

|T | = U T = T U = U |T |U

∗

(2.58) ∗

∗

on dom(|T |),

(2.59) (2.60) (2.61)

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where U ∗ U = Pran(|T |) = Pran(T ∗ ) ,

U U ∗ = Pran(|T ∗ |) = Pran(T ) .

(2.62)

In particular, U is a partial isometry with initial set ran(|T |) and final set ran(T ). Identifying V = U , A1 = |T |, A2 = |T ∗ |, Theorem 2.1 immediately implies the following generalized polar decomposition of T in (2.58): Theorem 2.7. Let T : dom(T ) → H2 , dom(T ) ⊆ H1 be a densely defined closed operator with polar decomposition as in (2.58). In addition, assume that φ and ψ are Borel functions on R such that φ(λ)ψ(λ) = λ, λ ∈ R, and dom(|T |) ⊆ dom(ψ(|T |)). Then T has the representation T = φ(|T ∗ |)U ψ(|T |) on dom(T ) = dom(|T |).

(2.63)

In particular, for each α ∈ [0, 1], T = |T ∗ |α U |T |1−α on dom(T ) = dom(|T |).

(2.64)

Remark 2.8. We note that in the case of a bounded operator T , (2.64) also follows from (2.58) and a Stone–Weierstrass-type approximation argument. More precisely, approximating the functions λ 7→ λα uniformly by a sequence of polynomials on a compact interval then yields (2.64) in analogy to the treatment in [22, p. 6–7] in connection with contractions and their associated defect operators. Closely related to this circle of ideas is the proof of the identity T f (|T |) = f (|T ∗ |)T,

(2.65)

for bounded operators T and continuous functions f on [0, ∞) in the proof of Lemma 9.5.1 in [2], using a Weierstrass approximation argument. Remark 2.9. The symmetric case α = 1/2 in (2.64) (which will play a special role in the following sections) permits a fairly simple and direct proof that we briefly sketch next: Define R = U ∗ |T ∗ |1/2 U . Then R ≥ 0 and R is densely defined since dom(R) = dom(|T ∗ |1/2 U ) ⊇ dom(|T ∗ |U ) = dom(T ). Thus one concludes that R is symmetric, R∗ ⊇ U ∗ |T ∗ |1/2 U = R. In addition, R∗ R ⊇ U ∗ |T ∗ |1/2 U U ∗ |T ∗ |1/2 U = U ∗ |T ∗ |U = |T |,

(2.66)

using the second relation in (2.62), U U ∗ = Pran(|T ∗ |) = Pran(|T ∗ |1/2 ) . Thus (R)∗ R ⊇ |T |, and since |T | is self-adjoint and hence maximal, one obtains (R)∗ R = |T |. In exactly the same manner one infers RR∗ ⊇ U ∗ |T ∗ |1/2 U U ∗ |T ∗ |1/2 U = U ∗ |T ∗ |U = |T |, ∗

∗

(2.67)

∗

hence, R(R) ⊇ |T | and thus, (R) R = R(R) = |T |. That is, R is normal and symmetric, and hence self-adjoint. Since in addition, R ≥ 0, R is the unique selfadjoint, nonnegative square root of |T |, R = (R)∗ = |T |1/2 .

(2.68)

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Introducing S = U |T |1/2 U ∗ , one obtains analogously, S = (S)∗ = |T ∗ |1/2 .

(2.69)

Thus, |T |1/2 ⊇ R = U ∗ |T ∗ |1/2 U,

|T ∗ |1/2 ⊇ S = U |T |1/2 U ∗ .

(2.70)

Next, using also the first relation in (2.62), U ∗ U = Pran(|T |) = Pran(|T |1/2 ) , one infers U ∗ |T ∗ |1/2 ⊇ U ∗ U |T |1/2 U ∗ = |T |1/2 U ∗ ,

|T |1/2 U ∗ ⊇ U ∗ |T ∗ |1/2 U U ∗ = U ∗ |T ∗ |1/2 , (2.71)

and hence U ∗ |T ∗ |1/2 = |T |1/2 U ∗ , implying U U ∗ |T ∗ |1/2 = |T ∗ |1/2 = U |T |1/2 U ∗ .

(2.72)

But then, |T ∗ |1/2 U |T |1/2 = U |T |1/2 U ∗ U |T |1/2 = U |T | = T,

(2.73)

as was to be proven.

3. Some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint operators The symmetric version T = |T ∗ |1/2 U |T |1/2 on dom(T ) = dom(|T |)

(3.1)

of equation (2.64) permits some applications to relatively (form) bounded and compact perturbations of a self-adjoint operator which we briefly discuss in this section. The first application concerns circumstances in which relatively bounded perturbations are also relatively form bounded perturbations of a self-adjoint operator. While, as noted in [11, Sect. VI.1.7], there seems to be no general connection between relative boundedness and relative form boundedness, such a connection does exist for symmetric perturbations of a self-adjoint operator (cf. [11, Sect. VI.1.7] and [17, Sect. X.2]). Here we add another result of this type. To set the stage, we briefly recall the notion of relatively bounded and relatively form bounded perturbations of an operator A in some complex separable Hilbert space H. For simplicity we will actually assume that A is a closed operator with nonempty resolvent set for the remainder of this section. We recall the following definition: Definition 3.1. (i) Suppose that A is a closed operator in H and ρ(A) 6= ∅. An operator B in H is called relatively bounded (resp., relatively compact ) with respect to A (in short, B is called A-bounded (resp., A-compact )), if dom(B) ⊇ dom(A) and B(A − zIH )−1 ∈ B(H) (resp., ∈ B∞ (H)),

z ∈ ρ(A). (3.2)

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(ii) Assume, in addition, that A is self-adjoint in H and bounded from below, that is, A > cIH for some c ∈ R. Then a densely defined and closed operator B in H is called relatively form bounded (resp., relatively form compact ) with respect to A (in short, B is called A-form bounded (resp., A-form compact )), if dom |B|1/2 ⊇ dom |A|1/2 (3.3) and |B|1/2 ((A + (1 − c)IH ))−1/2 ∈ B(H) (resp., ∈ B∞ (H)). In particular, B is A-form bounded (resp., A-form compact), if and only if |B| is. We emphasize that in Definition 3.1 (ii), since A1/2 and |B|1/2 are closed, dom |B|1/2 ⊇ dom A1/2 already implies |B|1/2 ((A + (1 − c)IH ))−1/2 ∈ B(H) (cf. [11, Remark IV.1.5]), and hence the first condition in (3.3) suffices in the relatively form bounded context. In this context we note that in the special case where B is self-adjoint, condition (i) in the definition used by Reed and Simon [17, p. 168] already implies their condition (ii). In fact, it implies a bit more, namely, the existence of α ≥ 0 and β ≥ 0, such that

|B|1/2 f, sgn(B)|B|1/2 f ≤ |B|1/2 f 2 ≤ α |A|1/2 f 2 + βkf k2H , H H H (3.4) f ∈ dom |A|1/2 . Similarly, if B is closed (in fact, closability of B suffices) in Definition 3.1 (i), then the first condition dom(B) ⊇ dom(A) in (3.2) already implies B(A−zIH )−1 ∈ B(H), z ∈ ρ(A), and hence the A-boundedness of B. Using the polar decomposition of B (i.e., B = U |B|), one observes that B is A-bounded (resp., A-compact) if and only if |B| is A-bounded (resp., A-compact). We recall that in connection with relative boundedness, (3.2) can be replaced by the condition dom(B) ⊇ dom(A), and there exist numbers a > 0, b > 0 such that kBf kH 6 akAf kH + bkf kH for all f ∈ dom(A),

(3.5)

or equivalently, by dom(B) ⊇ dom(A), and there exist numbers e a > 0, eb > 0 such that kBf k2H 6 e a2 kAf k2H + eb2 kf k2H for all f ∈ dom(A).

(3.6)

Clearly, (3.6) implies (3.5) with a = e a, b = eb and conversely, (3.5) implies (3.6) with e a2 = (1 + ε)a2 , eb2 = (1 + ε−1 )b2 for each ε > 0. We also note that if A is self-adjoint and bounded from below, the number α defined by

α = lim B(A + µIH )−1 B(H) = lim |B|(A + µIH )−1 B(H) (3.7) µ↑∞

µ↑∞

equals the greatest lower bound (i.e., the infimum) of the possible values for a in (3.5) (resp., for e a in (3.6)). This number α is called the A-bound of B. Similarly,

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−1

β = lim |B|1/2 |A|1/2 + µIH B(H)

(3.8)

µ↑∞

the A-form bound of B (resp., |B|). If α = 0 in (3.7) (resp., β = 0 in (3.8)) then B is called infinitesimally bounded (resp., infinitesimally form bounded ) with respect to A. We then have the following result: Theorem 3.2. Assume that A > 0 is self-adjoint in H. (i) Let B be a closed, densely defined operator in H and suppose that dom(B) ⊇ dom(A). Then B is A-bounded and hence (3.5) holds for some constants a ≥ 0, b ≥ 0. In addition, B is also A-form bounded, |B|1/2 (A + IH )−1/2 ∈ B(H).

(3.9)

1/2

|B| (A + IH )−1/2 6 (a + b)1/2 , B(H)

(3.10)

More specifically,

and hence, if B is A-bounded with A-bound α strictly less than one, 0 ≤ α < 1 (cf. (3.7)), then B is also A-form bounded with A-form bound β strictly less than one, 0 ≤ β < 1 (cf. (3.8)). In particular, if B is infinitesimally bounded with respect to A, then B is infinitesimally form bounded with respect to A. (ii) Suppose that B is closed and densely defined in H, that dom(B) ∩ dom(B ∗ ) ⊇ dom(A), and hence (3.5) holds for some constants a ≥ 0, b ≥ 0. Then also B ∗ is A-bounded, and hence (3.5) with B replaced by B ∗ holds for some constants a∗ ≥ 0, b∗ ≥ 0. In particular,

|B ∗ |1/2 (A + IH )−1/2 ∈ B(H) and |B ∗ |1/2 (A + IH )−1/2 6 (a∗ + b∗ )1/2 . B(H)

(3.11) Moreover, one has (A + IH )−1/2 B(A + IH )−1/2 , (A + IH )−1/2 B ∗ (A + IH )−1/2 ∈ B(H),

(A + IH )−1/2 B(A + IH )−1/2 6 (a∗ + b∗ )1/2 (a + b)1/2 , B(H)

(A + IH )−1/2 B ∗ (A + IH )−1/2 6 (a∗ + b∗ )1/2 (a + b)1/2 . B(H)

(3.12) (3.13) (3.14)

Proof. (i) Equation (3.5) implies

B(A + IH )−1 = |B|(A + IH )−1 B(H) 6 a + b, B(H)

(3.15)

using the polar decomposition B = U |B| of B (cf. (2.58)). Since also

∗

(A + IH )−1 |B| = |B|(A + IH )−1 B(H) 6 a + b, B(H)

(3.16)

the proof of Theorem X.18 in [17] yields that

(A + IH )−1/2 |B|(A + IH )−1/2 B(H)

1/2

−1/2 ∗ 1/2

= |B| (A + IH ) |B| (A + IH )−1/2 B(H) 6 a + b,

(3.17)

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T using complex interpolation. Indeed, one considers C ∞ (A) = n∈N dom(An ), introduces Hp (A), p ∈ R, as the completion of C ∞ (A) with respect to the norm kf kp = k(A + IH )p/2 f kH , f ∈ C ∞ (A). Then Hp∗ = H−p , p ∈ R. Given p0 , p1 ∈ R and pt = tp0 + (1 − t)p1 , t ∈ [0, 1], one can prove that Hpt are interpolating spaces between Hp0 and Hp1 . Given m, n ∈ N, an operator C : C ∞ (A) → H extends to a bounded operator from Hm to H−n , if and only if (A + IH )−n/2 C(A + IH )−m/2 ∈ B(H).

(3.18)

The case at hand in (3.17) then alludes to the special situation m = n = 1 in (3.18). Since kT ∗ T kB(H) = kT T ∗ kB(H) = kT k2B(H) = kT ∗ k2B(H) for all T ∈ B(H),

(3.19)

(3.17) yields the estimate (3.10). To prove the remaining assertions in item (i) one substitutes (A − µ)−1 f in place of f in (3.5) and obtains kB(A + µ)−1 f kH 6 akA(A + µ)−1 f kH + bk(A + µ)−1 f kH 6 (a + b/µ)kf kH , f ∈ dom(A), µ > 0,

(3.20)

f ∈ dom(A), µ > 0.

(3.21)

and hence, kBf kH ≤ [a + (b/µ)]kAf kH + (aµ + b)kf kH , Similarly, the inequality

1/2

|B| (A + µIH )−1/2

B(H)

≤ [a + (b/µ)]1/2

(3.22)

which follows from (3.20) in the same manner as (3.10) follows from (3.3) (i.e., by the same interpolation argument), implies

1/2

|B| f kH ≤ [a + (b/µ)]1/2 |A|1/2 f + (aµ + b)1/2 kf kH , H (3.23) f ∈ dom |A|1/2 , µ > 0. (ii) By symmetry of our hypotheses one obtains (3.11). Next, using the generalized polar decomposition B = |B ∗ |1/2 U |B|1/2 (cf. (3.1)), one thus obtains from (3.10) and (3.11) that

(A + IH )−1/2 B(A + IH )−1/2 B(H)

−1/2 ∗ 1/2 1/2

= (A + IH ) |B | U |B| (A + IH )−1/2 B(H)

∗ = |B ∗ |1/2 (A + IH )−1/2 U |B|1/2 (A + IH )−1/2 B(H) 6 (a∗ + b∗ )1/2 kU kB(H) (a + b)1/2 6 (a∗ + b∗ )1/2 (a + b)1/2 . Since h

(A + IH )−1/2 B(A + IH )−1/2

i∗

= (A + IH )−1/2 B ∗ (A + IH )−1/2 ,

this completes the proof of (3.12)–(3.14).

(3.24)

(3.25)

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Theorem 3.2 extends Theorem 1.38 in [11, Sect. VI.1.7] and Theorem X.18 in [17] since B is not assumed to be self-adjoint or symmetric. We note in connection with the hypotheses in Theorem 3.2, that if B is A-bounded, then B ∗ need not be A∗ -bounded nor A-bounded (we recall that A∗ = A > 0 in our present case). Indeed, the following simple example illustrates this point: Example 3.3. Consider the densely defined closed operators in L2 ((0, 1); dx): Tmin = −

d2 , dx2

dom(Tmin ) = {g ∈ L2 ((0, 1); dx) | g, g 0 ∈ AC([0, 1]); g(0) = g 0 (0) = g(1) = g 0 (1) = 0; g 00 ∈ L2 ((0, 1); dx)}

TF = −

d2 , dx2

dom(TF ) = {g ∈ L2 ((0, 1); dx) | g, g 0 ∈ AC([0, 1]);

(3.26)

g(0) = g(1) = 0; g 00 ∈ L2 ((0, 1); dx)} Tmax = −

d2 , dx2

dom(Tmax ) = {g ∈ L2 ((0, 1); dx) | g, g 0 ∈ AC([0, 1]); g 00 ∈ L2 ((0, 1); dx)},

where TF∗ = TF > 0 is the Friedrichs extension of the minimal operator Tmin . (Here AC(I) denotes the set of absolutely continuous functions on the interval I ⊂ R.) Then the maximal operator Tmax is TF -bounded since dom(Tmax ) ⊃ dom(TF ) ∗ and both operators are closed (cf. [11, Remark IV.1.5]), but Tmax = Tmin is not TF -bounded since dom(Tmin ) is strictly contained in dom(TF ). Before we turn to relatively (form) compact perturbations, we recall a useful interpolation result due to Heinz: Theorem 3.4. ( [7, Theorem 3], [13, Theorem IV.1.11]) Suppose A > 0 and B > 0 are self-adjoint operators in H with dom(B) ⊇ dom(A). If kBf kH 6 kAf kH for all f ∈ dom(A), α

(3.27)

α

then for all α ∈ [0, 1], one has dom(B ) ⊇ dom(A ) and

α

B f 6 Aα f for all f ∈ dom Aα . H H

(3.28)

Theorem 3.5. Assume that A > 0 is self-adjoint in H. (i) Let B be a densely defined closed operator in H and suppose that dom(B) ⊇ dom(A). In addition, suppose that B is A-compact. Then B is also A-form compact, |B|1/2 (A + IH )−1/2 ∈ B∞ (H). (3.29) (ii) Assume that B is densely defined and closed in H and suppose that dom(B) ∩ dom(B ∗ ) ⊇ dom(A). In addition, assume that B or B ∗ is A-compact. Then (A + IH )−1/2 B(A + IH )−1/2 , (A + IH )−1/2 B ∗ (A + IH )−1/2 ∈ B∞ (H).

(3.30)

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Proof. (i) An elementary computation shows that (3.6) implies

2 2 1/2 2 kBf k2H = kB ∗ f k2H = e a A + eb2 IH f H , f ∈ dom(A).

(3.31)

Replacing |B| by |B| + IH and A by A + IH , Theorem 3.4 implies dom |B|α ⊇ dom Aα , α ∈ [0, 1].

(3.32)

As a result, terms of the type (A + IH )−z |B|(A + IH )−1+z = (A + IH )−z |B|z |B|1−z (A + IH )−1+z ∗ = |B|z (A + IH )−z |B|1−z (A + IH )−1+z , (3.33) z ∈ C, Re(z) ∈ [0, 1], are well-defined as operators in B(H). Next we allude to the complex interpolation proof of the Lemma on p. 115 in [18]. The proof of this lemma and equation (3.33) yield that (A + IH )−z |B|(A + IH )−1+z ∈ B∞ (H) for all z ∈ C, Re(z) ∈ (0, 1).

(3.34)

Taking z = 1/2 in (3.34) one concludes (A + IH )−1/2 |B|(A + IH )−1/2 ∈ B∞ (H),

(3.35)

and the latter is then equivalent to |B|1/2 (A + IH )−1/2 ∈ B∞ (H)

(3.36)

(since T ∗ T ∈ B∞ (H) is equivalent to T ∈ B∞ (H)). (ii) Using again the generalized polar decomposition (3.1) of B, that is, B = ∗ 1/2 |B | U |B|1/2 , one obtains (A + IH )−1/2 B(A + IH )−1/2 = (A + IH )−1/2 |B ∗ |1/2 U |B|1/2 (A + IH )−1/2 ∗ = |B ∗ |1/2 (A + IH )−1/2 U |B|1/2 (A + IH )−1/2 ∈ B∞ (H), (3.37) since both square brackets in the last equality in (3.37) are bounded operators and by hypothesis at least one of them is compact. Employing (3.25) again completes the proof of (3.30). Equation (3.30) extends [18, Problem 73 (a), p. 373], since B is not assumed to be symmetric. In a completely analogous manner one proves membership of the operators in (3.30) in the Schatten–von Neumann classes Bp (H), p > 1; we omit further details. Remark 3.6. We conclude this section by recalling a well-known example, where A and B are self-adjoint, B is A-form bounded and even A-form compact, but B is not A-bounded (let alone A-compact): Denote by R the class of Rollnik potentials in R3 , that is, Z 3 3 3 |V (x)||V (y)| R = V : R → C d xd y <∞ , (3.38) |x − y|2 R6

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and by H0 the L2 (R; d3 x)-realization of (minus) the Laplacian −∆ defined on the Sobolev space H 2,2 (R3 ). Then there exist potentials 0 ≤ V0 ∈ L1 (R3 ; d3 x) ∩ R such that −1/2 1/2 ∈ B4 L2 (R3 ; d3 x) , (3.39) V0 H0 + IL2 (R3 ;d3 x) (cf. Simon [20], Theorem I.22 and Example 4 in Sect. I.6) and hence V0 is H0 -form compact, but dom(V0 ) ∩ dom(H0 ) = {0}, (3.40) and thus V0 is not H0 -bounded.

4. Some applications to maximally sectorial operators In this section we relax the condition that A is self-adjoint and study maximally sectorial operators A instead. We recall that A is called accretive if the numerical range of A (i.e., the set {(f, Af )H ∈ C | f ∈ dom(A), kf kH = 1}) is a subset of the closed right complex half-plane. A is called m-accretive if A is a closed and maximal accretive operator (i.e., A has no proper accretive extension). Moreover, A is called an m-sectorial operator with a vertex 0 and a corresponding semi-angle θ ∈ [0, π/2) if A is a maximal accretive, closed (and hence densely defined) operator, and the numerical range of A is contained in a sector | arg(z)| 6 θ < (π/2) in the complex z-plane. We also recall that an equivalent definition of an m-accretive operator A in H is 1 (A + ζIH )−1 ∈ B(H), k(A + ζIH )−1 k ≤ , Re(ζ) > 0. (4.1) Re(ζ) With A assumed to be m-sectorial, one associates the quadratic form t0A [f, g] = (f, Ag)H ,

f, g ∈ dom(t0A ) = dom(A).

(4.2)

t0A

The form is closable (cf. [11, Theorem VI.1.27]) and according to the first representation theorem (see, e.g., [11, Theorem VI.2.1]), A is associated with its closure tA = t0A , that is, dom(A) ⊆ dom(tA ) and tA [f, g] = (f, Ag)H ,

f ∈ dom(tA ), g ∈ dom(A). (4.3)

∗

Denoting by t the adjoint form of a sesqulinear form t in H, t∗ [f, g] = t[g, f ],

f, g ∈ dom(t∗ ) = dom(t),

(4.4)

t∗A )/2

the form tAR = (tA + is closed and nonnegative on dom(tAR ) = dom(tA ). We denote by AR > 0 the self-adjoint operator uniquely associated with tAR , that is, dom(AAR ) ⊆ dom(tA ) and tAR [f, g] = (f, AR g)H ,

f ∈ dom(tA ), g ∈ dom(AR ). (4.5) By the second representation theorem (cf. [11, Theorem VI.2.23]) 1/2 1/2 1/2 tAR [f, g] = AR f, AR g H , f, g ∈ dom(tAR ) = dom AR . (4.6)

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We denote by A1/2 the unique m-sectorial square root of A, and recall that ∗ (4.7) (A∗ )1/2 = A1/2 . It should be emphasized that in general, dom A1/2 6= dom (A∗ )1/2 . (4.8) 1/2 ∗ 1/2 However, if in fact, dom A = dom (A ) , then one can obtain the analog of the second representation theorem for densely defined closed, sectorial forms. For this purpose we next recall the following results: Theorem 4.1. ([11, Theorem VI.3.2]) Let A be m-sectorial in H with a vertex 0 and semi-angle θ ∈ [0, π/2). Then AR > 0 and there exists a bounded self-adjoint operator X ∈ B(H) such that kXkB(H) 6 tan(θ) and 1/2

1/2

A = AR (IH + iX)AR ,

1/2

1/2

A∗ = AR (IH − iX)AR .

(4.9)

Lemma 4.2. (cf. [8], [11, Theorem VI.3.2] and [5, Theorem IV.2.10]) Let A be m-sectorial in H with a vertex 0 and assume that dom A1/2 = dom (A∗ )1/2 . (4.10) Then the sesquilinear form f, g ∈ dom A1/2 = dom (A∗ )1/2 ,

(4.11)

is sectorial and closed. In particular, 1/2 dom(tA ) = dom A1/2 = dom (A∗ )1/2 = dom AR = dom(tAR ).

(4.12)

tA [f, g] = (A∗ )1/2 f, A1/2 g

H

,

Proof. Although this result is known (cf. Kato [8]), we thought it might be of some interest to present an alternative proof. Since tA+IH [f, g] = (A∗ + IH )1/2 f, (A + IH )1/2 g H = (A∗ )1/2 f, A1/2 g H + (f, g)H , = tA [f, g] + (f, g)H , f, g ∈ dom A1/2 = dom (A∗ )1/2 , (4.13) it suffices to consider tA+IH remainder of this proof. Since instead of tA in the 1/2 1/2 ∗ 1/2 dom A = dom (A ) , and the operators A and (A∗ )1/2 are closed, one concludes that the operator Y defined below, satisfies Y = (A + IH )1/2 (A∗ + IH )−1/2 ∈ B(H), Y −1 = (A∗ + IH )1/2 (A + IH )−1/2 ∈ B(H). (4.14) Next we show that the operator Y is accretive. Since A is m-accretive, one gets 2 Re(Y ) = Y + Y ∗ = (A + IH )1/2 (A∗ + IH )−1/2 + (A + IH )−1/2 (A∗ + IH )1/2 = (A + IH )1/2 (A∗ + IH )−1 + (A + IH )−1 (A∗ + IH )1/2 > 0. (4.15) Similarly one obtains 2i Im(Y ) = (Y − Y ∗ ) = (A + IH )1/2 (A∗ + IH )−1/2 − (A + IH )−1/2 (A∗ + IH )1/2 = 2i(A + IH )1/2 Im (A∗ + IH )−1 (A∗ + IH )1/2 . (4.16)

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Combining (4.15) with (4.16) one concludes that Y is a bounded θ-sectorial operator, because so is (A∗ + IH )−1 . Using this fact and 0 ∈ / σ(Y ), we next show that 0 ∈ / σ(Re(Y )). Indeed, assuming the contrary, 0 ∈ σ(Re(Y )), we get 0 ∈ σ (Re(Y ))1/2 . Then there exists a sequence {fn }n∈N ⊂ H, kfn k = 1, n ∈ N, such that (Re(Y ))1/2 fn −→ 0. n→∞

On the other hand, by Theorem 4.1, Y admits the representation Y = (Re(Y ))1/2 (I + iK)(Re(Y ))1/2 ,

(4.17)

∗

where K = K and kKk 6 tan(θ). Consequently,

kY fn k = (Re(Y ))1/2 (I + iK)(Re(Y ))1/2 fn 6 C (Re(Y ))1/2 fn −→ 0. (4.18) n→∞

Thus, 0 ∈ σ(Y ), a contradiction. Hence, the operator (Re(Y ))1/2 (A + I)1/2 is closed as (A + I)1/2 is and Re(Y ) is boundedly invertible. Moreover, using (4.14) one obtains tAR +IH [f, g] + 2−1 (A + IH )1/2 f, (A∗ + IH )1/2 g H = 2−1 (Y + Y ∗ )(A∗ + IH )1/2 f, (A∗ + IH )1/2 g H = (Re(Y ))1/2 (A∗ + IH )1/2 f, (Re(Y ))1/2 (A∗ + IH )1/2 g H , (4.19) 1/2 ∗ 1/2 f, g ∈ dom A = dom (A ) . = 2−1 (A∗ + IH )1/2 f, (A + IH )1/2 g

H

Since the operator (A∗ + IH )1/2 is closed and Re(Y ) is boundedly invertible, also the operator (Re(Y ))1/2 (A∗ +IH )1/2 is closed, and hence the form tAR +IH is closed too (cf. [11, Problem III.5.7, Example VI.1.13]). Remark 4.3. Let A = diag (it0 , −it0 ), t0 ∈ R\{0}. Then A is maximal accretive operator in C2 and 0 ∈ ρ(A), although AR = 0 and hence 0 ∈ σ(AR ). This simple example shows that the assumption on A to be m-sectorial is important in proving the implication 0 ∈ ρ(A) =⇒ 0 ∈ ρ(AR ). Corollary 4.4. Let A be m-sectorial in H with a vertex 0 and assume that dom A1/2 = dom (A∗ )1/2 (4.20) and that 0 ∈ ρ(A).

(4.21)

Then, in addition to (4.11) and (4.12), there exists an ε0 > 0 such that the following inequalities hold:

2

2 tAR [f, f ] = Re (A∗ )1/2 f, A1/2 f H > ε0 max A1/2 f H , (A∗ )1/2 f H , (4.22)

1/2 ∗ 1/2 ∗ 1/2 1/2

tA [f, f ] = Re (A ) f, A f > ε0 A f (A ) f , (4.23) R

H

H

H

f ∈ dom(tAR ).

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Proof. As in the proof of Lemma 4.2, we may write (A)1/2 = Y (A∗ )1/2 , where Y is m-accretive and 0 ∈ ρ(Re(Y )). In this context we note that we replaced A + IH and A∗ + IH by A and A∗ in the definition of Y in (4.14) to arrive at the operator Y = A1/2 (A∗ )−1/2 ∈ B(H),

(4.24) ∗

which is possible due to the hypothesis 0 ∈ ρ(A) (implying 0 ∈ ρ(A )). Therefore, tAR [f, f ] = Re (A∗ )1/2 f, (A)1/2 f H = (A∗ )1/2 f, Re(Y )(A∗ )1/2 f H (4.25)

2 > ε1 (A∗ )1/2 f H , f ∈ dom(tAR ), where ε1 = inf σ(Re(Y )) . Similarly, one gets

2 Re (A∗ )1/2 f, (A)1/2 f H > ε2 (A)1/2 f H , f ∈ dom(tAR ), (4.26) −1 where ε2 = inf σ Re Y . Setting ε0 = min{ε1 , ε2 } one arrives at (4.22). Inequality (4.23) is then immediate from (4.22). Remark 4.5. (i) Inequality (4.23) is mentioned in [10], and because of inequality (4.23), A1/2 and (A∗ )1/2 are said to have an acute angle. (ii) In general, if A is m-accretive (i.e., without assuming (4.20) and (4.21)), Kato [9] proved dom(Aα ) = dom((A∗ )α ), α ∈ (0, 1/2), (4.27) and that the (right-hand) inequality in (4.23) holds with 1/2 replaced by α (cf. also [10], [22, Theorem IV.5.1]), that is, there exists an ε0 (α) > 0 such that

Re (A∗ )α f, Aα f H > ε0 (α) Aα f H (A∗ )α f H , (4.28) f ∈ dom(Aα ), α ∈ (0, 1/2). (iii) We recall that ker(A) = ker(A∗ ) if A is m-accretive, in particular, ker(A) is a reducing subspace for A (cf., e.g., [22, p. 171]). Thus, one can write A = A0 ⊕ A1 with respect to the decomposition H = P0 H ⊕ [IH − P0 ]H, where P0 denotes the orthogonal projecton onto ker(A), such that A0 = P0 AP0 = 0 and ker(A1 ) = {0}. α Thus, also Aα = A0 ⊕ Aα 1 , α ∈ (0, 1], with ker(A1 ) = {0}. Hence, one actually obtains ker(A) = ker(Aα ) = ker(A∗ ), α ∈ (0, 1], (4.29) if A is m-accretive. Definition 4.6. Let A be an m-sectorial operator in H with a vertex 0 and B a densely defined closed operator in H. Then B is called A-form bounded (resp., A-form compact ) if dom |B|1/2 ⊇ dom A1/2 and |B|1/2 (A + IH )−1/2 ∈ B(H) (resp., ∈ B∞ (H)). (4.30) Again, B is A-form bounded (resp., A-form compact) if and only if |B| is. We also that due to the closedness of |B|1/2 and A1/2 , epmhasize1/2again 1/2 dom |B| ⊇ dom A alone implies that |B|1/2 is A1/2 -bounded (cf. [11,

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Remark IV.1.5]), and hence the first condition in (4.30) implies the second in connection with form boundedness. In the following, for simplicity of notation, we agree that for a densely defined linear operator C in H, the symbol C # either equals C or C ∗ .

(4.31)

If the symbol C # occurs twice in a formula, as it does occasionally below, both C # stand for C or both stand for C ∗ , so such a formula should be read in two possible ways (just as if C # occurs only once in a formula). Theorem 4.7. Let A be m-sectorial in H with a vertex 0 and assume that dom A1/2 = dom (A∗ )1/2 . (4.32) In addition, suppose that B is a densely defined and closed operator in H. Then the following assertions hold: (i) B is A-form bounded (resp., A-form compact ) if and only if it is AR -form bounded (resp., AR -form compact ), that is, −1/2 |B|1/2 A# + IH ∈ B(H) (resp., ∈ B∞ (H)) if and only if 1/2

|B|

(4.33) −1/2

(AR + IH )

∈ B(H) (resp., ∈ B∞ (H)).

(ii) The following conditions (α)–(δ) are equivalent: −1/2 −1/2 (α) (A# )∗ + IH B A# + IH is closable in H,

(γ)

(A∗ + IH )−1/2 B(A∗ + IH )−1/2 is closable in H, −1/2 (A + IH B(A + IH )−1/2 is closable in H,

(δ)

(AR + IH )−1/2 B(AR + IH )−1/2 is closable in H.

(β)

(4.34)

(iii) The following conditions (α)–(δ) are equivalent: (α)

(A# )∗ + IH

−1/2

B A# + IH

−1/2

∈ B(H) (resp., ∈ B∞ (H)),

(β)

(A∗ + IH )−1/2 B(A∗ + IH )−1/2 ∈ B(H) (resp., ∈ B∞ (H)),

(γ)

(A + IH )−1/2 B(A + IH )−1/2 ∈ B(H) (resp., ∈ B∞ (H)),

(δ)

(AR + IH )−1/2 B(AR + IH )−1/2 ∈ B(H) (resp., ∈ B∞ (H)).

(4.35)

1/2 1/2 Proof. (i) Since dom A1/2 = dom AR and the operators A1/2 and AR are closed, one concludes that the operator T# defined below, satisfies 1/2 T# = A# + IH (AR + IH )−1/2 ∈ B(H), (4.36) −1/2 −1 T# = (AR + IH )1/2 A# + IH ∈ B(H).

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Therefore, if |B|1/2 (AR + IH )−1/2 ∈ B(H), then also |B|1/2 A# + IH and the identity −1/2 −1 |B|1/2 (AR + IH )−1/2 T# = |B|1/2 A# + IH

103

−1/2

∈ B(H) (4.37)

holds. By (4.36), this argument can be reversed, proving the equivalence (4.33). −1/2 That |B|1/2 A# + IH ∈ B∞ (H) if and only if |B|1/2 (AR + IH )−1/2 ∈ B∞ (H) is proven in the same manner. (ii) Assume that (AR + IH )−1/2 B(AR + IH )−1/2 is closable in H. Then so −1 ∗ −1 is T# (AR + IH )−1/2 B(AR + IH )−1/2 T# due to (4.36). This follows from the following two facts: (1) If S1 ∈ B(H) and S2 is a closable (resp., closed) operator in H, then S2 S1 is closable (resp., closed) in H. (2) If T1 , T2 are closable (resp., closed) operators in H, and T2−1 ∈ B(H), then T2 T1 is closable (resp., closed) in H (cf. [23, p. 96]). Since −1/2 −1/2 −1 ∗ −1 T# (AR + IH )−1/2 B(AR + IH )−1/2 T# = (A# )∗ + IH B A# + IH , (4.38) −1/2 −1/2 this proves the closability of (A# )∗ + IH B A# + IH in H. Again by (4.36), this argument can be reversed, proving the equivalence of (α) and (δ) in (4.34). The remaining equivalences in (4.34) follow from (4.14) which permits one to individually exchange A and A∗ (or A∗ and A) in the most left and/or most right factor (. . . )−1/2 in (α). (iii) If (AR +IH )−1/2 B(AR +IH )−1/2 has a closure in B(H) (resp., in B∞ (B)), then (4.36) and (4.38) yield −1 −1 ∗ (AR + IH )−1/2 B(AR + IH )−1/2 T# T# −1 ∗ −1 (AR + IH )−1/2 B(AR + IH )−1/2 T# = T# −1 ∗ −1 = T# (AR + IH )−1/2 B(AR + IH )−1/2 T# −1/2 −1/2 = (A# )∗ + IH B A# + IH . (4.39) Here we used the following facts: (1) Let S be a bounded operator in H with domain dom(S). Then S is closable and the closure of S has domain dom(S) ⊆ H. (2) S 1 ∈ B(H), S2 ∈ B(H), dom(S1 S2 ) dense in H, then S1 S2 = S 1 S2 . (3) T1 ∈ B(H), T 2 ∈ B(H), then T1 T2 = T1 T 2 . −1/2 −1/2 Thus, (A# )∗ + IH B A# + IH has closure in B(H) (resp., in B∞ (B)). Once more by (4.36), this argument is reverseable, proving the equivalence of (α) and (δ) in (4.35). As in the final part of the proof of item (ii), the remaining equivalences in (4.35) follow from (4.14). To prove one of our main results on sectorial operators we next need a generalization of Theorem 3.4 to the sectorial case.

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First we recall that S1 is called subordinated to S2 (cf., e.g., [12, Sect. 14.5]) if dom(S1 ) ⊇ dom(S2 ), and for some C > 0, kS1 f kH 6 CkS2 f kH , f ∈ dom(S2 ). (4.40) Theorem 4.8. ([8, Theorem 1]) Let A, B be m-accretive operators in H and assume that T ∈ B(H). In addition, assume that there exists a constant C > 0 such that T dom(A) ⊆ dom(B) and kBT f kH 6 CkAf kH , f ∈ dom(A).

(4.41)

Then for all α ∈ (0, 1], there exists a constant Cα > 0 such that T dom(Aα ) ⊆ dom(B α ) and kB α T gkH 6 Cα kAα gkH , g ∈ dom(Aα ).

(4.42)

In the sequel we need the special case of Theorem 4.8 corresponding to T = IH . However, it turns out, that this special case is, in fact, equivalent to the general case displayed in Theorem 4.8, as will be shown subsequently. Corollary 4.9. Suppose A and B are m-accretive operators in H and B is subordinated to A. Then for all α ∈ (0, 1], B α is subordinated to Aα , that is, the inequality kBf kH 6 C1 kAf kH , f ∈ dom(A) ⊆ dom(B) (4.43) for some constant C1 > 0 independent of f ∈ dom(A), implies dom(Aα ) ⊆ dom(B α ) and kB α gkH 6 Cα kAα gkH , g ∈ dom(Aα )

(4.44)

for some constant Cα > 0 independent of g ∈ dom(Aα ). The following result was deduced in [8] from Theorem 4.8. (Actually, it is equivalent to Theorem 4.8 as we will show below.) For the sake of completeness we present a short proof based on the generalized polar decomposition (2.64) and on Corollary 4.9. Theorem 4.10. ([8, Theorem 2]) Let A and B be m-accretive operators in H and let Q be a densely defined closed linear operator in H such that dom(Q) ⊇ e 1 > 0 such that dom(A), dom(Q∗ ) ⊇ dom(B) and there exist constants D1 > 0, D kQgkH 6 D1 kAgkH , g ∈ dom(A),

e 1 kBf kH , f ∈ dom(B). (4.45) kQ∗ f kH 6 D

Then for each α ∈ (0, 1), there exists a constant Cα > 0 such that the following inequality holds: |(f, Qg)H | 6 Cα kB 1−α f kH kAα gkH ,

f ∈ dom(B), g ∈ dom(A).

(4.46)

Proof. By Corollary 4.9 and the fact that kQgkH = k|Q|gkH , kQ∗ f kH = k|Q∗ |f kH , the inequalities (4.45) yield for β, γ ∈ (0, 1],

β

|Q| g 6 Dβ kAβ gkH , g ∈ dom(A), H (4.47) ∗ γ e γ kB γ f kH , f ∈ dom(B) k|Q | f kH 6 D

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e γ > 0. On the other hand, by equation (2.64), for some constants Dβ > 0, D Q = |Q∗ |1−α U |Q|α , α ∈ [0, 1]. Combining these facts one arrives at |(f, Qg)H | = |(U ∗ |Q∗ |1−α f, |Q|α g)H | 6 k|Q∗ |1−α f kH k|Q|α gkH e 1−α kB 1−α f kH D e α kAα gkH , 6D

f ∈ dom(B), g ∈ dom(A),

(4.48)

completing the proof.

Next we show that Theorem 4.10, in fact, implies Theorem 4.8. This was stated (without proof) in Kato [8]): Deduction of Theorem 4.8 from Theorem 4.10. Let Q = BT . Then Q∗ ⊇ T ∗ B ∗ , and dom(Q∗ ) ⊇ dom(B ∗ ) and kQ∗ f k 6 kT ∗ kkB ∗ f kH , f ∈ dom(B ∗ ).

(4.49)

In addition, T dom(A) ⊆ dom(B) yields dom(Q) ⊇ dom(A). Therefore, by Theorem 4.10 (with B replaced by B ∗ ), for any α ∈ (0, 1),

|(f, Qg)H | 6 Cα (B ∗ )(1−α) f H kAα gkH , f ∈ dom(B ∗ ), g ∈ dom(A). (4.50) (The case α = 1 is obvious and needs not be considered.) Hence, |(f, Qg)H | = |(f, BT g)H | = |((B ∗ )1−α f, B α T g)H | 6 Cα k(B ∗ )1−α f kH kAα gkH , f ∈ dom(B ∗ ), g ∈ dom(A).

(4.51)

Clearly, |((B ∗ )1−α f, B α T g)H | = |(P (B ∗ )1−α f, B α T g)H | = |(P (B ∗ )1−α f, P B α T g)H |, f ∈ dom(B ∗ ), g ∈ dom(A),

(4.52)

∗

where P is the orthogonal projection onto the closure of ran(B ). Therefore, fixing g ∈ dom(A), inequality (4.51) yields kP B α T gkH 6 Cα kAα gkH ,

α ∈ (0, 1).

(4.53)

On the other hand, by (4.29), ker(B) = ker(B β ) = ker(B ∗ ), β ∈ (0, 1], since B is m-accretive. Therefore, ran(B) = ran(B β ) = ran(B ∗ ), α

β ∈ (0, 1].

(4.54)

α

Thus, P B T g = B T g, g ∈ dom(A), and hence finally, kB α T gkH = kP B α T gkH 6 Cα kAα gkH ,

g ∈ dom(A).

(4.55)

Thus we have shown Theorem 4.9 =⇒ Corollary 4.10 =⇒ Theorem 4.10 =⇒ Theorem 4.9 (4.56) and hence the equivalence of Theorem 4.9, Corollary 4.10, and Theorem 4.10 (illustrating the usefulness of the generalized polar decomposition (2.64) in this context). We conclude with the following two results:

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Theorem 4.11. Let A be m-sectorial in H with a vertex 0 and assume that B is densely defined and closed in H. (i) Suppose that B and B ∗ are AR -bounded. Then, −1 −1 |B|1/2 A# + IH |B|1/2 , |B|1/2 A# + IH |B ∗ |1/2 ∈ B(H), (4.57) −1 −1 |B ∗ |1/2 A# + IH |B|1/2 , |B ∗ |1/2 A# + IH |B ∗ |1/2 ∈ B(H). (ii) Suppose that B and B ∗ are AR -bounded and that dom A1/2 = dom (A∗ )1/2 . Then, −1/2 −1/2 |B|1/2 A# + IH , |B ∗ |1/2 A# + IH ∈ B(H), (4.58) and (A + IH )−1/2 B # (A + IH )−1/2 , (A + IH )−1/2 B # (A∗ + IH )−1/2 ∈ B(H), (A∗ + IH )−1/2 B # (A + IH )−1/2 , (A∗ + IH )−1/2 B # (A∗ + IH )−1/2 ∈ B(H). (4.59) In particular, B and B ∗ are A# -form bounded. Moreover, B and B ∗ are AR -form bounded, −1/2 |B # |1/2 AR + IH ∈ B(H), (4.60) and (AR + IH )−1/2 B # (AR + IH )−1/2 ∈ B(H). (4.61) 1/2 ∗ 1/2 (iii) Suppose that B is A-bounded and that dom A = dom (A ) . Then B is AR -form bounded. Moreover, if B ∗ is also A-bounded, then equation (4.61) and the relations (4.59) hold as well. Proof. (i) Since B and B ∗ are AR -bounded, Theorem 3.2 implies that |B # |1/2 (AR + IH )−1/2 , (AR + IH )−1/2 |B # |1/2 ∈ B(H).

(4.62)

Combining these inclusions with (4.9) one obtains |B|1/2 (A + IH )−1 |B|1/2 = |B|1/2 (AR + IH )−1/2 (IH + iX)−1 (AR + IH )−1/2 |B|1/2 ∈ B(H),

(4.63)

|B|1/2 (A∗ + IH )−1 |B|1/2 = |B|1/2 (AR + IH )−1/2 (IH − iX)−1 (AR + IH )−1/2 |B|1/2 ∈ B(H),

(4.64)

proving the first claim in assertion (i). The remaining three are proven in precisely the same manner. (ii) Since by hypothesis 1/2 dom A1/2 = dom (A∗ )1/2 = dom AR , (4.65) and A1/2 , (A∗ )1/2 , and (AR )1/2 are closed, one infers that (AR + IH )1/2 (A# + IH )−1/2 , (A# + IH )−1/2 (AR + IH )1/2 ∈ B(H).

(4.66)

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Hence, −1/2 |B|1/2 A# + IH −1/2 = |B|1/2 (AR + IH )−1/2 (AR + IH )1/2 A# + IH ∈ B(H), −1/2 ∗ 1/2 # |B | A + IH ∗ 1/2 −1/2 = |B | (AR + IH )−1/2 (AR + IH )1/2 A# + IH ∈ B(H),

(4.67)

(4.68)

applying Theorem 4.7 (i) (also with B replaced by B ∗ ). Using the generalized polar decomposition (3.1), B = |B ∗ |1/2 U |B|1/2 , one obtains from (4.62) and (4.66) that (A + IH )−1/2 B(A + IH )−1/2 = (A + IH )−1/2 (AR + IH )1/2 (AR + IH )−1/2 B(AR + IH )−1/2 × (AR + IH )1/2 (A + IH )−1/2 = (A + IH )−1/2 (AR + IH )1/2 (AR + IH )−1/2 |B ∗ |1/2 U × |B|1/2 (AR + IH )−1/2 (AR + IH )1/2 (A + IH )−1/2 ∈ B(H).

(4.69)

Precisely the same argument works for the remaining three operators in (4.59) (using also B ∗ = |B|1/2 U ∗ |B ∗ |1/2 ). Finally, since AR > 0 is self-adjoint, (4.60) and (4.61) follow from Theorem 3.2. (iii) By Corollary 4.9, |B|α is subordinated to (A + IH )α , α ∈ (0, 1]. In particular, the operator |B|1/2 is (A+IH )1/2 -bounded, that is, |B|1/2 (A+IH )−1/2 ∈ B(H). On the other hand, by (4.36), T = (A+IH )1/2 (AR +IH )−1/2 ∈ B(H). Thus, |B|1/2 (AR + IH )−1/2 = |B|1/2 (A + IH )−1/2 (A + IH )1/2 (AR + IH )−1/2 = |B|1/2 (A + IH )−1/2 T ∈ B(H),

(4.70)

and hence B is AR -form bounded. If, in addition, B ∗ is A-bounded, then again by Corollary 4.9, |B ∗ |1/2 (A+IH )−1/2 ∈ B(H) and hence also B ∗ is AR -form bounded, |B ∗ |1/2 (AR + IH )−1/2 = |B ∗ |1/2 (A + IH )−1/2 T ∈ B(H).

(4.71)

Combining (4.70) and (4.71) and using the generalized polar decomposition (3.1), one arrives at (AR + IH )−1/2 B # (AR + IH )−1/2 = (AR + IH )−1/2 |(B # )∗ |1/2 U |B # |1/2 (AR + IH )−1/2 ∈ B(H). Relations (4.59) then follow as in the proof of item (ii).

(4.72)

Finally, we state an analog of Theorem 4.11 in connection with relative (form) compactness:

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Theorem 4.12. Let A be m-sectorial in H with a vertex 0, assume that B is densely defined and closed in H. (i)Suppose that dom(B) ∩ dom(B ∗ ) ⊇ dom(AR ) and that B (resp., B ∗ ) is AR compact. Then, −1 −1 |B|1/2 A# + IH |B|1/2 , |B|1/2 A# + IH |B ∗ |1/2 ∈ B∞ (H), −1 |B|1/2 ∈ B∞ (H), (4.73) and |B ∗ |1/2 A# + IH −1 −1 resp., |B|1/2 A# + IH |B ∗ |1/2 , |B ∗ |1/2 A# + IH |B|1/2 ∈ B∞ (H), −1 and |B ∗ |1/2 A# + IH |B ∗ |1/2 ∈ B∞ (H). (ii) Suppose that dom(B) ∩ dom(B ∗ ) ⊇ dom(A B (resp., B ∗ ) is AR R ) and that 1/2 ∗ 1/2 compact. In addition, assume that dom A = dom (A ) . Then, −1/2 |B|1/2 (A# + IH )−1/2 ∈ B∞ (H) resp., |B ∗ |1/2 A# + IH ∈ B∞ (H) , (4.74) and (A + IH )−1/2 B # (A + IH )−1/2 , (A + IH )−1/2 B # (A∗ + IH )−1/2 ∈ B∞ (H), (4.75) (A∗ + IH )−1/2 B # (A + IH )−1/2 , (A∗ + IH )−1/2 B # (A∗ + IH )−1/2 ∈ B∞ (H). In particular, B (resp., B ∗ ) is A# -form compact. Moreover, B (resp., B ∗ ) is AR form compact, −1/2 |B|1/2 AR + IH ∈ B∞ (H) resp., |B ∗ |1/2 (AR + IH )−1/2 ∈ B∞ (H) (4.76) and (AR + IH )−1/2 B # (AR + IH )−1/2 ∈ B∞ (H).

(4.77)

(iii) Suppose that dom(B) ∩ dom(B ∗ ) ⊇ dom(A) and that B (resp., B ∗ ) is A1−ε compact for some ε ∈ (0, 1). In addition, assume that dom A1/2 = dom (A∗ )1/2 . Then B (resp., B ∗ ) is AR -form compact. Moreover, equation (4.77) and relations (4.75) hold as well. Proof. (i) Since by hypothesis B and B ∗ are AR -bounded and B (resp., B ∗ ) is AR -compact, Theorem 3.5 implies that |B|1/2 (AR + IH )−1/2 = (AR + IH )−1/2 |B|1/2 ∈ B∞ (H) resp., |B ∗ |1/2 (AR + IH )−1/2 = (AR + IH )−1/2 |B ∗ |1/2 ∈ B∞ (H) .

(4.78)

At this point one can follow the proof of Theorem 4.11 (i), noting that each operator in (4.63) and (4.64) contains at least one compact factor from (4.78). (ii) Again, one can follow the proof of Theorem 4.11 (ii), noting that the righthand side of (4.67) (resp., (4.68)) contains a compact factor from (4.78). Similarly, the right-hand side of (4.69) and the analogous equations with A replaced by A∗ (resp., B replaced by B ∗ ) contains at least one compact factor from (4.78). Relations (4.76) and (4.77) are clear from Theorem 3.5 since AR > 0 is self-adjoint.

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(iii) Since by hypothesis, dom(B) ∩ dom(B ∗ ) ⊇ dom(A), B and B ∗ are Abounded and hence Theorem 4.11 (iii) and the results (4.70)–(4.72) in its proof are at our disposal. Next, we first assume that B(A + IH )−1+ε ∈ B∞ (H). Then (using |B| = U ∗ B, cf. (2.60)), |B|(A + IH )−1+ε0 +iγ = |B|(A + IH )−1+ε (A + IH )−(ε−ε0 )+iγ ∈ B∞ (H), (4.79) 0 ≤ ε0 < ε, since (A + IH )−β+iγ ∈ B(H),

β ∈ (0, 1), γ ∈ R,

(4.80)

as is clear from the formula (cf. [11, Remark V.3.50], [12, Sect. 14.12]), Z sin(πz) ∞ dt t−z (S +(t+1)IH )−1 , z ∈ C, Re(z) ∈ (0, 1), (4.81) (S +IH )−z = π 0 for any m-accretive operator S in H. Since by hypothesis B(A + IH )−1+ε ∈ B∞ (H) ⊂ B(H), B is subordinated to (A + IH )1−ε , and hence by Corollary 4.9, |B|α is subordinated to (A + IH )(1−ε)α for all α ∈ (0, 1], |B|α (A + IH )−(1−ε)α ∈ B(H),

α ∈ (0, 1].

(4.82)

In the following we assume without loss of generality that ker(|B|) = ker(B) = {0}.

(4.83)

Thus, one obtains (A∗ + IH )−z |B|(A + IH )−1+z = (A∗ + IH )−z |B|z |B|1−z (A + IH )−1+z ∗ = |B|z (A + IH )−z |B|1−z (A + IH )−1+z ∈ B(H), Re(z) ∈ (0, 1),

(4.84)

since by (4.80) and (4.82), |B|α+iβ (A + IH )−α−iβ = |B|iβ |B|α (A + IH )−(1−ε)α (A + IH )−εα−iβ ∈ B(H), α ∈ (0, 1], β ∈ R, (4.85) as |B|iβ is unitary. Moreover, choosing a compact subinterval of (0, 1) containing 1/2 in its interior, for instance, [ε0 , 1 − ε0 ] for some ε0 ∈ (0, 1/2), one obtains for z = ε0 + iγ in (4.84), k(A∗ + IH )−ε0 −iγ |B|(A + IH )−(1−ε0 )+iγ k

∗ = |B|ε0 −iγ (A + IH )−ε0 +iγ |B|1−ε0 −iγ (A + IH )−(1−ε0 )+iγ

≤ |B|ε0 (A + IH )−ε0 +iγ

|B|1−ε0 (A + IH )−(1−ε0 )+iγ

≤ |B|ε0 (A + IH )−(1−ε)ε0

(A + IH )−εε0 +iγ

× |B|1−ε0 (A + IH )−(1−ε)(1−ε0 )

(A + IH )−ε(1−ε0 )+iγ

≤ |B|ε0 (A + IH )−(1−ε)ε0

|B|1−ε0 (A + IH )−(1−ε)(1−ε0 ) Ce2π|γ|

(4.86)

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for some C = C(ε, ε0 ) > 0 (cf. (4.87)). (In fact, using [10, Theorem 4], one can replace 2π by π in the exponent of (4.86), but this plays no role in our context.) Here we used the fact that by (4.1) and (4.81),

Z ∞

−z −1

(A + IH )−z = sin(πz) dt t (A + (t + 1)IH ) π 0 Z sin(πz) ∞

≤ dt t−Re(z) (A + (t + 1)IH )−1 π 0 Z sin(πz) ∞ t−Re(z) sin(πz) ≤ dt (4.87) = sin(πRe(z)) , Re(z) ∈ (0, 1). π 0 t+1 The same computation applies to z = 1 − ε0 + iγ in (4.84), and more generally, one has

sup (A∗ + IH )−α−iγ |B|(A + IH )−1+α+iγ e−2π|γ| < ∞, α ∈ (0, 1). (4.88) γ∈R

In addition, the map 2

z 7→ ez (A∗ + IH )−z |B|(A + IH )−1+z is analytic in the strip Re(z) ∈ (0, 1). (4.89) By the proof of the Lemma in [18, p. 115], (4.79), (4.84), (4.88) (for α = ε0 and α = 1 − ε0 ), and (4.89) imply, by complex interpolation, that 2

ez (A∗ + IH )−z |B|(A + IH )−1+z ∈ B∞ (H),

z ∈ C, Re(z) ∈ (ε0 , 1 − ε0 ). (4.90)

Since ε0 ∈ (0, 1/2) can be taken arbitrarily small, one finally concludes that (A∗ + IH )−z |B|(A + IH )−1+z ∈ B∞ (H),

z ∈ C, Re(z) ∈ (0, 1).

(4.91)

In particular, (A∗ + IH )−1/2 |B|(A + IH )−1/2 ∈ B∞ (H).

(4.92)

Thus, (AR + IH )−1/2 |B|(AR + IH )−1/2 = cl (AR + IH )−1/2 (A∗ + IH )1/2 (A∗ + IH )−1/2 |B|(A + IH )−1/2 × (A + IH )1/2 (AR + IH )−1/2 ∗ = (A + IH )1/2 (AR + IH )−1/2 (A∗ + IH )−1/2 |B|(A + IH )−1/2 × (A + IH )1/2 (AR + IH )−1/2 = T ∗ (A∗ + IH )−1/2 |B|(A + IH )−1/2 T ∈ B∞ (H),

(4.93)

where cl{·} abbreviates the operator closure (we made an exception to our usual notation due to lack of space), T = [(A + IH )1/2 (AR + IH )−1/2 ∈ B(H) (cf. (4.36)), and we used again the reasoning (1)–(3) as in the proof of (4.39). Relation (4.93) and the fact that an operator D is compact if and only if D∗ D is, then finally implies |B|1/2 (AR + IH )−1/2 ∈ B∞ (H). (4.94)

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In exactly the same manner, the assumption B ∗ (A + IH )−1+ε ∈ B∞ (H) then implies |B ∗ |1/2 (AR + IH )−1/2 ∈ B∞ (H). (4.95) In particular, since the operator in (4.70) (resp. in (4.71)) now lies in B∞ (H), B (resp., B ∗ ) is AR -form compact, that is, (4.76) holds. Equation (4.77) then follows as in (4.72) from (4.76). Finally, relations (4.75) again follow as in the proof of item (ii). Remark 4.13. We do not know if one can generally take ε = 0 in Theorem 4.12 (iii). Of course, if the condition

(4.96) sup (A + IH )iγ < ∞ γ∈R

holds, the proof of Theorem 4.12 (iii) (c.f., in particular, estimates (4.86)) shows that ε can indeed be taken equal to zero. In particular, (4.96) holds if A is similar to a self-adjoint operator S in some complex, separable Hilbert space H0 with S ≥ −IH0 and {−1} not an eigenvalue of S (by applying the spectral theorem to S). Conversely, suppose A is m-sectorial in H with a vertex 0 and consider i T = (A + IH )−i = (A + IH )−1 . Then by (4.96), T t , t ∈ R, is a uniformly bounded one-parameter commutative group of transformations, in fact, a C0 -group with generator i log (A + IH )−1 (cf. the discussion in [14, Corollary 5.4]),

−1 T t = (A + IH )−it = eit log((A+IH ) ) , T t ≤ C, t ∈ R, (4.97) for some fixed constant C > 0. Thus, by Sz.-Nagy’s theorem [21] (see also [1, Sect. I.6], [4, Lemma XV.6.1]), there exists an operator V ∈ B(H) with V −1 ∈ B(H), such that V −1 T t V = U (t) = eitH , t ∈ R, (4.98) where U (t), t ∈ R, is a strongly continuous unitary one-parameter group with a self-adjoint (possibly unbounded) generator H = H ∗ in H. Thus, T t = eit log((A+IH )

−1

)

= V eitH V −1 = eitV HV

−1

t ∈ R,

,

(4.99)

implying log (A + IH )−1 = V HV −1 .

(4.100) −1

) is also the generator

t ≥ 0,

(4.101)

On the other hand (cf. [14, Proposition 2.1]), log((A + IH ) of a C0 -semigroup of contractions in H, (A + IH )−t = et log((A+IH )

−1

)

,

and hence, (A + IH )−t = et log((A+IH )

−1

)

= etV HV

−1

= V etH V −1 ,

t ≥ 0.

(4.102)

Taking t = 1 in (4.102) then shows that A is similar to a self-adjoint operator in H. (Incidentally, we note that necessarily H ≤ cIH for some c ∈ R, since (4.101) represents a family of contractions.)

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Acknowledgments. We are indebted to Brian Davies, Nigel Kalton, Heinz Langer, Yuri Latushkin, Vladimir Ovchinnikov, Leiba Rodman, and Barry Simon for helpful correspondence. One of us (F.G.) gratefully acknowledges the extraordinary hospitality of the Faculty of Mathematics of the University of Vienna, Austria, and especially, that of Gerald Teschl, during his three month visit in the first half of 2008, where parts of this paper were written.

References [1] Ju. L. Dalecki˘ı and M. G. Kre˘ın, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monographs, 43, Amer. Math. Soc., Providence, RI, 1974. [2] E. B. Davies, Linear Operators and their Spectra, Cambridge Studies in Advanced Mathematics, Vol. 106, Cambridge University Press, Cambridge, 2007. [3] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Wiley– Interscience, New York, 1988. [4] N. Dunford and J. T. Schwartz, Linear Operators, Part III: Spectral Operators, Wiley–Interscience, New York, 1988. [5] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1989. [6] B. Fuglede, A commutativity theorem for normal operators, Proc. Nat. Acad. Sci., USA, 33, 35–40 (1950). [7] E. Heinz, Beitr¨ age zur St¨ orungstheorie der Spektralzerlegung, Math. Ann. 123, 415– 438 (1951). [8] T. Kato, A generalization of the Heinz inequality, Proc. Japan Acad. 37, 305–308 (1961). [9] T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13, 246–274 (1961). [10] T. Kato, Fractional powers of dissipative operators, II, J. Math. Soc. Japan, 14, 242–248 (1962). [11] T. Kato, Perturbation Theory for Linear Operators, corr. printing of the 2nd ed., Springer, Berlin, 1980. [12] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff, Leyden, 1976. [13] S. G. Krein, Ju. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators, Transl. Math. Monographs, 54, Amer. Math. Soc., Providence, RI, 1982. [14] N. Okazawa, Logarithms and imaginary powers of closed linear operators, Integral Equ. Oper. Theory 38, 458–500 (2000). [15] C. R. Putnam, On normal operators in Hilbert space, Amer. J. Math. 73, 357–362 (1951). [16] C. R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer, Berlin, 1967.

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[17] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. [18] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, New York, 1978. [19] M. Rosenblum, On a theorem of Fuglede and Putnam, J. London Math. Soc. 33, 376–377 (1958). [20] B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms, Princeton University Press, Princeton, NJ, 1971. [21] B. Sz.-Nagy, On uniformly bounded linear transformations in Hilbert space, Acta Sci. Math. (Szeged) 11, 152–157 (1947). [22] B. Sz.-Nagy and C. Foia¸s, Harmonic Analysis of Operators on Hilbert Space, NorthHolland, Amsterdam, 1970. [23] J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, 1980. Fritz Gesztesy Department of Mathematics, University of Missouri, Columbia, MO 65211, USA e-mail: [email protected] URL: http://www.math.missouri.edu/personnel/faculty/gesztesyf.html Mark Malamud Mathematics, Institute of Applied Mathematics and Mechanics, R. Luxemburg str. 74, Donetsk 83114, Ukraine e-mail: [email protected] Marius Mitrea Department of Mathematics, University of Missouri, Columbia, MO 65211, USA e-mail: [email protected] URL: http://www.math.missouri.edu/personnel/faculty/mitream.html Serguei Naboko Department of Mathematical Physics, Institute of Physics St. Petersburg State University, 1 Ulia- novskaia, St. Petergoff, St. Petersburg, 198504, Russia and Department of Mathematics, University of Alabama at Birmingham Birmingham, AL 35294-1170, USA e-mail: [email protected] [email protected] Submitted: February 20, 2009.

Integr. equ. oper. theory 64 (2009), 115–136 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010115-22, published online April 24, 2009 DOI 10.1007/s00020-009-1676-z

Integral Equations and Operator Theory

Norm Attaining Operators and Pseudospectrum S. Shkarin Abstract. It is shown that if 1 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that kI + T k > 1 and I + T does not attain its norm. Mathematics Subject Classification (2000). Primary 47A30; Secondary 47A10. Keywords. Norm of the resolvent, pseudospectrum, norm attaining operators.

1. Introduction All vector spaces in this paper are assumed to be over the field K, being either the field C of complex numbers or the field R of real numbers. As usual, N is the set of positive integers and R+ is the set of non-negative real numbers. The Banach space of all bounded linear operators from a Banach space X to a Banach space Y is denoted by L(X, Y ) and K(X, Y ) stands for the space of compact linear operators T : X → Y . We write L(X) instead of L(X, X), K(X) instead of K(X, X) and X ∗ instead of L(X, K). We say that T ∈ L(X, Y ) attains its norm on x ∈ X if kxk = 1 and kT xk = kT k. It is said that T attains its norm if there is an x ∈ X with kxk = 1 such that T attains its norm on x. We would like to mention a few classical results on the norm attaining property. The James theorem [10] says that a Banach space X is reflexive if and only if any f ∈ X ∗ attains its norm. As a corollary of the James theorem, we have that X is reflexive if and only if any T ∈ K(X) attains its norm. Indeed, if X is non-reflexive, then the James theorem provides a bounded rank one operator which does not attain its norm. On the other hand, if X is reflexive and T ∈ K(X), then the function x 7→ kT xk is weakly sequentially continuous and the closed unit ball of X is weakly sequentially compact and therefore the above The author would like to thank E. Shargorodsky for his interest and comments.

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function attains its maximum on the unit ball. Below it is shown that the situation with attaining of the norm for operators I +T with compact T is quite different. For further results on norm attaining operators we refer to [1, 2, 3, 4, 5, 12, 14, 16, 17] and references therein. The norm attaining property of operators is related to the concept of the pseudospectrum. Let X be a complex Banach space. For ε > 0 and T ∈ L(X), the ε-pseudospectrum of T is usually defined as σε (T ) = {λ ∈ C : k(T − λI)−1 k > ε−1 }

(1.1)

or as Σε (T ) = {λ ∈ C : k(T − λI)−1 k > ε−1 }, (1.2) −1 where k(T − λI) k is assumed to be infinite if λ belongs to the spectrum σ(T ) of T , see, for instance, [7, 8, 9, 13, 23, 24, 19, 20]. Recently Shargorodsky [19] demonstrated that the level set Σε (T ) \ σε (T ) = {λ ∈ C : k(T − λI)−1 k = ε−1 }

(1.3)

can have non-empty interior in general, while its interior is empty when the space X or the dual space X ∗ is complex uniformly convex. It is well-known that [ σε (T ) = σ(T + A), (1.4) kAk<ε

see, for instance, [9, 11]. This equality is one of the main reasons why many authors prefer (1.1) rather than (1.2) as the definition of pseudospectrum. We study the question whether the similar equality holds in the case of non-strict inequalities: [ Σε (T ) = Σ0ε (T ), where Σ0ε (T ) = σ(T + A). (1.5) kAk6ε

It is worth noting that the inclusion Σ0ε (T ) ⊆ Σε (T )

(1.6)

holds for any bounded operator T on any Banach space [9, 13]. It is proved by Finck and Ehrhardt, see [15], that the equality (1.5) holds if X is a Hilbert space. Shargorodsky [20] constructed a bounded linear operator T on the reflexive space X = `p × `q with 1 < p < q < ∞ and the norm k(x, y)k = kxkp + kykq for which (1.5) fails. He also constructed T ∈ K(`1 ) for which (1.5) fails. These examples naturally lead to the following question, raised in [20]. Question 1.1. Is it true that (1.5) holds for any compact operator on a reflexive complex Banach space? We show that, in general, the answer to Question 1.1 is negative and demonstrate that if 1 < p < ∞ and X is an `p -direct sum of finite dimensional Banach spaces, then (1.5) holds for each bounded operator T on X. In particular, it holds when X = `p with 1 < p < ∞. It turns out that the validity of (1.5) for any compact operator T on a Banach space X is closely related to the norm-attaining property.

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Proposition 1.2. Let X be a complex Banach space, T ∈ K(X), ε > 0 and z ∈ Σε (T ). Then the following conditions are equivalent: (1.2.1) z ∈ Σ0ε (T ); (1.2.2) if k(T − zI)−1 k = ε−1 > |z|−1 , then (T − zI)−1 attains its norm. We use the above proposition in order to prove the following result. Proposition 1.3. Let X be a complex Banach space. Then the following conditions are equivalent: (1.3.1) the equality (1.5) holds for any ε > 0 and any T ∈ K(X); (1.3.2) for any T ∈ K(X) such that I + T is invertible and kI + T k > 1, the operator I + T attains its norm. The above proposition motivates the introduction of the following class of Banach spaces. Definition 1. We say that a Banach space X belongs to the class W if for each T ∈ K(X) such that kI + T k > 1, I + T attains its norm. From Proposition 1.3 it follows that for any X ∈ W and any compact operator T on X, the equality (1.5) holds. It is also worth noting that the restriction kI + T k > 1 is natural. Indeed, the diagonal operator D on `2 with the diagonal entries {1 − 2−n }n∈N has norm 1 which is not attained and D is the sum of the identity operator and a compact operator. The following proposition provides a sufficient condition for a Banach space to belong to W. Definition 2. Let 1 < p < ∞. We say that a Banach space X is a p-space if X is reflexive and for any x ∈ X and any sequence {un }n∈N in X weakly converging to zero, lim kx + un k − (kxkp + kun kp )1/p = 0. (1.7) n→∞

It is easy to see that any Hilbert space is a 2-space and that any finite dimensional Banach space is a p-space for any p. Note that an infinite dimensional Banach space cannot be a p-space and a q-space for p 6= q. Recall that if 1 6 p < ∞ and {Xα }α∈Λ is a family of Banach spaces, then their `p -direct sum is the space Y X p X= x∈ Xα : kxα k < ∞ α∈Λ

α∈Λ

endowed with the norm kxk =

X

p

kxα k

1/p .

α∈Λ

When the family consists of just 2 spaces X and Y we denote its `p -direct sum by X ⊕p Y . We also denote X × Y with the norm k(x, y)k = max{kxk, kyk} by the symbol X ⊕∞ Y . Proposition 1.4. Let 1 < p < ∞. Then any closed linear subspace of an `p -direct sum of any family of finite dimensional Banach spaces is a p-space.

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The following two theorems provide, in particular, a partial affirmative answer to Question 1.1. The next one extends the validity of (1.5) for any bounded operator T from just Hilbert spaces to a wider class of Banach spaces. Theorem 1.5. Let 1 < p < ∞ and X be an `p -direct sum of a family of complex finite dimensional Banach spaces. Then (1.5) holds for any T ∈ L(X). In the case of compact operators, we can extend the last theorem. Theorem 1.6. Let 1 kJk, the operator J + T attains its norm. In particular, any p-space belongs to W. Theorem 1.6 and Proposition 1.3 imply the following corollary. Corollary 1.7. Let 1 < p < ∞ and X be a complex p-space. Then (1.5) holds for any T ∈ K(X). Even a slight perturbation of the norm destroys the above results. The following theorem provides a negative answer to Question 1.1. Theorem 1.8. Let 1 1 and I + T does not attain its norm. Proposition 1.3 and Theorem 1.8 imply that for 1 0 and T ∈ K(C ⊕q `p ) such that (1.5) fails. Since K ⊕q `p is isomorphic to `p , we see that belonging to W and validity of (1.5) are renorming sensitive properties. In particular, K ⊕p `2 is isomorphic to the Hilbert space `2 for any 1 6 p 6 ∞ and belongs to W if and only if p = 2. However, for the spaces K ⊕p `2 the situation improves if we consider finite rank operators instead of compact ones. Proposition 1.9. Let 1 6 p 6 ∞, Y be a finite dimensional Banach space and X = Y ⊕p `2 . Then for any bounded finite rank operator T on X, the operator I + T attains its norm. The last proposition suggests that the answer to Question 1.1 might be affirmative if we replace the compactness condition by the stronger one of T having finite rank. Unfortunately this is not the case. Proposition 1.10. There exists a norm k · k on `2 , equivalent to the original norm k · k2 , and a rank one operator T on `2 such that T 2 = 0, kI + T k = 2 (with respect to the norm k · k on `2 ) and the norm of I + T is not attained. The equality T 2 = 0 for the operator from the above proposition ensures invertibility of T − I and the equality I + T = −(T − I)−1 . Thus (T − I)−1 does not attain its norm and k(T − I)−1 k = 2 > 1. By Proposition 1.2, we have −1 ∈ Σ1/2 (T ) \ Σ01/2 (T ) and (1.5) fails for T with ε = 1/2.

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2. Proof of Propositions 1.2 and 1.3 The following lemma is a known fact [9, 13]. For convenience of the reader we reproduce its short proof. Lemma 2.1. Let X be a complex Banach space ε > 0 and T ∈ L(X). Assume also that z ∈ Σε (T ) \ σ(T ) and (T − zI)−1 attains its norm. Then there is A ∈ L(X) such that kAk 6 ε and z ∈ σ(T + A). Proof. Since (T − zI)−1 attains its norm, there exist x, y ∈ X such that kyk = kxk = 1 and (T − zI)−1 x = cy, where c = k(T − zI)−1 k. Using the Hahn-Banach theorem, we can pick f ∈ X ∗ for which kf k = f (y) = 1. Consider the operator A ∈ L(X),

Au = −c−1 f (u)x.

Clearly kAk 6 c−1 . Since z ∈ Σε (T ), we have c−1 6 ε. Thus kAk 6 ε. Moreover, Ay = −c−1 x. From the equality (T − zI)−1 x = cy it follows that T y = zy + c−1 x. Hence (T + A)y = zy and z ∈ σ(T + A). 2.1. Proof of Proposition 1.2 If X is finite dimensional, then any S ∈ L(X) attains its norm and according to Lemma 2.1, both (1.2.1) and (1.2.2) are satisfied. Thus for the rest of the proof, we can assume that X is infinite dimensional. Assume that (1.2.2) is satisfied. Since X is infinite dimensional and T is compact, k(T − zI)−1 k > |z|−1 . If the relation k(T − zI)−1 k = ε−1 > |z|−1 fails, then either k(T − zI)−1 k > ε−1 or k(T − zI)−1 k = ε−1 = |z|−1 . If k(T − zI)−1 k > ε−1 , then z ∈ σε (T ) and, according to (1.4), z ∈ Σ0ε (T ). If k(T − zI)−1 k = ε−1 = |z|−1 , then kAk = ε, where A = zI. Since X is infinite dimensional and T is compact, we have 0 ∈ σ(T ). Hence z ∈ σ(T + zI) = σ(T + A). Thus z ∈ Σ0ε (T ). It remains to consider the case when k(T − zI)−1 k = ε−1 > |z|−1 and (T − zI)−1 attains its norm. In this case, from Lemma 2.1 it follows that z ∈ Σ0ε (T ). The implication (1.2.2) =⇒ (1.2.1) is verified. Assume now that (1.2.1) is satisfied. That is, there exists A ∈ L(X) such that kAk 6 ε and z ∈ σ(T + A). Hence 0 ∈ σ(T − zI + A). Suppose that (1.2.2) fails. Then k(T − zI)−1 k = ε−1 > |z|−1 and the norm of (T − zI)−1 is not attained. Since kAk 6 ε and |z| > ε, the operator −zI + A is invertible. Then T − zI + A is a Fredholm operator of index zero as a sum of a compact operator T and an invertible operator −zI + A. Since 0 ∈ σ(T − zI + A), T − zI + A is non-invertible and therefore, being a Fredholm operator of index zero, it has non-trivial kernel. Thus we can pick x ∈ X such that kxk = 1 and (T − zI + A)x = 0. It follows that −Ax = (T − zI)x and therefore x = −(T − zI)−1 Ax. Using the relations k(T − zI)−1 k = ε−1 and kAk 6 ε, we obtain 1 = kxk = k − (T − zI)−1 Axk 6 ε−1 kAxk 6 ε−1 kAkkxk 6 kxk = 1. Obviously, all inequalities in the above display should be equalities which can only happen if kAk = kAxk = ε. Then kyk = 1, where y = −ε−1 Ax. Since −(T − zI)−1 Ax = x, we obtain (T − zI)−1 y = ε−1 x. Thus k(T − zI)−1 yk = ε−1 =

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k(T − zI)−1 k. That is, (T − zI)−1 attains its norm on y. This contradiction completes the proof of the implication (1.2.1) =⇒ (1.2.2) and that of Proposition 1.2. 2.2. Proof of Proposition 1.3 First, assume that (1.3.2) is satisfied. Let also T ∈ K(X), ε > 0 and z ∈ Σε (T ). According to (1.6), it suffices to show that z ∈ Σ0ε (T ). By Proposition 1.2, the latter happens if and only if (1.2.2) is satisfied. Assume that it is not the case. Then k(T − zI)−1 k = ε−1 > |z|−1 and the norm of (T − zI)−1 is not attained. On the other hand, (T − zI)−1 = −z −1 (I + S), where S = −z(T − zI)−1 − I is compact. Moreover kI + Sk > 1 since k(T − zI)−1 k > |z|−1 . By (1.3.2), I + S attains its norm and therefore so does (T − zI)−1 . This contradiction proves the implication (1.3.2) =⇒ (1.3.1). Next, assume that (1.3.1) is satisfied, T ∈ K(X), I + T is invertible and c = kI +T k > 1. Let S = (I +T )−1 −I. Clearly S is compact and I +T = (S +I)−1 . Let ε = c−1 . Since k(S + I)−1 k = kI + T k = ε−1 > 1, we have −1 ∈ Σε (S). According to (1.3.2), −1 ∈ Σ0ε (S). Proposition 1.2 implies now that (S + I)−1 = I + T attains its norm. This completes the proof of the implication (1.3.1) =⇒ (1.3.2) and that of Proposition 1.3.

3. `p -direct sums of finite dimensional Banach spaces Throughout this section 1 < p < ∞ and X is the `p -direct sum of a family {Xα : α ∈ Λ} of finite dimensional Banach spaces. For x ∈ X, the support of x is the set supp(x) = {α ∈ Λ : xα 6= 0}. From the definition of the `p -direct sum it follows that the support of any element of X is at most countable. For a subset B of Λ, we consider PB ∈ L(X) defined by the formula xα if α ∈ B, (PB x)α = (3.1) 0 if α ∈ / B. Clearly PB is a linear projection and kPB k = kI − PB k = 1 if B is non-empty and B 6= Λ. Lemma 3.1. Let {xn }n∈N be a sequence in X weakly convergent to zero and {εk }k∈N be a sequence of positive numbers. Then there exist a strictly increasing sequence {nk }k∈N of positive numbers and a sequence {uk }k∈N of elements of X such that kxnk − uk k < εk for each k ∈ N and the sets supp(uk ) are finite and pairwise disjoint. Proof. We construct the required sequences inductively. On the first step we take n1 = 1, pick a finite subset B of Λ such that kx1 −PB x1 k < ε1 and put u1 = PB x1 . Assume now that k > 2, n1 < . . . < nk−1 , u1 , . . . , uk−1 are vectors in X with pairwise disjoint finite supports such that kxnj − uj k < εj for 1 6 j 6 k − 1. Let now C be the union of supp(uj ) for 1 6 j 6 k−1. Since C is finite, PC is a compact

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operator and therefore kPC xn k → 0 as n → ∞. Thus we can pick nk > nk−1 such that kPC xnk k < εk /2. Next, choose a finite subset A of Λ such that C ⊆ A and kxnk − PA xnk k < εk /2 and put uk = PA xnk − PC xnk . Clearly supp(uk ) ⊆ A \ C and therefore uk has finite support and the supports of u1 , . . . , uk are pairwise disjoint. Finally, kxnk − uk k = kPC xnk + (xnk − PA xnk )k 6 kxnk − PA xnk k + kPC xnk k < εk /2 + εk /2 = εk . The description of the inductive construction of sequences {nk } and {uk } is now complete. Lemma 3.2. X is a p-space. Proof. Since 1 0. Pick a finite subset B of Λ such that kx − PB xk < ε. Since PB is a compact operator and {un } converges weakly to 0, we have kPB un k → 0 as n → ∞. Let xn = un − PB un . Then kxn − un k → 0 as k → ∞ and supports of xn do not meet B. Since the support of PB x is contained in B, the supports of xn do not intersect the support of PB x and from the definition of the norm on X it follows that kPB x + xn k = (kPB xkp + kxn kp )1/p . Since kx − PB xk < ε and kxn − un k → 0 as n → ∞, we see that |kPB x + xn k − kx + un k| < ε and (kxkp + kun kp )1/p − (kPB xkp + kxn kp )1/p < ε for all sufficiently large n. From the last two displays it follows that kx + un k − (kxkp + kun kp )1/p < 2ε for all sufficiently large n. Since ε > 0 is arbitrary, the equality (1.7) follows.

3.1. Proof of Proposition 1.4 By Lemma 3.2, the class of p-spaces contains `p -direct sums of finite dimensional Banach spaces. From the definition it immediately follows that (closed linear) subspaces of p-spaces are p-spaces. Hence closed linear subspaces of `p -direct sums of finite dimensional Banach spaces are p-spaces. 3.2. Operators on p-spaces Lemma 3.3. Let 1 < p 6 q < ∞, Y be a p-space, Z be a q-space, T ∈ L(Y, Z), {xn }n∈N be a sequence in Y such that kxn k → 1 and kT xn k → kT k as n → ∞ and {xn } is weakly convergent to x ∈ X. Then kT xk = kT kkxk. Proof. Let un = xn − x. Then {un } is weakly convergent to 0. Since T is linear and bounded, T is also continuous with respect to the weak topology and therefore

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{T un } is weakly convergent to 0. For brevity denote c = kT k. Since Y is a p-space and Z is a q-space, we have 1 = lim kxn k = lim kx + un k = lim (kxkp + kun kp )1/p ,

(3.2)

c = lim kT xn k = lim kT x + T un k = lim (kT xkq + kT un kq )1/q .

(3.3)

n→∞

n→∞

n→∞

n→∞

n→∞

n→∞

Clearly kT un k 6 ckun k for each n ∈ N. Assume that kT xk = 6 ckxk. Then kT xk < ckxk. Using these inequalities together with (3.2) and (3.3) and taking into account that p 6 q, we obtain c = lim (kT xkq + kT un kq )1/q < lim (cq kxkq + cq kun kq )1/q n→∞

n→∞

6 c lim (kxkp + kun kp )1/p = c. n→∞

This contradiction proves the equality kT xk = ckxk.

Recall that X is the `p -direct sum of the family {Xα : α ∈ Λ} of finite dimensional Banach spaces. Lemma 3.4. Let T ∈ L(X) be such that inf kT xk = c > 0.

kxk=1

(3.4)

Then there exists S ∈ L(X) such that kSk = c and inf k(T + S)xk = 0.

kxk=1

(3.5)

Proof. Pick a sequence {xn }n∈N in X such that kxn k → 1 and kT xn k → c as n → ∞. Since X is reflexive, we can choose such a sequence {xn } being weakly convergent to x ∈ X. Clearly kxk 6 1. Case x = 0. That is, {xn } weakly converges to 0. By Lemma 3.1, we can find a strictly increasing sequence {nk }k∈N of positive integers and a sequence {yk }k∈N of elements of X0 such that the supports of yk are pairwise disjoint and kxnk − yk k < 2−k for any k ∈ N.

(3.6)

Since the sequence {xnk } weakly converges to 0, formula (3.6) implies that {yk } also weakly converges to 0. Since T ∈ L(X), the sequence {T yk } weakly converges to 0. Using Lemma 3.1 once again, we see that there exist a strictly increasing sequence {km }m∈N of positive integers and a sequence {wm }m∈N in X0 such that the supports of wm are pairwise disjoint and kT ykm − wm k < 2−m for any m ∈ N.

(3.7)

From (3.6) it follows that kT xnk −T yk k 6 2−k kT k for any k ∈ N. Since kT xnk k → c as k → ∞, we have kT yk k → c as k → ∞. Now by (3.7) and (3.6) we obtain lim kykm k = 1

m→∞

and

lim kwm k = c.

m→∞

(3.8)

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For each m ∈ N let Am = supp(ykm ), Pm = PAm and Xm = Pm (X). By the ∗ Hahn-Banach theorem, for any m ∈ N, we can find ϕm ∈ Xm such that kϕm k = 1 and ϕm (ykm ) = kykm k. Consider the operator S ∈ L(X) defined by the formula Su = −c

∞ X ϕm (Pm u) wm . kwm k m=1

From the equalities kϕm k = 1, pairwise disjointness of Am and pairwise disjointness of supp(wm ) it immediately follows that kSk = c. On the other hand, by the definition of S ckykm k wm for any m ∈ N. Sykm = − kwm k According to (3.8) we have kSykm + wm k → 0 as m → ∞. Hence, by (3.7), k(T + S)ykm k → 0 as m → ∞. From (3.8) it follows that kykm k → 1 as m → ∞. Therefore, (3.5) is satisfied. Case x 6= 0. Let Y = T (X). According to (3.4), Y is a closed linear subspace of X and T : X → Y is invertible. Consider R ∈ L(Y, X) being the inverse of T : X → Y . From (3.4) it follows that kRk = c−1 . It is also clear that the sequence un = c−1 T xn is weakly convergent to c−1 T x and kun k → 1 as n → ∞. Moreover Run = c−1 xn for any n ∈ N and therefore Run weakly converges to c−1 x and kRun k → c−1 = kRk as n → ∞. By Proposition 1.4, X and Y are p-spaces. Hence, according to Lemma 3.3, kRkkc−1 T xk = kc−1 RT xk. Taking into account that RT x = x and kRk = c−1 , we have kT xk = ckxk. By the HahnBanach theorem, we can find ϕ ∈ X ∗ such that kϕk = 1 and ϕ(x) = kxk. Let now S ∈ L(X), Su = −kxk−1 ϕ(u)T x. Since kT xk = ckxk, we have kSk 6 c. Moreover (T + S)x = T x − T x = 0 and therefore T + S has non-trivial kernel. Hence (3.5) is satisfied. 3.3. Proof of Theorem 1.5 Let T ∈ L(X), ε > 0 and z ∈ Σε (T ). In view of (1.6), it suffices to show that z ∈ Σ0ε (T ). Since z ∈ Σε (T ), we have k(T − zI)−1 k > ε−1 . If k(T − zI)−1 k > ε−1 , the inclusion z ∈ Σ0ε (T ) follows from (1.4). It remains to consider the case k(T − zI)−1 k = ε−1 . In this case ε = inf k(T − zI)xk. kxk=1

By Lemma 3.4, we can find S ∈ L(X) such that kSk 6 ε and 0 = inf k(T − zI + S)xk. kxk=1

The last display implies that T + S − zI is not invertible. Hence z ∈ σ(T + S). Since kSk 6 ε, we obtain the required inclusion z ∈ Σ0ε (T ).

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3.4. Proof of Theorem 1.6 Lemma 3.5. Let Y and Z be Banach spaces, J ∈ L(Y, Z) and T ∈ K(Y, Z) be such that kJ + T k > kJk. Assume also that {xn }n∈N is a sequence of vectors in Y weakly convergent to x ∈ Y for which kxn k → 1 and k(J + T )xn k → kJ + T k as n → ∞. Then x 6= 0. Proof. Since T is compact, kT xn − T xk → 0 as n → ∞. Hence limn kJxn + T xk = kJ +T k. On the other hand, limn kJxn k 6 kJk. Thus using the triangle inequality, we obtain kT xk > kJ +T k−kJk. It follows that kxkkT k > kT xk > kJ +T k−kJk > 0. Hence, x 6= 0. We are ready to prove Theorem 1.6. Pick a sequence {xn }n∈N of elements of X such that kxn k = 1 for any n ∈ N and k(J + T )xn k → kJ + T k as n → ∞. Since X is reflexive, we, passing to a subsequence, if necessary, may assume that {xn } weakly converges to x ∈ X. By Lemma 3.5 x 6= 0. According to Lemma 3.3, kJx + T xk = kxkkJ + T k. Hence J + T attains its norm on the vector x/kxk.

4. Operators on K ⊕q `p We start by a series of elementary observations. Lemma 4.1. Let X and Y be Banach spaces and T ∈ L(X, Y ) be an operator attaining its norm. Then the dual operator T ∗ ∈ L(Y ∗ , X ∗ ) attains its norm. Proof. Since T attains its norm, there exists x ∈ X such that kxk = 1 and kT xk = kT k. By the Hahn-Banach theorem, we can pick ϕ ∈ Y ∗ such that kϕk = 1 and ϕ(T x) = kT xk. Since ϕ(T x) = (T ∗ ϕ)(x), we have ϕ(T x) 6 kT ∗ ϕkkxk = kT ∗ ϕk. Since kT xk = kT k = kT ∗ k, we see that kT ∗ ϕk > kT ∗ k and kϕk = 1. Thus kT ∗ ϕk = kT ∗ k and therefore T ∗ attains its norm at ϕ. The above lemma immediately implies the following corollary. Corollary 4.2. Let X be a reflexive Banach space and T ∈ L(X). Then T attains its norm if and only if T ∗ attains its norm. In particular, using the facts that an operator is compact if and only if its dual is compact and an operator is invertible if and only if its dual is invertible, we have the following result. Corollary 4.3. Let X be a reflexive Banach space. Then X ∈ W if and only if X ∗ ∈ W. Moreover, the following two statements are equivalent: (4.3.1) there is a compact operator T on X such that I +T is invertible, kI +T k > 1 and I + T does not attain its norm; (4.3.2) there is a compact operator S on X ∗ such that I +S is invertible, kI +Sk > 1 and I + S does not attain its norm.

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4.1. Proof of Theorem 1.8 Let 1 q 0 if p < q and p0 < q 0 if p > q. According to Corollary 4.3, it is enough to prove Theorem 1.8 in the case p < q. Thus from now on, we assume that p < q. We naturally interpret X as a space of sequences x = {xn }n>0 , where x0 and {xn }n∈N correspond to the K-component and the `p -component in the decomposition X = K ⊕q `p respectively. For any x ∈ X, we denote X 1/p ∞ α(x) = |x0 |, β(x) = |x1 | and γ(x) = |xn |p . (4.1) n=2

Clearly, for x ∈ X, kxk = f (α(x), β(x), γ(x)), where ( 1/q αq + (β p + γ p )q/p if q < ∞, f (α, β, γ) = p p 1/p max α, (β + γ ) if q = ∞.

(4.2) (4.3)

Consider the operator S ∈ L(X) defined by the formula (Sx)0 = x1 , (Sx)1 = x0 nxn if n > 2. That is, and (Sx)n = n+1 2x2 3x3 Sx = x1 , x0 , , ,... . 3 4 Clearly T = S − I is compact. Thus in order to verify that T satisfies the required conditions, it suffices to show that S = I + T is invertible, kSk > 1 and S does not attain its norm. Invertibility of S is obvious. Indeed, the operator R ∈ L(X) defined as Rx = x1 , x0 , 3x2 /2, 4x3 /3, . . . is the inverse of S. Next, let x ∈ X be such that kxk = 1 and let α(x), β(x) and γ(x) be the numbers defined in (4.1). It is clear that α(Sx) = β(x), β(Sx) = α(x), γ(Sx) 6 γ(x). Moreover, γ(Sx) < γ(x) if γ(x) > 0. Thus according to (4.2), f (α(x), β(x), γ(x)) = 1, kSxk = f (β(x), α(x), γ(Sx)) 6 f (β(x), α(x), γ(x)).

(4.4)

Moreover, since γ(Sx) < γ(x) when γ(x) > 0, we have kSxk < f (β(x), α(x), γ(x)) if q < ∞ and γ(x) > 0 and if q = ∞, γ(x) > 0 and β(x) < (α(x)p + γ(x)p )1/p .

(4.5)

According to (4.4), kSxk 6 C, where C = sup{f (β, α, γ) : (α, β, γ) ∈ K} and K = {(α, β, γ) ∈ R3+ : f (α, β, γ) = 1}. Since K is compact and f is continuous, the supremum in the definition of C is attained. Using, for instance, the Lagrange multipliers technique, one can easily see that the function (α, β, γ) 7→ f (β, α, γ) from K to R+ attains its maximal

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value C = 2 p − q in exactly one point being (2−1/q , 0, 2−1/q ). From (4.5) it now follows that 1 1 (4.6) kSxk < C = 2 p − q whenever kxk = 1. Now consider the sequence xn = 2−1/q e0 + 2−1/q en , n ∈ N, where {ek } is the canonical basis in the sequence space X. Clearly kxnk = 1 for each n ∈ N. On the n other hand, for any n > 2, Sxn = 2−1/q e1 + n+1 en and therefore n p 1/p 1 1 (4.7) → 2 p − q as n → ∞. kSxn k = 2−1/q 1 + n+1 1

1

From (4.6) and (4.7) it follows that kSk = 2 p − q > 1 and the norm of S is not attained. The proof of Theorem 1.8 is now complete.

5. Proper extensions of Hilbert spaces and finite rank operators In this section we prove a theorem slightly stronger then Proposition 1.9. We need some preparation. Throughout this section H is a Hilbert space and n ∈ N. We say that X = Kn × H is a proper extension of H if X is endowed with a norm such that k(t, x)k = ϕ(t, kxk) for any t ∈ Kn , x ∈ H, (5.1) where ϕ : Kn × R+ → R+ is a function and ϕ(0, 1) = 1. The fact that (t, x) 7→ ϕ(t, kxk) is a norm on X implies immediately that ϕ is Lipschitzian, convex, ϕ(t, a) > 0, whenever (t, a) 6= (0, 0) and ϕ(st, sa) = sϕ(t, a) for any s, a ∈ R+ and t ∈ Kn . The normalization condition ϕ(0, 1) = 1 implies that k(0, x)k = kxkH for any x ∈ H. Thus H is naturally isometrically embedded into X. Since H has finite codimension in X, we see that X is a Banach space and admits an equivalent norm which turns it into a Hilbert space. In particular, X is reflexive. Theorem 5.1. Let X = Kn × H be a proper extension of a Hilbert space H. Then for any bounded finite rank operator T on X, I + T attains its norm. Proof. Let ϕ : Kn × R+ → R+ be a function defining the norm on X according to (5.1). If H is finite dimensional, the result becomes trivial. Thus we can assume that H is infinite dimensional. Pick a sequence {ξk = (tk , xk )}k∈N of elements of X such that kξk k → 1 and k(I + T )ξk k → c as k → ∞. Since X is reflexive, we can, passing to a subsequence, if necessary, assume that {ξk } converges weakly to ξ = (t, x) ∈ X. Since T has finite rank, {T ξk } is norm convergent to T ξ = (s, y). Next, since weak and norm convergences on a finite dimensional Banach space coincide, we see that {tk } converges to t in Cn . Passing to a subsequence again, if necessary, we can assume that kxk − xk → α ∈ R+ . Since xk − x is weakly convergent to zero in the Hilbert space H and any Hilbert space is a 2-space, we see that lim kxk k = (kxk2 + α2 )1/2

k→∞

and

lim kxk + yk = (kx + yk2 + α2 )1/2 .

k→∞

(5.2)

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Since kξk k → 1, k(I + T )ξk k → c, tk → t and kT ξk − (s, y)k → 0, we have 1 = lim kξk k = lim k(tk , xk )k = lim k(t, xk )k, k→∞

k→∞

k→∞

c = lim k(I + T )ξk k = lim kξk + (s, y)k = lim k(t + s, xk + y)k. k→∞

k→∞

k→∞

Using (5.1), (5.2) and continuity of ϕ, we obtain ϕ(t, (kxk2 + α2 )1/2 ) = 1 and ϕ(t + s, (kx + yk2 + α2 )1/2 ) = c.

(5.3)

Since H is infinite dimensional and T has finite rank, the linear subspace L = {v ∈ H : T (0, v) = 0, hu, xi = hu, yi = 0} has finite codimension and therefore is non-trivial. Hence we can pick u ∈ L such that kuk = α. Since u is orthogonal to both x and y, we see that kx + uk = (kxk2 + α2 )1/2 and kx + y + uk = (kx + yk2 + α2 )1/2 . Hence, according to (5.1) and (5.3) k(t, x + u)k = ϕ(t, kx + uk) = 1 and k(t + s, x + y + u)k = ϕ(t, kx + y + uk) = c. Finally, since T (0, u) = 0, we have T (t, x + u) = T (t, x) = (s, y). Hence (I + T )(t, x + u) = (t + s, x + y + u). Since c = kI + T k, from the last two displays it follows that I + T attains its norm on the vector (t, x + u). Theorem 1.9 follows from Theorem 5.1 since Y ⊕p `2 for a finite dimensional Banach space Y is a particular case of a proper extension.

6. Examples with rank one operators As was already mentioned in the introduction, Shargorodsky [20] constructed T ∈ K(`1 ) such that (1.5) fails for T for one prescribed ε > 0. We shall demonstrate that for X = `1 and X = c0 one can find a rank 1 operator T for which (1.5) fails for any ε > 0. As usual, we denote the canonical basis in c0 or `1 by {en }n>0 . Example 6.1. Let T ∈ L(c0 ), Tx =

X ∞

−n

2

xn e0 .

n=1

Then T has rank 1, T 2 = 0 and for any z ∈ K, kT + zIk = 1 + |z| and the operator T + zI does not attain its norm. Proof. Obviously, T has rank 1, kT k = 1 and T 2 = 0. Let z ∈ K and r = |z|. Since kT k = 1, we have kT + zIk 6 1 + r. For n ∈ N, consider xn = (r/z)e0 + e1 + e2 + . . . + en . Clearly kxn k = 1 and the e0 -coefficient of (T + zI)xn equals r + 1 − 2−n . Hence kT + zIk > r + 1 − 2−n for any n ∈ N. Thus kT + zIk > 1 + r. Since the opposite inequality is also true, kT + zIk = 1 + r. It remains to show that T + zI does not attain its norm. Assume the contrary. Then there exists x ∈ c0 such that

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kxk = 1 and kyk = 1 + r, where y = zx + T x. Since T x is a scalar multiple of e0 , we have yn = zxn for n ∈ N. Hence |yn | 6 r forPn ∈ N. Thus 1 + r = kyk = |y0 |. ∞ Using the definition of T we obtain y0 = zx0 + n=1 2−n xn . Hence 1 + r = |y0 | 6 |z||x0 | +

∞ X

2−n |xn | 6 r +

n=1

∞ X

2−n = 1 + r.

n=1

The latter is possible only if |xj | = 1 for any j which contradicts the inclusion x ∈ c0 . Example 6.2. Let T ∈ L(`1 ), Tx =

X ∞

(1 − 2−n )xn e0 .

n=1 2

Then T has rank 1, T = 0 and for any z ∈ C, kT + zIk = 1 + |z| and the operator T + zI does not attain its norm. Proof. Obviously, T has rank 1, kT k = 1 and T 2 = 0. Let z ∈ K and r = |z|. For n ∈ N, we have (T + zI)en = (1 − 2n )e0 + zen . Hence k(T + zI)en k = 1 + r − 2−n . Since ken k = 1, we see that kT + zIk > 1 + r. Since the opposite inequality is also true, kT + zIk = 1 + r. It remains to show that T + zI does not attain its norm. Assume the contrary. Then there exists x ∈ `1 such that kxk = 1 and kyk = 1 + r, where y = zx + T x. By definition of T , ∞ ∞ ∞ X X X −n (1 − 2 )xn +r |xn | 6 r|x0 | + (1 + r − 2−n )|xn |. 1 + r = kyk = zx0 + n=1

n=1

n=1

−n

Since the coefficients r and 1 + r − 2 in the last sum are strictly less than 1 + r, we get 1 + r < (1 + r)kxk = 1 + r. This contradiction completes the proof. The following proposition clarifies the situation with the above two operators and formula (1.5). Proposition 6.3. Let T be the operator from either Example 6.1 or Example 6.2 in the case K = C. Then for any ε > 0, Σ0ε (T ) = σε (T ) 6= Σε (T ). Proof. Since T 2 = 0, we have σ(T ) = {0} and, for any z ∈ C \ {0}, (T − zI)−1 = (−z −2 )(T + zI). Thus (T − zI)−1 attains its norm if and only if so does T + zI and k(T − zI)−1 k = |z|−2 kT + zIk = |z|−1 + |z|−2 > |z|−1 . Since T + zI never attains its norm, we, applying Proposition 1.2, see that for any ε > 0, p Σ0ε (T ) = σε (T ) = z ∈ C : |z|−1 + |z|−2 > ε−1 = z : |z| < ε + 4ε + ε2 . On the other hand, for any ε > 0, p Σε (T ) = z ∈ C : |z|−1 + |z|−2 > ε−1 = z : |z| 6 ε + 4ε + ε2 . According to the last two displays, Σ0ε (T ) 6= Σε (T ) for each ε > 0.

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6.1. Proof of Proposition 1.10 Recall that a subset A of a vector space X is called balanced if λx ∈ A whenever x ∈ A, λ ∈ K and |λ| 6 1. A set is called absolutely convex if it is convex and balanced. By aconv(A) we denote the absolutely convex hull of A, being the minimal absolutely convex set containing A. Clearly X n n X aconv(A) = λj xj : n ∈ N, xj ∈ A, λj ∈ K, |λj | 6 1 . (6.1) j=1

j=1

For a subset A of a topological vector space X, aconv(A) stands for the closure of aconv(A). We recall two elementary properties of absolutely convex hulls. The proof of the first one can be found in virtually any book on topological vector spaces, see, for instance, [18]. The second one is proved in [6]. For a different proof see [22]. Lemma 6.4. Let n ∈ N and K1 , . . . , Kn be compact convex subsets of a Hausdorff topological vector space X. Then [ X [ n n n n X Kj = λj xj : λj ∈ K, xj ∈ Kj , |λj | 6 1 . Kj = aconv aconv j=1

j=1

j=1

j=1

Moreover, the above set is compact. Lemma 6.5. Let {xn }n∈N be a sequence of elements of a sequentially complete locally convex Hausdorff topological vector space X converging to x ∈ X as n → ∞. Then ∞ X aconv(A) = α0 x + αn xn : α ∈ `1 , kαk1 6 1 where A = {xn : n ∈ N}. n=1

Moreover, aconv(A) is metrizable and compact. From now on in this section, by k · k2 we denote the canonical norm on `2 . We use the same symbol to denote the standard Euclidean norm on K2 : k(t, s)k2 = (|t|2 + |s|2 )1/2 . Let also {en }n∈N be the canonical orthonormal basis in `2 . For x ∈ `2 we denote x0 = x − x1 e1 − x2 e2 . That is, x0 is the orthogonal projection of x onto the closed linear span of the vectors √ e3 , e4 , . . . . Fix a sequence {qn }n∈N of positive numbers such that 1/2 < qn < 1/ 2 for each n ∈ N and limn→∞ qn = 1/2. Consider the set B ⊂ `2 , ∞ X B = x+ (αn (e2 + en+2 ) + βn qn (e1 + e2 + en+2 )) : n=1

kx k2 + k(x1 , x2 )k2 + kαk1 + kβk1 6 1 , 0

(6.2)

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where x ∈ `2 , α, β ∈ `1 and k · k1 is the canonical norm in√ `1 . Obviously, B is absolutely convex. Taking into account that ke2 + en+2 k2 = 2 and kqn (e1 + e2 + p √ en+2 )k2 = qn 3 6 3/2, we see that √ kuk2 6 2 for any u ∈ B. (6.3) Now if kuk2 6 1/2, then ku0 k22 + |u1 |2 + |u2 |2 6 1/4. An elementary application of the Cauchy inequality gives ku0 k2 + k(u1 , u2 )k2 6 1. Taking α = β = 0 and x = u, we see then that u ∈ B. Thus u∈B

if kuk2 6 1/2.

(6.4)

We consider the norm k · k on `2 being the Minkowski functional of the set B. Formulae (6.3) and (6.4) imply that it is indeed a norm and that it is equivalent to the Hilbert space norm k · k2 : 2−1/2 kuk2 6 kuk 6 2kuk2 for all u ∈ `2 . In particular, `2 endowed with the norm k · k is a reflexive Banach space. Using the definition of the Minkowski functional, we have that for u ∈ `2 , kuk = inf kx0 k2 + k(x1 , x2 )k2 + kαk1 + kβk1 : u=x+

∞ X

(αn (e2 + en+2 ) + βn qn (e1 + e2 + en+2 )) .

(6.5)

n=1

We shall show that B coincides with the closed unit ball with respect to the norm k · k. Since B is bounded and absolutely convex, it suffices to show that B is closed in `2 . First, note that the set B1 = {x ∈ `2 : kx0 k2 + k(x1 , x2 )k2 6 1}

(6.6)

is weakly compact and B1 ⊆ B. Next, let B2 = aconv{e2 +en+2 : n ∈ N}. Since the sequence e2 + en+2 converges weakly to e2 , Lemma 6.5 implies that B2 is weakly compact and ∞ X αn (e2 + en+2 ) : |s| + kαk1 6 1 . (6.7) B2 = se2 + n=1

P∞ It follows that B2 ⊆ B. Indeed, for u = se2 + n=1 αn (e2 + en+2 ) ∈ B2 , one just has to take x = se2 and β = 0 to see that u ∈ B. Similarly, let B3 = aconv{qn (e1 + e2 + en+2 ) : n ∈ N}. Since the sequence qn (e1 + e2 + en+2 ) converges weakly to (e1 + e2 )/2, Lemma 6.5 implies that B3 is weakly compact and ∞ X t B3 = (e1 + e2 ) + βn qn (e1 + e2 + en+2 ) : |t| + kβk1 6 1 . (6.8) 2 n=1 As above, it is clear that B3 ⊆ B. Since B is absolutely convex, we have B0 = aconv(B1 ∪ B2 ∪ B3 ) ⊆ B.

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By Lemma 6.4, B0 is weakly compact and B0 = {ax + by + cw : x ∈ B1 , y ∈ B2 , w ∈ B3 , |a| + |b| + |c| 6 1}.

(6.9)

From formulae (6.6–6.9) and (6.2) it follows that B ⊆ B0 . Hence B = B0 and therefore B is weakly compact. Thus B is closed in `2 which ensures that B is the closed unit ball for the norm (6.5). It follows that the infimum in (6.5) is always attained and that we can write kuk = min kx0 k2 + k(x1 , x2 )k2 + kαk1 + kβk1 : u=x+

∞ X

(αn (e2 + en+2 ) + βn qn (e1 + e2 + en+2 )) .

(6.10)

n=1

Lemma 6.6. The norm on `2 defined by (6.5) satisfies the following conditions: (6.6.1) ke2 + en+2 k = 1 for any n ∈ N; (6.6.2) qn ke1 + e2 + en+2 k = 1 for any n ∈ N. Proof. Taking x = 0, β = 0 and α = en , we see that e2 + en+2 ∈ B. Hence ke2 + en+2 k 6 1. Assume that ke2 + en+2 k < 1. Then there exist x ∈ `2 and α, β ∈ `1 such that kx0 k2 + k(x1 , x2 )k2 + kαk1 + kβk1 < 1

(6.11)

and e2 + en+2 = x +

∞ X

(αk (e2 + ek+2 ) + βk qk (e1 + e2 + ek+2 )).

k=1

Taking the inner product of both sides of the above equality with e2 , we obtain 1 = x2 +

∞ X

(αk + qk βk ).

k=1

Hence |x2 | +

∞ X

(|αk | + qk |βk |) > 1,

k=1

which contradicts (6.11). This contradiction proves (6.6.1). Taking x = 0, α = 0 and β = en , we see that qn (e1 + e2 + en+2 ) ∈ B. Hence qn ke1 + e2 + en+2 k 6 1. Assume that qn ke1 + e2 + en+2 k < 1. Then there exist x ∈ `2 and α, β ∈ `1 such that qn (e1 + e2 + en+2 ) = x +

∞ X k=1

(αk (e2 + ek+2 ) + βk qk (e1 + e2 + ek+2 )).

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and (6.11) is satisfied. Taking the inner product of both sides of the above equality with en+2 , e1 and e2 we obtain the following equality in K3 : X qn (1 − βn )(1, 1, 1) = (xn+2 , x1 , x2 ) + αn (1, 0, 1) + (αk (0, 0, 1) + βk qk (0, 1, 1)) k6=n

= (xn+2 , τ, σ) + αn (1, 0, 1), where (τ, σ) = (x1 , x2 ) +

(6.12) X

(αk (0, 1) + βk qk (1, 1)).

k6=n

√ Note that k(0, 1)k2 = 1 and kqk (1, 1)k2 6 1 since qk 6 1/ 2. Using (6.12) and the triangle inequality, we obtain X k(τ, σ)k2 6 k(x1 , x2 )k2 + (|αk | + |βk |). k6=n

From the last display together with (6.11) and (6.12), it follows that qn (1 − βn )(1, 1, 1) = ((xn+2 + αn )en+2 , τ, σ + αn ) where |xn+2 | + |αn | + |βn | + k(τ, σ)k2 < 1. Dividing by 1−βn and denoting y = xn+2 /(1−βn ), a = αn /(1−βn ), r = τ /(1−βn ) and p = σ/(1 − βn ) we see arrive to the following equality in K3 : p (qn , qn , qn ) = (y + a, r, p + a), where |y| + |a| + |r|2 + |p|2 < 1. Hence r = qn , p = y = qn − a and |a| + |qn − a| +

p

|qn |2 + |qn − a|2 < 1.

(6.13)

On p the other hand, qn > 1/2 and therefore |a| + |qn − pa| > qn > 1/2 and |qn |2 + |qn − a|2 > qn > 1/2. Hence |a| + |qn − a| + |qn |2 + |qn − a|2 > 1 which contradicts (6.13). This contradiction completes the proof of (6.6.2). Remark. In a similar way one can show that kuk = kuk2 if either u1 = u2 = 0 or u belongs to the linear span of e1 and e2 . Now we consider the operator S ∈ L(`2 ) defined by the formula Su = u+u2 e1 . Clearly S is the sum of the identity operator and a bounded rank 1 operator T u = u2 e1 . Obviously, T 2 = 0. Proposition 1.10 will be proved if we verify that kSk = 2 and S does not attain its norm. Lemma 6.7. For any non-zero u ∈ `2 , kSuk < 2kuk. Proof. Let u ∈ `2 be such that kuk = 1. It suffices to show that kSuk < 2. Since kuk = 1, from (6.10) it follows that there are x ∈ `2 and α, β ∈ `1 such that u=x+

∞ X

(αn (e2 + en+2 ) + βn qn (e1 + e2 + en+2 )),

(6.14)

n=1

kx0 k2 + k(x1 , x2 )k2 + kαk1 + kβk1 = 1.

(6.15)

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Next, from (6.14) and the definition of S, we obtain that Su = x + τ e1 +

∞ X

qn (βn + qn−1 αn )(e1 + e2 + en+2 ),

∞ X

where τ = x2 +

n=1

qn βn .

n=1

Using (6.5), we see that kSuk 6 kx0 k2 + k(x1 + τ, x2 )k2 +

∞ X

|βn + qn−1 αn |.

n=1

From the definition of τ it follows that kSuk 6 kx0 k2 + k(x1 + x2 , x2 )k2 +

∞ X

((1 + qn )|βn | + qn−1 |αn |).

n=1

1 0

1 acting 1

Taking into account that the norm of the operator with the matrix √ 1/2 < 53 , we have k(x1 + on the 2-dimensional Hilbert space K2 equals 3+2 5 x2 , x2 )k2 6 35 k(x1 , x2 )k2 . Substituting this into the last display and taking into account that 1 + qn 6 1 + 2−1/2 < 74 , we obtain ∞ X 5 7 kSuk 6 kx0 k2 + k(x1 , x2 )k2 + kβk1 + qn−1 |αn |. 3 4 n=1

Since the coefficients in the above display in front of kx0 k2 , k(x1 , x2 )k2 , kβk1 and each |αn | are all strictly less than 2, formula (6.15) implies that kSuk < 2. Now, observe that S(e2 + en+2 ) = e1 + e2 + en+2 . By Lemma 6.6 we have ke2 + en+2 k = 1 and ke1 + e2 + en+2 k = qn−1 . Hence kSk > qn−1 for any n ∈ N. Since qn−1 → 2 as n → ∞, we have kSk > 2. Thus from Lemma 6.7 it follows that kSk = 2 and S does not attain its norm. This completes the proof of Proposition 1.10.

7. Concluding remarks 1. A more general approach to study the class W is to consider the following property. Definition 3. We say that a Banach space X is tame if for any y ∈ X, x ∈ X \ {0} and any sequence {un }n∈N in X weakly convergent to zero, ky + un k kyk lim 6 max 1, . (7.1) n→∞ kx + un k kxk It is easy to see that p-spaces for 1 < p < ∞ are tame. Proposition 7.1. Let X be a reflexive Banach space such that either X or X ∗ is tame. Then X ∈ W.

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Proof. According to Corollary 4.3, it is sufficient to consider the case, when X is tame. Let T ∈ K(X) and kI + T k = c > 1. Since X is reflexive, we can pick a sequence {xn }n∈N of elements of X such that kxn k → 1, k(I + T )xn k → c and {xn } is weakly convergent to x ∈ X. By Lemma 3.5, x 6= 0. Since the sequence un = xn − x is weakly convergent to 0 and T is compact, T xn is norm-convergent to T x. Hence kx + T x + un k = kxn + T xk → c. Since X is tame, we have kx + T xk kx + T x + un k 6 max 1, . c = lim n→∞ kx + un k kxk The inequality c > 1 and the last display imply that kx + T xk > ckxk. Taking into account that c = kI + T k, we see that I + T attains its norm on x/kxk. Unfortunately, it seems there are no known examples of tame Banach spaces which are not p-spaces. This naturally leads to the problem of characterizing the tame spaces. 2. Analyzing the proof of Theorem 1.6, one can easily see that if 1 1, then whenever {xn }n∈N is a sequence of elements of X weakly converging to x ∈ X and satisfying kxn k → 1, k(I + T )xn k → kI + T k as n → ∞, then {xn } is norm convergent to x and I + T attains its norm on x. 3. In a recent paper [21] Shargorodsky and the author constructed a strictly convex reflexive Banach space X and S ∈ L(X) such that for some ε > 0, the level set Σε (S) \ σε (S) has non-empty interior. The space X constructed in [21] is an `2 -direct sum of a countable family of finite dimensional Banach spaces. Thus by Theorem 1.5, (1.5) holds for any T ∈ L(X). This observation shows that there is no relation between validity of (1.5) and meagreness of the level sets of the norm of the resolvent. 4. Let 1 < p < ∞. As it follows from Proposition 1.4 and Theorem 1.6, any subspace of an `p -direct sum X of a family of finite dimensional Banach spaces belongs to W. Applying Corollary 4.3, one can easily see that same holds true for quotients of X as well. Indeed, X ∗ is naturally isometrically isomorphic to an `p0 -direct sum of a family of finite dimensional Banach spaces, where p1 + p10 = 1. Moreover, for any closed linear subspace Y of X, (X/Y )∗ is naturally isometrically isomorphic to a subspace of X ∗ . 5. It would be interesting to figure out which classical Banach spaces do belong to the class W. A good starting point would be to address the spaces Lp [0, 1] for 1 < p < ∞. There is a strong indication against their membership in W for p 6= 2. Namely, it is easy to show that Lp [0, 1] for p 6= 2 is not tame. 6. The compactness condition in Propositions 1.3 and 1.2 can be replaced by the weaker condition of T being strictly singular. The proofs work without any changes.

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References [1] M. Acosta, Denseness of norm-attaining operators into strictly convex spaces, Proc. Roy. Soc. Edinburgh Sect. A129 (1999), 1107–1114. [2] M. Acosta, F. Aguirre and R. Pay´ a, A new sufficient condition for the denseness of norm attaining operators, Rocky Mountain J. Math. 26 (1996), 407–418. [3] M. Acosta and G. Ruiz, Norm attaining operators and reflexivity, Rend. Circ. Mat. Palermo (2) Suppl. 1998, no. 56, 171–177. [4] M. Acosta and C. Ruiz, Norm attaining operators on some classical Banach spaces, Math. Nachr. 235 (2002), 17–27. [5] F. Aguirre, Norm-attaining operators into strictly convex Banach spaces, J. Math. Anal. Appl. 222 (1998), 431–437. [6] J. Bonet and P. P´erez-Carreras, Barrelled locally convex spaces, North-Holland Mathematics Studies 131, North-Holland Publishing Co., Amsterdam, 1987. [7] A. B¨ ottcher, Pseudospectra and singular values of large convolution operators, J. Integral Equations Appl. 6 (1994), 267–301. [8] A. B¨ ottcher, S. Grudsky and B. Silbermann, Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices, New York J. Math. 3 (1997), 1–31. [9] F. Chaitin-Chatelin and A. Harrabi, About definitions of pseudospectra of closed operators in Banach spaces, Tech. Rep. TR/PA/98/08, CERFACS. [10] J. Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics 485, Springer-Verlag, Berlin, 1975. [11] E. Gallestey, D. Hinrichsen and A. Pritchard, Spectral value sets of closed linear operators, Proc. R. Soc. London, Ser. A 456 (2000), 930–937. [12] B. Godun and S. Troyanski, Norm-attaining operators, and the geometry of the unit sphere of a Banach space, Soviet Math. Dokl. 42 (1991), 532–534. [13] A. Harrabi, Pseudospectre d’une suite d’op´erateurs born´es, RAIRO Mod´el. Math. Anal. Mum´er. 32 (1998) 671–680. [14] J. Partington, Norm attaining operators, Israel J. Math. 43 (1982), 273–276. [15] S. Roch and B. Silberman, C ∗ -algebra technique in numerical analysis, J. Operator Theory 35 (1996), 241–280. [16] W. Schachermayer, Norm attaining operators on some classical Banach spaces, Pacific J. Math. 105 (1983), 427–438. [17] W. Schachermayer, Walter Norm attaining operators and renormings of Banach spaces, Israel J. Math. 44 (1983), 201–212. [18] H. Sch¨ afer, Topological vector spaces, MacMillan, New York, 1966. [19] E. Shargorodsky, On the level sets of the resolvent norm of linear operators, Bull. Lond. Math. Soc. 40 (2008), 493–504. [20] E. Shargorodsky, On the definition of pseudospectra, Bull. Lond. Math. Soc. [to appear]. [21] E. Shargorodsky and S. Shkarin, The level sets of the resolvent norm and convexity properties of Banach spaces, [preprint].

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[22] S. Shkarin, Some results on solvability of ordinary linear differential equations in locally convex spaces, Math. USSR Sbornik 71 (1992), 29–40. [23] L. Trefethen, Pseudospectra of linear operators, SIAM Rev. 39 (1997), 383–406. [24] L. Trefethen and M. Embree, Spectra and pseudospectra: the behavior of non-normal matrices and operators, Princeton University Press, Priceton, NJ, 2005. S. Shkarin Queen’s University Belfast Department of Pure Mathematics University road, BT7 1NN Belfast, UK e-mail: [email protected] Submitted: August 20, 2008. Revised: November 6, 2008.

Integr. equ. oper. theory 64 (2009), 137–154 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/010137-18, published online April 24, 2009 DOI 10.1007/s00020-009-1677-y

Integral Equations and Operator Theory

Algebraic Properties of Toeplitz Operators with Radial Symbols on the Bergman Space of the Unit Ball Ze-Hua Zhou∗ and Xing-Tang Dong Abstract. In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in Cn . We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form ξ k ϕ is studied, where k ∈ Zn and ϕ is a radial function. Mathematics Subject Classification (2000). Primary 47B35; Secondary 32A36. Keywords. Toeplitz operator, Bergman space, Mellin transform, radial symbol, quasihomogeneous symbol.

1. Introduction Let dv denote the Lebesgue volume measure on the unit ball Bn of Cn , normalized so that the measure of Bn equals 1. The Bergman space L2a (Bn ) is the Hilbert space consisting of holomorphic functions on Bn that are also in L2 (Bn , dv). Each point evaluation is easily verified to be a bounded linear functional on L2a (Bn ). Hence, for each z ∈ Bn , there exists a unique function Kz ∈ L2a (Bn ) which has the following reproducing property f (z) = hf, Kz i for every f ∈ L2a (Bn ). It is known that the reproducing kernel Kw is given by Kz (w) = ∗

1 , w ∈ Bn . (1 − hw, zi)n+1

Corresponding author, supported in part by the National Natural Science Foundation of China (Grand Nos.10671141, 10371091).

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Let P be the orthogonal projection from L2 (Bn , dv) onto L2a (Bn ). For a function ϕ ∈ L∞ (Bn , dv), the Toeplitz operator Tϕ : L2a (Bn ) 7→ L2a (Bn ) with symbol ϕ is defined by Z Tϕ (f )(z) = P (ϕf )(z) = (1.1) f (w)ϕ(w)Kz (w)dv(w), f ∈ L2a (Bn ). Bn

The operators defined in this way are the simplest and most natural Toeplitz operators, but we prefer to consider a more general class of Toeplitz operators. Let µ be any finite complex measure on Bn , then we define an operator Tµ on L2a (Bn ) by Z Tµ f (z) = f (w)Kz (w)dµ(w), f ∈ L2a (Bn ). Bn

If dµ(w) = F (w)dv(w) for some F ∈ L1 (Bn , dv), then we simply write Tµ = TF . This operator is always defined on the polynomials and the image of any polynomial is always a holomorphic function on the unit ball. We are interested in the case where this densely defined operator is bounded in the L2a (Bn ) norm. This happens often. For example, if µ has compact support, then Tµ is not only bounded, but also compact. Thus, if F ∈ L1 (Bn , dv) and there is an r ∈ (0, 1) such that F is (essentially) bounded on the annulus {z : r < |z| < 1}, then TF is bounded on L2a (Bn ) because F can be written as an L1 function with compact support plus a bounded function. This motivates two of the following definitions, which are based on the definitions on the unit disk in [9]. Definition 1.1. Let F ∈ L1 (Bn , dv). (a) We say that F is a T-function if the equation (1.1), with ϕ = F , defines a bounded operator on L2a (Bn ). (b) If F is a T-function, we write TF for the continuous extension of the operator defined by equation (1.1). We say that TF is a Toeplitz operator if and only if TF is defined in this way. (c) If there is an r ∈ (0, 1) such that F is (essentially) bounded on the annulus {z : r < |z| < 1}, then we say that F is “nearly bounded”. Remark 1.2. Grudsky, Karapetyants and Vasilevski [8] gave sufficient and necessary conditions for boundedness of Toeplitz operators with radial symbols on the Bergman spaces of the unit ball. Thus those conditions give a characterization of the radial functions in L1 (Bn , dv) which correspond to bounded operators. Generally, the T-functions form a proper subset of L1 (Bn , dv) which contains all bounded and “nearly bounded” functions. In 1964, Brown and Halmos [4] gave simple conditions for the product of two Toeplitz operators on the Hardy space to be equal to a Toeplitz operator, and showed that two bounded Toeplitz operators Tϕ and Tψ commute if and only if: (I) both ϕ and ψ are analytic, or (II) both ϕ and ψ are analytic, or (III) one is a linear function of the other. On the Bergman space of the unit disk, Axler, Ahern and ˘ ckovi´c showed that a similar result holds for Toeplitz operators with bounded Cu˘

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harmonic symbols (see [2, 1]). In spite of several results obtained in recent years, it seems to be far from its solution. However, some results with special symbols ˘ ckovi´c and Rao [7] gave necessary and sufficient conditions for a were obtained. Cu˘ symbol that produces a Toeplitz operator commuting with another such operator whose symbol is a monomial. In [9] Louhichi, Strouse and Zakariasy gave necessary and sufficient conditions for the product of two Toeplitz operators whose symbols are quasihomogeneous (i.e., it is of the form eikθ φ, where φ is a radial function) to be a Toeplitz operator, then they characterized commuting Toeplitz operators with quasihomogeneous symbols (see [10]). On the Bergman space of several complex variables, the situation is much more complicated, and it is not clear whether they will have some interesting results. Zheng [18] studied commuting Toeplitz operators with pluriharmonic symbols on the unit ball in Cn . Recently, extending Vasilevski’s results in [15] and [16], Quiroga-Barranco and Vasilevski gave the description of many (geometrically defined) classes of commuting Toeplitz operators on the unit ball (see [11, 12]). Motivated by recent work on the disk of Louhichi, Strouse and Zakariasy, we discuss the same questions for more general symbols on the unit ball in this paper. However, we need different methods and some complex calculation skills in the proof. The remainder of the present paper is assembled as follows: In section 2, we introduce some basic properties of the Mellin transform and Mellin convolution which will be needed later. In Section 3, we first investigate some basic results concerning Toeplitz operators with radial symbols on the Bergman space of the unit ball in Cn , then use these results to characterize when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. As an application, in Section 3, an explicit formula for the symbol of the product in the certain cases is given. Then, we investigate the zero-product problem for several Toeplitz operators with radial symbols. In the last section, the corresponding commuting problem of Toeplitz operators with quasihomogeneous symbols is studied. Suppose ϕ and ψ are bounded radial functions. First we will see that a Toeplitz operator with a radial symbol may only commute with another such operator whose symbol is of s the form ξ p ξ ϕ in certain trivial cases. Then, we show that two Toeplitz operators s with symbols ξ p ϕ and ξ ψ commute only in the trivial case, i.e., if one of them is the constant operator. Finally, we will prove that the symbol ξ k ϕ must be a monomial if Tξk ϕ commutes with a Toeplitz operator whose symbol is a nonconstant bounded holomorphic function.

2. The Mellin transform and Mellin convolution One of the most useful tools in the following calculations will be the Mellin transform. The Mellin transform ϕˆ of a function ϕ ∈ L1 ([0, 1], r dr) is defined by the

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equation: Z ϕ(z) ˆ =

1

ϕ(s)sz−1 ds.

0

It is known that ϕˆ is a bounded analytic function in the half plane {z : Re(z) > 2}. The use of the Mellin transform in the study of Toeplitz operators was introduced for the first time in [7]. It is important and helpful to know that the Mellin transform is uniquely determined by its value on an arithmetic sequence of integers. In fact we have the following classical theorem (see [13, p. 102]). Theorem 2.1. Suppose that f is a bounded analytic function on {z : Re(z) > 0} which vanishes at the pairwise distinct points z1 , z2 , · · · , where i) P inf{|zn |} > 0 and 1 ii) n≥1 Re( zn ) = ∞. Then f vanishes identically on {z : Re(z) > 0}. Remark 2.2. We shall often use this theorem to show that if ϕ ∈ L1 ([0, 1], r dr) and if there exists a sequence (nk )k≥0 ⊂ N such that X 1 ϕ(n b k ) = 0 and = ∞, nk k≥0

then ϕ(z) b = 0 for all z ∈ {z : Re(z) > 2} and so ϕ = 0. When considering the product of two Toeplitz operators we need a known fact about the Mellin convolution of their symbols. If f and g are defined on [0, 1), then their Mellin convolution is defied by Z 1 dt r (f ∗M g)(r) = g(t) , 0 ≤ r < 1. f t t r The Mellin convolution theorem states that f\ ∗M g(s) = fb(s)b g (s),

(2.1)

and that, if f and g are in L1 ([0, 1], r dr) then so is f ∗M g.

3. Products of Toeplitz operators with radial symbols For any multi-index α = (α1 , · · · , αn ), where each αi is a nonnegative integer, we write |α| = α1 + · · · + αn and α! = α1 ! · · · αn !. We will also write z α = z1α1 · · · znαn for z = (z1 , · · · , zn ) ∈ Bn .

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For two multi-indexes α = (α1 , · · · , αn ) and β = (β1 , · · · , βn ), the notation α β means that αi ≥ βi ,

i = 1, · · · , n,

and α⊥β means that α1 β1 + · · · + αn βn = 0. We also define α − β = (α1 − β1 , · · · , αn − βn ). Moreover, if α β then we obtain |α − β| = |α| − |β|. Recall that a function ϕ on Bn is radial if ϕ(z) depends only on |z|. It is known that ϕ is radial if and only if ϕ(U z) = ϕ(z) for any unitary transform U of Cn . Then for each radial function ϕ, we define the function ϕ e on [0, 1) by ϕ(r) e = ϕ(re) where e is a unit vector in Cn . It is obvious that ϕ e is well-defined. In the following, we shall often identify an integrable radial function ϕ on the unit ball with the corresponding function ϕ e defined on the interval [0, 1). First, we will give some basic results concerning Toeplitz operators with radial symbols on the Bergman space of the unit ball. Let ϕ ∈ L1 (Bn , dv) be a radial T-function, then for any multi-index α, it follows from formula (4.1) of [8] that Tϕ (z α ) = (2n + 2|α|)ϕ(2n ˆ + 2|α|)z α .

(3.1)

Thus, the Toeplitz operator with a radial symbol on the Bergman space of the unit ball acts in a very simple way. In fact, we have the following very simple but essential theorem. Theorem 3.1. Let ϕ ∈ L1 (Bn , dv). Then the following assertions are equivalent: (a) For each multi-index α there exists λ|α| ∈ C which depends only on |α| such that Tϕ (z α ) = λ|α| z α . (b) ϕ is a radial function. Proof. The implication (b) ⇒ (a) is obvious by equation (3.1). Next, we prove the converse implication (a) ⇒ (b). Assume Tϕ (z α ) = λ|α| z α , then for any unitary transformation U of Cn with U −1 = (a1 , · · · , an )T = (aij )1≤i,j≤n ,

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we have Tϕ (U −1 z)α = Tϕ [(a11 z1 + · · · + a1n zn )α1 · · · (an1 z1 + · · · + ann zn )αn ] X 1 1 α1 ! = Tϕ (a11 z1 )m1 · · · (a1n zn )mn × · · · 1 m ! |m1 |=α1 X αn ! mn mn 1 · · · (a n (a z ) z ) × n1 1 nn n mn ! n |m |=αn

=

X |m1 |=α1

= λ|α|

X

···

|mn |=αn

X

···

|m1 |=α1

1 n 1 n α1 ! αn ! · · · n (a1 )m · · · (an )m Tϕ (z m · · · z m ) m1 ! m !

X |mn |=αn

1 n 1 n αn ! α1 ! · · · n (a1 )m · · · (an )m (z m · · · z m ) 1 m ! m !

= λ|α| (a11 z1 + · · · + a1n zn )α1 · · · (an1 z1 + · · · + ann zn )αn = λ|α| (U −1 z)α , where m1 , · · · , mn are multi-indexes. It follows that Z ϕ(U u)uα α Tϕ◦U (z )(w) = dv(u) n+1 Bn (1 − hw, ui) Z ϕ(u)(U −1 u)α dv(u) = −1 ui)n+1 Bn (1 − hw, U Z ϕ(u)(U −1 u)α = dv(u) n+1 Bn (1 − hU w, ui) = Tϕ (U −1 z)α (U w) = λ|α| (U −1 z)α (U w) = λ|α| wα = Tϕ (z α )(w), which implies Tϕ◦U = Tϕ . Thus ϕ ◦ U = ϕ and ϕ is a radial function.

Corollary 3.2. Let ϕ1 and ϕ2 be two radial T-functions on Bn . If Tϕ1 Tϕ2 = Tψ , then ψ is a radial T-function. Proof. Using equation (3.1) to calculate Tϕ1 Tϕ2 (z α ), we can get Tϕ1 Tϕ2 (z α ) = (2n + 2|α|)2 ϕ c1 (2n + 2|α|)c ϕ2 (2n + 2|α|)z α .

(3.2)

It follows from Theorem 3.1 that ψ is a radial function. Moreover, Tψ is clearly a bounded operator. The question of when the product of two Toeplitz operators is equal to a Toeplitz operator is more complicated and still open. Grudsky, Karapetyants and Vasilevski [8] gave a particular answer to this question considering the Toeplitz operators with radial symbols. Let ϕ1 and ϕ2 be two radial T-functions on Bn , and let further √ √ A1 (t) = ϕ1 ( e−t )e−nt , A2 (t) = ϕ2 ( e−t )e−nt .

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The formal construction (inverse Fourier-Laplace transform) Z 1 (F −1 A)(z) = √ A(t)eizt dt, z ∈ Π ∪ R, 2π R defines a holomorphic function in the upper half-plane Π (⊂ C) which coincides on the real axis with the inverse Fourier transform (F −1 A)(ζ) of the function A(t). Theorem 3.7 of [8] shows that if the function √ 2π(n − iζ)(F −1 A1 )(ζ)(F −1 A2 )(ζ), ζ ∈ R, belongs to Wiener ring W0 of the inverse Fourier transforms of sumable functions, then there exists a Toeplitz operator with the radial symbol ψ such that Tϕ1 Tϕ2 = Tψ . The following theorem will give another condition for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator. Theorem 3.3. Let ϕ1 and ϕ2 be two radial T-functions on Bn . Then Tϕ1 Tϕ2 is equal to the Toeplitz operator Tψ if and only if ψ is a solution of the equation I ∗M ψ = ϕ1 ∗M ϕ2 ,

(3.3)

where I denotes the constant function with value one. Proof. For each multi-index α, it follows from (3.1) and (3.2) that Tϕ1 Tϕ2 (z α ) = Tψ (z α ) if and only if 1 b ψ(2n + 2|α|) = ϕ1\ ∗M ϕ2 (2n + 2|α|). 2n + 2|α| A direct calculation gives bI(2n + 2|α|) = χ[ [0,1] (2n + 2|α|) =

(3.4)

1 , 2n + 2|α|

by (2.1) we have 1 b b ψ(2n + 2|α|) = bI(2n + 2|α|)ψ(2n + 2|α|) = I\ ∗M ψ(2n + 2|α|). 2n + 2|α| So (3.4) is equivalent to I\ ∗M ψ(2n + 2|α|) = ϕ1\ ∗M ϕ2 (2n + 2|α|). By Remark 2.2, (3.5) is equivalent to (3.3).

(3.5)

Remark 3.4. Since the Mellin transform is closely related, using the change of variables s = e−t , to the inverse Fourier-Laplace transform, the arguments of the both theorems are equivalent. About some products of Toeplitz operators, we have the following fun calculation.

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Corollary 3.5. Let l and m be two real numbers greater than or equal to −1. Then l m m if l 6= m l−m T|z|l − l−m T|z| T|z|l T|z|m = T|z|l (1+l ln |z|) if l = m. Proof. By Theorem 3.3, T|z|l T|z|m is equal to the Toeplitz operator Tψ if and only if ψ is a solution of the equation Z 1 Z 1 −1 ψ(t)t dt = rl tm−l−1 dt, r

r

from which we can get ψ(r) =

l l l−m |r| l

m − l−m |r|m |r| (1 + l ln |r|)

if l 6= m if l = m.

Then the desired result is obvious.

˘ ckovi´c solved the zero-product problem for two Toeplitz In [1] Ahern and Cu˘ operators with harmonic symbols. On the Bergman space of the unit ball, Choe and Koo [5] gave a similar result with an assumption about the continuity of the symbols on an open subset of the boundary, and solved the zero-product problem for several Toeplitz operators with harmonic symbols that have Lipschitz continuous extensions to the whole boundary. This problem for Toeplizt operators with arbitrary symbols still remains open, even for two Toeplitz operators. Next, we will study the zero-product problem for several Toeplitz operators with radial symbols acting on the Bergman space of the unit ball. Theorem 3.6. Let ϕ1 , · · · , ϕN be radial T-functions on Bn . If Tϕ1 · · · TϕN = 0, then ϕi = 0 for some i. Proof. Suppose Tϕ1 · · · TϕN = 0, then for any multi-index α, using equation (3.1), we can easily see that ϕ c1 (2n + 2|α|) · · · ϕc N (2n + 2|α|) = 0. Let Ei = {|α| ∈ N : ϕbi (2n + 2|α|) = 0} . Notice that E1 ∪ · · · ∪ EN = N, hence there exists some i such that X 1 = ∞, |α| |α|∈Ei

then by Remark 2.2 ϕi = 0.

The following corollary states that the only idempotent Toeplitz operators with radial symbols are 0 and I. Corollary 3.7. Let ϕ be a radial T-function on Bn . Then Tϕ2 = Tϕ if and only if either ϕ = 0 or ϕ = 1.

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Proof. If Tϕ2 = Tϕ , then Tϕ Tϕ−1 = 0, and by Theorem 3.6, ϕ = 0 or ϕ = 1. The converse implication is clear.

4. Commuting Toeplitz operators with quasihomogeneous symbols In this section, we characterize commuting Toeplitz operators with bounded quasihomogeneous symbols on the Bergman space of the unit ball. The definition of the quasihomogeneous function on the unit disk has been given in many papers (see [9] or [17]), and we will give a similar definition on the unit ball. Definition 4.1. Let k ∈ Zn and let f be a function in L1 (Bn , dv). Then we say that f is a quasihomogeneous function of quasihomogeneous degree k if f is of the form ξ k ϕ where ϕ is a radial function, i.e., f (rξ) = ξ k ϕ(r) for any ξ in the unit sphere Sn and r ∈ [0, 1). Remark 4.2. It is obvious that any k ∈ Zn , can be uniquely written as p − s, where p and s are two multi-indexes such that p ⊥ s. Thus in this paper, we always define the function s ξ k = ξ p ξ , ξ ∈ Sn , for any k ∈ Zn . A direct calculation gives the following lemma which we shall use often. Lemma 4.3. Let p, s be two multi-indexes and let ϕ be a bounded radial function on Bn . Then, for any multi-index α, Tξp ϕ (z α ) = 2(n + |α| + |p|)ϕ(2n b + 2|α| + |p|)z α+p ;  0 if α 6 s,  2α!(n + |α| − |s|)! Tξs ϕ (z α ) = α−s ϕ(2n b + 2|α| − |s|)z if α s;  (α − s)!(n − 1 + |α|)! Tξp ξs ϕ (z α )  0 if α + p 6 s,  2(α + p)!(n + |α| + |p| − |s|)! = α+p−s ϕ(2n b + 2|α| + |p| − |s|)z if α + p s.  (α + p − s)!(n − 1 + |α| + |p|)! Proof. Here we only give the proof of the third equation. For all multi-indexes α and β, Z s α β s hTξp ξ ϕ (z ), z i = ϕ(z)ξ p ξ z α z β dv(z) Bn

Z = 2n ( =

0

1

r2n−1 dr

Z

ϕ(r)r|α|+|β| ξ p+α ξ

s+β

dσ(ξ)

Sn

0 if α + p 6 s, R p+α s+β 2nϕ(2n b + |α| + |β|) Sn ξ ξ dσ(ξ) if α + p s.

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If α + p s, then by the well-known equations (see Proposition 1.4.9 of [14])  Z 0 if β 6= α + p − s,  s+β (n − 1)!(α + p)! ξ p+α ξ dσ(ξ) = if β = α + p − s  Sn (n − 1 + |α| + |p|)! and hz α+p−s , z β i =

 

0 n!(α + p − s)!  (n + |α| + |p| − |s|)!

if β 6= α + p − s, if β = α + p − s,

we obtain hTξp ξs ϕ (z α ), z β i =

2(α + p)!(n + |α| + |p| − |s|)!ϕ(2n b + 2|α| + |p| − |s|) α+p−s β hz , z i. (α + p − s)!(n − 1 + |α| + |p|)!

Note that hTξp ξs ϕ (z α ), z β i = 0 if α + p 6 s, then we can get Tξp ξs ϕ (z α )  0  2(α + p)!(n + |α| + |p| − |s|)! = ϕ(2n b + 2|α| + |p| − |s|)z α+p−s  (α + p − s)!(n − 1 + |α| + |p|)!

if α + p 6 s, if α + p s.

This completes the proof.

We are now ready to discuss the commuting problem of Toeplitz operators with quasihomogeneous symbols. Theorem 4.4. Let p, s be two multi-indexes and let ψ and φ be two bounded radial functions on Bn . If ψ is nonconstant, then Tψ Tξ p ξ s φ = Tξ p ξ s φ Tψ if and only if either |p| = |s| or φ = 0. Proof. If Tξp ξs φ and Tψ commute, then for any multi-index α such that α + p s, it follows from (3.1) and Lemma 4.3 that b b (2n + 2|α| + 2|p| − 2|s|)ψ(2n + 2|α| + 2|p| − 2|s|)φ(2n + 2|α| + |p| − |s|) b b = (2n + 2|α|)ψ(2n + 2|α|)φ(2n + 2|α| + |p| − |s|).

(4.1)

Assume |p| = 6 |s|, without loss of generality, we can also assume that |p| > |s|, for otherwise we could take the adjoins. Let n o b E = |α| ∈ N : φ(2n + 2|α| + |p| − |s|) = 0 , P 1 then we will show that |α| = ∞, which implies φ = 0 by Remark 2.2. |α|∈E

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Toeplitz operators with radial symbols

P |α|∈E

1 |α|

147

< ∞, then we will derive a contradiction. Let

E 0 = E c ∩ |α| ∈ N : α ∈ Zn+ , α + p s , where E c is the complement of E in N, then X 1 = ∞. |α| 0 |α|∈E

On the other hand, for any |α| ∈ E 0 , (4.1) gives b b (2n + 2|α| + 2|p| − 2|s|)ψ(2n + 2|α| + 2|p| − 2|s|) = (2n + 2|α|)ψ(2n + 2|α|). (4.2) Denote b + 2n + 2|p| − 2|s|) − F (z) = ψ(z

z + 2n b + 2n), ψ(z z + 2n + 2|p| − 2|s|

then F is analytic and bounded on {z : Re(z) > 0} since ψ is bounded. Moreover, (4.2) implies that F (2|α|) = 0, ∀|α| ∈ E 0 . According to Theorem 2.1, F must be zero, thus b +2n+2|p|−2|s|) = (z +2n)ψ(z b +2n), z ∈ {z : Re(z) > 0} . (z +2n+2|p|−2|s|)ψ(z For any integer n0 greater than 2n, the above equation gives that b 0 + 2m(|p| − |s|)) = n0 ψ(n b 0 ), ∀m ∈ N. (n0 + 2m(|p| − |s|))ψ(n b 0 ), we obtain If we denote by C the constant n0 ψ(n b 0 + 2m(|p| − |s|)) = ψ(n

C = C bI(n0 + 2m(|p| − |s|)) n0 + 2m(|p| − |s|)

for any m ∈ N, by Remark 2.2 again, and then clearly ψ is constant, which is a contradiction. Thus we conclude that either |p| = |s| or φ = 0. Conversely, if |p| = |s| or φ = 0, then we can easily show that the equation (4.1) holds, and consequently Tξp ξs φ Tψ (z α ) = Tψ Tξp ξs φ (z α ) for each multi-index α, which implies Tξp ξs φ and Tψ commute.

It was shown in [7], that a Toeplitz operator with a radial symbol on the Bergman space of the unit disk D may only commute with another such operator with a radial symbol, but it is not true in the higher dimension by the theorem above. It is known that every function f ∈ L2 (D, dA) has the decomposition f (reikθ ) =

+∞ X k=−∞

eikθ fk (r),

(4.3)

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where fk (r) are square integrable in [0, 1] with respect to the measure r dr. More details can be found in [7]. Similarly, let Z 1 2n−1 2 r |ϕ(r)| dr < ∞ . < = ϕ : Bn → C radial 0

k

n

Denote
0

Sn

It is also clear that
but we still might study a P function being of that form. In the one-dimensional case, the function f (rξ) = ξ k fk (r) is exactly the same as (4.3). Moreover, if k∈Zn

f (rξ) =

X

ξ k fk (r) ∈ L∞ (Bn , dv),

k∈Zn n

then for each k ∈ Z , |fk (r)| =

Z k ≤ sup |f (z)|, f (rξ)ξ dσ(ξ) z∈B k 2 ||ξ || Sn n 1

and so the functions fk are bounded on Bn . Lemma 4.5. Let p be a multi-index and let φ be a bounded radial function on Bn . If X f (rξ) = ξ k fk (r) ∈ L∞ (Bn , dv), k∈Zn

then Tf Tξp φ = Tξp φ Tf ⇐⇒ Tξk fk Tξp φ = Tξp φ Tξk fk , ∀k ∈ Zn . P k Proof. Suppose f (rξ) = ξ fk (r) ∈ L∞ (Bn , dv), then for any multi-index α, a k∈Zn

direct calculation by Lemma 4.3 gives that X Tf Tξp φ (z α ) =

T ξ k fk T ξ p φ z α

k+α+p0

and Tξp φ Tf (z α ) =

X

T ξ p φ T ξ k fk z α .

k+α0

If Tf and T

commute, the identity of the above two series implies that 0 if k + α + p 0 and k + α 6 0 α p Tξk fk Tξ φ (z ) = Tξp φ Tξk fk (z α ) if k + α 0. ξp φ

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Recall that Tξp φ Tξk fk (z α ) is equal to zero if k + α 6 0, then for each multi-index α and for all k ∈ Zn , Tξk fk Tξp φ (z α ) = Tξp φ Tξk fk (z α ). Thus Tξk fk Tξp φ = Tξp φ Tξk fk , ∀k ∈ Zn . The converse implication is clear.

Corollary 4.6. Let ϕ be a nonconstant bounded radial function on Bn . If X f (rξ) = ξ k fk (r) ∈ L∞ (Bn , dv), k∈Zn

then Tϕ Tf = Tf Tϕ if and only if f (eiθ z) = f (z) for almost all θ ∈ R and z ∈ Bn . Proof. It follows from Lemma 4.5 that Tϕ commutes with Tf if and only if Tϕ Tξk fk = Tξk fk Tϕ for all k ∈ Zn . Suppose k = p − s, where p and s be two multi-indexes such that p ⊥ s, then by Theorem 4.4, the above equation is equivalent to |p| = |s| or fk = 0.

(4.4)

Obviously, (4.4) is equivalent to (ξ k fk )(eiθ z) = (ξ k fk )(z), ∀θ ∈ R and z ∈ Bn . Therefore Tϕ commutes with Tf if and only if f (eiθ z) = f (z) for almost all θ ∈ R and z ∈ Bn . Remark 4.7. In the one-dimensional case, the function such that f (eiθ z) = f (z) is exactly a radial function, so this corollary coincides with Theorem 6 of [7]. There are lots of examples of functions with the form of ξ p φ, which are the symbols of commuting Toeplitz operators (see [7]), but the following theorem will show that two Toeplitz operators with quasihomogeneous symbols of degrees p and −s respectively commute only in a trivial case. Theorem 4.8. Suppose p and s be two nonzero multi-indexes, and let ψ and φ be two bounded radial functions on Bn . If Tξ p φ Tξ s ψ = Tξ s ψ Tξ p φ then, φ = 0 or ψ = 0. Proof. For any multi-index α, it follows from Lemma 4.3 that, if α s, then Tξp φ Tξs ψ (z α ) =

4α!(n + |α| − |s|)!(n + |α| + |p| − |s|) (α − s)!(n − 1 + |α|)! b b × φ(2n + 2|α| + |p| − 2|s|)ψ(2n + 2|α| − |s|)z α+p−s ,

and if α 6 s, then Tξp φ Tξs ψ (z α ) = 0.

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Similarly, if α + p s, then Tξs ψ Tξp φ (z α ) =

4(α + p)!(n + |α| + |p| − |s|)!(n + |α| + |p|) (α + p − s)!(n − 1 + |α| + |p|)! b b × φ(2n + 2|α| + |p|)ψ(2n + 2|α| + 2|p| − |s|)z α+p−s ,

and if α + p 6 s, then Tξs ψ Tξp φ (z α ) = 0. For two nonzero multi-indexes p and s, to prove this theorem we need to consider two cases. Case 1. Suppose pi0 6= 0 and si0 6= 0 for some i0 ∈ {1, · · · , n}. If Tξp φ and Tξs ψ commute, then b b (a) φ(2n + 2|α| + |p|)ψ(2n + 2|α| + 2|p| − |s|) = 0 if α + p s and α 6 s; b b (b) φ(2n + 2|α| + |p|)ψ(2n + 2|α| + 2|p| − |s|) b b = Cα φ(2n + 2|α| + |p| − 2|s|)ψ(2n + 2|α| − |s|) if α s, α!(α + p − s)!(n + |α| − |s|)!(n − 1 + |α| + |p|)! . (α + p)!(α − s)!(n − 1 + |α|)!(n + |α| + |p| − |s| − 1)!(n + |α| + |p|) 0 0 Suppose multi-index α = (s1 , · · · , si0 −1 , si0 −1, si0 +1 , · · · , sn ), then αi0 < si0 0 0 and α + p s since pi0 6= 0 and si0 6= 0. Denote a0 = |α |, then it follows from (a) that b b φ(2n + 2a0 + |p|)ψ(2n + 2a0 + 2|p| − |s|) = 0. b If φ(2n + 2a0 + |p|) = 0, we will let a1 = a0 + |s|, otherwise, let a1 = a0 + |p|, then a direct calculation from (b) shows that

where Cα =

b b φ(2n + 2a1 + |p|)ψ(2n + 2a1 + 2|p| − |s|) = 0. So we can find a sequence {am }m∈N , which is defined by am+1 = am + |s| or am + |p|, such that b b φ(2n + 2am + |p|)ψ(2n + 2am + 2|p| − |s|) = 0. n o P 1 b It is clear that m ∈ N : φ(2n + 2am + |p|) = 0 and am = ∞. Let E1 = m∈N n o b E2 = m ∈ N : ψ(2n + 2am + 2|p| − |s|) = 0 . Since X 1 X 1 X 1 ≤ + , am am am m∈E1 m∈E2 m∈N P P 1 we know that at least one of the series am and m∈E1

m∈E2

1 am

diverges, then it

follows from Remark 2.2 that φ = 0 or ψ = 0. Case 2. Suppose either pi = 0 or si = 0 for all i ∈ {1, · · · , n}. Without loss of generality, we can also assume |s| ≥ |p| > 0. Obviously, for any multi-index α s, the fact either pi = 0 or si = 0 for all i ∈ {1, · · · , n} implies α!(α + p − s)! = (α + p)!(α − s)!.

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Thus if Tξp φ and Tξs ψ commute, it follows from (b) that b b (n + |α| + |p| − |s|)ψ(2n + 2|α| − |s|)φ(2n + 2|α| + |p| − 2|s|) (n + |α| − |s| + 1) · · · (n + |α| − |s| + |s| − 1) b b (n + |α| + |p|)ψ(2n + 2|α| + 2|p| − |s|)φ(2n + 2|α| + |p|) = . (n + |α| + |p| − |s| + 1) · · · (n + |α| + |p| − |s| + |s| − 1) As in the proof of Theorem 4.3, the above equation implies that b ψ(2z + |s|) b (z + |p|)φ(2z + |p|) (z + 1) · · · (z + |s| − 1) b ψ(2z + 2|p| + |s|) b (z + |p| + |s|)φ(2z + |p| + 2|s|) = (z + |p| + 1) · · · (z + |p| + |s| − 1) for z ∈ {z : Re(z) > 0}. Let

f (z) =

b ψ(2z + |s|) b and g(z) = (z + |p|)φ(2z + |p|), (z + 1) · · · (z + |s| − 1)

thus the above equation can be written as f (z)g(z) = f (z + |p|)g(z + |s|).

(4.5)

Next, denote H(z) = f (z)f (z + 1) · · · f (z + |p| − 1)g(z)g(z + 1) · · · g(z + |s| − 1). Obviously, H(z) is analytic on {z : Re(z) > 0}. It follows from equation (4.5) and |s| ≥ |p| > 0 that H(z) = f (z + |p|)g(z + |s|) · · · f (z + |p| − 1 + |p|)g(z + |p| − 1 + |s|) × g(z + |p|) · · · g(z + |s| − 1) = f (z + |p|)f (z + |p| + 1) · · · f (z + 2|p| − 1) × g(z + |p|)g(z + |p| + 1) · · · g(z + |p| + |s| − 1) = H(z + |p|), which implies H(z) is periodic function with period |p| on {z : Re(z) > 0}. Thus, the function H(z) can be extended to whole plane C, so we can think of the function H(z) as an entire function. By the definition of the Mellin transform, we

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can get |H(z)| = |f (z)f (z + 1) · · · f (z + |p| − 1)g(z)g(z + 1) · · · g(z + |s| − 1)| Z 1 |s| t2z+|s|−1 dt × · · · ≤ kψk|p| kφk |(z + |p|) · · · (z + |p| + |s| − 1)| ∞ ∞ 1

1

0 1

t2z+2(|s|−1)+|p|−1 dt 0 0 0 1 1 × ··· (z + 1) · · · (z + |s| − 1) (z + |p|) · · · (z + |p| + |s| − 1) Z

×

t2z+2(|p|−1)+|s|−1 dt

Z

t2z+|p|−1 dt · · ·

Z

= O(|z|−|p||s| ). Noting that −|p||s| < 0, we obtain H(z) = 0, which implies φ = 0 or ψ = 0.

˘ ckovi´c [6] showed that if Tzn and Tψ commute, then ψ is an analytic Cu˘ ˘ ckovi´c and Rao [3] extended Cu˘ ˘ ckovi´c’s result function on the unit disk. Axler, Cu˘ by replacing the disk with an arbitrary bounded domain and more importantly by replacing z n with an arbitrary bounded analytic function, then they asked a question what is the situation on Bergman spaces in higher dimensions. The following result will give an answer in a special situation on the unit ball. Theorem 4.9. Let f be a nonconstant bounded holomorphic function and g be a nonzero bounded quasihomogeneous function respectively. If Tf and Tg commute, then g is a monomial. P Proof. Suppose f = fβ z β is a power series representation of f and g = ξ k ϕ. If β0

Tf and Tg commute, then for any multi-index β, it is easy to see that fβ Tzβ Tξk ϕ = fβ Tξk ϕ Tzβ . Recall that f is nonconstant and therefore fγ 6= 0 for some γ with |γ| ≥ 1. Then it yields T z γ T ξ k ϕ = Tξ k ϕ T z γ . Suppose k = p−s, where p and s be two multi-indexes such that p ⊥ s. The equality Tzγ Tξp ξs ϕ (z α ) = Tξp ξs ϕ Tzγ (z α ) for each multi-index α together with Lemma 4.3 gives (a) ϕ(2n b + 2|α| + 2|γ| + |p| − |s|) = 0, if α + γ s and α 6 s; (b) ϕ(2n b + 2|α| + 2|γ| + |p| − |s|) = Cα ϕ(2n b + 2|α| + |p| − |s|), if α s, (α + p)!(α + γ + p − s)!(n + |α| + |p| − |s|)!(n − 1 + |α| + |γ| + |p|)! . (α + γ + p)!(α + p − s)!(n + |α| + |γ| + |p| − |s|)!(n − 1 + |α| + |p|)! First, we claim that s = 0. Otherwise, for two nonzero multi-indexes s and γ, we can consider two cases: si0 γi0 6= 0 for some i0 and si γi = 0 for all i ∈ {1, · · · , n}.

where Cα =

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Analogously to the proof of Theorem 4.8, we can prove that ϕ = 0 in both cases, which is a contradiction. Then, it follows from (b) that (2n + 2|α| + 2|p|)ϕ(2n b + 2|α| + |p|) = (2n + 2|α| + 2|γ| + 2|p|)ϕ(2n b + 2|α| + 2|γ| + |p|), which implies that (z + 2n + |p|)ϕ(z b + 2n) = (z + 2n + |p| + 2|γ|)ϕ(z b + 2n + 2|γ|) for z ∈ {z : Re(z) > 0}. Using the similar argument as in the proof of Theorem 4.4, we can conclude that (z + 2n + |p|)ϕ(z b + 2n) = c for some constant c. Then the uniqueness of the Mellin transform implies that ϕ(r) = cr|p| , and consequently g = cz p . P k Corollary 4.10. Let p be a nonzero multi-index and let f (rξ) = ξ fk (r) ∈ L∞ (Bn , dv). If Tf and Tzp commute, then f is holomorphic.

k∈Zn

Proof. If Tf and Tzp commute, it follows from Lemma 4.5 that T ξ k fk T z p = T z p T ξ k fk for each k ∈ Zn . Hence the desired result is obvious by Theorem 4.9.

Remark 4.11. In the one-dimensional case, every bounded function can be written ˘ ckovi´c [6] mentioned above. as (4.3), so this corollary recover the result of Cu˘ Acknowledgment The authors would like to thank the referee for his (or her) excellent suggestions.

References ˇ Cu˘ ˘ ckovi´c, A theorem of Brown-Halmos type for Bergman space [1] P. Ahern and Z. Toeplitz operators, J. Funct. Anal. 187 (2001), 200–210. ˇ Cu˘ ˘ ckovi´c, Commuting Toeplitz operators with harmonic symbols, [2] S. Axler and Z. Integral Equations Operator Theory. 14 (1991), 1–12. ˘ Cu˘ ˘ ckovi´c and N.V. Rao, Commutants of analytic Toeplitz operators on [3] S. Axler, Z. the Bergman space, Proc. Amer. Math. Soc. 128 (1999), 1951–1953. [4] A. Brown and P.R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963), 89–102. [5] B. Choe, H. Koo, Zero products of Toeplitz operators with harmonic symbols, J. Funct. Anal. 233 (2006), 307–334. ˘ Cu˘ ˘ ckovi´c, Commutants of Toeplitz operators on the Bergman space, Pacific J. [6] Z. Math. 162 (1994), 277–285. ˘ Cu˘ ˘ ckovi´c and N.V. Rao, Mellin transform, monomial symbols, and commuting [7] Z. Toeplitz operators, J. Funct. Anal. 154(1) (1998), 195–214.

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[8] S. Grudsky, A. Karapetyants and N. Vasilevski, Toeplitz operators on the unit ball in Cn with radial symbols, Journal of Operator Theory. 49 (2003), 325–346. [9] I. Louhichi, E. Strouse and L. Zakariasy, Products of Toeplitz operators on the Bergman space, Integral Equations Operator Theory. 54 (2006), 525–539. [10] I. Louhichi and L. Zakariasy, On Toeplitz operators with quasihomogeneous symbols, Arch. Math. 85 (2005), 248–257. [11] R. Quiroga-Barranco and N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators on the unit ball, I. Bargman-type transforms and spectral representations of Toeplitz operators, Integral Equations Operator Theory. 59 (2007), 379–419. [12] R. Quiroga-Barranco and N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators on the unit ball, II. Geometry of the level sets of symbols, Integral Equations Operator Theory. 60 (2008), 89–132. [13] R. Remmert, Classical Topics in Complex Function Theory. Graduate Texts in Methematics, Springer, New York, 1998. [14] W. Rudin, Function Theory in the Unit Ball of Cn . Springer-Verlag, BerlinHeidelberg-New York, 1980. [15] N. Vasilevski, Toeplitz operator on the Bergman spaces: Inside-the-Dommain Effects, Contemp. Math. 289 (2001), 79–146. [16] N. Vasilevski, Bergman space structure, commutative algebras of Toeplitz operators and hyperbolic geometry, Integral Equations Operator Theory. 46 (2003), 235–251. [17] L. Zakariasy, The rank of Hankel operators on harmonic Bergman spaces, Proc. Amer. Math. Soc. 131(4) (2003), 1177–1180. [18] D. Zheng, Commuting Toeplitz operators with pluriharmonic symbols, Trans. Amer. Math. Soc. 350 (1998), 1595–1618. Ze-Hua Zhou and Xing-Tang Dong Department of Mathematics Tianjin University Tianjin 300072 P.R. China e-mail: [email protected] [email protected] Submitted: July 19, 2008. Received: February 4, 2009.

Integr. equ. oper. theory 64 (2009), 155–175 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020155-21, published online May 12, 2009 DOI 10.1007/s00020-009-1683-0

Integral Equations and Operator Theory

Projection-Iterative Methods for a Class of Difference Equations Petru A. Cojuhari and Michal A. Nowak Abstract. We develop a theoretical framework for projection-iterative methods to solve operator equations of the form Au + Bu = f, where A is a Toeplitz operator in a Banach space lp (N) (1 ≤ p < ∞), B is considered as a perturbation (of general form) of A, and f is a given element in this space. The methods are adopted for application to general situations, in particular, to the equations in which A need not be a Fredholm operator. The idea to involve iteration procedures and the technique which we apply allow to obtain conditions on perturbations for convergence and effective error estimates in terms of some weighted spaces (without any restrictions on the norms for perturbations). Based on established evaluations we derive further information about decaying properties of the solutions. The obtained results are illustrated by considering concrete classes of equations as, for instance, equations corresponding to Jacobi type operators. Mathematics Subject Classification (2000). Primary 65J10; Secondary 65Q05. Keywords. Projection methods, iterative methods, Toeplitz operators, difference equations.

1. Introduction Let lp (N) (1 ≤ p < ∞) P stand for the usual Banach space of complex-valued ∞ sequences (xn ) such that n=1 |xn |p < ∞, and let A denote a Toeplitz operator defined on lp (N) by the matrix ∞ A = aj−k j,k=1 , where aj ∈ C (j = 0, ±1, . . . ). For the sake of simplicity we assume that ∞ X j=−∞

|aj | < ∞ .

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The function defined on the complex unit circle T by A(z) =

∞ X

an z n

(z ∈ T)

n=−∞

is referred to as the symbol of the operator ∞ A. We take another operator B = bjk j,k=1 bounded on lp (N) and consider the following equation Au + Bu = f , (1.1) where f is a given element in lp (N). Equations of the form (1.1) occur in a wide variety of concrete applications and they were objects of study for many authors. In particular, various approximation methods have been developed for these equations. Among them the projection methods are distinguished eloquently; cf. [2], [29], [11], [3], [16], [23], [28] (see also the references cited therein). Generally, solving an equation (1.1) by a projection method means to give two sequences (Pn ) and (Qn ) of projections on lp (N) converging strongly to the identity operator, and pass to the approximate equation Qn (A + B)Pn un = Qn f

(un ∈ Pn (lp (N))) .

(1.2)

If for each f ∈ lp (N) the equation (1.2) has a unique solution un ∈ Pn (lp (N)) for sufficiently large n, and if the sequence of the approximate solutions (un ) converges in lp (N) to a solution u of the initial equation, then one says that the projection method is convergent in lp (N). Note that the problem of the convergence of a projection method for difference equations of the form (1.1) is in general a delicate question. The analysis of convergence of projection methods is referred mostly to the case when A is a Fredholm operator (the elliptic case) while the perturbation B is a compact operator or of small norm (cf. [11], [3], [23]). However, in case that the operator A is not Fredholm, that means that its symbol A(z) vanishes on some points of the unit circle, the projection method in general is not convergent (even for B = 0). For the first this fact was observed by Gohberg and Levcenko in [12], [13], [14], [20], [21]. It turns out that the projection method (with Pn = Qn being the canonical projections) applied in a space lp (N) (1 ≤ p < ∞) to the equation Au = f (i.e. when B = 0) is always divergent and it is convergent in the space c0 of all sequences convergent to zero if and only if the zeros of the symbol A(z) lying on the unit circle are simple. The investigations on this topic were then continued and extended to other classes of convolution equations by I. Gohberg and S. Pr¨ ossdorf in [15], S. Pr¨ ossdorf [24], S. Pr¨ossdorf and B. Silbermann [25], [26], etc. An account of this research area has been reviewed in [23] and [27] (see also the references cited therein). In this paper we propose another approach to solve operator equations of the form (1.1). The basic idea is to combine projection and iterative procedures with a view to obtain an approximation method suitable for applications to (1.1) in nonelliptic cases. As far as we know iteration processes were not involved practically for the equations under discussion. The technique which we apply allows to describe

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conditions on perturbations for the convergence of the method and give effective error estimates for solutions. The obtained conditions turned out to depend essentially on the maximal multiplicity of zeros (those lying on the unit circle) of the symbol of the unperturbed operator (see Theorem 4.3). As a rule in the presence of an iteration procedure perturbations need to be relatively small (in sense of the norms). We avoid such restrictions by involving a family of weighted spaces in terms of which the evaluations are made. Namely for this purpose Assumptions 3.7 and 3.8 were introduced (see Remark 3.9) being ensured by estimations (4.23) and (4.24) in our applications with c(τ ) → 0 as τ → ∞ via Hardy type inequalities. Besides, we derive further information about decaying properties of the solutions. The problems are treated within the framework of operator-theoretical methods. Then the obtained results are applied to difference equations of the form (1.1). The methods developed in the present paper can be applied to other equations involving block Toeplitz operators, Wiener-Hopf integral operators, differential and integro-differential operators, etc.. Note that our approaches somewhat differ of those generally used in successive approximation procedures. We would like to mention the works [1], [5], [18], [22] for standard texts on iterated projection methods and the more recent overview [4] where a theoretical framework, mostly related to our approaches, is developed for the analysis of convergence of the iterated Petrov-Galerkin method with applications to Fredholm integral equations of the second kind. The paper is organized as follows: in Sections 2 and 3 we develop projectioniterative methods for abstract operator equations. At this level of generality, we give convergence analysis for our approximate algorithms. Effective error estimates are also presented. We apply these approximation methods in Section 4 to the class of difference equations of the form (1.1). In Section 5 we illustrate the methods by considering perturbed second-order difference equations. In particular, equations corresponding to Jacobi and discrete Schr¨odinger operators are examined. The results of Section 5 are of interest by themselves.

2. Projection-iterative methods Let X be a Banach space and let T be a bounded linear operator on X. We consider an equation, in an unknown u, of the form u − Tu = g ,

(2.1)

where g is a given element of X. We are concerned with the approximate solution of the equation (2.1). The method which we apply can be described as follows. Let (Pn ) and (Qn ) be sequences of bounded projections acting on X with the property that Pn → I and Qn → I strongly (I designates the identity operator in X). For convenience we denote Pn0 = I − Pn and Q0n = I − Qn . We take an arbitrary element u0 ∈ X, and define a sequence (un ) of elements of X, successively, by un = Tn un−1 + Qn g

(n = 1, 2, . . . ) ,

(2.2)

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where Tn = Qn T Pn (n = 1, 2, . . . ) . This is simply a combination of the general projection method and the iterative one. We call the procedure defined by means of (2.2) the general projectioniterative method. Conditions under which the method (2.2) is convergent can be obtained by standard arguments. To this end we assume (throughout this section) that the equation (2.1) has a solution u, u ∈ X, for a given g. According to (2.1) and (2.2), we have un − u = Tn un−1 + Qn g − u = Tn un−1 + Qn u − Qn T u − u = Tn un−1 − Tn u − Q0n u − Qn T Pn0 u = Tn (un−1 − u) + Sn u , i.e. un − u = Tn (un−1 − u) + Sn u ,

(2.3)

where Sn := −Q0n − Qn T Pn0 Further, we have from (2.3)

(n = 1, 2, . . .) .

un − u = Tn (un−1 − u) + Sn u = Tn (Tn−1 (un−2 − u) + Sn−1 u) + Sn u = Tn Tn−1 (un−2 − u) + Tn Sn−1 u + Sn u, and repeating this substitution, we obtain, for each n = 1, 2, . . . , un − u = Tn1 (u0 − u) + Tn2 S1 u + . . . + Tnn Sn−1 u + Sn u ,

(2.4)

where Tnk := Tn Tn−1 . . . Tk

(k = 1, 2, . . . , n) .

Next we put tnk := kTn Tn−1 . . . Tk k

(k = 1, 2, . . . , n; n = 1, 2, . . .) ,

(2.5)

and make the following assumption. Assumption 2.1. The numbers tnk (k = 1, 2, . . . , n; n = 1, 2, . . .) defined by (2.5) satisfy the following conditions: (i) tnk → 0 as n → ∞ (for each fixed k); (ii) There is a constant M such that n X

tnk ≤ M

k=1

for each n = 1, 2, . . . . Let sn := kSn uk According to (2.4), we obtain

(n = 1, 2, . . .) .

kun − uk ≤ tn1 ku0 − uk + tn2 s1 + . . . + tnn sn−1 + sn .

(2.6)

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Since Pn → I and Qn → I strongly, it follows that sn → 0 as n → +∞, and then, by Assumption 2.1 (applying a theorem of Toeplitz ([10], Chapter 11, [391])) we derive that the right side of (2.6) tends to zero as n → +∞. Thus the sequence (un ) converges in X to the solution u of the equation (2.1). Remark 2.2. An analysis of the formula (2.4) shows that the convergence of (un ) does not depend on the initial approximation u0 , and also that (under the assumptions which we made) the equation (2.1) has a unique solution. In order to obtain an a posteriori error estimate of the method we need the following additional assumption. Assumption 2.3. I − T and also I − Tn for all sufficiently large n are (boundedly) invertible operators on X. Again from (2.3) we have (I − Tn )(un − u) = Tn (un−1 − un ) + Sn u or, talking into account Assumption 2.3, we can write un − u = (I − Tn )−1 Tn (un−1 − un ) + (I − Tn )−1 Sn (I − T )−1 g

(2.7)

for sufficiently large n. In this way we obtain for all sufficiently large n the following estimation kun − uk ≤ cn kun−1 − un k + kRn gk ,

(2.8)

where cn = k(I − Tn )−1 Tn k and Rn = (I − Tn )−1 Sn (I − T )−1

(n = 1, 2, . . . ) .

The following theorem summarizes the discussion above. Theorem 2.4. Under conditions of Assumption 2.1 the sequence (un ) determined by (2.2) is convergent to a solution u, u ∈ X, of the equation (2.1). Moreover, the equation (2.1) has a unique solution. If the conditions of Assumption 2.1 are satisfied together with Assumption 2.3, then the error estimate (2.8) holds. Remark 2.5. In case T is a compact operator in X, the strong convergence Qn → I implies the uniform convergence Qn T → T . This fact implies that the operators I − T and I − Tn for sufficiently large n are (boundedly) invertible, simultaneously. Remark 2.6. In case Pn = Qn = I the general method (2.2) reduces to the standard iteration un = T un−1 + g (n = 1, 2, . . . ; u0 ∈ X) , (2.9) which converges to the solution u of (2.1) if kT k < 1. In this case Rn = 0 and the error estimate (2.8) becomes kun − uk ≤ ckun−1 − un k

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with c = k(I − T )−1 T k, or by a less precise but a more practical estimate kun − uk ≤

kT kn ku0 − u1 k . 1 − kT k

3. A realization of the method In this section we discuss situations in which the approximation method considered before can be applied. We are still in the abstract framework of a class of perturbed operator equations Au + Bu = f , (3.1) where A and B are linear bounded operators in a Banach space E, and f is a given element in E. We will give conditions on A and B such that the corresponding equation (3.1) can be reduced to a suitable equation of the form (2.1) for which the method (2.2) is applicable. To this end, we introduce several assumptions which are listed below depending on the requirement of the exposition of the material. Assumption 3.1. The operator A is invertible on the left in the sense that there is an operator A(−1) , in general unbounded in E but Dom(A(−1) ) ⊃ Ran(A) (Dom(A(−1) ) denotes the domain of A(−1) and Ran(A) is the range of A) such that A(−1) A = I. In the sequel we consider a family of operators (in general unbounded) Lτ (τ ≥ 0) with the following property (cf. [6], Assumption 4). Assumption 3.2. For each τ ≥ 0 the operator Lτ is one-to-one, i.e. KerLτ = 0, and L0 = I. On the domain Dτ = Dom(Lτ ) of the operator Lτ (τ ≥ 0) we introduce a new norm |u|τ = kLτ uk (u ∈ Dτ ) (k · k stands for the norm of the space E). The norm |u|τ turns the linear manifold Dτ into a normed space which we denote by Eτ . Clearly E0 = E. The next assumption is the following. Assumption 3.3. For τ 0 ≥ τ ≥ 0 there holds |u|τ ≤ |u|τ 0

(u ∈ Eτ 0 ) .

(3.2)

Remark 3.4. Assumption 3.3 states that Eτ 0 ⊂ Eτ for τ 0 ≥ τ ≥ 0 and that the embedding operator from Eτ 0 into Eτ is bounded. Of course, instead of (3.2) it can be required that |u|τ ≤ c|u|τ 0 (u ∈ Eτ 0 ) with an arbitrary constant c.

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Remark 3.5. Eτ in general is not complete. But if Lτ is a closed operator and there is a constant c = c(τ ) such that kLτ uk ≥ c(τ )kuk

(u ∈ Dτ ) ,

then the space Eτ is complete. We connect the family (Lτ ) with the operators A and B by the next assumptions. Assumption 3.6. There exists a number m > 0 and a constant a > 0 independent of τ such that for every τ ≥ 0 the estimate |A(−1) u|τ ≤ a|u|τ +m

(u ∈ Eτ +m )

(3.3)

holds. We note that from (3.3) it follows a−1 |u|τ ≤ |Au|τ +m

(3.4)

for each τ ≥ 0 and u ∈ Eτ . However, in general, (3.4) does not imply (3.3). It is true that from (3.4) it follows that the operator A is one-to-one, and its inverse A−1 considered as an operator from Eτ +m to Eτ is bounded for τ ≥ 0, i.e. A−1 ∈ B(Eτ +m , Eτ ) for τ ≥ 0. Of course, from (3.4) it follows |A−1 v|τ ≤ a|v|τ +m for each v ∈ A(Eτ ) and τ ≥ 0. These arguments show that the operator A(−1) considered in Assumption 3.1 is in fact an exstention of A−1 defined above. Assumption 3.7. There exists τ ≥ 0 such that |A(−1) Bu|τ ≤ c(τ )|u|τ

(u ∈ Eτ )

(3.5)

with 0 ≤ c(τ ) < 1. We need the following supplementary assumption. Assumption 3.8. There exists a constant b > 0 independent of τ such that |A(−1) Bu|τ ≤ b|u|τ −

(u ∈ Eτ − )

(3.6)

if τ ≥ > 0. Remark 3.9. Assumptions 3.6–3.8, a prototype of which was introduced in [7] (see also [8]), play an essential role in our applications considered in the next sections. On the basis of them we will obtain conditions on the perturbed operator B in order to make applicable the method (2.2). In Assumption 3.7, a relative boundedness of the perturbation with a constant c(τ ) less than 1 in the norm of Eτ allows the perturbing operator B to have an unrestricted norm in the basic space E.

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Theorem 3.10. Let Assumptions 3.1–3.3, 3.6 and 3.8 be satisfied. (i) If f ∈ Eτ0 with τ0 ≥ m, then each solution of the equation (3.1) belongs to Eτ for τ ≤ τ0 − m. (ii) In addition, if Assumption 3.7 is also fulfilled with τ ≤ τ0 − m, then for each f ∈ Eτ0 (τ0 ≥ m) the equation (3.1) can have only a unique solution u and u ∈ Eτ for every τ ≤ τ0 − m. Proof. (i) Choose a positive integer n such that τ0 − m ≤ n, where is as in (3.6), and put 0 = (τ0 − m)/n. Let u, u ∈ E, satisfy the equation (3.1), that is Au + Bu = f . Then we can write |u|τ = |A(−1) Au|τ = |A(−1) (f − Bu)|τ ≤ |A(−1) f |τ + |A(−1) Bu|τ ≤ a|f |τ +m + b|u|τ − ≤ a|f |τ0 + b|u|τ −0 for each τ , 0 ≤ τ ≤ τ0 − m. Thus we have |u|τ ≤ a|f |τ0 + b|u|τ −0

(0 ≤ τ ≤ τ0 − m) .

(3.7)

For τ = 0 from (3.7) it follows |u|0 ≤ a|f |τ0 + b|u|0 = a|f |τ0 + bkuk < ∞ . Then, for τ = 20 we have |u|20 ≤ a|f |τ0 + b|u|0 < ∞ , and, continuing this process, we finally obtain for τ = n0 = τ0 − m |u|τ0 −m ≤ a|f |τ0 + b|u|(n−1)0 < ∞ , that is u ∈ Eτ0 −m , and hence u ∈ Eτ for τ ≤ τ0 − m. This proves the first part of the theorem. In order to prove (ii), first we observe that the equation (3.1) has a unique solution in E, because if the homogeneous equation of (3.1) has a nontrivial solution u, i.e. if Au + Bu = 0 (u 6= 0; u ∈ E) , then by Assumption 3.7 we have |u|τ = |A(−1) Bu|τ ≤ c(τ )|u|τ , and since, by what was proved in (i), u ∈ Eτ for each τ ≥ 0, it follows 1 ≤ c(τ ), a contradiction. From the above arguments it is seen that under Assumptions 3.1–3.3 and 3.6–3.8 for f ∈ Eτ with τ ≥ m the equation (3.1) can be reduced to u − Tu = g ,

(3.8)

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where g = A(−1) f and T = −A(−1) B . (3.9) We apply the projection-iterative method (2.2) to the equation (3.8) by supposing that the sequences (Pn ) and (Qn ) of projections on E are connected with the corresponding topologies of the norms | · |τ according to the following assumption. Assumption 3.11. Pn and Qn (n = 1, 2, . . .) are bounded operators in each space Eτ , and the sequences (Pn ) and (Qn ) converge strongly to the identity operator on Eτ , τ ≥ 0. Remark 3.12. Let (Pn ) and (Qn ) be sequences of bounded projections on E and let Pn → I and Qn → I strongly on E. If Pn Lτ = Lτ Pn and Qn Lτ = Lτ Qn for each n = 1, 2, . . . and τ ≥ 0, then Assumption 3.11 is fulfilled. For the sake of simplicity we require the following. Assumption 3.13. kPn k = kQn k = 1 (n = 1, 2, . . . ). Now, using Theorem 2.4, we prove that the projection-iterative process (2.2) for the equation (3.8) converges to a solution of the equation (3.1). For that purpose let τ ≥ 0 be fixed satisfying Assumption 3.7 and let us choose an initial element u0 in (2.2) belonging to Eτ . Theorem 3.14. Let Assumptions 3.1–3.3, 3.6 and 3.8 be satisfied together with Assumptions 3.11 and 3.13, and let (un ) be the approximating sequence determined by the process (2.2) for the equation (3.8), with the initial element u0 chosen as above, i.e. u0 ∈ Eτ for τ as in Assumption 3.7. If f ∈ Eτ0 (τ0 ≥ τ + m), then (un ) converges in the norm of Eτ to the solution u of the (3.1) with the error estimate |un − u|τ ≤

c(τ ) |un−1 − un |τ + |Rn g|τ , 1 − c(τ )

(3.10)

where c(τ ) is defined by (3.5) and Rn = (I − Tn )−1 Sn (I − T )−1

(n = 1, 2, . . . ) .

Proof. For convenience we preserve the notations used in Section 2. In particular, let tnk denote the values expressed in (2.5) with Tn = Qn T Pn for T defined by (3.9). By hypothesis u0 ∈ Eτ and, since T maps Eτ boundedly in Eτ , it follows that un ∈ Eτ for each n = 1, 2, . . . . We prove the convergence of (un ) to the solution of (3.8) by applying Theorem 2.4. To this end we evaluate the values tnk (k = 1, . . . , n; n = 1, 2, . . .). By virtue of Assumptions 3.11, 3.13 and (3.5) we have |Tn u|τ ≤ |T Pn u|τ ≤ c(τ )|u|τ for every u ∈ Eτ . Thus |Tn |τ ≤ c(τ )

(n = 1, 2, . . . ) ,

and hence tnk = |Tn Tn−1 . . . Tk |τ ≤ |Tn |τ |Tn−1 |τ . . . |Tk |τ ≤ (c(τ ))n−k+1 ,

(3.11)

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i.e. tnk ≤ (c(τ ))n−k+1 (k = 1, . . . , n; n = 1, 2, . . . ) , where c(τ ) is as in Assumption 3.7. By hypothesis 0 ≤ c(τ ) < 1, hence tnk → 0 as n → ∞ for each fixed k, and also n n X X c(τ ) . tnk ≤ (c(τ ))n−k+1 ≤ 1 − c(τ ) k=1

k=1

Thus the conditions (i) and (ii) of Assumption 2.1 are satisfied and, in view of Theorem 2.4, we may conclude that (un ) converges in Eτ to the solution u of the equation (3.8) or, which is the same, of the equation (3.1). The error estimate (3.10) follows immediately from (2.8). The proof of Theorem 3.14 is complete. According to Remark 2.6 we can formulate the following assertion. Corollary 3.15. In the particular case Pn = Qn = I (n = 1, 2, . . . ), under Assumptions 3.1–3.3, 3.6, 3.7 and 3.8, the standard iteration (2.9) for the equation (3.8), where f ∈ Eτ0 (τ0 ≥ τ + m; τ as in Assumption 3.7 ), with an arbitrary initial element u0 belonging to Eτ converges in the norm Eτ with the error estimate |un − u|τ ≤

(c(τ ))n |u0 − u1 |τ , 1 − c(τ )

(3.12)

As before c(τ ) is determined by (3.5).

4. Perturbed discrete convolution equations In this section we illustrate the applicability of the projection-iterative method developed in Sections 2 and 3, by considering a concrete class of discrete equations. Let lp (N) for 1 ≤ p < ∞ be the Banach space of all sequences u = (un ) of complex numbers satisfying ∞ X kukpp = |un |p < ∞ . n=1

By V we denote the elementary shift in lp (N), i.e. (V u)n = un−1 The operator V

(−1)

defined by

(V

(−1)

u)n = un+1

(n = 1, 2, . . . ; u0 = 0) . (n = 1, 2, . . . ; u = (un ) ∈ lp (N))

is obviously an inverse on the left of V , i.e. V (−1) V = I. In the sequel E denotes one of the Banach spaces lp (N) (1 ≤ p < ∞), and A stands for a Toeplitz operator defined on E by ∞ X (Au)n = an−k uk (u = (un ) ∈ E) , (4.1) k=1

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where an (n = 0, ±1, . . . ) are complex numbers such that ∞ X

|an | < ∞ .

(4.2)

n=−∞

According to the theory of Toeplitz operators (or, in other terms, discrete Wiener-Hopf operators) [3] (see also [11] and [23]) the operator A can be regarded (in a certain sense) to be the value of some function A (we preserve the same notation as for the operator) of the operator V , i.e. A = A(V ). Namely, the function A is given on the complex unit circle T = {z ∈ C : |z| = 1} and is represented in the form A(z) =

∞ X

an z n

(z ∈ T) .

n=−∞

The function A(z), z ∈ T, is called the symbol of the operator A. Note that the numbers an (n = 0, ±1, . . . ) are the Fourier coefficients of A, so that, in case A is given, they can be computed by the formula Z 2π 1 A(eiφ )e−inφ dφ (n = 0, ±1, . . . ) . an = 2π 0 Clearly, the function A is continuous on T. Next we assume that the function A(z) has only a finite number of zeros on T, and that each of them has finite multiplicity. In this setting it will be convenient to write the symbol A(z), z ∈ T, as follows: A(z) =

r Y

(z −1 − αj )mj A0 (z)

(z ∈ T) ,

(4.3)

j=1

where αj (j = 1, . . . , r) are pairwise distinct points on the unit circle T, mj ≥ 0 (j = 1, . . . , r) and A0 (z) (z ∈ T) is a continuous function on T such that A0 (z) 6= 0 (z ∈ T). In order to avoid supplementary prudence, henceforth we consider that mj (j = 1, . . . , r) are non-negative integers. Let 2π 1 κ = ind A0 (z) = arg A0 (eiφ ) φ=0 2π 2π ( [ ]φ=0 means the increment of the function on the interval [0, 2π]). Then A0 (z), z ∈ T, can be written in the form (the Wiener-Hopf factorization) [19] A0 (z) = A− (z)z κ A+ (z) ,

(4.4)

where A+ (z) and A− (z) are functions holomorphic inside and continuous up to the boundary in domains |z| ≤ 1 and |z| ≥ 1, respectively, and A+ (z) 6= 0 (|z| ≤ 1) and A− (z) 6= 0 (|z| ≥ 1). Note that the functions A− (z) and A+ (z) can be considered in turn as symbols of some Toeplitz operators; let us denote them by A− and A+ , respectively.

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Thus, in view of (4.3) and (4.4), we may express the operator A in the form A = RA− V (κ) A+ ,

(4.5)

where R=

r Y

(V (−1) − αj I)mj ,

j=1

and where V (κ) stands for V κ if κ ≥ 0 and (V (−1) )−κ if κ < 0. We note that an operator V (−1) − αI, where α ∈ T, is one-to-one, and there exists a closed and unbounded operator (V (−1) − αI)−1 . It is easy to see that (V (−1) − αI)−1 u = (−

∞ X

αn−j−1 uj )

(u = (un ) ∈ Ran(V (−1) − αI)) . (4.6)

j=n

In the sequel we assume that κ > 0. In this case the operator A admits an inverse operator on the left A(−1) in the sense of Assumption 3.1, and via (4.5), we can write (−κ) −1 −1 A(−1) u = A−1 A− R u + V

(u ∈ Ran(A)) .

(4.7)

Next, let B denote a bounded operator defined on the space E by B = [bnk ]∞ n,k=1 , i.e. (Bu)n =

∞ X

bnk uk

(n = 1, 2, . . . ; u = (un ) ∈ E) ,

(4.8)

k=1

where bnk ∈ C (n, k = 1, 2, . . . ). Our problem is to apply the general procedure discussed in Sections 2 and 3 for the approximate solution of the operator equation (A + B)u = f ,

(4.9)

where f is a given element in E, and A and B are operators defined on E by (4.1) and (4.8), respectively. We first consider the case where the factor A+ (z) in factorization (4.4) is a constant. This means in particular that the symbol A(z) is not equal to zero on the domain |z| > 1. In other words, we study the case where the symbol A(z) can be written in the form r Y (z −1 − αj )mj A− (z) (z ∈ T) , (4.10) A(z) = z κ j=1

where κ > 0 and A− (z) is a function holomorphic inside and continuous up to the boundary in |z| ≥ 1, and A− (z) 6= 0 (|z| ≥ 1). In order to apply the results from Section 3 we define the family of operators (Lτ )τ ≥0 by (Lτ u)n = nτ un (n = 1, 2, . . . ; u ∈ Dom(Lτ )) ,

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where Dom(Lτ ) = {u ∈ E :

∞ X

|nτ un |p < ∞}.

n=1

For the introduced family (Lτ )τ ≥0 Assumption 3.2 is trivially fulfilled, the corresponding spaces Eτ , τ ≥ 0, are complete and Assumption 3.3 is also obvious. We now proceed to the verification of Assumption 3.6. To this end, we let m = max{mj : j = 1, . . . , r} (recall that mj are multiplicities of the zeros of the symbol A(z) belonging to the unit circle T, cf. (4.3)). The following auxiliary assertions will be used. Lemma 4.1 ([6]).

(i) For every τ ≥ 0 the operator V (−1) is bounded on Eτ , and |V (−1) u|τ ≤ |u|τ

(u ∈ Eτ ) .

(4.11)

(ii) For τ > 0 and > 0 there exists a0 (τ, ) > 0 such that a0 (τ, ) → 0 as τ → ∞, and |V (−1) u|τ ≤ a0 (τ, )|u|τ +

(u ∈ Eτ + ) .

(4.12)

(iii) For every τ ≥ 0 and every α ∈ T the estimate |(V (−1) − αI)−1 u|τ ≤ c|u|τ +1

(u ∈ Eτ +1 )

(4.13)

holds with c = p when E = lp (N). Proof. The assertions (i)–(iii) were pointed out in [6] (see Lemmas 1 and 3 [6]). For the sake of the completeness we give the proofs of them. Let τ and τ 0 be arbitrary non-negative numbers and u ∈ Eτ 0 . Then it is easily verified that |V (−1) u|τ ≤ a(τ, τ 0 )|u|τ 0

(u ∈ Eτ 0 ) ,

(4.14)

where a(τ, τ 0 ) = max nτ (n + 1)−τ n

0

.

For τ = τ 0 we have a(τ, τ 0 ) = 1, and hence the assertion (i) follows. For τ 0 = τ + we obtain a(τ, τ 0 ) = max nτ (n + 1)−τ − ≤ τ τ (τ + )−τ − . n

Clearly a(τ, τ + ) → 0 as τ → ∞, and thus the assertion (ii) is verified. The assertion (iii) is less trivial. The estimate (4.13) is a kind of Hardy’s inequality with weights (cf. [9]). It can also be obtained directly by using the following classical Hardy’s inequality (see [17], Theorem 331, pag. 246) ∞ X ∞ X X ∞ −1 p j u j ≤ pp |un |p n=1 j=n

n=1

(u = (un ) ∈ lp (N)) .

(4.15)

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Indeed, for p when E = lp (N) and u ∈ Eτ +1 denote v = Lτ +1 u. Then, in view of (4.6) and (4.15), we can write ∞ ∞ X τX p p1 (−1) −1 |(V − αI) u|τ = n αn−j−1 uj n=1

≤

j=n

∞ ∞ X X n=1

∞ X p1 p p1 ≤p = p|u|τ +1 . |vn |p j −1 |vj | n=1

j=n

Remark 4.2. In (4.13) the constant is the best possible [17]. Next we note that, by virtue of a Wiener theorem [19] (see also [3]), the Toeplitz operator A−1 − like A− is upper triangular. In terms of its symbol this means that b− (z) = 1/A− (z) can be represented in the form b− (z) =

0 X

n b− nz

(z ∈ T)

with

kb− kW =

n=−∞

0 X

|b− n| < ∞.

n=−∞

(Here the Wiener norm is denoted as usual by k · kW ). Taking into account this fact, by Lemma 4.1 (i), we conclude that the operator A−1 is bounded in each space Eτ , and − |A−1 − u|τ ≤ kb− kW |u|τ .

(4.16)

In order to estimate R−1 we observe that, since the roots αj−1 (j = 1, . . . , r) of A(z) are pairwise distinct, there exist numbers ajk (k = 1, . . . , mj ; j = 1, . . . , r) such that mj r r X Y X (z −1 − αj )−mj = ajk (z −1 − αj )−k (z ∈ T) . (4.17) j=1

j=1 k=1

From (4.17) it follows that the operator R−1 can be represented as R−1 =

mj r X X

ajk (V (−1) − αj I)−k .

(4.18)

j=1 k=1

In view of (4.13) we can obtain |(V (−1) − αj I)−k u|τ ≤ pk |u|τ +k

(u ∈ Eτ +k ; k = 1, . . . , mj , j = 1, . . . , r)

in case E = lp (N). Therefore, from (4.18) we arrive at the following estimate (recall that m is the maximal multiplicity of zeros αj−1 (j = 1, . . . , r)) |R−1 u|τ ≤ c1 |u|τ +m where c1 =

mj r X X j=1 k=1

(u ∈ Eτ +m ) , pk |ajk | .

(4.19)

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Now, let u ∈ Eτ +m (τ ≥ 0). In view of (4.11), (4.16) and (4.19) we can estimate −1 −1 |A(−1) u|τ ≤ |V (−κ) A−1 u|τ ≤ |A−1 u|τ − R − R

≤ kb− kW |R−1 u|τ ≤ c1 kb− kW |u|τ +m , i.e. |A(−1) u|τ ≤ a|u|τ +m

(u ∈ Eτ +m )

(4.20)

with a = c1 kb− kW . Thus the Assumption 3.6 is verified. We point out the estimate |A(−1) u|τ ≤ a|V (−κ) u|τ +m

(u ∈ Eτ +m )

(4.21)

which will be used later. Obviously, (4.21) can be obtained by using similar arguments as above. The constant a is the same as in (4.20). Next, we find conditions on B such that Assumptions 3.7 and 3.8 are also verified. Let u ∈ Eτ , τ ≥ 0. By virtue of (4.21) and (4.12) we have |A(−1) Bu|τ ≤ a|V (−κ) Bu|τ +m ≤ aa0 (τ, )|V (−κ+1) Bu|τ +m+ ≤ aa0 (τ, )kBτ, k|u|τ , where Bτ, = Lτ +m+ V (−κ+1) BL−1 τ . Note that the operator Bτ, is induced on E by the matrix ∞ Bτ, = j τ +m+ k −τ bj+κ−1 k j,k=1 . It is seen that the operator Bτ, is generally speaking unbounded in the basic space E, and even if it is bounded, its norm could increase as the parameter τ increases. However, if we let bjk = 0 for j >k+κ−1 and require that the operator ∞ B1 = j m+ | bj+κ−1 k | j,k=1

(4.22)

is bounded on the space E, then we have kBτ, k ≤ kB1 k . In this way we obtain conditions for which estimates |A(−1) Bu|τ ≤ c(τ )|u|τ

(4.23)

hold with c(τ ) = aa0 (τ, )kB1 k. Clearly c(τ ) → 0 as τ → +∞, and thus the fulfilment of Assumption 3.7 is ensured. It turns out that under the same conditions Assumptions 3.8 is also satisfied. Indeed, let u ∈ Eτ − , τ ≥ > 0. Then, again by virtue of (4.21) and by (4.11), we have |A(−1) Bu|τ ≤ a|V (−κ+1) Bu|τ +m ≤ akLτ +m V (−κ+1) BL−1 τ − k|u|τ − .

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As before, the operator m+we show that ∞ under conditions bjk = 0 for j > k + κ − 1 if(−κ+1) B1 = j bj+κ−1 k j,k=1 is bounded on E, then the operator Lτ +m V BL−1 τ − is bounded on E and its norm does not exceed kB1 k. Therefore |A(−1) Bu|τ ≤ b|u|τ − (τ ≥ > 0) (4.24) with b = akB1 k. Assumption 3.7 is verified. According to the discussion in Section 3, the equation (4.9) for a given element f in Eτ with τ enough large can be reduced to an equation of the form (3.8), namely u − Tu = g ,

(4.25)

where g = A(−1) f and T = −A(−1) B. We apply to the equation (4.25) the projection-iterative method (2.2) by taking Pn and Qn canonical projections in the space E, that is Pn = Qn

(n = 1, 2, . . . )

and Pn u = (u1 , . . . , un , 0, 0, . . . )

(u = (un ) ∈ E) .

Evidently, Pn → I strongly in E, kPn k = 1 and Pn Lτ = Lτ Pn for each n = 1, 2, . . . . Hence, Assumptions 3.11 and 3.13 are fulfilled. We have thus verified all the required assumptions, and Theorem 3.14 can be applied. We denote Tn = Pn T Pn ,

gn = Pn g

(n = 1, 2, . . . ),

take an arbitrary element u0 ∈ Eτ (τ ≥ 0) and define the approximating sequence (un ) by un = Tn un−1 + gn (n = 1, 2, . . . ). (4.26) We have the following result. Theorem 4.3. Let A and B be operators defined by (4.1) and (4.8), respectively. Assume that the symbol A(z) of A can be factorized as in (4.10), κ > 0 and m = max{mj : j = 1, . . . , r}. Furthermore, assume that for any f ∈ Eτ0 with sufficiently large τ0 (for instance, τ0 ≥ τ +m; τ being as in Assumption 3.7) the equation (4.9) possesses a solution u in Eτ . If under the conditions bjk = 0 for j > k + κ − 1 the operator B1 defined by (4.22) is bounded on E, then the approximating sequence (un ) determined by the process (4.26) converges in the norm Eτ (τ ≤ τ0 ) to the solution u of the equation (4.9) for any initial approximation u0 in Eτ . The error estimate is given by |un − u|τ ≤

1 + c(τ ) c(τ ) |un−1 − un |τ + |hn |τ , 1 − c(τ ) 1 − c(τ )

where c(τ ) is as in (4.23) and hn = Pn0 (I − T )−1 g.

(4.27)

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Proof. That the approximating sequence (un ) converges to the solution of the equation (4.9) follows immediately from Theorem 3.14. Furthermore, since f ∈ Eτ0 with τ0 sufficiently large, by Assumptions 3.3 and 3.6 it follows that g ∈ Eτ and also hn ∈ Eτ by virtue of Theorem 3.10. The error estimate (4.27) is a consequence of (3.10). Corollary 4.4. In the particular case Pn = I (n = 1, 2, . . . ) under the hypotheses of Theorem 4.3 the sequence (un ) determined by the iterative process un = T un−1 + g

(n = 1, 2, . . . )

converges in the norm of Eτ (τ ≤ τ0 ) to the solution u of the equation (4.9) for any initial approximation u0 in Eτ . The error estimate is given by |un − u|τ ≤

c(τ )n |u0 − u1 |τ 1 − c(τ )

(4.28)

with c(τ ) given as in (4.23). Finally, we note that the general case when the symbol A(z) of A contains a non-constant factor A+ (z) (cf. (4.4)) can be reduced to the previous one by denoting v = A+ u (recall that A+ is the Toeplitz operator corresponding to the symbol A+ (z)) and reducing the equation (4.9) to the following e + Bv e =f Av

(4.29)

e is a Toeplitz operator associated to a symbol of the form (4.10) and in which A −1 e B = BA+ . Note that, due to the fact that A+ is a boundedly invertible operator in E, u is a solution of the equation (4.9) if and only if v = A+ u is a solution of the corresponding equation (4.29).

5. Examples. Perturbed second-order difference equations In this section we illustrate the obtained results by considering perturbed secondorder difference equations. We are concentrated mainly on the singular cases, that is when the symbols of the unperturbed operators degenerate on the unit circle. Let A denote the Toeplitz operator generated on the space E (we preserve notations used in the previous section) by the function (the symbol of A) A(z) = a−1 z −1 + a1 z

(z ∈ T) ,

where a−1 , a1 ∈ C. For determination let us consider that |a1 | ≥ |a−1 | > 0. Note that the set of all values of the symbol A(z), z ∈ T, represents an ellipse E on the complex plane. In what follows E+ denotes the interior of this ellipse. Let B be a bounded operator of the form (4.8) and consider the following equation Au + Bu − λu = f (f ∈ E) , (5.1) where λ is a given scalar parameter.

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In dependence of the parameter λ we describe conditions under which the scheme given in the previous section can be applied to the equation (5.1). First, we consider the case |a1 | > |a−1 |. For this case it is easy to observe that if λ ∈ E the symbol A(z) − λ has a root on the unit circle T, another one lying inside of the unit disk. In other words we have the factorization A(z) − λ = a−1 z(z −1 − α)(z −1 − β)

(z ∈ T)

(5.2)

with, for instance, |α| = 1 and |β| > 1. Obviously, a similar factorization (5.2) holds for λ ∈ E+ but with |α| > 1 and |β| > 1. It is seen that Theorem 4.3 can be applied to the pair A − λI and B. We have κ = 1, m = 1 for λ ∈ E and κ = 1, m = 0 for λ ∈ E+ . Therefore, if the perturbation ∞ B is such that bjk = 0 for j > k and the operator defined by Bδ = j δ bjk j,k=1 is bounded on E with δ > 1 for λ ∈ E and δ > 0 for λ ∈ E+ , then the equation (5.1) can be reduced to u − T (λ)u = g (5.3) with g = (A − λI)(−1) f and T (λ) = −(A − λI)(−1) B, where (A − λI)(−1) is a left inverse of A − λI the existence of which follows from the representation (5.2). Actually, as in the hypotheses of Theorem 4.3 we assume that the equation (5.1) possesses a solution u in E for f ∈ Eτ0 with sufficiently large τ0 . Similarly as in Section 4 we denote Tn (λ) = Pn T (λ)Pn ,

gn = Pn g ,

take an arbitrary element u0 ∈ Eτ (τ > 0) and define the approximating sequence (un ) by un = Tn (λ)un−1 + gn (n = 1, 2, . . . ) . (5.4) According to Theorem 4.3 the equation (5.1) has a solution u, u ∈ Eτ , the procedure (5.4) converges in the norm of Eτ to u. Moreover, it can be computed that the corresponding constant c(τ ) in (4.27) is c(τ ) = p kB1+ k |a−1 |−1 (|β| − 1)−1 a0 (τ, ) −1

c(τ ) = kB k |a−1 |

−1

(|α| − 1)

−1

(|β| − 1)

a0 (τ, )

for

λ∈E,

for

λ ∈ E+

with a0 (τ, ) = τ τ (τ + )−τ − as in (4.12). Therefore, we can conclude the following result. Theorem 5.1. Under the above conditions ifthe perturbation B is such that bjk = 0 ∞ for j > k and the operator defined by Bδ = j δ bjk j,k=1 is bounded on E with δ > 1 for λ ∈ E and δ > 0 for λ ∈ E+ , then the approximating sequence (un ) determined by (5.4) converges in the norm of Eτ (τ ≤ τ0 ) to the solution u, which belongs to Eτ , of the equation (5.1). The error estimate is given by a formula like (4.27), with c(τ ) = O(τ − ).

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Corollary 5.2. In the particular case Pn = I (n = 1, 2, . . .) under the hypotheses of Theorem 5.1 the sequence (un ) determined by the iterative process un = T (λ)un−1 + g

(n = 1, 2, . . . )

(5.5)

converges in the norm of Eτ (τ ≤ τ0 ) to the solution u, u ∈ Eτ , of the equation (5.1) for any initial approximation u0 in Eτ , and the effective error estimate | un − u |τ = O(τ −n )

(5.6)

holds. In case |a−1 | = |a1 | the ellipse E degenerates to a segment, so that E+ = ∅. In this case both roots of the symbol A(z) − λ for λ ∈ E are situated on T. Moreover, 2 it is observed that for λ2 6= 4a−1 a1 these roots are simple, √ but for λ √= 4a−1 a1 −1 the symbol A(z) − λ has only a root lying on T (either − a−1 a1 or a−1 a−1 1 ) with multiplicity m = 2. Therefore, for this case the following result holds. Theorem 5.3. The procedure (5.4) converges in the norm of Eτ (τ ≤ τ0) to the ∞solution of the equation (5.1) if bjk = 0 for j > k and the operator Bδ = j δ bjk j,k=1 is bounded on E with δ > 1 for λ ∈ E and λ2 6= 4a−1 a1 , and δ > 2 for λ2 = 4a−1 a1 . Similar error estimates as in Theorem 4.1 hold with c(τ ) = O(τ − ). Remark 5.4. Under the conditions assumed in Theorem 5.3 the iterative process (5.5) (i.e. when Pn = I, n = 1, 2, ...) is convergent in the space Eτ (τ ≤ τ0 ) to the solution u, u ∈ Eτ , of the equation (5.1) with error estimates similar to (5.6). Remark 5.5. The requirement that B be an∞upper-triangular matrix is essential. For instance, let a−1 = a1 = 1, and B = bjk j,k=1 with b23 = b32 = −1 and bjk 6= 0 otherwise. In this case E = [−2, 2] and it is easy to calculate that the numbers λ = ±1 are eigenvalues for the operator A+B. The corresponding eigenvectors are u1 = (1, 1, 0, 0, . . . ) and u2 = (1, −1, 0, 0, . . . ). Therefore, for this case the equation (5.1) in general can have no solutions. Remark ∞ 5.6. In the particular case when B is a diagonal matrix, that is B = bj δjk j,k=1 (δjk denotes the Kronecker symbol: δjk = 1 for j = k and δjk = 0 for j 6= k), the above conditions on the perturbation (in case that |a−1 | = |a1 |) are equivalent with sup j δ |bj | < ∞ j 2

for δ > 1 if λ ∈ E and λ 6= 4a−1 a1 , and δ > 2 for λ2 = 4a−1 a1 . A similar remark can be made for the previous case |a−1 | < |a1 |. The results in this section represent interest by themselves. In this connection we note that in case B is a diagonal matrix the corresponding perturbed operator A + B is referred to Jacobi operators, or often if a−1 = a1 = 1 to discrete Schr¨ odinger operators (see, for instance, [30] and the references quoted there).

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References [1] Atkinson K., Han W., Theoretical numerical analysis. A functional analysis framework, Springer-Verlag, New York, 2005. [2] Baxter G., A norm inequality for a “finite-section” Wiener-Hopf equation, Illinois J. Math. 7 (1963), 97–103. [3] B¨ ottcher A., Silbermann B., Analysis of Toeplitz Operators, Springer-Verlag, Berlin, 1990. [4] Chen Z., Xu Y., The Petrov-Galerkin and iterated Petrov-Galerkin methods for second-kind integral equations, SIAM J. Numer. Anal., 35 (1998), 406–434. [5] Chatelin F., Spectral approximation of linear operators, Academic Press, New York, 1983. [6] Cojuhari P.A., The absence of eigenvalues for operators that are close to operators generated by infinite-dimensional Jacobi matrices [Russian], Izv. Akad. Nauk Moldav. SSR Mat. 1990, no. 2, 15–21. [7] Cojuhari P.A., The absence of eigenvalues in a perturbed discrete Wiener-Hopf operator [Russian], Izv. Akad. Nauk Moldav. SSR Mat. 1990, no. 3, 26–35 [8] Cojuhari P.A., On the spectrum of singular nonselfadjoint differential operators. Operator extensions, interpolation of functions and related topics, Oper. Theory Adv. Appl., 61, Birkh¨ auser, Basel, 1993, 47–64. [9] Cojuhari P.A., Generalized Hardy type inequalities and some applications to spectral theory. Operator theory, operator algebras and related topics, 79–99, Theta Found., Bucharest, 1997. [10] Fichtenholz G.M., Differential- und Integralrechnung II, Hochschulb¨ ucher f¨ ur Mathematik, Band 62, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. [11] Gohberg I., Feldman I.A., Convolution equations and projection methods for their solution, Transl. of Math. Monographs, Vol. 41. Amer. Math. Soc., Providence, R.I., 1974. [12] Gohberg I., Levcenko V.I., The projection method of solution of degenerate discrete Wiener-Hopf equations [Russian], Funkcional. Anal. i Priloˇzen. 5 (1971) no. 4, 69–70. [13] Gohberg I., Levcenko V.I., The convergence of the projection method for the solution of the degenerate discrete Wiener-Hopf equation [Russian], Mat. Issled. 6 (1971), no. 4(22), 20–36. [14] Gohberg I., Levcenko V.I., The projection method for a degenerate discrete WienerHopf equation [Russian], Mat. Issled. 7 (1972), no. 3(25), 238–253. [15] Gohberg I., Pr¨ ossdorf S., Ein Projektionsverfahren zur L¨ osung entarteter System von diskreten Wiener-Hopf-Gleichungen, Math. Nachr., 65 (1975), 19–45. [16] Hagen R., Roch S., Silbermann B., Spectral theory of approximation methods for convolution equations, Birkh¨ auser Verlag, Basel, 1995. [17] Hardy G.H., Littlewood J.E., P´ olya G., Inequalities, Cambridge University Press, Cambridge 1952. [18] Krasnosel’skii M.A., Vainikko G.M., Zabreiko P.P., Rutitskii Ya.B., Stetsenko V.Ya., Approximate solution of operator equations, Wolters-Noordhoff, Groningen, 1972. [19] Krein M.G., Integral equations on the half-line with a kernel depending on the difference of the arguments [Russian], Uspehi Mat. Nauk 13 (1958) no. 5 (83), 3–120.

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[20] Levcenko V.I., The projection method for operators that are defined by degenerate Toeplitz matrices [Russian], Mat. Issled. 7 (1972), no. 4(26), 123–140. [21] Levcenko V.I., The projection method for the solution of degenerate dual discrete Wiener-Hopf equations [Russian], Mat. Issled. 8 (1973), no. 3(29), 26–45. [22] Luchka A. Ju., Projective-iterative methods for solving differential and integral equations [Russian], “Naukova Dumka”, Kiev, 1980 [23] Pr¨ ossdorf S., Einige Klassen singul¨ arer Gleichungen, Akademie Verlag, Berlin 1974. [24] Pr¨ ossdorf S., Systeme einiger singul¨ arer Gleichungen vom nicht normalen Typ und Projektionsverfahren zur ihrer L¨ osung, Studia Math., 53 (1975), 225–252. [25] Pr¨ ossdorf S., Silbermann B., Ein Projektionsverfahren zur L¨ osung abstrakter singular Gleichungen vom nicht normalen Typ und einige seiner Anwendungen, Math. Nachr., 61 (1974), 133–155. [26] Pr¨ ossdorf S., Silbermann B., Projektionsverfahren zur L¨ osung von System singul¨ arer Gleichungen vom nicht normalen Typ, Revue Roum. Math. Pures Appl., 22:7 (1977), 965–991. [27] Pr¨ ossdorf S., Silbermann B., Projektionsverfahren und die n¨ aherungsweise L¨ osung singul¨ arer Gleichungen, Teubner-Verlag, Leipzig, 1977. [28] Pr¨ ossdorf S., Silbermann B., Numerical analysis for integral and related operator equations., Birkh¨ auser Verlag, Basel, 1991. [29] Reich E., On non-Hermitian Toeplitz matrices, Math. Scand. 10 (1962), 145–152. [30] Teschl G., Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs, Volume 72, Amer. Math. Soc., Providence 2000. Petru A. Cojuhari AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30 30-059 Krak´ ow Poland e-mail: [email protected] Michal A. Nowak AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30 30-059 Krak´ ow Poland e-mail: [email protected] Submitted: October 5, 2008. Revised: February 12, 2009.

Integr. equ. oper. theory 64 (2009), 177–192 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020177-16, published online May 12, 2009 DOI 10.1007/s00020-009-1681-2

Integral Equations and Operator Theory

Multipliers of Fractional Cauchy Transforms Evgueni Doubtsov Abstract. Let Bn denote the unit ball of Cn , n ≥ 2. Given an α > 0, let Kα (n) denote the class of functions defined for z ∈ Bn by integrating the kernel (1 − hz, ζi)−α against a complex-valued measure on the sphere {ζ ∈ Cn : |ζ| = 1}. Let Hol(Bn ) denote the space of holomorphic functions in the ball. A function g ∈ Hol(Bn ) is called a multiplier of Kα (n) provided that f g ∈ Kα (n) for every f ∈ Kα (n). In the present paper, we obtain explicit analytic conditions on g ∈ Hol(Bn ) which imply that g is a multiplier of Kα (n). Also, we discuss the sharpness of the results obtained. Mathematics Subject Classification (2000). Primary 32A26; Secondary 32A37, 42B35, 46E15, 46J15. Keywords. Fractional Cauchy transform, Hardy–Sobolev space, holomorphic Lipschitz space, pointwise multiplier.

1. Introduction Let B = Bn denote the unit ball of Cn and let M = M(n) denote the space of complex-valued Borel measures on the sphere ∂B = ∂Bn . 1.1. Fractional Cauchy transforms Let α > 0. Given a measure µ ∈ M(n), its fractional Cauchy transform of order α is defined by the formula Z 1 dµ(ζ), z ∈ Bn . Kα [µ](z) = (1 − hz, ζi)α ∂Bn Here and in what follows we use the principal branch of the logarithm. Put Kα = Kα (n) = {Kα [µ] : µ ∈ M(n)} . This research was supported by RFBR (grant no. 08-01-00358-a), by the Russian Science Support Foundation and by the programme “Key scientific schools NS 2409.2008.1”.

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1.2. Multipliers Let Hol(Bn ) denote the space of holomorphic functions in the ball Bn . A function g ∈ Hol(Bn ) is called a (pointwise) multiplier of Kα (n) provided that f g ∈ Kα (n) for every f ∈ Kα (n). Let Mα (n) denote the set of all multipliers of Kα (n). 1.3. The families Kα (n) and Mα (n) as Banach spaces Standard arguments show that Kα (n), α > 0, is a Banach space with respect to the norm defined by kf kKα (n) = inf kµkM(n) : f = Kα [µ] , f ∈ Kα (n). Also, note that the above infimum is attained. The set Mα (n), α > 0, is a Banach algebra with the natural norm defined by kgkMα (n) = sup kf gkKα (n) : kf kKα (n) ≤ 1 , g ∈ Mα (n). 1.4. The problem under consideration The following abstract characterization of Mα (n) is known. Lemma 1.1. ([5]) Assume that n ∈ N and α > 0. Then the following properties are equivalent: (i) g ∈ Mα (n); (ii) g(z)(1 − hz, ζi)−α ∈ Kα (n) for all ζ ∈ ∂Bn , and ) (

g(z)

: ζ ∈ ∂Bn < ∞. sup

(1 − hz, ζi)α Kα (n)

(1.1)

In the present paper, we are interested in explicit analytic conditions on a function g ∈ Hol(Bn ) which imply g ∈ Mα (n) when n ≥ 2 and α > 0. 1.5. Comments 1.5.1. The spaces Kα (n) generalize the standard family Kn (n) of Cauchy integrals. Investigations of the family K1 (1) as a Banach space were initiated by V. P. Havin [7, 8]. Main results about M1 (1) were obtained in [20, 19, 11]. The classical families K1 (1) and M1 (1) are investigated in the recent monograph [4]. The spaces Kα (1), α > 0, were introduced by T. H. MacGregor [13]. Main results about Mα (1), α > 0, were obtained in [9, 6, 12, 10]. Various properties of the spaces Kα (1) and Mα (1) are collected in the recent monograph [10]. See [4] and [10] for historical remarks, further references and motivations. 1.5.2. Certain basic properties of Mα (n), n ∈ N, are proved in [5]. In particular, Mα (n) ⊂ Kα (n) and Mα (n) ⊂ H ∞ (Bn ) for all n ∈ N and α > 0. Here H ∞ (Bn ) denotes the space of bounded holomorphic functions in the ball. To the best knowledge of the author, the spaces Mα (n), n ≥ 2, have not been investigated systematically.

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1.5.3. Families of fractional Cauchy transforms are closely related with holomorphic Sobolev spaces in the ball. The multipliers of holomorphic Besov spaces were investigated in [14]; see also [15]. The multipliers of Hardy–Sobolev spaces were recently studied in [16, 3, 18]. 1.6. Organization of the paper Auxiliary results are collected in Section 2. General properties of the families Mα (n), n ∈ N, α > 0, are obtained in Sections 3 and 4. The main results of this paper are presented in Sections 5 and 6. It is proved in Section 5 that various Lipschitz and Hardy–Sobolev restrictions on g ∈ Hol(Bn ) guarantee that g ∈ Mα (n). Further results are obtained in Section 6. For example, the following assertion is a particular case of Theorem 6.1. Theorem 1.2. Let n ∈ N. Suppose that n > α > n − 1, g ∈ Hol(Bn ) and Z 1 |gξ0 (r)|(1 − r)α−n dr < +∞, sup ξ∈∂Bn

0

where gξ (w) = g(wξ) for w ∈ B1 . Then g ∈ Mα (n). For n = 1, Theorem 1.2 is proved in [12, Theorem 2]; it generalizes several results from [6].

2. Auxiliary results 2.1. Basic results Let σ = σn denote the normalized Lebesgue measure on the sphere ∂B = ∂Bn . In the following lemma, the notation a(z) ≈ b(z) means that the quotient a(z)/b(z) has a finite positive limit as |z| → 1−. Lemma 2.1. ([17, Proposition 1.4.10]) Let n ∈ N and let c ∈ R. Put Z dσn (ζ) , z ∈ Bn. Jc (z) = n+c ∂Bn |1 − hz, ζi| (i) If c < 0, then Jc (z) is bounded in B n . (ii) If c > 0, then Jc (z) ≈ (1 − |z|2 )−c . (iii) Finally, J0 (z) ≈ log

1 . (1 − |z|2 )

Recall that the radial derivative R : Hol(Bn ) → Hol(Bn ) is defined by the identity n X ∂f Rf (z) = zj (z), z ∈ Bn . ∂z j j=1 The radial differential operator of order 1 is defined as R1 = R + I. For 0 < p ≤ +∞, let H p (Bn ) denote the Hardy class in the ball Bn .

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Lemma 2.2. ([1, Chapter 2, Theorem 1.3.3], see also [2, Theorem 5.13]) Assume that n ∈ N, p1 , p2 ∈ (0, +∞) and n n − = 1. p2 p1 If f ∈ Hol(Bn ) and R1 f ∈ H p2 (Bn ), then f ∈ H p1 (Bn ). 2.2. Families of fractional Cauchy transforms For n ∈ N, denote by K0 = K0 (n) the family of functions f such that Z 1 f (z) − f (0) = dµ(ζ), z ∈ Bn , log 1 − hz, ζi ∂Bn for a measure µ ∈ M(n). Lemma 2.3. ([5]) Let n ∈ N. (i) If α > 0, β > 0, then Kα (n) · Kβ (n) ⊂ Kα+β (n). (ii) If 0 ≤ α < β, then Kα (n) ⊂ Kβ (n). Lemma 2.4. ([5]) Assume that n ∈ N, f ∈ Hol(Bn ) and α ≥ 0. Then f ∈ Kα (n) if and only if Rf ∈ Kα+1 (n). Assume that n ∈ N, j ∈ N and γ > 0. Denote by Bγj (Bn ) the space of functions f ∈ Hol(Bn ) for which Z 1Z X ∂mf kf kBγj (Bn ) = |Rj f (rζ)|(1 − r)γ−1 dσn (ζ) dr < +∞, ∂z m (0) + |m|≤j−1

0

∂Bn

where m = (m1 , . . . , mn ) ∈ Zn+ and |m| = m1 + · · · + mn . The Besov space Bγj (Bn ) with the norm k · kBγj (Bn ) is a Banach space (cf. [21, Proposition 6.2]). Lemma 2.5. ([5]) Suppose that n ∈ N, j ∈ {1, . . . , n} and α > n − j. If f ∈ j Bα−n+j (Bn ), then kf kKα (n) ≤ Ckf kBj (Bn ) , α−n+j

where the constant C > 0 does not depend on f . 2.3. Multiplier spaces Lemma 2.6. ([5]) If n ∈ N and 0 < α < β, then Mα (n) ⊂ Mβ (n). 2.4. Lipschitz spaces Fix β ∈ (0, 1). By definition, a function f : B n → C is in the real Lipschitz space LipR β (Bn ) if there exists a constant C > 0 such that |f (z) − f (w)| ≤ C|z − w|β ,

z, w ∈ B n .

Fix α ∈ (0, 1/2). By definition, a function f : B n → C is in the complex Lipschitz space LipC α (Bn ) if there exists a constant C > 0 such that |f (z) − f (w)| ≤ C|1 − hz, wi|α ,

z, w ∈ B n .

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Lemma 2.7. Let n ∈ N. Assume that 0 < β < 1, f ∈ Hol(Bn ) and |Rf (rζ)| ≤

C , (1 − r)1−β

r ∈ [0, 1), ζ ∈ ∂Bn .

(2.1)

Then f ∈ LipC γ (Bn ), where 0 < γ < min{1/2, β}. Proof. Given an f ∈ Hol(Bn ) with property (2.1), let the same symbol denote the continuous extension of f to the closed ball B n . Such extension exists by [17, Theorem 6.4.10]; moreover, (2.1) guarantees that f ∈ LipR β (Bn ). Therefore, |f (z) − f (w)| ≤ Cγ |1 − hz, wi|γ ,

z, w ∈ B n ,

where 0 < γ < 1/2 and 0 < γ ≤ β (see [17, Theorem 6.4.10] and [1, Chapter 3, Corollary 3.2.4]).

3. Multipliers and the family K0 (n) Proposition 3.1. Let n ∈ N. Then Mβ (n) ∩ K0 (n) = Mα (n) ∩ K0 (n) for all β ≥ α > 0. Proof. Assume that α ≤ β ≤ βe and βe−α ∈ N. We have Mα (n) ⊂ Mβ (n) ⊂ Mβe(n) by Lemma 2.6. Thus, it suffices to prove the proposition under the additional assumption β − α ∈ N. Hence, we may assume, without loss of generality, that β = α + 1. So, suppose that g ∈ Mα+1 (n) ∩ K0 (n). We have to prove that g ∈ Mα (n). In other words, given a function f ∈ Kα (n), we must prove that f g ∈ Kα (n). By Lemma 2.4, f g ∈ Kα (n) if and only if f Rg + gRf = R(f g) ∈ Kα+1 (n). On the one hand, g ∈ K0 (n) implies Rg ∈ K1 (n) by Lemma 2.4. Therefore, f Rg ∈ Kα (n) · K1 (n) ⊂ Kα+1 (n) by Lemma 2.3. On the other hand, f ∈ Kα (n) implies Rf ∈ Kα+1 (n) by Lemma 2.4. Thus, gRf ∈ Mα+1 (n) · Kα+1 (n) ⊂ Kα+1 (n) by the definition of the multiplier space. Finally, we obtain R(f g) ∈ Kα+1 (n) and f g ∈ Kα (n), as required.

4. General sufficient conditions The results of this section are motivated by [12, Theorem 1], where the families Mα (1), 0 < α < 1, are investigated. If n = 1, then the corresponding sufficient conditions are known to be quite close to related necessary conditions (see [12] for further details). Theorem 4.1. Let n ∈ N. Assume that j ∈ {1, . . . , n}, α > n − j, g ∈ Hol(Bn ) ∩ LipC ε (Bn )

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for some ε ∈ (0, 1/2) and Z

1

Z

Iα (g, j, k) = sup ζ∈∂Bn

0

∂Bn

|Rk g(rξ)|(1 − r)α+j−n−1 dσn (ξ) dr < ∞ |1 − rhξ, ζi|α+j−k

for k = 1, . . . , j. Then g ∈ Mα (n). Proof. First, we prove the following claim. Claim. For ζ ∈ ∂Bn , put hζ (z) =

g(z) − g(ζ) , (1 − hz, ζi)α

z ∈ Bn .

(4.1)

Then khζ kKα (n) ≤ C,

(4.2)

where the constant C > 0 does not depend on ζ ∈ ∂Bn . Proof of the claim. The definition of hζ (z) implies that Rj hζ (z), z ∈ Bn , is a linear combination of the following terms: 1 k j−k Qk (ζ, z) = R g(z) · R , z ∈ Bn , k = 1, . . . , j, (1 − hz, ζi)α 1 j , z ∈ Bn . Q0 (ζ, z) = (g(z) − g(ζ)) · R (1 − hz, ζi)α Below we estimate the integrals Z 1Z |Qk (ζ, rξ)|(1 − r)(α+j−n)−1 dσn (ξ) dr, 0

k = 0, . . . , j.

∂Bn

1. By the definition of Qj (ζ, z), z ∈ Bn , we have Z 1Z |Qj (ζ, rξ)|(1 − r)(α+j−n)−1 dσn (ξ) dr 0

∂Bn 1Z

|Rj g(rξ)|(1 − r)α+j−n−1 dσn (ξ) dr |1 − rhξ, ζi|α 0 ∂Bn ≤ Iα (g, j, j). Z

≤

2. Note that R

1 (1 − hz, ζi)α

=

−α α + . (1 − hz, ζi)α (1 − hz, ζi)α+1

Hence, 1 R (1 − hz, ζi)α is a linear combination of the fractions 1 , m = 0, . . . , j. (1 − hz, ζi)α+m j

(4.3)

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To estimate the integral (4.3) with k = 0, it suffices to consider the case m = j, since |1 − hz, ζi| ≤ 2. Recall that g ∈ LipC ε (Bn ). Assume, without loss of generality, that α+j −ε 6= n. Then the definition of the space LipC ε (Bn ) and Lemma 2.1 imply the following chain of inequalities: Z 1Z |g(rξ) − g(ζ)| dσn (ξ)(1 − r)α+j−n−1 dr α+j |1 − rhξ, ζi| 0 ∂Bn Z 1Z C ≤ dσn (ξ)(1 − r)α+j−n−1 dr |1 − rhξ, ζi|α+j−ε 0 ∂Bn Z 1 Z 1 (1 − r)α+j−n−1 dr, (1 − r)α+j−n−1−α−j+ε+n dr ≤ C max 0

0

≤C because α + j − n − 1 > −1 and −1 + ε > −1. So, we have Z 1Z |Q0 (ζ, rξ)|(1 − r)(α+j−n)−1 dσn (ξ) dr ≤ C, 0

∂Bn

where the constant C > 0 does not depend on ζ ∈ ∂Bn . 3. If j ≥ 2, then we have to consider the integrals (4.3) with k = 1, . . . , j − 1. The derivative 1 Rj−k (1 − hz, ζi)α is a linear combination of the fractions 1 , (1 − hz, ζi)α+m

m = 0, . . . , j − k.

As above, to estimate the integral (4.3), it suffices to consider the case m = j − k. Now, note that Z 1Z |Rk g(rξ)|(1 − r)α+j−n−1 dσn (ξ) dr ≤ Iα (g, j, k). |1 − rhξ, ζi|α+j−k 0 ∂Bn So, we have Z 1Z 0

|Qk (ζ, rξ)|(1 − r)(α+j−n)−1 dσn (ξ) dr ≤ C,

k = 1, . . . , j − 1,

∂Bn

where the constant C > 0 does not depend on ζ ∈ ∂Bn . So, all integrals (4.3) are bounded by a universal constant C > 0. Thus, Z 1Z |Rj hζ (rξ)|(1 − r)(α+j−n)−1 dσn (ξ) dr ≤ C, 0

∂Bn

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where C > 0 does not depend on ζ ∈ ∂Bn . Also, we have  X X ∂ m hζ  ∂z m (0) ≤ C kgkH ∞ (Bn ) +

|m|≤j−1

|m|≤j−1

Therefore, we obtain khζ kBj

 m ∂ g  ∂z m (0) .

≤ C. Finally, since α > n − j, Lemma 2.5 guar-

α+j−n

antees that (4.2) holds.

Now, we return to the proof of the theorem. Let ζ ∈ ∂Bn . We have g(z) g(ζ) = + hζ (z), (1 − hz, ζi)α (1 − hz, ζi)α

z ∈ Bn ,

where hζ (z) is defined by (4.1). Note that

g(ζ)

= |g(ζ)| ≤ kgkH ∞ (Bn ) .

(1 − hz, ζi)α Kα (n)

Therefore, applying the claim, we obtain

g(z)

(1 − hz, ζi)α

≤ C,

Kα (n)

where the constant C > 0 does not depend on ζ ∈ ∂Bn . Finally, Lemma 1.1 guarantees that g ∈ Mα (n). Corollary 4.2. Let n ∈ N. (i) Assume that α > n − 1 and g ∈ Hol(Bn ) ∩ LipC ε (Bn ) for some ε ∈ (0, 1/2). If Iα (g, 1, 1) < +∞, then g ∈ Mα (n). (ii) Assume that j ∈ {2, . . . , n}, α > n − j, g ∈ Hol(Bn ) and Rj−1 g ∈ H ∞ (Bn ). If Iα (g, j, j) < +∞, then g ∈ Mα (n). Proof. Part (i). Theorem 4.1 applies. Part (ii). Lemma 2.7 yields Rk g ∈ H ∞ (Bn ) for k = 1, . . . , j − 1. Also, Lemma 2.7 guarantees that g ∈ LipC ε (Bn ), 0 < ε < 1/2. So, to prove part (ii), it suffices to verify that Iα (g, j, k) < +∞ for k = 1, . . . , j − 1. Let ζ ∈ ∂Bn . By Lemma 2.1, Z 1Z |Rk g(rξ)|(1 − r)α+j−n−1 dσn (ξ) dr |1 − rhξ, ζi|α+j−k 0 ∂Bn Z 1Z kRk gkH ∞ (Bn ) dσn (ξ) ≤ (1 − r)α+j−n−1 dr |1 − rhξ, ζi|α+j−k ∂Bn 0 Z 1 Z 1 1 ≤ C max (1 − r)α+j−n−1 dr, (1 − r)α+j−n−1−α−j+k− 2 +n dr 0

0

≤C because α + j − n − 1 > −1 and k − 3/2 ≥ −1/2. In other words, Iα (g, j, k) < +∞ and Theorem 4.1 is applicable.

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5. Smoothness conditions and Mα (n) 5.1. Holomorphic Lipschitz functions Below we assume that R0 = I. Corollary 5.1. Let n ∈ N. Suppose that j ∈ {1, . . . , n}, n + 1 − j > α > n − j, g ∈ Hol(Bn ) and Rj−1 g ∈ LipR β (Bn )

for some β ∈ (n + 1 − j − α, 1).

Then g ∈ Mα (n). Proof. Note that β > 0. Since Rj−1 g ∈ LipR β (Bn ), we obtain |Rj g(rξ)| ≤

C , (1 − r)1−β

r ∈ [0, 1), ξ ∈ ∂Bn ,

by [17, Theorem 6.4.9]. So, on the one hand, Lemma 2.7 guarantees that g ∈ LipC ε (Bn ) for some ε ∈ (0, 1/2). On the other hand, applying Lemma 2.1, we have Z 1Z |Rj g(rξ)|(1 − r)α+j−n−1 dσn (ξ) dr |1 − rhξ, ζi|α 0 ∂Bn Z 1Z Cdσn (ξ) ≤ (1 − r)α+β+j−n−2 dr |1 − rhξ, ζi|α 0 ∂Bn Z 1 ≤C (1 − r)α+β+j−n−2 dr 0

≤C because n > α and α + β + j − n − 2 > −1. Hence, Iα (g, j, j) < +∞. So, all hypotheses of Corollary 4.2 are satisfied. Finally, g ∈ Mα (n) by Corollary 4.2. Corollary 5.2. Let n ∈ N. Suppose that g ∈ Hol(Bn )∩LipR β (Bn ) for some β ∈ (0, 1). Then g ∈ Mn (n). Proof. Put j = 1 in Corollary 5.1 and apply Lemma 2.6.

5.2. Lipschitz functions and Hardy–Sobolev functions Let j ∈ N. The fractional differential operator of order j is defined as Rj = (R+I)j . Also, we assume that R0 = I. Given 0 < p < ∞, the Hardy–Sobolev space Hjp (Bn ) is defined by the identity Hjp (Bn ) = g ∈ Hol(Bn ) : Rj g ∈ H p (Bn ) . Corollary 5.3. Let n ∈ N and let β ∈ (0, 1). Assume that one of the following properties holds: (i) g ∈ K0 (Bn ) ∩ LipR β (Bn ); 1 (ii) g ∈ Hn (Bn ) ∩ LipR β (Bn ); (iii) g ∈ Hnp (Bn ) for some p > 1. Then g ∈ Mα (n) for all α > 0.

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Proof. Assume that (i) holds. By Corollary 5.2, we obtain g ∈ Mn (n). Hence, g ∈ Mα (n) for all α > 0 by Proposition 3.1. Next, assume that Rn g ∈ H 1 (Bn ). By Lemma 2.2, we have Rn g ∈ H 1 (Bn ) ⊂ Kn (n). Therefore, repeated application of Lemma 2.4 guarantees that g ∈ K0 (n). So, (ii) implies (i). Now, assume that (iii) holds. By induction, Lemma 2.2 guarantees that Rk g ∈ H pk (Bn ) for some pk > n/k, where k = n, . . . , 1. Also, we may assume, without loss of generality, that pk ≥ pk+1 for k = 1, . . . , n − 1 when n ≥ 2. So, we obtain Rg ∈ H p1 (Bn ) with p1 > n. Now, [17, Theorem 7.2.5] yields |Rg(rξ)| ≤

C , (1 − r)n/p1

r ∈ [0, 1), ξ ∈ ∂Bn .

Since n/p1 < 1, Lemma 2.7 guarantees that g ∈ LipC ε (Bn ), where 0 < ε < min{1/2, 1−n/p1 }. So, (iii) implies (ii). The proof of the corollary is complete. 5.3. Comments R 5.3.1. It is well-known that Hn1 (Bn ) 6⊂ LipR β (Bn ) and Hol(Bn ) ∩ Lipβ (Bn ) 6⊂ Hn1 (Bn ) for any β ∈ (0, 1) (cf. property (ii) from Corollary 5.3). 5.3.2. Assume that n − 1 < α < n. Then Corollary 5.1 with j = 1 guarantees that g ∈ Mα (n) if g ∈ LipR β (Bn ) for some β ∈ (n − α, 1). For n = 1, this result was obtained in [6, Corollary 4]. The nature of Corollary 5.3 is different. Namely, the additional assumption g ∈ K0 (n) guarantees that g ∈ Mα (n) for all α > 0 if g ∈ LipR β (Bn ) for an arbitrarily small β ∈ (0, 1). 5.3.3. It would be interesting to know whether g ∈ Hn1 (Bn ) implies that g ∈ Mα (n) for all α > 0 when n ≥ 2 (cf. properties (ii) and (iii) from Corollary 5.3). Note that the above implication is known to be true when n = 1 (see [19] and [9]). 5.3.4. If q < 1, then Hnq (Bn ) 6⊂ Mα (n) for all α > 0 (cf. property (iii) from Corollary 5.3). Indeed, by Lemmas 2.3 and 2.4, we have Rn K0 (n) ⊂ Kn (n) ⊂ H p (Bn ) for all p ∈ (0, 1). Hence, K0 (n) ⊂ Hnq (Bn ). It remains to observe that Mα (n) ⊂ H ∞ (Bn ) for all α > 0 (see Section 1.5.2) but K0 (n) 6⊂ H ∞ (Bn ); thus, K0 (n) 6⊂ Mα (n) and Hnq (Bn ) 6⊂ Mα (n) for all α > 0. 5.4. Hardy–Sobolev functions Corollary 5.4. Let n ∈ N. Suppose that j ∈ {1, . . . , n}, α > n − j and g ∈ Hjp (Bn ) for some p > n/j. Then g ∈ Mα (n). Proof. By hypothesis, Rj g ∈ H pj (Bn ) for some pj > n/j. By induction, Lemma 2.2 guarantees that Rk g ∈ H pk (Bn ) for some pk > n/k, where k = j, . . . , 1. Also, we may assume, without loss of generality, that pk ≥ pk+1 for k = 1, . . . , j − 1 when j ≥ 2. So, we obtain Rk g ∈ H pk (Bn ) for some pk > n/k, where k = 1, . . . , j.

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1. We have Rg ∈ H p (Bn ) for some p = p1 > n. So, [17, Theorem 7.2.5] yields |Rg(rξ)| ≤

C , (1 − r)n/p

r ∈ [0, 1), ξ ∈ ∂Bn .

Since n/p < 1, Lemma 2.7 guarantees that g ∈ LipC ε (Bn ), where 0 < ε < min{1/2, 1 − n/p}. −1 2. Let k ∈ {1, . . . , j}. Given pk ∈ (n/k, +∞), assume that p−1 k +qk = 1. Then k pk k 1 < qk < n/(n − k). If R g ∈ H (Bn ) for some pk > n/k, then R g ∈ H pek (Bn ) for all pek ∈ (n/k, pk ). Therefore, without loss of generality, we may assume that (α + j − k)qk 6= n. Let ζ ∈ ∂Bn . Put 1/qk Z dσn (ξ) . I(r, α, qk ) = (α+j−k)qk ∂Bn |1 − rhξ, ζi| Lemma 2.1 guarantees that n o n −α−j+k I(r, α, qk ) ≤ C max 1, (1 − r) qk . Indeed, if (α + j − k)qk < n, then I(r, α, qk ) ≤ C; if (α + j − k)qk > n, then n −α−j+k . I(r, α, qk ) ≤ C(1 − r) qk Thus, applying H¨ older’s inequality, we obtain Z 1 Z |Rk g(rξ)| dσn (ξ) dr (1 − r)α+j−n−1 α+j−k 0 ∂Bn |1 − rhξ, ζi| Z 1 ≤ (1 − r)α+j−n−1 kRk gkH pk (Bn ) I(r, α, qk ) dr 0 Z 1 Z 1 α+j−n−1+ qn −α−j+k α+j−n−1 k dr ≤ C max (1 − r) dr, (1 − r) 0

0

≤C because α + j − n − 1 > −1 and qnk + k − n − 1 > −1, respectively. In other words, Iα (g, j, k) < +∞ for k = 1, . . . , j. To finish the proof of the corollary, it remains to apply Theorem 4.1. 5.5. Comments 5.5.1. If n = 1 or j = n, then Corollary 5.4 is a particular case of Corollary 5.3. So, assume that n ≥ 2. Then, by Lemma 2.2, the assumption g ∈ H1p (Bn ) for some p > n is the weakest among the following properties: p

g ∈ Hj j (Bn )

for some pj > n/j, j = 1, . . . , n.

5.5.2. Corollary 5.4 does not imply Corollary 5.1 and vice versa.

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6. Further results 6.1. Radial variations Theorem 6.1. Let n ∈ N. Suppose that j ∈ {1, . . . , n}, n − j + 1 > α > n − j, g ∈ Hol(Bn ) and Z 1 |Rj g(rξ)|r−1 (1 − r)α+j−n−1 dr < +∞. Jα (g, j) = sup ξ∈∂Bn

0

Then g ∈ Mα (n). Proof. Fix ζ ∈ ∂Bn . Assume that ξ ∈ ∂Bn and ζ 6= ξ. For r ∈ [0, 1], put Z r u(r) = |Rj g(ρξ)|(1 − ρ)α+j−n−1 dρ and 0

v(r) = |1 − rhξ, ζi|−α . On the one hand, we have u0 (r) = |Rj g(rξ)|(1 − r)α+j−n−1 . On the other hand, v(r) = |f (r)|, where the function f (λ) =

1 (1 − λhξ, ζi)α

is holomorphic in a neighborhood of B1 ∪ [0, 1] ⊂ C. Since f (r) 6= 0, we obtain ∂|f (λ)| C 0 (r) ≤ |f 0 (r)| ≤ . |v (r)| = ∂r |1 − rhξ, ζi|α+1 So, integrating by parts, we have Z 1 j |R g(rξ)|(1 − r)α+j−n−1 dr |1 − rhξ, ζi|α 0 R1 j Z 1 Rr j |R g(ρξ)|(1 − ρ)α+j−n−1 dρ |R g(ρξ)|(1 − ρ)α+j−n−1 dρ 0 0 ≤ + dr |1 − hξ, ζi|α |1 − rhξ, ζi|α+1 0 Z 1 Jα (g, j) dr + Jα (g, j) . ≤ α α+1 |1 − hξ, ζi| 0 |1 − rhξ, ζi| We consider integrals of non-negative functions, thus, Fubini’s theorem and Lemma 2.1 guarantee that Z Z 1 j |R g(rξ)|(1 − r)α+j−n−1 dr dσn (ξ) |1 − rhξ, ζi|α ∂Bn 0 Z Z 1Z dσn (ξ) dσn (ξ) ≤ Jα (g, j) + J (g, j) dr α α |1 − hξ, ζi| |1 − rhξ, ζi|α+1 ∂Bn 0 ∂Bn Z 1 n−α−1 ≤ CJα (g, j) 1 + max 1, (1 − r) dr 0

≤ CJα (g, j)

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because n > α. Since the above constant C > 0 does not depend on ζ ∈ ∂Bn , applying Fubini’s theorem, we obtain Iα (g, j, j) < +∞.

(6.1)

Now, put h(z) = Rj−1 g(z), z ∈ Bn . Also, for ζ ∈ ∂Bn , put hζ (w) = h(wζ), w ∈ B1 . Recall that hζ is said to be a slice function. Observe that Rj g(rλζ) = Rh(rλζ) = rλh0ζ (rλ) for r ∈ [0, 1) and λ ∈ ∂B1 . Therefore, Z 1 Jα (hζ ) = sup |h0ζ (rλ)|(1 − r)α+j−n−1 dr ≤ Jα (g, j) < +∞. (6.2) λ∈∂B1

0

By [10, Theorem 6.13], for every ζ ∈ ∂Bn , the function hζ extends continuously to B1 and hζ satisfies the Lipschitz condition hζ (ei(θ+t) ) − hζ (eiθ ) ≤ C(α)Jα (hζ )tn+1−α−j ≤ C(α)Jα (g, j)tn+1−α−j for h > 0 and θ ∈ R. Also, the hypothesis Jα (g, j) < +∞ implies Rj−1 g ∈ H ∞ (Bn ) because n + 1 − α − j > 0. If j 6= 1, then, applying Corollary 4.2, we complete the proof of the theorem. If j = 1, then g = h ∈ H ∞ (Bn ) and (6.2) holds with g in place of h. Hence, [17, Theorem 6.4.9] guarantees that C |Rg(rξ)| ≤ , r ∈ [0, 1), ξ ∈ ∂Bn . (1 − r)α+j−n Therefore, by Lemma 2.7, we have g ∈ LipC ε (Bn ) for 0 < ε < min{1/2, n+1−α−j}. By (6.1), all hypotheses of Corollary 4.2 are fulfilled. So, to finish the proof, it remains to apply Corollary 4.2. 6.2. Comments 6.2.1. For n = 1, Theorem 6.1 was obtained in [12, Theorem 2]. 6.2.2. Assume that n ∈ N, j ∈ {1, . . . , n}, n − j + 1 > α > n − j and g ∈ Hol(Bn ). Observe that Jα (g, j) < +∞ if and only if Z 1 sup |Rj g(rξ)|(1 − r)α+j−n−1 dr < +∞. ξ∈∂Bn

0

6.2.3. Theorem 6.1 implies Corollary 5.1. Indeed, suppose that n ∈ N, j ∈ {1, . . . , n}, n − j + 1 > α > n − j, g ∈ Hol(Bn ) and Rj−1 g ∈ LipR β (Bn )

for some β ∈ (n + 1 − j − α, 1).

Then β > 0 and |Rj g(rξ)| ≤

C , (1 − r)1−β

r ∈ [0, 1), ξ ∈ ∂Bn ,

by [17, Theorem 6.4.9]. Hence, Z 1 Z 1 sup |Rj g(rξ)|(1 − r)α+j−n−1 dr ≤ C (1 − r)α+j+β−n−2 dr < +∞ ξ∈∂Bn

0

0

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because α + j + β − n − 2 > −1. 6.2.4. Let α > 0. It is proved in [5] that g ∈ Mα (n) implies Z 1 sup |Rg(rξ)| dr < +∞. ξ∈∂Bn

0

On the other hand, Section 6.2.2 and Lemma 2.6 guarantee that g ∈ Mn (n) if Z 1 sup |Rg(rξ)|(1 − r)−β dr < +∞ ξ∈∂Bn

0

for some β > 0. 6.3. Slice functions The last result concerns the second differences of slice functions. Given f : ∂B1 → C and t ∈ R, let D(f, t) = f (eit ) − 2f (1) + f (e−it ). Corollary 6.2. Let n ∈ N. Suppose that j ∈ {1, . . . , n}, n − j + 1 > α > n − j, g ∈ Hol(Bn ) and h = Rj−1 g ∈ H ∞ (Bn ). For ξ ∈ ∂Bn , put hξ (w) = h(wξ), w ∈ B 1 . If Z π |D(hξ , t)| Dα (h) = sup dt < +∞, n+2−α−j ξ∈∂Bn 0 t then g ∈ Mα (n). Proof. Fix ξ ∈ ∂Bn . Note that hξ ∈ H ∞ (B1 ). Hence, as shown in the proof of [12, Theorem 3], we have Z 1 Z π |D(hξ , t)| |h0ξ (r)|(1 − r)α+j−n−1 dr ≤ C dt. n+2−α−j 0 0 t Since |Rj g(rξ)| = |Rh(rξ)| = |rh0ξ (r)|, we obtain Z Jα (g, j) = sup ξ∈∂Bn

1

|Rj g(rξ)|r−1 (1 − r)α+j−n−1 dr ≤ CDα (h) < +∞.

0

Finally, g ∈ Mα (n) by Theorem 6.1.

6.4. Comments 6.4.1. For n = 1, Corollary 6.2 was obtained in [12, Theorem 3]; see also [20, 6]. 6.4.2. Corollary 6.2 implies Corollary 5.1.

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References [1] A. B. Aleksandrov, Function theory in the ball, Current problems in mathematics. Fundamental directions, vol. 8, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 115–190, 274 (Russian); English transl.: G. M. Khenkin and A. G. Vitushkin (Eds.), Several Complex Variables II, Encyclopaedia Math. Sci., vol. 8, Springer–Verlag, Berlin, 1994, pp. 107–178. [2] F. Beatrous and J. Burbea, Holomorphic Sobolev spaces on the ball, Dissertationes Math. (Rozprawy Mat.) 276 (1989), 60pp. [3] F. Beatrous and J. Burbea, On multipliers for Hardy–Sobolev spaces, Proc. Amer. Math. Soc. 136 (2008), no. 6, 2125–2133. [4] J. A. Cima, A. L. Matheson, and W. T. Ross, The Cauchy transform, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, 2006. [5] E. S. Dubtsov, Families of fractional Cauchy transforms in the ball, Algebra i Analiz, to appear. [6] D. J. Hallenbeck, T. H. MacGregor, and K. Samotij, Fractional Cauchy transforms, inner functions and multipliers, Proc. London Math. Soc. (3) 72 (1996), no. 1, 157– 187. [7] V. P. Havin, On analytic functions representable by an integral of Cauchy–Stieltjes type, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13 (1958), no. 1, 66–79 (Russian). [8] V. P. Havin, Relations between certain classes of functions regular in the unit circle, Vestnik Leningrad. Univ. 17 (1962), no. 1, 102–110 (Russian). [9] R. A. Hibschweiler and T. H. MacGregor, Multipliers of families of Cauchy–Stieltjes transforms, Trans. Amer. Math. Soc. 331 (1992), no. 1, 377–394. [10] R. A. Hibschweiler and T. H. MacGregor, Fractional Cauchy transforms, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 136, Chapman & Hall/CRC, Boca Raton, FL, 2006. [11] S. V. Hruˇsˇcev and S. A. Vinogradov, Inner functions and multipliers of Cauchy type integrals, Ark. Mat. 19 (1981), no. 1, 23–42. [12] D. Luo and T. MacGregor, Multipliers of fractional Cauchy transforms and smoothness conditions, Canad. J. Math. 50 (1998), no. 3, 595–604. [13] T. H. MacGregor, Analytic and univalent functions with integral representations involving complex measures, Indiana Univ. Math. J. 36 (1987), no. 1, 109–130. [14] J. M. Ortega and J. F` abrega, Pointwise multipliers and decomposition theorems in analytic Besov spaces, Math. Z. 235 (2000), no. 1, 53–81. [15] J. M. Ortega and J. F` abrega, Pointwise multipliers and decomposition theorems in Fs∞,q , Math. Ann. 329 (2004), no. 2, 247–277. [16] J. M. Ortega and J. F` abrega, Multipliers in Hardy–Sobolev spaces, Integral Equations Operator Theory 55 (2006), no. 4, 535–560. [17] W. Rudin, Function theory in the unit ball of Cn , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York, 1980. [18] P. Ryan and M. Stoll, Hardy–Sobolev spaces and algebras of holomorphic functions on the unit ball in Cn , Complex Var. Elliptic Equ. 53 (2008), no. 6, 565–584.

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[19] S. A. Vinogradov, Properties of multipliers of integrals of Cauchy–Stieltjes type, and some problems of factorization of analytic functions, Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974), Theory of ` functions and functional analysis (Russian), Central Ekonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 5–39; English transl.: Amer. Math. Soc. Transl. (2) 115 (1980), 1–32. [20] S. A. Vinogradov, M. G. Goluzina, and V. P. Havin, Multipliers and divisors of Cauchy–Stieltjes type integrals, Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 19 (1970), 55–78 (Russian); English transl.: Seminars in Math., V. A. Steklov Math. Inst., Leningrad 19 (1972), 29–42. [21] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005. Evgueni Doubtsov St. Petersburg Department of V.A. Steklov Mathematical Institute Fontanka 27 St. Petersburg 191023 Russia e-mail: [email protected] Submitted: September 1, 2008.

Integr. equ. oper. theory 64 (2009), 193–201 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020193-9, published online May 12, 2009 DOI 10.1007/s00020-009-1684-z

Integral Equations and Operator Theory

Spectral Radius Algebras of Idempotents John David Herron Abstract. We show that the spectral radius algebras of certain quadratic operators possess nontrivial invariant subspaces. Additionally, such algebras properly contain the operator’s commutant, so that the invariant subspaces are in some sense beyond hyperinvariant. The spectral radius algebras of idempotents are completely described and, as a consequence, it is shown that every intransitive collection of operators must be contained in a norm-closed proper spectral radius algebra. Mathematics Subject Classification (2000). Primary 47A15. Keywords. Spectral radius algebra, idempotent, invariant subspace.

1. Introduction Let H be a separable infinite-dimensional complex Hilbert space, and denote by L(H) the algebra of bounded linear operators on H. By a subspace of H we mean a closed linear manifold in H, and a subspace M is said to be invariant under an operator T if T M ⊆ M . If A is a subset of L(H), we say that M is invariant under A if it is invariant under every operator in A. A nontrivial invariant subspace is one that is not equal to H nor {0}, and the lattice of all invariant subspaces of a collection A is denoted Lat A. The invariant subspace problem is the question as to whether every bounded linear operator on H has a nontrivial invariant subspace. Among the central results concerning this open problem is a 1973 result by V.I. Lomonosov: 0

Theorem 1.1. [6] If K ∈ L(H) is a nonzero compact operator, then {K} has a nontrivial invariant subspace. In other words, every operator commuting with a nonzero compact operator must have a nontrivial invariant subspace. More recently, a new class of operator algebra was introduced in [5] that allowed for a generalization of Lomonosov’s result. In [5] Alan Lambert and Srdjan Petrovic define the spectral algebra (or

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spectral radius algebra as it is called in [2]) of a bounded linear operator T with spectral radius r(T ) to be the unital subalgebra

−1 <∞ , BT = S ∈ L(H) : sup Rm SRm m

where, for each positive integer m, Rm = Rm (T ) =

∞ X

!1/2 ∗ n n d2n m (T ) T

n=0

and

1 . 1/m + r(T ) The study of such algebras is motivated by the fact that the spectral radius al0 gebra BT of an operator T contains the operator’s commutant {T } . Consequently, every invariant subspace of BT is a hyperinvariant subspace of T (i.e., invariant 0 0 under {T } ); and if {T } is properly contained in BT , such a subspace would be, as the title of [5] suggests, “beyond hyperinvariant”. Among the principal results in this line of inquiry is the following: dm = dm (T ) =

Theorem 1.2. [5] If K ∈ L(H) is a nonzero compact operator, then BK has a 0 nontrivial invariant subspace. Moreover, BK 6= {K} whenever r(K) > 0. Characterizations of spectral radius algebras of normal operators can be found in [2]: Theorem 1.3. [2] Let N ∈ L(H) be a nonzero normal operator and let A = N/ kN k. 1. If A is unitary, then BN = L(H). 2. If A is completely nonunitary, then BN is a weakly dense proper subalgebra of L(H). 3. If A is neither unitary nor completely nonunitary, then BN has a nontrivial invariant subspace. In this paper we add to the list of these types of results the following: Theorem 1.4. Let T ∈ L(H) be a nonzero operator for which T 2 = αT , where 0 T 6= αI. Then, BT has a nontrivial invariant subspace and BT 6= {T } . Operators for which T 2 = αT belong to the class of quadratic operators and include idempotents (α = 1). In the case of idempotent operators, we will also obtain (in Theorem 2.2) a complete description of BT in terms of the invariant subspace guaranteed in Theorem 1.4. Therefore, in addition to the types of compact and normal operators studied in [5] and [2], we shall find that certain types of quadratic operator have subspaces associated with them that are “beyond hyperinvariant”. Our interest in idempotents stems from a 1976 paper by E. Nordgren, H. Radjavi and P. Rosenthal, wherein the following equivalence is proved:

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Theorem 1.5. [8] Every bounded linear operator on H has a nontrivial invariant subspace if and only if every pair of idempotents has a common nontrivial invariant subspace. This equivalence was later broadened to include pairs of quadratic operators in [7]. In 1998, G. Allan and J. Zemanek published several results providing conditions under which two idempotents will have a common nontrivial invariant subspace (see [1]), and further work along these lines can be found in [4] and [9]. Having a complete description of the spectral radius algebras of idempotents allows us to show that every collection of bounded operators leaving a given nontrivial subspace invariant must be contained in the spectral radius algebra of a nontrivial idempotent. In light of Theorem 1.5 this suggests the following question: Is there a pair of idempotents that is not contained in a proper normed-closed spectral radius algebra?

2. Idempotents A bounded linear operator T is said to be quadratic if there are complex numbers α and β such that (T − αI)(T − βI) = 0. The class of quadratic operators includes involutions (where α = −1 and β = 1), nilpotent operators of order 2 (where α = β = 0) and idempotents (where α = 1 and β = 0). The spectral radius algebras of involutions can be determined by an immediate application of a result in [5]: BT = L(H) whenever T n = λI for any positive integer n and nonzero complex number λ. In the case when T is an involution, n = 2 and λ = 1. 2 Suppose T is nilpotent. Let u be any nonzero vector in ker T . Then, Rm u = u; −1 u. Let v be any nonzero vector in H and consider the that is, Rm u = Rm rank-one operator

−1 u

⊗ v. It is shown

−1 in [5] that u ⊗ v ∈ BT if and only if

≤ 1, we have supm kRm uk Rm v < ∞. Since Rm

−1

−1 2

kuk kvk < ∞. sup kRm uk Rm v ≤ sup Rm m

m

Lastly, T ∈ QT , where n o

−1 QT = S ∈ BT : lim Rm SRm =0 . m−→∞

In [5] it is shown that, for any T , QT is a two-sided ideal consisting entirely of quasinilpotent operators, and if QT 6= {0} and BT contains a compact operator, then BT has a nontrivial invariant subspace. Since T need not commute with u ⊗ v (take v 6∈ (ran T )⊥ ), we have that if T is nilpotent, then BT has a nontrivial 0 invariant subspace and BT 6= {T } .

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We begin our study of idempotents by considering the slightly more general case when {α, β} = 6 {0}. We will show that in this case the spectral radius algebra of the corresponding quadratic operator leaves the operator’s nullspace invariant. Lemma 2.1. Let T ∈ L(H) be a quadratic operator and let α and β be complex numbers such that (T − αI)(T − βI) = 0. If at least one of the α or β is nonzero, then ker T is an invariant subspace of BT . Proof. Let T ∈ L(H) be a quadratic operator and let α and β be scalars defined as above. Suppose α 6= 0 and β 6= 0. Let x ∈ ker T . Then, 0 = T 2 x = (α + β)T x − αβx = −αβx. Thus, ker T = {0}, and the statement holds trivially in this case. We consider, then, the case when α 6= 0 and β = 0. Then, T 2 = αT . Moreover, by induction it can be shown that T n = αn−1 T for any n ∈ N. By the spectral radius formula, r(T ) = |α|, and we have dm =

m 1 = , 1/m + r(T ) 1 + |α| m

for each positive integer m. From this we have 2 Rm

=

∞ X n=0

∗ n n d2n m (T ) T

=I+

∞ X

! d2n m

2(n−1)

|α|

T ∗T = I +

n=1

m2 T ∗ T. 1 + 2 |α| m

2 −1 Let S ∈ BT and let x ∈ ker T . Since Rm x = x, it follows that Rm x = x = Rm x. Therefore,

−1 −1 −1 2

Rm SRm x, Rm SRm x x = Rm SRm

2 = Rm Sx, Sx m2 ∗ = I+ T T Sx, Sx 1 + 2 |α| m m2 = hSx, Sxi + hT Sx, T Sxi 1 + 2 |α| m m2 2 2 = kSxk + kT Sxk . 1 + 2 |α| m

−1 Since S ∈ BT , supm Rm SRm < ∞. Therefore,

m2 2 2 −1 2 −1 2 kxk < ∞ kT Sxk ≤ Rm SRm x ≤ sup Rm SRm 1 + 2 |α| m m from which we conclude that T Sx = 0; that is, Sx ∈ ker T .

In addition to obtaining facts about the spectral radius algebras of more general types of quadratic operators, we will specifically be interested in describing completely the spectral radius algebras of idempotents. To this end, we shall need the following result from [5]. Recall that an operator B is power-bounded if supn kB n k < ∞.

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Theorem 2.1. [5] If T ∈ L(H) is nonzero, B is a power-bounded operator in {T } , and S ∈ L(H) such that T S = BST , then S ∈ BT . Lemma 2.1, together with our earlier discussion of nilpotent operators, guarantees the existence of a nontrivial invariant subspace for BT whenever T 2 = αT . Our next result states that this invariant subspace is invariant in the sense of 0 [5]; that is, {T } ⊂ BT ⊂ L(H), where the inclusions are proper, and Lat BT 6= {{0} , H}. Lemma 2.2. Let T ∈ L(H) such that T 2 = αT for some nonzero α ∈ C. Then, 0 ST ∈ BT for all S ∈ L(H). Moreover, if T 6= αI, then BT 6= {T } . Proof. If T is a scalar multiple of the identity, then its spectral radius algebra is all of L(H). Therefore, suppose T 2 = αT with T 6= αI and α 6= 0. Let B = α−1 T . −1 Note that for any n ∈ N, kB n k = |α| kT k, so that B is a power-bounded 0 operator in {T } . Let S ∈ L(H) and consider the operator A = ST . Then, BAT = (α−1 T )(ST )T = T A, and so ST ∈ BT by Theorem 2.1. Since T 2 = αT , it follows that ker T is nontrivial if and only if T 6= αI. So, let u be a nonzero vector in ker T and let v be a nonzero vector in ran T . Let S be the rank-one operator u ⊗ v. We have already shown that ST ∈ BT . We now 0 show that ST is not in {T } . Let x ∈ H. Then, (ST )T x = (u ⊗ v)T 2 x = α hT x, vi u. Clearly, this is not the zero operator. However, T (ST )x = T (u ⊗ v)T x = hT x, vi T u = 0 for all x. Hence, 0 ST 6∈ {T } . The proof of Theorem 1.4 now proceeds as follows: Let T 2 = αT , where T is not a scalar multiple of the identity. If α = 0, then T is nilpotent of order 2 and the theorem follows from the application of various results in [5] as outlined at the start of this section. If α 6= 0, then Lemma 2.1 guarantees that ker T is a nontrivial invariant subspace of BT . Furthermore, Lemma 2.2 states that the 0 commutant {T } is a proper subalgebra of BT . As noted in [5] explicit descriptions of the operators in a given spectral radius algebra are generally difficult to obtain. In the case of a bounded idempotent, however, we have a simple description of the spectral radius algebra in terms of the operator’s nullspace. Theorem 2.2. Let T ∈ L(H) be an idempotent. Then, BT = {S ∈ L(H) : ker T ∈ Lat S} . Proof. Let T ∈ L(H) with T 2 = T . If T ∈ {0, I}, then BT = L(H) and the nullspace of T is a trivial subspace, invariant under every operator in L(H). Assume, then, that T is a nontrivial idempotent. Let S ∈ L(H) such that ker T is an invariant subspace of S. Let x ∈ H. Then, T Sx = T S((x − T x) + T x) = T S(x − T x) + T ST x = T ST x. Therefore, T S = T ST , and since T is a power 0 bounded element of {T } , we have S ∈ BT . If S ∈ BT , then it follows from Lemma 2.1 that ker T ∈ Lat S.

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The following are immediate consequences of Theorem 2.2. Corollary 2.1. Let T ∈ L(H) be an idempotent. 1. BT = L(H) if and only if T ∈ {0, I}. 2. BT is closed in the norm topology on L(H). Proof. Statement (1) follows from the fact that if T 2 = T , then ker T = {{0} , H} if and only if T ∈ {0, I}. Statement (2) is a consequence of the fact that in the norm topology, if {Sn } is a sequence of bounded operators such that Sn leaves invariant a given subspace M for each n, then the norm-limit of {Sn } (if one exists) must likewise leave M invariant. A collection A of operators is said to be transitive if Lat A = {{0} , H}. If A has a nontrivial invariant subspace, A is said to be intransitive. Theorem 2.3. Every intransitive collection of bounded operators is contained within a norm-closed proper spectral radius algebra. Proof. Let A ⊂ L(H) with M ∈ Lat A for some nontrivial subspace M of H. Let P be the orthogonal projection of H onto M ⊥ . Then, P is a nontrivial idempotent. Moreover, if S ∈ A, then ker P = (ran P )⊥ = M ∈ Lat S, which implies that S ∈ BP . In [1] it is shown that every pair of orthogonal projections (self-adjoint idempotents) has a common nontrivial invariant subspace. By Theorem 2.3 this implies that every such pair is in the proper norm-closed spectral radius algebra of some idempotent. Of course, were this to hold for pairs of more general quadratic operators, the invariant subspace problem would be answered affirmatively by a result in [7], which states that every bounded operator on H has a nontrivial invariant subspace if and only if every pair of quadratic operators has a common nontrivial invariant subspace (a generalization of Theorem 1.5). This leads to the following question: Question 2.1. Is there a pair of quadratic operators that is not contained in any proper norm-closed spectral radius algebra? It is interesting that the question has a negative answer in the case of three quadratic operators, even when the operators in question are orthogonal projections [3]. It is easy to show, however, that every quadratic operator must be in the spectral radius algebra of some nontrivial idempotent. Corollary 2.2. Every quadratic operator on H is in the spectral radius algebra of some nontrivial idempotent. Proof. Let T ∈ L(H) be a quadratic operator. If T is a scalar multiple of the identity, then T is in the spectral radius algebra of every bounded operator (since such algebras are unital). Therefore, suppose T is not a multiple of the identity. Let α, β ∈ C such that (T −αI)(T −βI) = 0. If α 6= β, then S = (α −β)−1 (T −βI) 0 is an idempotent and T ∈ {S} ⊂ BS .

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If α = β, then one can show that α is an eigenvalue of T . Let M = ker(T −αI) and let P be the orthogonal projection of H onto M ⊥ . Then, by Theorem 2.2, T ∈ BP .

3. Orthogonal Projections and Reducing Subspaces An idempotent P ∈ L(H) is an orthogonal projection if and only if it is selfadjoint; and this, of course, is equivalent to ker P = (ran P )⊥ . Therefore, if P is the orthogonal projection of H onto M = ran P , then BP = S ∈ L(H) : M ⊥ ∈ Lat S . As to be expected, the conditions P ∗ = P = P 2 allow for a relatively easy calcu−1 lation of the operators Rm and Rm . For each positive integer m, 2 Rm

=

∞ X

∗ n n d2n m (P ) P = I +

n=0

m2 P. 2m + 1

A calculation shows that Rm = Q + γm P where Q = I − P (the orthogonal projection onto ker P ) and 1/2 m2 . γm = 1 + 2m + 1 Furthermore, it can be verified that −1 −1 Rm = Q + γm P.

Recall that M is said to be a reducing subspace for an operator S ∈ L(H) (or M reduces S) if M and its orthogonal complement M ⊥ are invariant subspaces of S. It is well-known that M reduces S if and only if P S = SP , where P is the orthogonal projection of H onto M . If we consider the direct sum decomposition H = M ⊕ M ⊥ , then S ∈ L(H) has a 2 × 2 matrix representation W X S= , Y Z where W = (P S)|M , X = (P S)|M ⊥ , Y = [(I − P )S]|M and Z = [(I − P )S]|M ⊥ . The subspace M then reduces T if and only if X = 0 and Y = 0 between the appropriate spaces. Moreover, S ∈ BP if and only if X = 0. Theorem 3.1. Let M be a subspace of H, P the orthogonal projection of H onto M , and S ∈ L(H). The following statements are equivalent: −1 1. Rm SRm = S for some m ∈ Z+ ; 2. M reduces S; −1 3. Rm SRm = S for all m ∈ Z+ , where Rm = Rm (P ).

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−1 Proof. (1) =⇒ (2) Suppose Rm0 SRm = S for some positive integer m0 . Let 0 W X S= Y Z

be the matrix representation of S with respect to the direct sum decomposition −1 H = M ⊕ M ⊥ . Then, Rm0 SRm = S implies 0 W γm0 X W X = −1 γm Y Z Y Z 0 from which we conclude X = 0 and Y = 0, with 0 being defined on the appropriate spaces in each case. Hence, M reduces S. (2) =⇒ (3) If M reduces S, then X and Y are both the zero operator (again, −1 on the appropriate spaces) and we have Rm SRm = S for every m ∈ Z+ . (3) =⇒ (1) Trivial. The following is an example of how, in addition to providing information regarding invariant subspaces, spectral radius algebra techniques can be applied to the study of structural features of an operator algebra. Example 3.1. For a subalgebra A of L(H), the Jacobson radical of A is the set R = {S ∈ A : r(SX) = 0 for all X ∈ A}. The algebra A is said to be semisimple if its Jacobson radical contains only the zero operator; it is a radical algebra if A = R. As an illustration of some of our earlier ideas, we will use spectral radius algebra techniques to show that the algebra of all operators leaving a specific subspace invariant is never semisimple nor is it a radical algebra. Let M be a nontrivial subspace of H and let Alg(M ) denote the set of all bounded operators leaving M invariant. Then, Alg(M ) = BP , where P is the orthogonal projection of H onto M ⊥ . Recall that o n

−1 =0 QP = S ∈ BP : lim Rm SRm m−→∞

is a two-sided ideal consisting entirely of quasinilpotent operators. Therefore, QP ⊆ R and it is enough to show that QP 6= {0}. Consider the nonzero operator QSP , where Q = I − P and S is any operator for which M 6∈ Lat S. Consider the following:

−1 −1 −1

Rm (QSP )Rm kQSP k −→ 0 P ) = γm = (Q + γm P )(QSP )(Q + γm as m −→ ∞. Therefore, QSP ∈ QP and Alg(M ) cannot be semisimple. On the other hand, it is clear that BP is never a radical algebra, since P itself is not quasinilpotent. Of course in the example above it need not be the case that QP contains every quasinilpotent operator in BP [5]. For instance, if S is a nonzero quasinilpotent operator possessing a nontrivial reducing subspace M , then S ∈ BP , where P is −1 the orthogonal projection of H onto M ⊥ . By Theorem 3.1, Rm SRm = kSk for every m, and so S 6∈ QP .

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References [1] G. Allan; J. Zemanek, Invariant subspaces for pairs of projections. J. London Math. Soc. (2) 57 (1998), 449–468. [2] A. Biswas, A. Lambert, S. Petrovic, On spectral radius algebras and normal operators. Indiana Univ. Math. J. 56, No. 4 (2007), 1661–1674. [3] C. Davis, Generators of the ring of bounded operators. Proc. Amer. Math. Soc. 6 (1955), 970–972. [4] R. Drnovsek, H. Radjavi, P. Rosenthal, A characterization of commutators of idempotents. Linear Algebra and its Applications 347 (2002), 91–99. [5] A. Lambert, S. Petrovic, Beyond hyperinvariance for compact operators. J. Funct. Anal. 219 (2005), 93–108. [6] V.I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator. Funkcional. Anal. i Prilozen. 7 (1973), 55–56. [7] E.A. Nordgren, M. Radjablipour, H. Radjavi, P. Rosenthal, Quadratic operators and invariant subspaces. Studia Math. 88 (1988), 263–268. [8] E.A. Nordgren, H. Radjavi, P. Rosenthal, A geometric equivalent of the invariant subspace problem. Proc. Amer. Math. Soc. 61 (1976), 66–68. [9] H. Radjavi, P. Rosenthal, On commutators of idempotents. Linear and Multilinear Algebra 50 (2002), No. 2, 121–214. John David Herron 224 Harman Hall Department of Biology, Chemistry and Mathematics University of Montevallo Montevallo, Alabama 35115-6000 USA e-mail: [email protected] Submitted: August 29, 2008.

Integr. equ. oper. theory 64 (2009), 203–237 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020203-35, published online May 12, 2009 DOI 10.1007/s00020-009-1685-y

Integral Equations and Operator Theory

Wiener-Hopf Operators with Slowly Oscillating Matrix Symbols on Weighted Lebesgue Spaces Yu.I. Karlovich and J. Loreto Hern´andez Abstract. Fredholm conditions and an index formula are obtained for WienerHopf operators W (a) with slowly oscillating matrix symbols a on weighted Lebesgue spaces LpN (R+ , w) where 1 < p < ∞, w is a Muckenhoupt weight on R and N ∈ N. The entries of matrix symbols belong to a Banach subalgebra of Fourier multipliers on Lp (R, w) that are continuous on R and have, in general, different slowly oscillating asymptotics at ±∞. To define the Banach algebra SOp,w of corresponding slowly oscillating functions, we apply the theory of pseudodifferential and Calder´ on-Zygmund operators. Established sufficient conditions become a Fredholm criterion in the case of Muckenhoupt weights with equal indices of powerlikeness, and also for Muckenhoupt weights with different indices of powerlikeness under some additional condition on p, w and a. Mathematics Subject Classification (2000). Primary 47B35; Secondary 42A45, 47A53, 47G10, 47G30. Keywords. Wiener-Hopf operator, pseudodifferential operator, limit operator, weighted Lebesgue space, slowly oscillating matrix function, local principle, symbol, Fredholmness, index.

1. Introduction Given 1 ≤ p ≤ ∞, let Lp (R) be the usual Lebesgue space with norm denoted by k · kp . A measurable function w : R → [0, ∞] is called a weight if w−1 ({0, ∞}) has Lebesgue measure zero. For 1 ≤ p < ∞ and a weight w, we denote by Lp (R, w) the weighted Lebesgue space with the norm Z 1/p kf kp,w := |f (x)|p wp (x)dx . R

Work was supported by the SEP-CONACYT Project No. 25564 (M´ exico). The second author was also sponsored by the CONACYT scholarship No. 163480.

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Given N ∈ N, let LpN (R, w) be the Banach space of vector-functions f = (fk )N 1/p k=1 PN p . If A kf k with entries fk ∈ Lp (R, w) and the norm kf kLpN (R,w) = k p,w k=1 is a subalgebra of L∞ (R), then AN ×N or [A]N ×N denote the matrix functions a : R → CN ×N whose entries belong to A. Let B(X) be the Banach algebra of all bounded linear operators on a Banach space X, and K(X) the closed two-sided ideal of all compact operators in B(X). An operator A ∈ B(X) is called Fredholm if Im A is closed in X and the numbers n(A) := dim Ker A and d(A) := dim(X/Im A) are finite. In that case Ind A := n(A) − d(A). In what follows we assume that 1 < p < ∞ and w is a Muckenhoupt weight on R (w ∈ Ap (R)), which means that 1/p Z 1/q Z 1 1 p −q w (x) dx w (x) dx < ∞, cp,w := sup |I| I |I| I I where 1/p + 1/q = 1, I ranges over all bounded intervals I ⊂ R, and |I| is the length of I. Then the Cauchy singular integral operator SR given by Z 1 f (t) (SR f )(x) = lim dt, x ∈ R, (1.1) ε→0 πi R\(x−ε,x+ε) t − x is bounded on the space Lp (R, w) (see [18] and also [15]). Let F : L2 (R) → L2 (R) denote the Fourier transform, Z ˆ (Ff )(x) := f (x) := f (t)eitx dt, x ∈ R. R

A function a ∈ L∞ (R) is called a Fourier multiplier on Lp (R, w) if the convolution operator W 0 (a) := F −1 aF maps L2 (R) ∩ Lp (R, w) into itself and extends to a bounded linear operator on Lp (R, w) (notice that L2 (R) ∩ Lp (R, w) is dense in Lp (R, w) if w ∈ Ap (R)). Let Mp,w stand for the set of all Fourier multipliers on Lp (R, w). One can show that Mp,w is a Banach algebra under the norm kakMp,w := kW 0 (a)kB(Lp (R,w)) . Let χ+ be the characteristic function of R+ = (0, ∞). By Lp (R+ , w) we understand the space Lp (R+ , w|R+ ). For a ∈ Mp,w , the Wiener-Hopf operator W (a) is defined on the space Lp (R+ , w) by W (a)f = χ+ W 0 (a)χ+ f, for f ∈ Lp (R+ , w). ˙ = R ∪ {∞}, R = [−∞, +∞], and let P C be the C ∗ -algebra of all Let R ˙ By Stechkin’s functions on R having finite one-sided limits at every point t ∈ R. inequality (see, e.g., [7, Theorem 17.1]), if a ∈ P C has finite total variation V1 (a), then a ∈ Mp,w and kakMp,w ≤ kSR kB(Lp (R,w)) kak∞ + V1 (a) , (1.2) ˙ (resp. Cp,w (R), P Cp,w ) the where SR is given by (1.1). We denote by Cp,w (R) ˙ closure in Mp,w of the set of all functions a ∈ C(R) (resp. a ∈ C(R), a ∈ P C) with

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˙ ⊂ C(R), ˙ Cp,w (R) ⊂ C(R), P Cp,w ⊂ P C. finite total variation. Obviously, Cp,w (R) ˙ Cp (R), P Cp and so on. In the case w = 1, we will omit w and write Cp (R), Fredholm theories for Wiener-Hopf operators W (a) with symbols a ∈ P Cp on Lebesgue spaces Lp (R+ ) and for the algebras alg {P C, W 0 (P Cp )} ⊂ B(Lp (R)) generated by the multiplication operators aI (a ∈ P C) and by the convolution operators W 0 (b) (b ∈ P Cp ) were constructed by R. V. Duduchava (see [12] and [13]). Fredholmness and index formulas for such operators on weighted Lebesgue spaces and algebras generated by these operators were studied in [30] and [28] in the case of power weights, and in [9] and [10] for general Muckenhoupt weights. These results were generalized to a bigger algebra in [21]. For the theory of Wiener-Hopf operators with almost periodic and semi-almost periodic matrix symbols, see [7] and the references therein. We intend to study Wiener-Hopf operators with slowly oscillating matrix symbols on weighted Lebesgue spaces. Let Cb (R) be the C ∗ -algebra of all bounded continuous functions a : R → C. Following [26] we denote by SO the C ∗ -algebra of slowly oscillating at ∞ functions, n o SO := f ∈ Cb (R) : lim sup |f (t) − f (s)| = 0 . (1.3) x→+∞ t,s∈[−2x,−x]∪[x,2x]

Fredholmness for Banach algebras of convolution type operators generated by the multiplication operators aI (a ∈ [SO, P C]N ×N ) and by the convolution operators W 0 (b) with symbols b ∈ [SOp , P Cp ]N ×N on unweighted Lebesgue spaces LpN (R) was studied in [2]–[3]. Consider the Banach algebra [SOp,w , Cp,w (R)]N ×N generated by all matrix functions in [SOp,w ]N ×N and [Cp,w (R)]N ×N , where SOp,w is the Banach algebra of slowly oscillating functions in Mp,w defined in Section 3. In the present paper we establish Fredholm conditions and an index formula for Wiener-Hopf operators W (a) with matrix symbols a ∈ [SOp,w , Cp,w (R)]N ×N on weighted Lebesgue spaces LpN (R+ , w) with Ap (R) weights w. Appearance of weights leads to a joint influence of oscillations of symbols and weights. To define an Mp,w analogue of slowly oscillating functions, we first study convolution operators from the viewpoint of pseudodifferential and Calder´on-Zygmund operators. A further investigation of Wiener-Hopf operators with mentioned oscillating symbols on weighted Lebesgue spaces is based on the Allan-Douglas local principle (see, e.g., [8]) and on the techniques of limit operators (see [25], [6], [27]). The paper is organized as follows. In Section 2 we estimate the norms of convolution operators W 0 (a) on weighted Lebesgue spaces in terms of the quantities kDγ akL∞ (R) (γ = 0, 1, 2, 3) where (Da)(x) := xa0 (x) for x ∈ R. To this end we apply the pointwise estimates from [1] and the theory of pseudodifferential and Calder´ on-Zygmund operators on weighted Lebesgue spaces with Muckenhoupt weights (see [11], [14], [19], [32], [5]). In Section 3 we define the Banach subalgebras SOp,w and [SOp,w , Cp,w (R)] of Mp,w that consist, respectively, of the functions slowly oscillating at ∞ and the functions having different slowly oscillating behavior at ±∞, and then we characterize their maximal ideal spaces M(SOp,w ) and M([SOp,w , Cp,w (R)]).

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In Section 4, applying the techniques of limit operators, we calculate basic limit operators for the operators wW 0 (a)w−1 I ∈ B(Lp (R)) with symbols a ∈ [SOp,w , Cp,w (R)] in the case of symmetric weights w = ev ∈ Ap (R) for which the function σ : x 7→ xv 0 (x) belongs to Cb (R+ ) and slowly oscillates at 0. In Section 5 we prove that all compact operators on the space Lp (R+ , w) are contained in the Banach subalgebra of B(Lp (R+ , w)) generated by the Wiener˙ establish the compactness of the comHopf operators with symbols in Cp,w (R), mutators W (a)W (b) − W (b)W (a) for a, b ∈ [SOp,w , Cp,w (R)] and, making use of the Allan-Douglas local principle, study the invertibility of corresponding cosets in local quotient algebras. Here we also obtain a Fredholm criterion for the WienerHopf operators W (a) with symbols a ∈ [SOp,w ]N ×N on the spaces LpN (R+ , w). Section 6 contains the main results of the paper. Applying results of Sections 3–5, we establish here sufficient Fredholm conditions and an index formula for Wiener-Hopf operators W (a) with matrix symbols a ∈ [SOp,w , Cp,w (R)]N ×N on weighted Lebesgue spaces LpN (R+ , w) with Ap (R) weights w. These sufficient conditions become a Fredholm criterion for the operators W (a) on the spaces LpN (R+ , w) in the case of Muckenhoupt weights with equal indices of powerlikeness (see definition of such indices in [5, Section 3.6]), and also for weights w ∈ Ap (R) with different indices of powerlikeness under some additional condition on p, w and symbols a (see Theorem 6.2 and Corollary 6.3). In these cases the regularizers of Fredholm Wiener-Hopf operators belong to the Banach algebra generated by all operators W (a) with symbols a ∈ [SOp,w , Cp,w (R)]N ×N .

2. Convolution operators from the viewpoint of pseudodifferential and Calder´on-Zygmund operators Let C n (R) be the set of all n times continuously differentiable functions a : R → C, and let K denote the distribution whose Fourier transform is a. We suppose that a ∈ C 3 (R \ {0}) and Aγ := kDγ akL∞ (R) < ∞ (γ = 0, 1, 2, 3)

(2.1)

0

where (Da)(x) = xa (x) for x ∈ R. Below we need the following dyadic decomposition (see, e.g., [32, Chapter VI, Subsection 4.1]). Let η be a C ∞ function of compact support and such that η(λ) = 1 for |λ| ≤ 1, η(λ) = 0 for |λ| ≥ 2, and η is monotone for 1 ≤ |λ| ≤ 2. Put δ(λ) = η(λ) − η(2λ). Then we have the following “partition of unity”: X+∞ 1= δ(2−j λ) for all λ ∈ R \ {0}, (2.2) j=−∞

where δ(2−j λ) are supported in the shells 2j−1 ≤ |λ| ≤ 2j+1 , and for each λ ∈ R \ {0} there are at most two non-zero terms in (2.2). The series (2.2) converges pointwise. For every j ∈ Z, put Z 1 aj (λ)e−ixλ dλ, aj (λ) := a(λ)δ(2−j λ). (2.3) Kj (x) := 2π R

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Following [32, p. 244] we prove the next assertion. Lemma 2.1. If a ∈ C 3 (R \ {0}) and eγ := sup xγ (dγx a)(x) < ∞ A

(γ = 0, 1, 2, 3),

(2.4)

x∈R

then for all α = 0, 1 and all γ = 0, 1, 2, 3 there are constants Aγ,α < ∞ such that γ α x (dx Kj )(x) ≤ Aγ,α 2j(1+α−γ) (2.5) for all x ∈ R and all j ∈ Z. Proof. Applying integration by parts, we obtain Z (−1)α aj (λ)(ix)γ ∂xα {e−iλx }dλ (−1)α (ix)γ (dα K )(x) = j x 2π R Z Z 1 1 γ α γ −iλx = (−1) aj (λ)(iλ) ∂λ {e dγ {aj (λ)(iλ)α }e−iλx dλ. }dλ = 2π R 2π R λ

(2.6)

If α = 0, then according to the Leibniz rule we deduce from (2.5) that γ X γ dβλ a (λ)2−j(γ−β) dλγ−β δ (2−j λ). (2.7) dγλ aj (λ) = β β=0

Hence, taking into account (2.4) and the lower estimate |λ| ≥ 2j−1 of the support β eβ 2−β(j−1) and therefore of δ(2−j λ), we infer from (2.7) that dλ a ∞ ≤ A ! γ X γ

γ−β γ β eβ 2 d eγ,0 2−jγ . d aj (λ) ≤ A δ ∞ 2−jγ =: A (2.8) λ λ β β=0

Since the integrand in (2.6) is supported in the interval |λ| ≤ 2j+1 of length 4 · 2j , we deduce from (2.6) and (2.8) that 4 e |x|γ |Kj (x)| ≤ Aγ,0 2j(1−γ) =: Aγ,0 2j(1−γ) , (2.9) 2π which gives (2.5) in the case α = 0. By analogy, if α = 1, then γ 1 X X γ γ − β −j(γ−β−k) γ−β−k γ β dλ {aj (λ)iλ} = (dλ a)(λ) 2 (dλ δ)(2−j λ)iλ1−k , β k β=0 k=0

β eβ 2−β(j−1) and |λ| ≤ 2j+1 , we obtain

whence, applying the estimate dλ a ∞ ≤ A ! γ X γ

γ−β−1 i j(1−γ) γ e βh γ−β ∂ {aj (λ)iλ} ≤

Aβ 2 2 dλ δ ∞ + (γ − β) dλ δ ∞ 2 λ β β=0

eγ,1 · 2j(1−γ) . =: A

(2.10)

Finally, by (2.6) and (2.10), |x|γ |Kj0 (x)| ≤

4 e Aγ,1 2j(2−γ) =: Aγ,1 2j(2−γ) , 2π

(2.11)

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which is consistent with (2.5) for α = 1.

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We see from (2.8) to (2.11) that for all α = 0, 1 and all γ = 0, 1, 2, 3, eβ : β = 0, 1, . . . , γ Aγ,α ≤ Cγ,α max A (2.12) where the constants Cγ,α depend only on γ and α. Since D0 a = a, D1 a = xa0 , D2 a = xa0 + x2 a00 , D3 a = xa0 + 3x2 a00 + x3 a000 , and hence x2 a00 = D2 a − D1 a, x3 a000 = D3 a − 3D2 a + 2D1 a, we conclude that 1 e0 , A e1 , A e2 , A e3 } ≤ 6 max{A0 , A1 , A2 , A3 }. (2.13) max{A0 , A1 , A2 , A3 } ≤ max{A 5 As a result, (2.12) and (2.13) imply the estimates Aγ,α ≤ 6Cγ,α max{A0 , A1 , A2 , A3 } (α = 0, 1; γ = 0, 1, 2, 3).

(2.14)

Following the proof of Proposition 2(a) in [32, p. 245] we get the next result. Lemma 2.2. If a ∈ C 3 (R \ {0}) satisfies (2.1), then the distribution K = F −1 a agrees with a function K(·) differentiable in R \ {0} and such that |(Dα K)(x)| ≤ A0α |x|−1 f or all x ∈ R \ {0} and all α = 0, 1, where the constants

A0α

are estimated by A0α ≤ Cα0 max{A0 , A1 , A2 , A3 }

and the constants

Cα0

(2.15)

(2.16)

∈ (0, ∞) depend only on α.

Proof. Applying the dyadic decomposition (2.2), we get X+∞ X+∞ a(λ) = a(λ)δ(2−j λ) = aj (λ). j=−∞ j=−∞ Pn Since j=−n aj (λ) converges to a(λ) uniformly on compacts of R \ {0}, it follows Pn from (2.3) that Kj converges to K in the sense of distributions. So it j=−n P suffices to estimate j |dα x Kj (x)| for x 6= 0. Using (2.5) for γ = 0 and α = 0, 1 gives X X α 2j0 (x)(1+α) dx Kj (x) ≤ A0,α 2j(1+α) = A0,α j −1 j −1 2 ≤|x| 2 ≤|x| 1 − 2−(1+α) A0,α e0α |x|−1−α , ≤ |x|−1−α =: A (2.17) 1 − 2−(1+α) where j0 (x) is the maximal integer j satisfying the inequality 2j ≤ |x|−1 . Similarly, by (2.5) with γ = 3 and α = 0, 1, we get X 2j >|x|−1

X α A3,α 2j1 (x)(α−2) dx Kj (x) ≤ A3,α 2j(1+α−3) = 3 j −1 2 >|x| |x| |x|3 1 − 2α−2 A3,α e00α |x|−1−α , ≤ |x|−1−α =: A (2.18) 1 − 2α−2

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where j1 (x) is the minimal integer j satisfying the inequality 2j > |x|−1 . Hence, for α = 0, 1 we obtain α X+∞ −1 e0 e00 (D K)(x) ≤ |x|α (dα =: A0α |x|−1 x Kj )(x) ≤ (Aα + Aα )|x| j=−∞

where the constants

A0α

are estimated by (2.16) due to (2.17), (2.18) and (2.14).

It follows from (2.15) that K(·) is a classical Calder´on-Zygmund kernel with K(x) ≤ A00 |x|−1 , K 0 (x) ≤ A01 |x|−2 for all x ∈ R \ {0}, (2.19) where the constants A0α (α = 0, 1) are estimated by (2.16). It is well known (see, e.g., [14, p. 248]) that the classical Calder´on-Zygmund operator given by Z (T f )(x) = v.p. K(x − y)f (y)dy for x ∈ R, (2.20) R

where K is a Calder´ on-Zygmund kernel satisfying (2.19), is bounded on every weighted Lebesgue space Lp (R, w) (1 < p < ∞, w ∈ Ap (R)). We want to estimate kT kB(Lp (R,w)) via max{A0 , A1 , A2 , A3 }. In what follows, let |Ω| be the Lebesgue measure of a measurable set Ω ⊂ R. As is well known (see, e.g., [14], [32]), the Calder´on-Zygmund operator (2.20) is of weak-type (1, 1), that is, there exists a finite constant C1,1 > 0 such that kT f k∼ := sup λ x ∈ R : |(T f )(x)| > λ ≤ C1,1 kf kL1 (R) (2.21) λ>0

1

for all f ∈ L (R). To determine C1,1 , we need the following two results on the Calder´ on-Zygmund decomposition (cf. [11] and [5, Lemma 5.4, Theorem 5.5]). Let I(x, ε) := (x − ε, x + ε) for x ∈ R and ε > 0. Lemma 2.3. If f ∈ L1 (R), then for every λ > 0 there exists an at most countable (and possible empty) set of intervals I(xk , εk ) centered at points xk ∈ R such that Z 1 |f (τ )|dτ ≤ 2λ for all k; (a) λ < |I(xk , εk )| I(xk ,εk ) S (b) |f (x)| ≤ λ for all x ∈ R \ k I(xk , εk ); (c) every point x ∈ R is contained in at most θ1 = 2 of the intervals I(xk , εk ). Theorem 2.4. Let f, λ and {I(xk , εk )} be as in the preceding lemma. Then there exist functions g and hk ∈ L1 (R) such that X f =g+ hk ; (2.22) k

|g(x)| ≤ 4λ f or almost all x ∈ R, kgkL1 (R) ≤ kf kL1 (R) ;

(2.23)

hk (x) = 0 f or all x ∈ R \ I(xk , εk ) and all k; X hk (τ )dτ = 0 f or all k, khk kL1 (R) ≤ 2kf kL1 (R) .

(2.24)

Z R

k

(2.25)

Applying Lemma 2.3 and Theorem 2.4, we can determine the constant C1,1 in (2.21) following the scheme of the proof in [5, Theorem 5.6].

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Theorem 2.5. If the operator T is given by (2.20) with a kernel K satisfying (2.19), (2.16) and (2.1), then (2.21) is fulfilled for all f ∈ L1 (R) with C1,1 := (20 + 16C10 ) max{A0 , A1 , A2 , A3 }. C0∞ (R)

1

(2.26) 1

Proof. Because is dense in L (R) and convergence in L (R) implies convergence in measure, it is sufficient to prove (2.21) for all functions f ∈ C0∞ (R). Since the operator T is bounded on the space L2 (R), we conclude that the set {x ∈ R : |(T f )(x)| > λ} is measurable for every f ∈ C0∞ (R), and kT kB(L2 (R)) = kakL∞ (R) = A0 .

(2.27)

Without loss of generality we may assume that max{A0 , A1 , A2 , A3 } = 1. (2.28) P Fix λ > 0 and let f = g + h, h = k hk be the Calder´on-Zygmund decomposition of f in accordance with Theorem 2.4. Then T f = T g + T h and hence n λ o λ o n + x ∈ R : |(T h)(x)| > . x ∈ R : |(T f )(x)| > λ ≤ x ∈ R : |(T g)(x)| > 2 2 (2.29) S Put Y := R \ k I(xk , 2εk ). From assertions (a) and (c) of Lemma 2.3 we obtain Z Z X 1 X 2 1X |f (τ )|dτ = χk (τ )|f (τ )|dτ ≤ kf kL1 (R) , (2.30) |I(xk , εk )| ≤ λ λ λ I(xk ,εk ) R k k k S and hence the set Rλ := k I(xk , εk ) is bounded. Since f has a compact support, we deduce from (2.22) and (2.24) that the set supp g ⊂ supp f ∪Rλ is also bounded. Consequently, from (2.23) it follows that kgk2L2 (R) ≤ 4λkgkL1 (R) ≤ 4λkf kL1 (R) < ∞,

(2.31)

whence, by Chebyshev’s inequality (see, e.g., [5, p. 145]) and by (2.31) and (2.27), Z n 2 λ o 2 2 (T g)(x) dx ≤ x ∈ R : |(T g)(x)| > 2 λ R 2 2 16 2 2 2 kT kB(L2 (R)) kgkL2 (R) ≤ A kf kL1 (R) . (2.32) ≤ λ λ 0 On the other hand, (2.30) implies that n λ o n λ o [ x ∈ R : |(T h)(x)| > ≤ x ∈ Y : |(T h)(x)| > + I(xk , 2εk ) 2 2 k n λ o 4 ≤ x ∈ Y : |(T h)(x)| > + kf kL1 (R) . (2.33) 2 λ R As supp hk ⊂ [xk − εk , xk + εk ] and R hk (τ )dτ = 0 in view of (2.24) and (2.25), we see that for almost all x ∈ Y , Z Z (T hk )(x) = K(x − y)hk (y)dy = K(x − y) − K(x − xk ) hk (y)dy. I(xk ,εk )

I(xk ,εk )

(2.34)

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Since |x − ξ| ≥ 2−1 |x − xk | for any ξ in the interval with the endpoints xk and y ∈ I(xk , εk ), we infer by the mean value theorem and (2.19) that |y − xk | 4A01 εk ≤ . (2.35) K(x − y) − K(x − xk ) ≤ |K 0 (x − ξ)||y − xk | ≤ A01 |x − ξ|2 |x − xk |2 Hence from (2.34) and (2.35) it follows that Z Z Z K(x − y) − K(x − xk ) hk (y) dydx (T hk )(x) dx ≤ Y I(xk ,εk ) Y Z Z dx hk (y) dy ≤ 4A01 εk 2 Y |x − xk | I(xk ,εk ) Z

εk dx 0 = 4A01 hk L1 (R) . (2.36) ≤ 4A1 hk L1 (R) 2 |x−xk |≥2εk |x − xk | Consequently, by (2.36) and (2.25), Z XZ X (T h)(x) dx ≤ (T hk )(x) dx ≤ 4A01 khk kL1 (R) ≤ 8A01 kf kL1 (R) , Y

k

Y

k

which implies that Z n 0 λ o 2 (T h)(x) dx ≤ 16A1 kf kL1 (R) . x ∈ Y : |(T h)(x)| > ≤ 2 λ Y λ

(2.37)

Combining (2.33) and (2.37), we obtain n λ o 16A01 + 4 kf kL1 (R) , x ∈ R : |(T h)(x)| > ≤ 2 λ which together with (2.32) and (2.29) shows that (2.21) holds if C1,1 = 16A20 + 16A01 + 4. On the other hand, from (2.16) and (2.28) it follows that 16A20 + 16A01 + 4 ≤ 16A20 + 4 + 16C10 max{A0 , A1 , A2 , A3 } ≤ 20 + 16C10 , and therefore (2.21) holds with C1,1 = 20 + 16C10 if max{A0 , A1 , A2 , A3 } = 1. Obviously, the latter implies (2.21) with (2.26) for any max{A0 , A1 , A2 , A3 }. Following [1], let us now estimate kT kB(Lp (R,w)) via kM kB(Lp (R,w)) where M is the Hardy-Littlewood maximal operator, Z 1 (M f )(x) = sup |f (τ )|dτ, x ∈ R, (2.38) I3x |I| I the supremum is taken over all intervals I ⊂ R containing x, |I| is the length of I. As is well known (see, e.g., [15, Section 6.6] or [5, Corollary 5.3]), the operator M is bounded on every space Lp (R, w) with 1 < p < ∞ and w ∈ Ap (R). Let M ] be the sharp maximal operator of C. Fefferman and E. Stein, Z Z 1 1 (M ] f )(x) = sup |f (τ ) − fI |dτ, where fI = f (τ )dτ (2.39) |I| I I3x |I| I

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and the supremum is taken over all intervals I ⊂ R containing x. Consider the s-sharp maximal operator Ms] (0 < s < 1) given by 1/s Ms] (g) = M ] (|g|s ) . (2.40) For each interval I = I(x0 , r) ⊂ R, let ZZ 1 K(z − y) − K(x − y) dxdz. (DI K)(y) = 2 |I| I×I According to [1] we assume that (D) There are constants CD , N ∈ (0, ∞) such that Z sup (DI K)(y)|f (y)|dy ≤ CD (M f )(x0 ) r>0

(2.41)

(2.42)

|y−x0 |>N r

for all f ∈ C0∞ (R) and all x0 ∈ R. From [1, Theorem 2.1] we infer the following. Theorem 2.6. Let T be the Calder´ on-Zygmund operator (2.20) with a kernel K satisfying (2.42) and (2.19) with (2.1) and (2.16). Then for every s ∈ (0, 1) there is the finite constant Cs := 22/s−1 N (1 − s)−1/s C1,1 + CD (2.43) such that for all f ∈ C0∞ (R) and all x0 ∈ R, ] Ms (T f ) (x0 ) ≤ Cs (M f )(x0 ).

(2.44)

Proof. Fix x0 ∈ R. It is easily seen from (2.39) that, for every f ∈ L1loc (R), Z Z 2 1 ] (M f )(x0 ) = sup |f (τ ) − fI |dτ ≤ sup inf |f (τ ) − c|dτ. (2.45) I3x0 |I| I I3x0 c∈C |I| I Fix f ∈ C0∞ (R). Hence, by (2.40) and (2.45), for every s ∈ (0, 1) we obtain Z 1/s ] 1/s 1 |(T f )(τ )|s − c dτ . Ms (T f ) (x0 ) = M ] (|T f |s ) (x0 ) ≤ 21/s sup inf I3x0 c∈C |I| I (2.46) Let f = f1 + f2 where f1 = f χI(x0 ,N r) and N is given by condition (D). Since s |a| − |b|s ≤ |a − b|s , (|a| + |b|)1/s ≤ 21/s−1 (|a|1/s + |b|1/s ) for 0 < s < 1, setting cI := (T f2 )I , we obtain 1/s Z Z 1/s 1 1 |T f |s − |(T f2 )I |s dτ T f − (T f2 )I s dτ ≤ |I| I |I| I 1/s Z s 1 s |T f1 | + T f2 − (T f2 )I dτ (2.47) ≤ |I| I Z 1/s Z s 1/s 1 1 1/s−1 s ≤2 |T f1 | dτ + T f2 − (T f2 )I dτ =: 21/s−1 (I1 + I2 ). |I| I |I| I

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By Theorem 2.5, T possesses the property (2.21) where C1,1 is given by (2.26). Hence, applying [14, Lemma 1.4] (also see [5, (5.48)]), (2.21) and (2.38), we get Z 1/s 1 1/s 1 1 s |T f1 | dτ ≤ kT f1 k∼ I1 = |I| I 1−s |I| −1/s −1/s C1,1 (M f )(x0 ). (2.48) C1,1 |I|−1 kf1 kL1 (R) ≤ N 1 − s ≤ 1−s Further, Jensen’s inequality for the concave function ϕ(x) = xs (0 < s < 1), Fubini’s theorem, (2.41) and condition (D) yield according to [1] that Z 1/s Z s 1 1 T f2 − (T f2 )I dx ≤ T f2 − (T f2 )I dx I2 = |I| I |I| I Z Z Z 1 K(x − y) − K(z − y) |f (y)|dydzdx ≤ 2 |I| I I R\I(x0 ,N r) Z (DI K)(y)|f (y)|dy ≤ CD (M f )(x0 ). (2.49) = |y−x0 |>N r

Finally, (2.46), (2.47) with cI := (T f2 )I , (2.48) and (2.49) imply the relation (2.44) with the constant Cs given by (2.43). Lemma 2.7. If T is the Calder´ on-Zygmund operator (2.20) with a kernel K satisfying (2.19), (2.16) and (2.1), then (2.42) holds for all f ∈ C0∞ (R), N = 2 and CD := 32C10 max{A0 , A1 , A2 , A3 }.

(2.50)

Proof. Let us estimate the constant CD in (2.42) on the basis of (2.19). Put N = 2 and fix y ∈ / I(x0 , 2r). Then, by the mean value theorem, there is a point ξ in the interval with the endpoints z, x ∈ I(x0 , r) such that 0 A01 2r K(z − y) − K(x − y) = |K 0 (ξ − y)||z − x| ≤ A1 |z − x| ≤ , |ξ − y|2 (|y − x0 | − r)2 which implies in view of (2.41) that (DI K)(y) ≤

A01 2r (|y − x0 | − r)2

if |y − x0 | > 2r.

(2.51)

Hence, for I = I(x0 , r), we infer from (2.51) that for all f ∈ C0∞ (R), Z X∞ Z |f (y)| 0 (DI K)(y)|f (y)|dy ≤ 2A1 r dy n=0 (|y − x0 | − r)2 n+1 n+2 |y−x0 |>2r 2 r<|y−x0 |≤2 r Z X∞ 1 ≤ 2A01 r |f (y)|dy n=0 (2n+1 r − r)2 I(x ,2n+2 r) 0 Z X∞ 1 |f (y)|dy ≤ 16A01 2−n n=0 |I(x0 , 2n+2 r)| I(x0 ,2n+2 r) ≤ 32A01 (M f )(x0 ), which together with (2.16) gives (2.42) with N = 2 and CD defined by (2.50).

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Theorems 2.5 and 2.6 and Lemma 2.7 immediately imply the following. Corollary 2.8. If the conditions of Lemma 2.7 are fulfilled, s ∈ (0, 1) and (2.52) Cs = 22/s−1 2(1 − s)−1/s (20 + 16C10 ) + 32C10 max{A0 , A1 , A2 , A3 }, then Ms] (T f ) (x0 ) ≤ Cs (M f )(x0 ) for every x0 ∈ R and every f ∈ C0∞ (R). By [19, p. 41], for each p ∈ (1, ∞), each w ∈ Ap (R) and each s ∈ (0, 1) there exists a constant Cp,w,s ∈ (0, ∞) such that Z Z p/s p ] p/s p (M f )(x) w (x)dx ≤ Cp,w,s (M f )(x) w (x)dx (f ∈ C0∞ (R)). R

R

(2.53) Since M ∈ B(Lp (R, w)) for all 1 < p < ∞ and all w ∈ Ap (R), the Lebesgue differentiation theorem, (2.53) and Corollary 2.8 yield, as in [1], that for every f ∈ C0∞ (R), Z h Z ip/s p (T f )(x) p wp (x)dx ≤ M (|T f |s ) (x) w (x)dx R R Z h Z h ip ip/s p w (x)dx = Cp,w,s Ms] (T f ) (x) wp (x)dx ≤ Cp,w,s M ] (|T f |s ) (x) R R Z p ≤ Cp,w,s Csp (M f )(x) wp (x)dx ≤ Cp,w,s Csp kM kpB(Lp (R,w)) kf kpLp (R,w) . (2.54) R

Thus, taking s = 1/2 in (2.54) and (2.52), we obtain the following. Theorem 2.9. If T is the Calder´ on-Zygmund operator given by (2.20), with a kernel K satisfying (2.19), (2.16) and (2.1), then T is bounded on every weighted Lebesgue space Lp (R, w) with 1 < p < ∞ and w ∈ Ap (R), and ep,w max{Aγ : γ = 0, 1, 2, 3}, kT kB(Lp (R,w)) ≤ C 0 ep,w := 20 · 64 C 1/p p C p,w,1/2 (1 + C1 )kM kB(L (R,w)) < ∞.

Corollary 2.10. If a ∈ C 3 (R \ {0}) and kDγ akL∞ (R) < ∞ for all γ = 0, 1, 2, 3, then the convolution operator W 0 (a) is bounded on every weighted Lebesgue space Lp (R, w) with 1 < p < ∞ and w ∈ Ap (R), and ep,w max kDγ akL∞ (R) : γ = 0, 1, 2, 3 < ∞, kW 0 (a)kB(Lp (R,w)) ≤ C ep,w ∈ (0, ∞) depends only on p and w. where the constant C

3. Slowly oscillating symbols and Fourier multipliers on Lp (R, w) Consider the commutative C ∗ -algebra SO of slowly oscillating functions defined by ˙ (1.3). Clearly, SO is a subalgebra of L∞ (R) which contains all functions in C(R). ˙ ˙ Identifying the points t ∈ R with the evaluation functionals δt on R, δt (f ) = f (t),

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where δ∞ (f ) = lim f (x), we see that the maximal ideal space M(SO) of SO is x→∞

of the form M(SO) = R ∪ M∞ (SO) where n o M∞ (SO) = ξ ∈ M(SO) : ξ|C(R) ˙ = δ∞ is the fiber of M(SO) over ∞. By [4, Proposition 5], M∞ (SO) = (clos SO∗ R) \ R where clos SO∗ R is the weak-star closure of R in SO∗ , the dual space of SO. Thus, any functional ξ ∈ M∞ (SO) is the limit of a net tα ∈ R that does not converge to functionals t ∈ R, that is, f (ξ) := ξ(f ) = lim f (tα ) for every f ∈ SO. α

{ak }∞ k=1

Proposition 3.1. [4, Proposition 6] Let be a countable subset of SO. If ξ ∈ M∞ (SO), then there exists a sequence g = {gn } ⊂ R such that gn → ∞ and ξ(ak ) = lim ak (gn ), n→∞

k ∈ N.

(3.1)

Conversely, if gn ∈ R, gn → ∞, and the limits lim ak (gn ) exist for all k, then n→∞

there is a ξ ∈ M∞ (SO) such that (3.1) holds. Consider the commutative Banach algebras n o 3 g := a ∈ Cb (R) ∩ C 3 (R) : lim (Dγ a)(x) = 0, γ = 1, 2, 3 SO |x|→∞

(3.2)

3

g ∩ SO equipped with the norm and SO3 := SO kakSO3 := max kDγ akL∞ (R) : γ = 0, 1, 2, 3 = max Aγ : γ = 0, 1, 2, 3 .

(3.3)

3

g ⊂ Mp,w . Let Let 1 < p < ∞ and w ∈ Ap (R). By Corollary 2.10, SO3 ⊂ SO 3 SOp,w denote the closure of SO in Mp,w . Clearly, SOp,w is a commutative Banach subalgebra of Mp,w . Since Mp,w ⊂ M2 = L∞ (R), we conclude that SOp,w ⊂ SO. To determine the maximal ideal space M(SOp,w ) of SOp,w , we take three unital commutative Banach algebras homomorphically embedded one into another, SO3 ⊂ SOp,w ⊂ SO,

(3.4)

and, by analogy with [2], apply the following result (see [31, Theorem 3.10]). Theorem 3.2. Let Bi (i = 1, 2, 3) be commutative Banach algebras with the same unit which are homomorphically embedded one into another, B1 ⊂ B2 ⊂ B3 . Suppose that B1 is dense in B2 and every multiplicative linear functional defined on B1 extends to a multiplicative linear functional on B3 . Then every multiplicative linear functional on B2 also extends to a multiplicative linear functional on B3 . Lemma 3.3. Every multiplicative linear functional defined on SO3 extends to a multiplicative linear functional on SO. Proof. As SO3 ⊂ SO, it follows that M(SO) ⊂ M(SO3 ). Indeed, let η ∈ M(SO) and a ∈ SO3 . Then, a ∈ SO and, by (3.3), kak∞ ≤ kakSO3 . Hence η(a) ≤ kηkSO∗ kak∞ ≤ kηkSO∗ kakSO3 = kakSO3 , and thus η is a (non-zero) multiplicative linear functional on SO3 .

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Suppose that ξ ∈ M(SO3 ) is not extendable to a multiplicative linear functional on SO, that is, ξ ∈ M(SO3 ) \ M(SO). Clearly, there exists a function a ∈ SO3 such that η(a) 6= 0 for all η ∈ M(SO) and ξ(a) = 0. Since a ∈ SO and SO is a C ∗ -algebra, we infer that the function a is invertible in SO. But the inverse function 1/a automatically belongs to SO3 , that is, SO3 is inverse closed in SO. Then from the Gelfand theory (see e.g., [29, Chapter 11]) it follows that ξ(a) 6= 0, and we arrive at a contradiction, which completes the proof. Choosing the commutative Banach algebras B1 = SO3 , B2 = SOp,w and B3 = SO that satisfy (3.4) and have the same unit, and taking into account the density of SO3 in SOp,w in the norm of Mp,w , we deduce from Theorem 3.2 and Lemma 3.3 that every multiplicative linear functional on SOp,w extends to a multiplicative linear functional on SO, that is, M(SOp,w ) ⊂ M(SO). On the other hand, M(SO) ⊂ M(SOp,w ) because SOp,w ⊂ SO. Thus we get the following. Lemma 3.4. If 1 < p < ∞ and w ∈ Ap (R), then the maximal ideal spaces of SOp,w and SO coincide as sets, that is, M(SOp,w ) = M(SO). Lemma 3.4 and the Gelfand theory immediately give the following assertion. Corollary 3.5. If 1 < p < ∞ and w ∈ Ap (R), then the Banach algebra SOp,w is inverse closed in the C ∗ -algebras SO and L∞ (R). Let V (R) be the Banach algebra of all functions a : R → C of bounded total variation equipped with the norm kakV := kakL∞ (R) + V1 (a), where V1 (a) is the total variation of a. Then V (R) ⊂ P C (see, e.g., [23, Chapter VI, § 2]). ˙ ⊂ SOp,w and, for every Lemma 3.6. If 1 < p < ∞ and w ∈ Ap (R), then Cp,w (R) ˙ vanishing at infinity and every a ∈ SOp,w , the product ba belongs to b ∈ Cp,w (R) ˙ and (ba)(∞) = 0. the algebra Cp,w (R) ˙ with b(∞) = 0 can be approximated Proof. Since every function b ∈ Cp,w (R) ˙ ∩ V (R) with compact support and since every in Mp,w by functions bn ∈ C(R) function a ∈ SOp,w can be approximated in Mp,w by functions an ∈ SO3 , we infer ˙ and (ba)(∞) = 0 because all the functions bn an that ba = limn→∞ bn an ∈ Cp,w (R) ˙ ∩ V (R). have compact support and belong to C(R) ˙ On the other hand, every function c ∈ C(R)∩V (R) with compact support can be approximated in Mp,w by continuous piecewise linear functions with compact support (see [12, Lemma 2.10 and Theorem 2.11]). Hence the function c can be ˙ ∩ V (R) and therefore in Mp,w by functions cn ∈ C 3 (R) ˙ approximated in C(R) with compact support. Obviously, cn ∈ SO3 , which implies that c ∈ SOp,w . Let ˙ Since the function b−b(∞) can be approximated in Mp,w by functions b ∈ Cp,w (R). ˙ in C(R) ∩ V (R) with compact support and these functions belong to SOp,w , we ˙ ⊂ SOp,w . conclude that Cp,w (R) If u± (x) := [1 ± tanh x]/2 for x ∈ R, then u± ∈ Cp,w (R) and u+ (+∞) = 1, u+ (−∞) = 0, u− = 1 − u+ .

(3.5)

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Clearly, from Lemma 3.6 it follows that the set a+ u+ + a− u− : a± ∈ SOp,w , u± ∈ Cp,w (R) and (3.5) holds is an algebra. Consider the Banach subalgebra [SOp,w , Cp,w (R)] of Mp,w generated 3 gp,w be the closure of SO g in Mp,w , by all functions in SOp,w ∪ Cp,w (R), and let SO 3

g is given by (3.2). Obviously, SO gp,w is a Banach subalgebra of Mp,w . where SO gp,w and Lemma 3.7. If 1 < p < ∞ and w ∈ Ap (R), then the Banach algebras SO [SOp,w , Cp,w (R)] coincide, and every function a in [SOp,w , Cp,w (R)] is of the form a = a+ u+ + a− u− ,

(3.6)

where a± ∈ SOp,w and u± (x) = [1 ± tanh x]/2 for x ∈ R. Proof. By [12, Lemma 2.10 and Theorem 2.11], every function c ∈ C(R) ∩ V (R) can be approximated in Mp,w by continuous piecewise linear functions. Hence the function c can be approximated in C(R)∩V (R) and therefore in Mp,w by functions 3 g , which cn ∈ C 3 (R) whose derivative c0n have compact supports. Clearly, cn ∈ SO gp,w . Thus, C(R) ∩ V (R) ⊂ SO gp,w . Since Cp,w (R) implies that c = limn→∞ cn ∈ SO gp,w too. is the closure of C(R) ∩ V (R) in Mp,w , we conclude that Cp,w (R) ⊂ SO g g Obviously, SOp,w ⊂ SOp,w and, consequently, [SOp,w , Cp,w (R)] ⊂ SOp,w . gp,w . Put e Conversely, let a ∈ SO a+ (x) := a(|x|) and e a− (x) := a(−|x|) for x ∈ R. Obviously, e a± ∈ SOp,w , which implies that ˙ a0 := a − e a+ u+ − e a− u− ∈ Cp,w (R)

and a(∞) = 0.

(3.7)

gp,w . Hence, a ∈ [SOp,w , Cp,w (R)], and therefore [SOp,w , Cp,w (R)] = SO ˙ Finally, since Cp,w (R) ⊂ SOp,w according to Lemma 3.6, we deduce from (3.7) that a is represented in the form (3.6) where a± := e a± + a0 ∈ SOp,w . Consider the homomorphic embeddings of the commutative Banach algebras [SO3 , C 3 (R) ∩ V (R)] ⊂ [SOp,w , Cp,w (R)] ⊂ [SO, C(R)], where [SO3 , C 3 (R) ∩ V (R)] is the Banach algebra with the norm n o kf k = max kf kL∞ (R) , V1 (f ), kDγ f kL∞ (R) , γ = 1, 2, 3 . Representing functions a ∈ [SO, C(R)] in the form (3.6) due to Lemma 3.7, one can identify the fibers M± ∞ (SO) of M([SO, C(R)]) with the fiber M∞ (SO) of M(SO) by the rule: for every ξ± ∈ M± ∞ (SO) there is a ξ ∈ M∞ (SO) and for every ξ ∈ M∞ (SO) there are ξ± ∈ M± (SO) such that ξ± (a+ u+ + a− u− ) = ξ(a± ). ∞ Analogously to Lemma 3.4 and Corollary 3.5 one can prove the following. Lemma 3.8. For every p ∈ (1, ∞) and every w ∈ Ap (R), the Banach algebra [SOp,w , Cp,w (R)] is inverse closed in the C ∗ -algebras [SO, C(R)] and L∞ (R), and + M([SOp,w , Cp,w (R)]) = M([SO, C(R)]) = M− ∞ (SO) ∪ R ∪ M∞ (SO).

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4. Limit operators To study Wiener-Hopf operators with symbols in the algebra [SOp , Cp,w (R)] on weighted Lebesgue spaces Lp (R+ , w), we need to apply the limit operators techniques (see [4], [6], [27]) to corresponding operators acting on the space Lp (R). Given p ∈ (1, ∞), for every k > 0 we consider the isometric dilation operators Uk ∈ B(Lp (R)),

(Uk f )(x) = k 1/p f (kx)

p

(x ∈ R).

(4.1)

p

Let A ∈ B(L (R)). The operator Ah ∈ B(L (R)) is called the limit operator of A with respect to a sequence h = {hm } of positive real numbers tending to +∞ if Ah = s-lim Uhm AUh−1 . m m→∞

By [4] and [6], for every function a ∈ SO and every sequence h ⊂ R+ tending to +∞ there is a subsequence g of h such that there exists the limit operator (aI)g := s-lim Uhm aUh−1 = a g I and ag is a constant. Thus, if a ∈ SO and m m→∞

ξ ∈ M∞ (SO), then (aI)g = ξ(a)I under the conditions of Proposition 3.1. Following [5, Section 1.5], for every continuous function ψ : R+ → R+ , we consider the submultiplicative functions ψ(xR) ψ(xR) , (W00 ψ)(x) := lim sup (x ∈ R+ ). (4.2) (W0 ψ)(x) := sup ψ(R) ψ(R) R→0 R>0 If the function W0 ψ is regular, that is, if W0 ψ is bounded from above in some open neighborhood of the point 1, then, by [5, Lemma 1.16], the function W00 ψ is also regular and has the same with W0 ψ lower and upper indices log(W00 ψ)(x) log(W00 ψ)(x) , βψ := lim (4.3) x→∞ x→0 log x log x where −∞ < αψ ≤ βψ < +∞ (see, e.g., [5, Theorem 1.13]). A slight modification of the proof in [5, Theorem 2.33] gives the following. αψ := lim

Theorem 4.1. If 1 < p < ∞, w : R+ → R+ is a continuous function and W0 w is regular, then w ∈ Ap (R+ ) if and only if −1/p < αw ≤ βw < 1 − 1/p. We say that σ ∈ SO0 if σ ∈ Cb (R+ ) and lim max σ(y) − σ(z) = 0. x→0 y,z∈[x,2x]

(4.4)

Obviously, SO0 is a C ∗ -subalgebra of L∞ (R+ ). If w = ev ∈ Ap (R+ ) for some p ∈ (1, ∞) where the function v : R+ → R is continuously differentiable (v ∈ C 1 (R+ )) and the function σ : t 7→ tv 0 (t) belongs to the C ∗ -algebra SO0 , then Z xR dt lim sup v(xR) − v(R) = lim sup tv 0 (t) t R→0 R→0 R  0 (4.5)  lim inf [tv (t)] ln x if x ∈ (0, 1), t→0 =  lim sup [tv 0 (t)] ln x if x ∈ (1, ∞). t→0

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Hence the function W0 w is regular in view of the estimate (W0 w)(x) = sup ev(xR)−v(R) ≤ ekσkL∞ (R+ ) | ln x|

(x ∈ R+ ),

R>0

and from (4.2), (4.3) and (4.5) it follows that αw = lim inf [tv 0 (t)], t→0

βw = lim sup [tv 0 (t)].

(4.6)

t→0

Consequently, by Theorem 4.1 and [5, Corollary 1.14], for every ε > 0 for which −1/p < αw − ε < βw + ε < 1/q there are x0 ∈ (0, 1) and Cw ∈ (0, ∞) such that  α −ε x w for x ∈ (0, x0 ), w(kx)  Cw for x ∈ [x0 , x−1 sup (4.7) ≤ 0 ],  βw +ε k>0 w(k) x for x ∈ (x−1 , ∞). 0 We now consider the Banach space L := χ[0,1] Lp (R+ , xαw −ε ) + χ[1,∞) Lp (R+ , xβw +ε )

(4.8)

equipped with the norm kf kL = kχ[0,1] f kLp (R+ ,xαw −ε ) + kχ[1,∞) f kLp (R+ ,xβw +ε ) . Obviously, from (4.7) it follows that the operators w(kx) I : L → Lp (R+ ) (k > 0) (4.9) w(k) w +ε are uniformly bounded and kAk k ≤ Cw max x−α , 1, xβ0 w +ε . 0 Since for every function σ ∈ SO0 the function σ e(x) := σ(1/|x|) e−1/|x| belongs to SO, we can identify the fibers M0 (SO0 ) and M∞ (SO) by the rule: ξ0 (σ) := ξ(e σ ) for every ξ ∈ M∞ (SO). Hence, if a± ∈ SOp,w and the function σ : t 7→ tv 0 (t) belongs to the C ∗ -algebra SO0 , then, by Proposition 3.1, for every ξ ∈ M∞ (SO) there is a sequence {kn } ⊂ R+ such that Ak :=

lim kn = 0, γξ := ξ(e σ ) = lim kn v 0 (kn ), a± (ξ) := ξ(a± ) = lim a± (1/kn ).

n→∞

n→∞

n→∞

(4.10) Lemma 4.2. If a = a+ u+ + a− u− where a± ∈ SO3 and u± (x) = 2−1 [1 ± tanh x] on R, and {kn } ⊂ R+ is a sequence satisfying the first and the last relations in (4.10) for some ξ ∈ M∞ (SO), then for every p ∈ (1, ∞) and every γ ∈ (−1/p, 1 − 1/p), s-lim Ukn |x|γ W 0 (a)|x|−γ Uk−1 = |x|γ W 0 (a+ (ξ)χ+ + a− (ξ)χ− )|x|−γ I n n→∞

(4.11)

on the space Lp (R), where χ± are the characteristic functions of R± . Proof. Because a± ∈ SO3 and u± ∈ V (R), the operator W 0 (a) is bounded on every space Lp (R, |x|γ ) with p ∈ (1, ∞) and γ ∈ (−1/p, 1 − 1/p). Since Ukn |x|γ W 0 (a)|x|−γ Uk−1 = |x|γ W 0 (a(·/kn ))|x|−γ I ∈ B(Lp (R)), n to prove (4.11), it remains to show that s-lim W 0 (a± (·/kn ) − a± (ξ)) = 0, n→∞

s-lim W 0 (u± (·/kn ) − χ± ) = 0 n→∞

(4.12)

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on the spaces Lp (R, |x|γ ). By analogy with [20, Lemma 2.5], one can prove that if 1 < p < ∞ and w ∈ Ap (R), then the set Y of all functions ψ ∈ L2 (R)∩Lp (R, w) for which Fψ has compact support in R \ {0} is dense in Lp (R, w). Therefore, taking any function f ∈ Lp (R, |x|γ ) for which Ff has a compact support in R \ {0}, we infer that W 0 (a± (·/kn ) − a± (ξ))f = W 0 (a± (·/kn ) − a± (ξ))χl f, (4.13) W 0 (u± (·/kn ) − χ± )f = W 0 (u± (·/kn ) − χ± )χl f, −1 where χl is the characteristic function of a set l = [−t0 , −t−1 0 ] ∪ [t0 , t0 ] containing supp Ff for some t0 > 1. Since a± , u± ∈ SO, it follows that

lim (u± (·/kn ) − χ± )χl = 0. (4.14) lim (a± (·/kn ) − a± (ξ))χl = 0, ∞

n→∞

∞

n→∞

On the other hand, we obtain Z

V1 (a± (·/kn ) − a± (ξ))χl ≤ kn−1 a0± (t/kn ) dt + 2 (a± (·/kn ) − a± (ξ))χl ∞ , Zl

V1 (u± (·/kn ) − χ± )χl ≤ kn−1 u0± (t/kn ) dt + 2 (u± (·/kn ) − χ± )χl ∞ . l

(4.15) As a± ∈ SO3 and u± (x) = 2−1 [1 ± tanh x], we conclude that lim kn−1 a0± (t/kn ) = 0,

n→∞

lim kn−1 u0± (t/kn ) = 0

n→∞

uniformly on the compact l, which implies in view of (4.14) and (4.15) that lim V1 (a± (·/kn ) − a± (ξ))χl = 0, lim V1 (u± (·/kn ) − χ± )χl = 0. (4.16) n→∞

n→∞

Finally, from (4.14) and (4.16) it follows due to Stechkin’s inequality (1.2) that

lim (a± (·/kn ) − a± (ξ))χl M γ = 0, p,|x| n→∞

lim (u± (·/kn ) − χ± )χl M γ = 0, n→∞

p,|x|

which according to (4.13) gives (4.12) and hence proves (4.11).

If w ∈ Ap (R), then w|R+ ∈ Ap (R+ ). On the other hand, with every weight w ∈ Ap (R+ ) we will identify the symmetric weight w(| · |) ∈ Ap (R). Let Uk (k > 0) denote both the dilation operators (4.1) on Lp (R) and their restrictions on Lp (R+ ). Theorem 4.3. If a = a+ u+ + a− u− ∈ [SOp,w , Cp,w (R)] where a± ∈ SOp,w and u± (x) = 2−1 [1 ± tanh x] on R, 1 < p < ∞, w = ev ∈ Ap (R+ ), v ∈ C 1 (R+ ), the function σ : t 7→ tv 0 (t) belongs to the C ∗ -algebra SO0 , and {kn } ⊂ R+ is a sequence satisfying (4.10) for some ξ ∈ M∞ (SO), then s-lim Ukn ev W (a)e−v Uk−1 = xγξ W (a+ (ξ)χ+ + a− (ξ)χ− )x−γξ I (4.17) n n→∞

where γξ = lim (kn v 0 (kn )), a± (ξ) = lim a± (1/kn ), and χ± are the characteristic n→∞ n→∞ functions of R± .

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Proof. Put α := αw , β := βw . Since w ∈ Ap (R+ ), then by Theorem 4.1 and (4.6), −1/p < α = lim inf [tv 0 (t)] ≤ β = lim sup[tv 0 (t)] < 1 − 1/p. t→0

t→0

Taking a sequence {kn } ⊂ R+ satisfying (4.10), let us show that for the operators Ak : L → Lp (R+ ) given by (4.9), where L is defined by (4.8), we get s-lim Akn = xγξ I n→∞

where

γξ = ξ0 (σ) = ξ(e σ ).

(4.18)

Note that for every γ ∈ [α, β] the operator xγ I is bounded from the space L into the space Lp (R+ ) and kxγ Ik ≤ 1. Indeed, Z ∞ Z Z 1 (γ−α+ε)p α−ε p γp p x(γ−β−ε)p |xβ+ε f (x)|p dx x |x f (x)| dx + x |f (x)| dx = 1

0

R+

1

Z

|xα−ε f (x)|p dx +

≤ 0

Z

∞

|xβ+ε f (x)|p dx = kf kpL .

1

(4.19)

Since σ ∈ SO, from (4.4) it follows that for every x0 > 1, lim

n→∞

|σ(t) − γξ | = 0.

sup t∈[kn x−1 0 ,kn x0 ]

Hence the equality Z

kn x

v(kn x) − v(kn ) − γξ ln x =

[tv 0 (t) − γξ ]

kn

dt t

implies that lim v(kn x) − v(kn ) − γξ ln x = 0

(4.20)

n→∞

uniformly with respect to x ∈ [x−1 0 , x0 ]. Further, for every segment l ⊂ R+ and every f ∈ L, from (4.19) it follows that

Akn − xγξ I χl f p = ev(kn x)−v(kn )−γξ ln x − 1 χl xγξ f Lp (R+ ) L (R+ )

≤ ev(kn x)−v(kn )−γξ ln x − 1 L∞ (l) kxγξ f kLp (R+ ) ≤ esupx∈l |v(kn x)−v(kn )−γξ ln x| − 1 kf kL . (4.21) γξ Relations (4.20) and (4.21) imply that lim Akn − x I χl f = 0 for every l ⊂ R+ n→∞

and every f ∈ L, which gives (4.18) because the operators Akn −xγξ I are uniformly bounded in B(L, Lp (R+ )). Clearly, for every a ∈ Cp,w (R), Ukn ev W (a)e−v Uk−1 = ev(kn x)−v(kn ) W (a(x/kn ))e−v(kn x)+v(kn ) I, n and these operators are uniformly bounded on the space Lp (R+ ). Thus, to prove (4.17), it remains to show that for every segment l = [x−1 0 , x0 ] ⊂ R+ and every function f ∈ Lp (R+ ),

γξ −γξ I χl f p = 0. lim Akn W (a(x/kn ))A−1 kn − x W (a+ (ξ)χ+ + a− (ξ)χ− )x n→∞

L (R+ )

(4.22)

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Since every function a ∈ Cp,w (R) can be approximated in Mp,w by functions in V (R), it is sufficient to prove (4.22) for functions a ∈ V (R). By Stechkin’s inequality (1.2) the operators W (a(x/kn )) are uniformly bounded on every Banach space Lp (R+ , w) with 1 < p < ∞ and w ∈ Ap (R+ ). Thus, to prove (4.22) for a ∈ V (R), it remains to show that every summand on the right of the following estimate tends to zero as n → ∞:

γξ −γξ − x W (a (ξ)χ + a (ξ)χ ) x I χ f

Akn W (a(x/kn ))A−1 + + − − l p kn L (R+ )

−1 −γξ

≤ kAkn kB(L,Lp (R+ )) W (a(x/kn )) Akn − x I χl f L

+ kAkn kB(L,Lp (R+ )) W (a(x/kn )) − W (a+ (ξ)χ+ + a− (ξ)χ− ) x−γξ χl f L

+ Akn − xγξ I W (a+ (ξ)χ+ + a− (ξ)χ− ) x−γξ χl f Lp (R+ ) . (4.23) Since W (a+ (ξ)χ+ + a− (ξ)χ− )x−γξ χl f ∈ L in view of the estimate

W (a+ (ξ)χ+ + a− (ξ)χ− ) x−γξ χl f L

≤ W (a+ (ξ)χ+ + a− (ξ)χ− ) B(Lp (R+ ,xα−ε )) x−γξ χl f Lp (R+ ,xα−ε )

+ W (a+ (ξ)χ+ + a− (ξ)χ− ) B(Lp (R+ ,xβ+ε )) x−γξ χl f Lp (R+ ,xβ+ε ) < ∞,

(4.24)

we infer from (4.18) that

lim Akn − xγξ I W (a+ (ξ)χ+ + a− (ξ)χ− )x−γξ χl f Lp (R+ ) = 0.

(4.25)

By analogy with (4.24), we obtain

W (a(x/kn )) − W (a+ (ξ)χ+ + a− (ξ)χ− ) x−γξ χl f L

≤ W (a(x/kn )) − W (a+ (ξ)χ+ + a− (ξ)χ− ) x−γξ χl f Lp (R+ ,xα−ε )

+ W (a(x/kn )) − W (a+ (ξ)χ+ + a− (ξ)χ− ) x−γξ χl f Lp (R+ ,xβ+ε )

(4.26)

n→∞

and

W (a(x/kn )) A−1 − x−γξ I χl f kn

L

−γξ ≤ W (a(x/kn )) B(Lp (R+ ,xα−ε )) A−1 I χl f Lp (R+ ,xα−ε ) kn − x

−γξ + W (a(x/kn )) B(Lp (R ,xβ+ε )) A−1 I χl f Lp (R ,xβ+ε ) . kn − x +

+

(4.27)

Since x−γξ χl f ∈ Lp (R+ , xα−ε ) ∩ Lp (R+ , xβ+ε ) and since, by Lemma 4.2, s-lim W (a(x/kn )) = W (a+ (ξ)χ+ + a− (ξ)χ− ) n→∞

on both the spaces Lp (R+ , xα−ε ) and Lp (R+ , xβ+ε ), we conclude from (4.26) that

lim W (a(x/kn )) − W (a+ (ξ)χ+ + a− (ξ)χ− ) x−γξ χl f L = 0. (4.28) n→∞

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Applying (4.20), we infer from the estimate

−1

A − x−γξ I χl f p kn L (R+ ,xα−ε )

−(v(k x)−v(k )−γ ln x)

n n ξ = e − 1 x−γξ χl f Lp (R ,xα−ε ) + ≤ esupx∈l |v(kn x)−v(kn )−γξ ln x| − 1 kx−γξ χl f kLp (R+ ,xα−ε ) that

−γξ lim A−1 I χl f Lp (R kn − x

= 0.

(4.29)

−γξ lim A−1 I χl f Lp (R+ ,xβ+ε ) = 0. kn − x

(4.30)

n→∞

+ ,x

α−ε )

Analogously, n→∞

Since the operators W (a(x/kn )) are uniformly bounded for a ∈ V (R) on both the spaces Lp (R+ , xα−ε ) and Lp (R+ , xβ+ε ), we deduce from (4.27), (4.29) and (4.30) that

−γξ lim W (a(x/kn )) A−1 I χl f L = 0. (4.31) kn − x n→∞

Finally, as the operators Akn ∈ B(L, Lp (R+ )) are uniformly bounded, we infer from (4.25), (4.28) and (4.31) that every summand on the right of (4.23) tends to zero as n → ∞, which completes the proof.

5. Local Fredholm study Consider the Laguerre polynomials Ln (x) :=

n X n (−1)k k 1 x dn n −x e (x e ) = x n n! dx k k!

(n = 0, 1, 2, . . .).

k=0

As is known (see, e.g. [33, Chapter 5]), the functions √ ψn (x) := 2 Ln (2x) e−x (n = 0, 1, 2, . . .)

(5.1)

form an orthogonal basis in L2 (R+ ). By [8, p. 487] and [16, Chapter I, § 8.3], ψn = W n (x − i)/(x + i) ψ0 for all n ∈ N. (5.2) Obviously, ψn ∈ Lp (R+ , w) for n = 0, 1, 2, . . ., all p ∈ (1, ∞) and all w ∈ Ap (R+ ). By analogy with [23, p. 404], one can prove the following. Lemma 5.1. If 1 < p < ∞ and w ∈ Ap (R+ ), then the linear hull of the system {ψ0 , ψ1 , ψ2 , . . .} given by (5.1) is dense in the space Lp (R+ , w). Proof. According to (5.1), it is sufficient to prove that the linear hull of the system {xn e−x : n = 0, 1, 2, . . .} ⊂ Lp (R+ , w) is dense in Lp (R+ , w). Suppose this is false. Then, by the Hahn-Banach theorem, there exists a non-zero function h ∈ Lq (R+ , w−1 ) such that Z xn e−x h(x)dx = 0 for all n = 0, 1, 2, . . . . (5.3) R+

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Let f be the extension of the function x 7→ e−x h(x) by zero to the whole R. Since eδ|x| f ∈ L1 (R) for every δ ∈ [0, 1), the function Z g(λ) = f (x)e−iλx dx (λ ∈ R) R

admits an analytic extension to the strip Π := {z ∈ C : |Im z| < 1} (see [23, Chapter VIII, § 4]). By (5.3), for all n = 0, 1, 2, . . . it follows that Z (n) n g (0) = (−1) xn f (x)dx = 0. R

Hence, the analytic function g equals zero for all z ∈ Π, and therefore h(x) = 0 for all x ∈ R+ , which is impossible. Thus, the span of the system {ψ0 , ψ1 , ψ2 , . . .} is dense in the space Lp (R+ , w). ˙ denote the smallest Given 1 < p < ∞ and w ∈ Ap (R+ ), let alg p,w W (C(R)) p ˙ closed subalgebra of B(L (R+ , w)) containing the set {W (c) : c ∈ Cp,w (R)}. Taking into account (5.2), applying Lemma 5.1 and literally repeating the proof of [8, p. 487] for Lp (R+ , w) in place of Lp (R+ ), we obtain the following. ˙ Lemma 5.2. For p ∈ (1, ∞) and w ∈ Ap (R+ ), the Banach algebra alg p,w W (C(R)) p p contains the ideal K(L (R+ , w)) of all compact operators in B(L (R+ , w)). We also need the following compactness results. Lemma 5.3. If a ∈ P C and b ∈ SOp,w where 1 < p < ∞ and w ∈ Ap (R), then the commutator [aI, W 0 (b)] = aW 0 (b) − W 0 (b)aI is compact on the space Lp (R, w). Proof. Since the algebra SO3 is dense in every Banach algebra SOp,w (1 < p < ∞, w ∈ Ap (R)), it is sufficient to prove the compactness of the commutator [aI, W 0 (b)] on the spaces Lp (R, w) for a ∈ P C and b ∈ SO3 . By [2, Theorem 4.2], for such a and b, the commutator [aI, W 0 (b)] is compact on every space Lp (R) (1 < p < ∞). Since this operator is bounded on all the spaces Lp (R, w) with w ∈ Ap (R), we infer by analogy with [24, Theorem 3.10] and [22, Theorem 3.2] that the commutator [aI, W 0 (b)] is compact on all the spaces Lp (R, w) (1 < p < ∞, w ∈ Ap (R)). Lemma 5.4. If a, b ∈ [SOp,w , Cp,w (R)] where 1 < p < ∞ and w ∈ Ap (R), then the commutator [W (a), W (b)] is compact on the space Lp (R+ , w). Proof. If b ∈ SOp,w where 1 < p < ∞ and w ∈ Ap (R), from Lemma 5.3 it follows that the operator χ+ W 0 (b) − W 0 (b)χ+ I is compact on the space Lp (R, w). Hence, for every a ∈ [SOp,w , Cp,w (R)] and every b ∈ SOp,w , we obtain χ+ W 0 (a)χ+ W 0 (b)χ+ I ' χ+ W 0 (ab)χ+ ' χ+ W 0 (b)χ+ W 0 (a)χ+ I,

(5.4)

where A ' B means that A − B is a compact operator. Obviously, (5.4) implies that for every a ∈ [SOp,w , Cp,w (R)] and every b ∈ SOp,w , W (a)W (b) − W (ab), W (b)W (a) − W (ab) ∈ K(Lp (R+ , w)), p

W (a)W (b) − W (b)W (a) ∈ K(L (R+ , w)).

(5.5) (5.6)

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e where c, d, e If both a, b ∈ [SOp,w , Cp,w (R)], then a = c + du and b = e c + du c, de ∈ SOp,w and u(x) = tanh x for x ∈ R. Consequently, (5.5) and (5.6) imply that e (u) W (a)W (b) ' W (c) + W (d)W (u) W (e c) + W (d)W e (u) + W (dd)[W e ' W (ce c) + W (de c + cd)W (u)]2 e (u) W (c) + W (d)W (u) ' W (b)W (a) ' W (e c) + W (d)W for all a, b ∈ [SOp,w , Cp,w (R)], which completes the proof.

Given p ∈ (1, ∞), w ∈ Ap (R) and N ∈ N, we consider the Banach subalgebras n o A = alg W (a) : a ∈ [SOp,w , Cp,w (R)]N ×N , n o Z = alg W (bIN ) : b ∈ SOp,w of B(LpN (R+ , w)) generated by the operators W (a) with a ∈ [SOp,w , Cp,w (R)]N ×N and a = bIN (b ∈ SOp,w ), respectively, where IN is the N × N identity matrix. We also consider the quotient Banach algebras Aπ := A/K and Z π := (Z + K)/K with elements W π (a) := W (a) +K, where K = K(LpN (R+ , w)) ⊂ A by Lemma 5.2. Obviously, (5.6) implies that Z π is a central subalgebra of Aπ . Let Λ = Λ(Z) stand for the Banach subalgebra of B(LpN (R+ , w)) consisting of all operators of local type with respect to Z, that is, n o Λ := A ∈ B(LpN (R+ , w)) : W (bIN )A − AW (bIN ) ∈ K for all b ∈ SOp,w . For every ξ ∈ M(SO) = M(SOp,w ), let Jξπ and Jeξπ be the smallest closed two-sided ideals of the Banach algebra Aπ and Λπ , respectively, that contain the maximal ideal n o Iξπ := W π (bIN ) : b ∈ SOp,w , ξ(b) = 0 (5.7) of the commutative algebra Z π . Consider the quotient Banach algebras Aπξ := Aπ /Jξπ and Λπξ := Λπ /Jeξπ . Then for a ∈ [SOp,w , Cp,w (R)]N ×N it follows that Wξπ (a) := W π (a) + Jξπ ∈ Aπξ ,

fξπ (a) := W π (a) + Jeξπ ∈ Λπξ . W

The Banach algebra Λπ := Λ/K is inverse closed in the Calkin algebra B π := B/K, Aπ ⊂ Λπ , and Z π is a central subalgebra of Λπ . Thus, the operator W (a) ∈ A with a symbol a ∈ [SOp,w , Cp,w (R)]N ×N is Fredholm on the space LpN (R+ , w) if and only if the coset W π (a) is invertible in the quotient algebra Λπ . By the AllanDouglas local principle (see, e.g. [8, Theorem 1.35]), the coset W π (a) is invertible f π (a) is invertible in the quotient algebra in the algebra Λπ if and only if the coset W ξ π Λξ for every ξ ∈ M(SO), which gives a Fredholm criterion for the operator W (a). On the other hand, the coset W π (a) is invertible in the quotient algebra Aπ if and only if the coset Wξπ (a) is invertible in the quotient algebra Aπξ for every ξ ∈ M(SO), which gives sufficient Fredholm conditions for the operators W (a) and guarantees that the regularizers for such operators belong to the algebra A.

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Lemma 5.5. If a ∈ [SOp,w , Cp,w (R)]N ×N where 1 < p < ∞, w ∈ Ap (R) and N ∈ N, then for every ξ ∈ R the following assertions are equivalent: (i) the coset Wξπ (a) is invertible in the quotient algebra Aπξ , f π (a) is invertible in the quotient algebra Λπ , (ii) the coset W ξ ξ (iii) det a(ξ) 6= 0. Proof. Given a ∈ [SOp,w , Cp,w (R)]N ×N and ξ ∈ R, let us show that W π (a) − W π (a(ξ)) = W π (a − a(ξ)) ∈ Jξπ ,

(5.8)

is represented in the form n Xn o Jξπ = closAπ Aπi Biπ : Aπi ∈ Aπ , Biπ ∈ Iξπ , n ∈ N .

(5.9)

where the ideal

Jξπ

i=1

Because the matrix function e a := a − a(ξ) belongs to [SOp,w , Cp,w (R)]N ×N along with a, we conclude that W π (e a) ∈ Aπ . Moreover, e a(ξ) = 0N where 0N is the N × N zero matrix. By (3.6), e a is represented in the form e a = a+ u+ + a− u− , where a± ∈ [SOp,w ]N ×N and the functions u± : x 7→ 2−1 [1 ± tanh x] are in Cp,w (R) and satisfy (3.5). Hence we obtain e a = [a+ − a+ (ξ)]u+ + [a− − a− (ξ)]u− + a+ (ξ)[u+ − u+ (ξ)] + a− (ξ)[u− − u− (ξ)], = [a+ − a+ (ξ)]u+ + [a− − a− (ξ)]u− + [a+ (ξ) − a− (ξ)] [c+ u+ − c− u− ], where the matrix functions a+ − a+ (ξ) and a− − a− (ξ) belong to [SOp,w ]N ×N and equal 0N at the point ξ, and the functions c± =

u2 u− u+ u2− u+ (ξ)u− (ξ) − u+ u− + + − u± (ξ) u+ (ξ) u− (ξ)

˙ ⊂ SOp,w and vanish at the point ξ. Hence are in Cp,w (R) W π (e a) = W π (a+ − a+ (ξ))W π (u+ ) + W π (a− − a− (ξ))W π (u− ) + W π (a+ (ξ) − a− (ξ)) W π (c+ )W π (u+ ) − W π (c− )W π (u− ) .

(5.10)

As c± (ξ) = 0, the cosets W π (c± ) belong to the ideal Iξπ given by (5.7). Let Ei,j be the N × N matrix whose (i, j)-entry is 1 and all other entries equal zero. Since for every matrix function b = (bi,j )N i,j=1 ∈ [SOp,w ]N ×N with b(ξ) = 0N , XN XN W π (b) = W π (Ei,j )W π (bi,j ) i=1

j=1

where W π (Ei,j ) ∈ Aπ and W π (bi,j ) ∈ Iξπ , we conclude from (5.9) that W π (b) ∈ Jξπ . Hence the cosets W π (a± − a± (ξ)) are in the ideal Jξπ . Therefore, from (5.10) it follows that W π (e a) ∈ Jξπ too, which gives (5.8). Since W (a(ξ)) = χ+ F −1 a(ξ)F = a(ξ)I, we have W π (a(ξ)) = [a(ξ)I]π . Hence we infer from (5.8) and Jξπ ⊂ Jeξπ that for every ξ ∈ R, Wξπ (a) = W π (a(ξ)) + Jξπ = [a(ξ)I]π + Jξπ , f π (a) = W π (a(ξ)) + Jeπ = [a(ξ)I]π + Jeπ , W ξ ξ ξ

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which immediately implies the assertion of the lemma.

227

Let GA denote the group of invertible elements in a unital algebra A. Lemma 5.6. If a ∈ [SOp,w , Cp,w (R)]N ×N and the Wiener-Hopf operator W (a) is Fredholm on the space LpN (R+ , w), then a ∈ G[SOp,w , Cp,w (R)]N ×N . Proof. Fix a ∈ [SOp,w , Cp,w (R)]N ×N . By Lemma 3.7, a = a+ u+ + a− u− where a± ∈ [SOp,w ]N ×N and u± (x) = 2−1 [1 ± tanh x] for x ∈ R. Then for every ξ ∈ M∞ (SO) there is a sequence {hn } ⊂ R+ such that hn → +∞ as n → ∞ and lim a± (hn ) = a± (ξ), which implies that lim a(±hn ) = lim a± (hn ) = a± (ξ). n→∞ n→∞ n→∞ Hence, by analogy with Lemma 4.2, we infer that (5.11) s-lim e±ixhn W (a)e∓ixhn I = s-lim W (a(· ± hn )) = a± (ξ)I. n→∞

n→∞

Suppose the operator W (a) is Fredholm on the space LpN (R+ , w). Then, by analogy with [7, Corollary 18.11], from (5.11) it follows that for every ξ ∈ M∞ (SO) the limit operators a± (ξ)I are invertible on the space LpN (R+ , w), and therefore + det a(ξ) 6= 0 for all ξ ∈ M+ ∞ (SO)∪M∞ (SO) (see Lemma 3.8). On the other hand, f π (a) = a(ξ)I π +Jeπ by the Allan-Douglas local principle, for every ξ ∈ R the coset W ξ ξ π is invertible in the quotient algebra Λξ , and hence, by Lemma 5.5, det a(ξ) 6= 0 for all ξ ∈ R. Consequently, by Lemma 3.8, det a(ξ) 6= 0 for all ξ ∈ M([SO, C(R)]), which implies that a is invertible in the algebra [SOp,w , Cp,w (R)]N ×N . Corollary 5.7. Let b ∈ [SOp,w ]N ×N . Then the following assertions are equivalent: (i) (ii) (iii) (iv)

the Wiener-Hopf operator W (b) is Fredholm on the space LpN (R+ , w), the matrix function b is invertible in the Banach algebra [SOp,w ]N ×N , inf x∈R | det b(x)| > 0, det b(ξ) 6= 0 for every ξ ∈ M(SOp,w ) = M(SO).

Proof. The inverse closedness of the algebra SOp,w in L∞ (R) implies the equivalence (ii)⇔(iii). The equivalence (ii)⇔(iv) follows from the Gelfand theory for det b ∈ SOp,w . Lemma 5.6 and the inverse closedness of [SOp,w ]N ×N in the algebra [SOp,w , Cp,w (R)]N ×N (see Corollary 3.5) give the implication (i)⇒(ii). Finally, if b is invertible in [SOp,w ]N ×N , then the operator W (b−1 ) is a regularizer of the operator W (b) in view of (5.5), which proves the implication (ii)⇒(i). Let 1 < p < ∞ and w ∈ Ap (R). Then each of the sets n o λ Ix (p, w) := λ ∈ R : (ξ − x)/(ξ + i) w(ξ) ∈ Ap (R) (x ∈ R), n o I∞ (p, w) := λ ∈ R : |ξ + i|−λ w(ξ) ∈ Ap (R) is an open interval of length not greater than 1 which contains the origin. Thus, ˙ Ix (p, w) = (−νx− (p, w), 1 − νx+ (p, w)) (x ∈ R)

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˙ where 0 < νx− (p, w) ≤ νx+ (p, w) < 1 (see, e.g., [9, Theorem 2.10]). For every x ∈ R, the numbers νx± (p, w) are closely related to the indices of powerlikeness αx (w) and βx (w) of the weight w ∈ Ap (R) (see [5, Section 3.6]): νx− (p, w) = 1/p + αx (w),

νx+ (p, w) = 1/p + βx (w).

In particular, if w = ev ∈ Ap (R+ ) where p ∈ (1, ∞) and the function σ : t 7→ tv 0 (t) belongs to the C ∗ -algebra SO0 , then in view of (4.6) we get ν0− (p, w) = 1/p + lim inf [tv 0 (t)], t→0

ν0+ (p, w) = 1/p + lim sup [tv 0 (t)].

(5.12)

t→0

Further, for all ν ∈ (0, 1), we consider the circular arcs o n Lν := µ = 2−1 (1 + coth[π(x + iν)]) : x ∈ R with the endpoints 0 and 1, where L1/2 = [0, 1], and also the horn [ Hp,w := H(0, 1; ν0− (p, w), ν0+ (p, w)) := Lν .

(5.13)

(5.14)

ν∈[ν0− (p,w),ν0+ (p,w)]

By [17, Vol. 2] (also see [8, p. 255]), with every θ ∈ (−π, π) we associate the number p(θ) := 2π/(π − θ) and the homeomorphism ϕθ : [0, 1] → L1/p(θ) given by ( 2iθt sin(θt) iθ(t−1) −1 if θ ∈ (−π, π) \ {0}, e = ee2iθ −1 sin θ ϕθ (t) = (5.15) t if θ = 0. Taking ν = 1/p(θ), we get θ = π − 2πν, and therefore ϕπ−2πν maps [0, 1] onto Lν and ϕπ−2πν (0) = 0, ϕπ−2πν (1) = 1. By (5.15), we obtain 1 − ϕπ−2πν (1 − t) = ϕ−π+2πν (t)

for all t ∈ [0, 1],

(5.16)

where ϕ−π+2πν maps [0, 1] onto L1−ν and ϕ−π+2πν (j) = j for j ∈ {0, 1}. Hence, for every ν ∈ (0, 1), 1 − Lν = L1−ν , (5.17) and the arc Lν in contrast to L1−ν is traced in the opposite direction: from 1 to 0. Let ξ ∈ M∞ (SO) and let a = a+ u+ + a− u− ∈ [SOp,w , Cp,w (R)]N ×N where a± ∈ [SOp,w ]N ×N and u± (x) = 2−1 [1 ± tanh x] on R. Since W π (a± − a± (ξ)) ∈ Jξπ and W π (u± ) ∈ Aπ , we infer that W π (a) − W π a+ (ξ)u+ − a− (ξ)u− = W π (a+ − a+ (ξ))W π (u+ ) + W π (a− − a− (ξ))W π (u− ) ∈ Jξπ ⊂ Jeξπ . Thus, we get Wξπ (a) = Wξπ (a+ (ξ)u+ + a− (ξ)u− ),

f π (a) = W f π (a+ (ξ)u+ + a− (ξ)u− ). W ξ ξ

Lemma 5.8. If a ∈ [SOp,w , Cp,w (R)]N ×N where 1 < p < ∞, w ∈ Ap (R) and N ∈ N, then for every ξ ∈ M∞ (SO) the assertions (i),(ii) below are equivalent and imply (iii),(iv):

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˙ N ×N such that a0 (∞) = 0N and the operator (i) there is an a0 ∈ [Cp,w (R)] W (a+ (ξ)u+ + a− (ξ)u− + a0 ) is Fredholm on the space LpN (R+ , w); 1 − coth[π(x + iν)] 1 + coth[π(x + iν)] (ii) det a+ (ξ) + a− (ξ) 6= 0 2 2 − + for every x ∈ R and every ν ∈ [ν0 (p, w), ν0 (p, w)]; (iii) the coset Wξπ (a+ (ξ)u+ + a− (ξ)u− ) is invertible in the quotient algebra Aπξ ; f π (a+ (ξ)u+ + a− (ξ)u− ) is invertible in the quotient algebra Λπ . (iv) the coset W ξ ξ Proof. (i)⇔(ii). If the operator W (a+ (ξ)u+ + a− (ξ)u− + a0 ) is Fredholm on the ˙ with a0 (∞) = 0N , then [7, Theorem 17.10] space LpN (R+ , w) for some a0 ∈ Cp,w (R) implies assertion (ii) according to (5.13) and (5.14). ˙ Conversely, if (ii) holds, then there exists a matrix function a0 ∈ [C p,w (R)]N ×N such that a0 (∞) = 0N and the function det a+ (ξ)u+ + a− (ξ)u− + a0 is separated from zero. In this case [7, Theorem 17.10] implies that W (a+ (ξ)u+ +a− (ξ)u− +a0 ) is a Fredholm operator on the space LpN (R+ , w), that is, assertion (i) holds. (i)⇒(iii)⇒(iv). It follows from [7, Theorem 17.9] that the Banach algebra alg p,w W π (C(R)) ⊂ Aπ generated by all cosets W π (b) with b ∈ [Cp,w (R)]N ×N is inverse closed in the algebra Λπ . Hence, if the operator W (a+ (ξ)u+ +a− (ξ)u− +a0 ) ˙ N ×N with a0 (∞) = is Fredholm on the space LpN (R+ , w) for some a0 ∈ [Cp,w (R)] 0N , then the coset W π (a+ (ξ)u+ + a− (ξ)u− + a0 ) is invertible in the quotient algebra Aπ . By the Allan-Douglas local principle, the latter is equivalent to the invertibility of the cosets Wηπ (a+ (ξ)u+ + a− (ξ)u− + a0 ) in the quotient algebras Aπη for all η ∈ M(SO). In particular, for ξ ∈ M∞ (SO) the coset Wξπ (a+ (ξ)u+ + a− (ξ)u− ) = Wξπ (a+ (ξ)u+ + a− (ξ)u− + a0 ) is invertible in the algebra Aπξ . Finally, as Aπ ⊂ Λπ and Jξπ ⊂ Jeξπ , the invertibility of the coset Wξπ (a+ (ξ)u+ + a− (ξ)u− ) in the algebra Aπξ implies the invertibility f π (a+ (ξ)u+ + a− (ξ)u− ) in the algebra Λπ . of the coset W ξ ξ

6. Fredholm theory Lemmas 5.5 and 5.8 and the Allan-Douglas local principle applied to the algebra Λπ immediately give the following sufficient Fredholm condition. Theorem 6.1. Let 1 < p < ∞, w ∈ Ap (R), N ∈ N and a ∈ [SOp,w , Cp,w (R)]N ×N . Then the Wiener-Hopf operator W (a) is Fredholm on the space LpN (R+ , w) if a ∈ G[SOp,w , Cp,w (R)]N ×N and 1 − coth[π(x + iν)] 1 + coth[π(x + iν)] det a+ (ξ) + a− (ξ) 6= 0 (6.1) 2 2 for every ξ ∈ M∞ (SO), every x ∈ R, and every ν ∈ [ν0− (p, w), ν0+ (p, w)]. For special Muckenhoupt weights we obtain the following Fredholm criterion.

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Theorem 6.2. Let a ∈ [SOp,w , Cp,w (R)]N ×N where 1 < v ∈ C 1 (R+ ), the function σ : t 7→ tv 0 (t) belongs to the and let n [ a+ (ξ)(1 − µ) + a− (ξ)µ : ξ∈M∞ (SO) n [ = a+ (ξ)(1 − µ) + a− (ξ)µ :

p < ∞, w = ev ∈ Ap (R), C ∗ -algebra SO0 , N ∈ N,

ξ∈M∞ (SO)

µ ∈ Lνξ

o

o µ ∈ Hp,w ,

(6.2)

where νξ := 1/p + ξ(e σ ) ∈ [ν0− (p, w), ν0+ (p, w)], the function σ e : x 7→ σ(1/|x|) e−1/|x| is in SO, and Lν and Hp,w are given by (5.13) and (5.14), respectively. Then the Wiener-Hopf operator W (a) is Fredholm on the space LpN (R+ , w) if and only if a ∈ G[SOp,w , Cp,w (R)]N ×N and, for every ξ ∈ M∞ (SO) and every x ∈ R, 1 + coth[π(x + iνξ )] 1 − coth[π(x + iνξ )] + a− (ξ) 6= 0. (6.3) det a+ (ξ) 2 2 If W (a) is Fredholm, then all its regularizers belong to the algebra A. Proof. Sufficiency follows from Theorem 6.1 because the relation (6.3), fulfilled for every ξ ∈ M∞ (SO) and every x ∈ R, implies in view of (6.2) that (6.1) holds for every ξ ∈ M∞ (SO), every x ∈ R, and every ν ∈ [ν0− (p, w), ν0+ (p, w)]. Necessity. Suppose the operator W (a) with symbol a ∈ [SOp,w , Cp,w (R)]N ×N is Fredholm on the space LpN (R+ , w). By Lemma 3.7, a = a+ u+ + a− u− , where a± ∈ [SOp,w ]N ×N and u± (x) = [1 ± tanh x]/2 for x ∈ R. Identifying the fibers M0 (SO0 ) and M∞ (SO) as in Section 4, for every ξ ∈ M∞ (SO) we can choose a sequence {kn } ⊂ R+ satisfying (4.10). Then from Theorem 4.3 it follows that s-lim Ukn ev W (a)e−v Uk−1 = xγξ W (a+ (ξ)χ+ + a− (ξ)χ− )x−γξ I, n n→∞

where v = ln w, γξ := ξ(e σ ) and a± (ξ) = ξ(a± ) for ξ ∈ M∞ (SO), and χ± are the characteristic functions of R± . By analogy with [7, Corollary 18.11], the Fredholmness of the operator ev W (a)e−v I on the space LpN (R+ ) implies that the operator W (a+ (ξ)χ+ + a− (ξ)χ− ) is invertible on the space LpN (R+ , xγξ ) for every ξ ∈ M∞ (SO), which gives (6.3) for every ξ ∈ M∞ (SO) and every x ∈ R by the matrix analogue of [7, Theorem 17.7] with νξ = 1/p + γξ . On the other hand, by Lemma 5.6, a ∈ G[SOp,w , Cp,w (R)]N ×N , which completes the proof of necessity. Finally, if a ∈ G[SOp,w , Cp,w (R)]N ×N and (6.3) holds for all ξ ∈ M∞ (SO) and all x ∈ R, then Lemmas 5.5 and 5.8 imply due to (6.2) that for each ξ ∈ M(SO) the coset Wξπ (a) is invertible in the quotient algebra Aπξ . Applying now the Allan-Douglas local principle to the algebra Aπ , we deduce that the regularizers of any Fredholm operator W (a) with symbol a ∈ [SOp,w , Cp,w (R)]N ×N belong to the algebra A. S Since Hp,w = ν∈[ν − (p,w),ν + (p,w)] Lν due to (5.14), we conclude that (6.2) 0

0

always holds if a ∈ [Cp,w (R)]N ×N . On the other hand, if ν0− (p, w) = ν0+ (p, w), then for any a ∈ [SOp,w , Cp,w (R)]N ×N the equality (6.2) also holds.

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Consider now less trivial examples related to (6.2). Let w = ev ∈ Ap (R), where v ∈ C 1 (R+ ), the function σ : t 7→ tv 0 (t) in SO0 is such that the function σ e(x) = σ(1/|x|) e−1/|x| belongs to SOp,w , and 0 < ν0− (p, w) := 1/p + lim inf [tv 0 (t)] < 1/p + lim sup [tv 0 (t)] =: ν0+ (p, w) < 1. t→0

t→0

Taking a = a+ u+ + a− u− ∈ [SOp,w , Cp,w (R)] such that u± (x) = [1 ± tanh x]/2 and a± (x) = ±1 + i(1/p + σ e(x)) ∈ SOp,w , we infer that for every ξ ∈ M∞ (SO), o n o n (6.4) a+ (ξ)(1 − µ) + a− (ξ)µ : µ ∈ Lνξ = N (νξ , νξ , x) : x ∈ R , where νξ = 1/p + ξ(e σ ) ∈ [ν0− (p, w), ν0+ (p, w)], Lν is given by (5.13) and 1 − coth[π(x+iν)] 1 + coth[π(x+iν)] + (−1 + iλ) 2 2 = iλ − coth[π(x+iν)]. (6.5)

N (λ, ν, x) := (1 + iλ)

Applying (6.5), we conclude that Im N (ν1 , ν1 , x) ≤ min Im N (ν1 , ν2 , x), Im N (ν2 , ν1 , x) ≤ max Im N (ν1 , ν2 , x), Im N (ν2 , ν1 , x) ≤ Im N (ν2 , ν2 , x) for ν0− (p, w) ≤ ν1 < ν2 ≤ ν0+ (p, w) and x ∈ R, which implies in view of (6.4) that (6.2) holds. On the other hand, if a± (x) = ±1 − i(1/p + σ e(x)), then n o n o a+ (ξ)(1 − µ) + a− (ξ)µ : µ ∈ Lνξ = N (−νξ , νξ , x) : x ∈ R for every ξ ∈ M∞ (SO), and therefore (6.2) does not hold because by (6.5), Im N (−ν2 , ν1 , x) < min Im N (−ν1 , ν1 , x), Im N (−ν2 , ν2 , x) ≤ max Im N (−ν1 , ν1 , x), Im N (−ν2 , ν2 , x) < Im N (−ν1 , ν2 , x) for ν0− (p, w) ≤ ν1 < ν2 ≤ ν0+ (p, w) and x ∈ R. If ν0− (p, w) = ν0+ (p, w), then ν0± (p, w) = 1/p + lim [tv 0 (t)] in view of (5.12), t→0

and Theorem 6.2 implies the following result. Corollary 6.3. Let a ∈ [SOp,w , Cp,w (R)]N ×N where 1 < p < ∞, w = ev ∈ Ap (R), v ∈ C 1 (R+ ), the function σ : t 7→ tv 0 (t) belongs to the C ∗ -algebra SO0 , N ∈ N, and ν0− (p, w) = ν0+ (p, w). Then the Wiener-Hopf operator W (a) is Fredholm on the space LpN (R+ , w) if and only if a ∈ G[SOp,w , Cp,w (R)]N ×N and 1 − coth[π(x + iν0 )] 1 + coth[π(x + iν0 )] det a+ (ξ) + a− (ξ) 6= 0 2 2 for every ξ ∈ M∞ (SO) and every x ∈ R, where ν0 := ν0± (p, w) = 1/p + lim [tv 0 (t)]. t→0

If W (a) is Fredholm, then all its regularizers belong to the algebra A. We now proceed to establishing an index formula for operators W (a).

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Lemma 6.4. If 1 < p < ∞, w ∈ Ap (R+ ), N ≥ 1, and a is an even matrix function in [SOp,w ]N ×N , then the operator W (a) is Fredholm on the space LpN (R+ , w) if and only if a ∈ G[SOp,w ]N ×N . If W (a) is Fredholm, then Ind W (a) = 0. Proof. By Corollary 5.7, the operator W (a) is Fredholm on the space LpN (R+ , w) if and only if a ∈ G[SOp,w ]N ×N . Thus, it remains to prove that Ind W (a) = 0. With the operator W (a) ∈ B(LpN (R+ , w)) we associate the operator A := χ+ F −1 aFχ+ I + χ− I ∈ B(LpN (R, w)) e

(6.6)

where the weight w e ∈ Ap (R) is defined by w(x) e = w(|x|) for x ∈ R. As is well known, the operator A is Fredholm along with W (a), and Ind A = Ind W (a). B(LpN (R, w)) e

Consider the operator V ∈ Since a(−x) = a(x), we deduce that

(6.7)

given by (V f )(x) = f (−x) for x ∈ R.

e := V AV −1 = V (χ+ F −1 aFχ+ I + χ− I)V −1 = χ− F −1 aFχ− I + χ+ I. A

(6.8)

e is Fredholm along with A, and Hence the operator A e = Ind A. Ind A

(6.9)

Further, we deduce from (6.6) and (6.8) that e = F −1 aF − (χ+ F −1 aFχ− I + χ− F −1 aFχ+ I). AA

(6.10)

By [2, Theorem 4.2], the operators χ+ F −1 aFχ− I and χ− F −1 aFχ+ I are compact on all the spaces LpN (R) (1 < p < ∞). Since these operators are bounded on all the spaces LpN (R, w) with 1 < p < ∞ and w ∈ Ap (R), we infer by analogy with [24, Theorem 3.10] and [22, Theorem 3.2] that the operators χ+ F −1 aFχ− I and χ− F −1 aFχ+ I are compact on all the spaces LpN (R, w) (1 < p < ∞, w ∈ Ap (R)). Hence, by (6.10), the operator W 0 (a) = F −1 aF is Fredholm, and e Ind W 0 (a) = Ind A + Ind A.

(6.11)

Because W 0 (a) is invertible in case of Fredholmness and therefore Ind W 0 (a) = 0, ˜ = 0, which together with we infer from (6.9) and (6.11) that 2 Ind A = Ind (AA) (6.7) completes the proof. With every matrix function a ∈ [SOp,w , Cp,w (R)]N ×N and every r > 0 we associate the matrix functions ar ∈ [Cp,w (R)]N ×N given by ( a(x) if |x| ≤ r, ar (x) = (6.12) a(±r) if ± x > r, Let us calculate Ind W (a) in terms of Ind W (ar ).

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Theorem 6.5. Let 1 0 the Wiener-Hopf operators W (ar ) with symbols ar ∈ [Cp,w (R)]N ×N given by (6.12) are Fredholm on the space LpN (R+ , w) along with W (a), and Ind W (a) = lim Ind W (ar ). r→∞

(6.13)

Proof. By Theorem 6.1, the operator W (a) is Fredholm on the space LpN (R+ , w) if the following conditions hold: inf | det a(t)| > 0, t∈R det a+ (ξ)(1 − µ) + a− (ξ)µ 6= 0 for all (ξ, µ) ∈ M∞ (SO) × Hp,w ,

(6.14) (6.15)

where Hp,w is given by (5.14). By (6.14), the even matrix function e a given on R by e a(x) = a(|x|) belongs to G[SOp,w ]N ×N . Hence, by Lemma 6.4, the operator W (e a) is Fredholm on the space LpN (R+ , w) and Ind W (e a) = 0. Let b := e a−1 a. By (6.14) and (6.15), b ∈ G[SOp,w , Cp,w (R)]N ×N and det IN (1 − µ) + b− (ξ)µ 6= 0 for all (ξ, µ) ∈ M∞ (SOp,w ) × Hp,w ,

(6.16)

(6.17)

where b− (ξ) = a−1 + (ξ)a− (ξ) and IN is the N × N identity matrix. Then, by Theorem 6.1, the operator W (b) is Fredholm on the space LpN (R+ , w). Moreover, as W (a) − W (e a)W (b) is a compact operator, from (6.16) it follows that Ind W (a) = Ind W (b).

(6.18)

Since the set [

det IN (1 − µ) + b− (ξ)µ

(ξ,µ)∈M∞ (SO)×Hp,w

is compact on C

N ×N

, in view of (6.17) there exists an ε > 0 such that det IN (1 − µ) + (b− (ξ) + c)µ 6= 0

(6.19)

for all ξ ∈ M∞ (SO), all µ ∈ Hp,w and all matrices c ∈ CN ×N with kckCN ×N < ε. Further, in view of (6.19) and Proposition 3.1, there exists an x0 < 0 such that det IN (1 − µ) + b(x)µ 6= 0 for all x ≤ x0 and all µ ∈ Hp,w . (6.20) Let ϕ := ϕ−π+2πν00 (p,w) , where ν00 (p, w) = 2−1 ν0− (p, w) + ν0+ (p, w) and ϕθ is defined by (5.15). Hence ϕ is a homeomorphism of [0, 1] onto L1−ν00 (p,w) such that ϕ(0) = 0 and ϕ(1) = 1. For every z ∈ [0, 1], we define the continuous matrix function   if x < x0 − 1, IN ϕ(z) + b(x)(1 − ϕ(z)) b(z) (x) := IN ϕ(z|x − x0 |) + b(x)(1 − ϕ(z|x − x0 |)) if x ∈ [x0 − 1, x0 ],   b(x) if x > x0 ,

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where b(x) = IN for all x ≥ 0. Obviously, the matrix functions b(z) belong to [SOp,w , Cp,w (R)]N ×N for every z ∈ [0, 1]. Moreover, b(z) ∈ G[SOp,w , Cp,w (R)]N ×N for all z ∈ [0, 1] according to (6.20), (5.16) and (5.17). We claim that the operator function [0, 1] → B(LpN (R+ , w)), z 7→ W (b(z) ), is continuous. Indeed, for all z, ζ ∈ [0, 1] we get W (b(z) ) − W (b(ζ) ) = W (IN − b)ηz,ζ , (6.21) where, for x ∈ R, ηz,ζ (x) := ϕ(z) − ϕ(ζ) χ(−∞,x0 −1) (x) + ϕ(z|x − x0 |) − ϕ(ζ|x − x0 |) χ[x0 −1,x0 ] (x) and χγ is the characteristic function of a set γ. If ϕ = ϕ0 , then from (5.15) it follows that kηz,ζ kL∞ (R) = |z − ζ|, V1 (ηz,ζ ) = |z − ζ|, and, by Stechkin’s inequality (1.2), kηz,ζ kMp,w ≤ 2 kSR kB(Lp (R,w)) |z − ζ|.

(6.22)

If ϕ = ϕθ where θ ∈ (−π, 0) ∪ (0, π), then 2iθzt e − e2iθζt 2θ|z − ζ| = max kηz,ζ k = max ϕ(zt) − ϕ(ζt) ≤ , e2iθ − 1 |e2iθ − 1| t∈[0,1] t∈[0,1] Z 1 Z 1 0 1 2θze2iθzt − 2θζe2iθζt dt zϕ (zt) − ζϕ0 (ζt) dt ≤ V1 (ηz,ζ ) = 2iθ |e − 1| 0 0 Z 1 2θ 2θ(1 + θ) ≤ 2iθ |z − ζ|, (1 + 2θt)|z − ζ| dt = 2iθ |e − 1| 0 |e − 1| L∞ (R)

and therefore there is a constant C ∈ (0, ∞) such that kηz,ζ kMp,w ≤ 2 CkSR kB(Lp (R,w)) |z − ζ|.

(6.23)

Finally, (6.21)–(6.23) imply the continuity of the function z 7→ W (b(z) ) on [0, 1]. ˙ the stability of operator indices with respect to small perSince b(1) ∈ Cp,w (R), turbations gives 1 arg det b(1) (x) x∈R . (6.24) Ind W (b) = Ind W (b(0) ) = Ind W (b(1) ) = − 2π On the other hand, replacing the functions e a, b and b(z) by the functions e ar (x) := ar (|x|), br (x) := e a−1 r (x)ar (x) (x ∈ R),   if x < x0 − 1, IN ϕ(z) + br (x)(1 − ϕ(z)) b(z) (x) := I ϕ(z|x − x |) + b (x)(1 − ϕ(z|x − x |)) if x ∈ [x0 − 1, x0 ], N 0 r 0 r   br (x) if x > x0 , (z)

respectively, where the function ar is given by (6.12) and the functions br are invertible in [Cp,w (R)]N ×N for all z ∈ [0, 1], and repeating the previous arguments,

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we infer that, for every sufficient large r > 0, Ind W (ar ) = Ind W (br ) = Ind W (b(0) r ) 1 = Ind W (b(1) {arg det b(1) r )=− r (x)}x∈R . 2π

(6.25) (1)

But if r > 0 is sufficiently large, then the matrix functions b(1) and br coincide. Consequently, for these r the formulas (6.18), (6.24) and (6.25) imply that Ind W (a) = Ind W (b(1) ) = Ind W (b(1) r ) = Ind W (ar ), which gives (6.13).

Acknowledgment The authors are grateful to the referee for useful comments and suggestions.

References [1] J. Alvarez and C. P´erez, Estimates with A∞ weights for various singular integral operators. Boll. Un. Mat. Ital. (7) 8-A (1994), 123–133. [2] M. A. Bastos, A. Bravo, and Yu. I. Karlovich, Convolution type operators with symbols generated by slowly oscillating and piecewise continuous matrix functions. Operator Theory: Advances and Applications 147 (2004), 151–174. [3] M. A. Bastos, A. Bravo, and Yu. I. Karlovich, Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data. Math. Nachrichten 269–270 (2004), 11–38. [4] M. A. Bastos, Yu. I. Karlovich, and B. Silbermann, Toeplitz operators with symbols generated by slowly oscillating and semi-almost periodic matrix functions. Proc. London Math. Soc. 89 (2004), 697–737. [5] A. B¨ ottcher and Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics 154, Birkh¨ auser, Basel, 1997. [6] A. B¨ ottcher, Yu. I. Karlovich, and V. S. Rabinovich, The method of limit operators for one-dimensional integrals with slowly oscillating data. J. Operator Theory 43 (2000), 171–198. [7] A. B¨ ottcher, Yu. I. Karlovich, and I. M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions. Operator Theory: Advances and Applications 131, Birkh¨ auser, Basel, 2002. [8] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz Operators, 2nd ed. Springer, Berlin, 2006. [9] A. B¨ ottcher and I. M. Spitkovsky, Wiener-Hopf integral operators with P C symbols on spaces with Muckenhoupt weight. Revista Matem´ atica Iberoamericana 9 (1993), 257–279. [10] A. B¨ ottcher and I. M. Spitkovsky, Pseudodifferential operators with heavy spectrum. Integral Equations and Operator Theory 19 (1994), 251–269. [11] R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homog´enes. Lect. Notes in Math. 242, Springer-Verlag, Berlin, 1971.

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[12] R. V. Duduchava, Integral Equations with Fixed Singularities. B. G. Teubner Verlagsgesellschaft, Leipzig, 1979. [13] R. V. Duduchava, On algebras generated by convolutions and discontinuous functions. Integral Equations and Operator Theory 10 (1987), 505–530. [14] E. M. Dyn’kin, Methods of the theory of singular integrals: Hilbert transform and Calderon-Zygmund theory. Commutative Harmonic Analysis I: General Surveys, Classical Aspects. V. P. Khavin, N. K. Nikol’skij (Eds.). Encyclopaedia of Mathematical Sciences 15, Springer-Verlag, Berlin, 1991, pp. 167-259. Russian original: VINITI, Moscow, 1987. [15] J. B. Garnett, Bounded Analytic Functions. Academic Press, New York, 1981. [16] I. Gohberg and I. A. Feldman, Convolution Equations and Projection Methods for Their Solutions. Transl. of Math. Monographs 41, Amer. Math. Soc., Providence, R.I., 1974. Russian original: Nauka, Moscow, 1971. [17] I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations, Vols. 1 and 2, Birkh¨ auser, Basel, 1992. Russian original: Shtiintsa, Kishinev, 1973. [18] R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227– 251. [19] J.-L. Journ´e, Calder´ on-Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calder´ on. Lect. Notes in Math. 994, Springer-Verlag, Berlin, 1983. [20] Yu. I. Karlovich and J. Loreto Hern´ andez, Wiener-Hopf operators with matrix semialmost periodic symbols on weighted Lebesgue spaces. Integral Equations and Operator Theory 62 (2008), 85–128. [21] Yu. I. Karlovich and E. Ram´ırez de Arellano, A shift-invariant algebra of singular integral operators with oscillating coefficients. Integral Equations and Operator Theory 39 (2001), 441–474. [22] Yu. I. Karlovich and E. Ram´ırez de Arellano, Singular integral operators with fixed singularities on weighted Lebesgue spaces. Integral Equations and Operator Theory 48 (2004), 331–363. [23] A. N. Kolmogorov and S. V. Fomin, Elements of Function Theory and Functional Analysis. Nauka, Moscow, 1972 [Russian]. [24] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions. Noordhoff I. P., Leyden, 1976. Russian original: Nauka, Moscow, 1966. [25] B. V. Lange and V. S. Rabinovich, Pseudo-differential operators on Rn and limit operators. Math. USSR-Sb. 57 (1987), 183–194. [26] S. C. Power, Fredholm Toeplitz operators and slow oscillation. Can. J. Math. 32 (1980), 1058–1071. [27] V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory, Birkh¨ auser, Basel, 2004. [28] S. Roch and B. Silbermann, Algebras of Convolution Operators and Their Image in the Calkin Algebra. Report R-Math-05/90, Akad. Wiss. DDR, Karl-WeierstrassInstitut f. Mathematik, Berlin, 1990. [29] W. Rudin, Functional Analysis. 2nd ed., McGraw-Hill Inc., New York 1991.

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[30] R. Schneider, Integral equations with piecewise continuous coefficients in Lp spaces with weight. J. Integral Equations 9 (1985), 135–152. [31] I. B. Simonenko and Chin Ngok Min, Local Method in the Theory of One-Dimensional Singular Integral Equations with Piecewise Continuos Coefficients. Noetherity. University Press, Rostov on Don, 1986 [Russian]. [32] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ, 1993. [33] G. Szeg¨ o, Orthogonal Polynomials. 4th ed., Amer. Math. Soc., Providence, R.I., 1975. Yu.I. Karlovich Facultad de Ciencias Universidad Aut´ onoma del Estado de Morelos Av. Universidad 1001, Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos M´exico e-mail: [email protected] J. Loreto Hern´ andez Instituto de Matem´ aticas Universidad Nacional Aut´ onoma de M´exico Av. Universidad 1001, Col. Chamilpa, C.P. 62210 Cuernavaca, Morelos M´exico e-mail: [email protected] Submitted: October 29, 2008. Revised: February 23, 2009.

Integr. equ. oper. theory 64 (2009), 239–249 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020239-11, published online June 5, 2009 DOI 10.1007/s00020-009-1687-9

Integral Equations and Operator Theory

Boundary Integral Solution of the Time-Fractional Diffusion Equation J. Kemppainen and K. Ruotsalainen Abstract. Here we consider initial boundary value problem for the time– fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces. Mathematics Subject Classification (2000). Primary 31A10; Secondary 26A33. Keywords. Boundary integral equation, time-fractional diffusion, fundamental solution, single layer operator.

1. Introduction In this paper we discuss the boundary integral solution of the fractional diffusion equation ∂tα Φ − ∆Φ = 0 in QT = Ω × (0, T ), B(Φ) = g on ΣT = Γ × (0, T ),

(1.1)

Φ(x, 0) = 0 for x ∈ Ω, where the boundary operator B(Φ) = Φ|ΣT , and ∂tα is the Caputo time derivative of the fractional order 0 < α ≤ 1. For α = 1 we get the ordinary diffusion equation, and for α = 0 we have the Helmholtz equation. Here we assume that the domain Ω is open and bounded with smooth boundary. Hilbert space methods to study the initial boundary value problems has been well known by now for the heat and wave equations [9], [10].

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The boundary integral equation method for elliptic, parabolic and hyperbolic equations has been extensively studied by several authors (see [2] and references therein). The idea to represent the solution of these equations with boundary potentials is well known already for decades (centuries). In this way the problem will be converted to an equivalent integral equation on the boundary of the domain. The method has been well studied in the papers [2], [3], [6]. The functional framework has been the interpretation of the boundary integral operators as anisotropic pseudo-differential operators acting on anisotropic Sobolev spaces [4]. In this way the boundary integral method is closely connected with the Hilbert space approach of the initial boundary value problems studied in [9] and [10]. In this paper we present the fundamental solution by means of the Fox Hfunctions, and represent the solution of (1.1) as the single layer potential. By the jump relations of the potential we derive the appropriate boundary integral operator. We give detailed mapping properties of the single-layer operator in anisotropic Sobolev spaces, which yields the unique solution of the boundary integral equation and thus the unique solution of the initial boundary value problem as well.

2. Function spaces Let r, s be non–negative real numbers. The anisotropic Sobolev space H r,s (Rn ×R) contains those distributions u ∈ S 0 (Rn+1 ) for which the norm Z 12 − n+1 2 {(1 + |ξ|2 )r + (1 + |η|2 )s }|b u(ξ, η)|2 dξ dη kukH r,s = (2π) Rn+1

is finite. The spaces H r,s (QT ) consist of restrictions of elements in H r,s (Rn × R) to QT equipped with the norm kukr,s;T = inf{kU kH r,s : u = U |QT }. Furthermore, the space H0r,s (QT ) is defined as the closure of C0∞ (QT ) in H r,s (QT ) and H −r,−s (QT ) is defined by duality H −r,−s (QT ) = (H0r,s (QT ))0 . For r, s ≥ 0 the space H r,s (Γ × R) is defined by H r,s (Γ × R) = L2 (R; H r (Γ)) ∩ H s (R; L2 (Γ)) with the norm kuk2H r,s (Γ×R) = kuk2L2 (R;H r (Γ)) + kuk2H s (R;L2 (Γ)) . The spaces H r,s (ΣT ) and H −r,−s (ΣT ) are defined analogously with H r,s (QT ) and H −r,−s (QT ). e r,s (Rn × R) which takes In the following we need the anisotropic Sobolev space H the vanishing initial condition at t = 0 into account, e r,s (Rn × R) = {u ∈ H r,s (Rn × R) : supp(u) ⊂ Rn × [0, ∞[ }. H For a finite time interval we denote Rn+1 := Rn × (0, T ) for T > 0, and define T e r,s (Rn+1 ) = {u = U |Rn ×(−∞,T ) : U ∈ H e r,s (Rn × R)}, H T

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equipped with the norm kukr,s;T = inf{kU kH r,s : u = U |Rn ×(−∞,T ) }. e r,s (ΣT ) are defined analogously [9], [10]. The spaces H

3. Boundary integral formulation of the problem 3.1. The fundamental solution In order to formulate the boundary integral equation corresponding to (1.1), we need to calculate the fundamental solution E(x, t). It is constructed by taking the Laplace-transform in the time and the Fourier-transform in the spatial variable of the fractional diffusion equation (∂tα − ∆)E(x, t) = δ(x, t), where δ(x, t) is the Dirac’s delta function. The transformed equation is then be (|ξ|2 + sα )E(ξ, s) = 1, where the Fourier-transform is defined by Z u b(ξ, t) = e−ihx,ξi u(x, t) dx Rn

and the Laplace-transform by Z u e(x, s) =

∞

e−st u(x, t) dt.

0

Hence the Fourier-Laplace-transform of the fundamental solution is be E(ξ, s) =

|ξ|2

1 . + sα

(3.1)

Using the Laplace-transform of the Mittag-Leffler functions [8] Z ∞ k!sµ−β (k) e−st tµk+β−1 Eµ,β (−atµ ) dt = a + sµ 0 we find out that the Fourier-transform of the fundamental solution is (0) b t) = F(E)(ξ, t) = tα−1 Eα,α E(ξ, (−|ξ|2 tα ).

By taking the inverse Fourier-transform of the Mittag-Leffler function we notice that the fundamental solution is (α,α) 20 1 E(x, t) = π −n/2 tα−1 |x|−n H12 ( |x|2 t−α |(n/2,1),(1,1) ), 4 where H is the Fox H-function [8], [13].

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3.2. The mapping properties of the single layer potential Once the fundamental solution is known, we now define the single layer potential Z Z t Φ(x, t) = Sσ(x, t) = σ(y, τ )E(x − y, t − τ ) dsy dτ, x ∈ Ω, t ∈ (0, T ) Γ

0

for a given boundary distribution σ(x, t) ∈ C ∞ (ΣT ). The potential is the solution of the fractional diffusion equation both in interior domain Ω × (0, T ) and on the exterior domain QcT := [Rn \ Ω] × (0, T ) with the zero initial condition. We denote the direct value of Sσ on the boundary by V σ. This leads us to the boundary relation γ(Sσ)(x, t) = γ(Φ)(x, t) = V σ(x, t). In other words we have converted the initial boundary value problem of the fractional diffusion equation (1) to a boundary integral equation V σ(x, t) = γ(Φ)(x, t) = g(x, t), (x, t) ∈ ΣT .

(3.2)

In our analysis we need the mapping properties of the single layer potential in Sobolev-spaces. The single layer potential can be written as Sφ = E ∗ γ 0 (φ).

(3.3)

By (3.1) we have F(E ∗ f )(ξ, η) =

|ξ|2

1 fˆ(ξ, η) + (iη)α

(3.4)

for smooth f with suppf ⊂ Rn × [0, ∞[. It follows that the map α

α

r, 2 r e r+2, 2 (r+2) (Rn × (0, T )) e comp (Rn × (0, T )) → H ψ 7→ E ∗ ψ : H loc

(3.5)

is continuous for any r ∈ R, where comp means compact support and loc local behaviour in space variables. Since the trace map γ : H r,s (QT ) → H λ,µ (ΣT ) is continuous and surjective for every λ = r − 12 , µ = rs λ, r > 12 and s ≥ 0 ([9], Theorem 4.2 of Chapter 1, and [10], Theorem 2.1 of Chapter 4), by duality we have −r,−s γ 0 : H −λ,−µ (ΣT ) → Hcomp (Rn × (0, T )).

(3.6)

Using trace theorem once again , combining (3.5) and (3.6) and noting that the e r,s (ΣT ) coincide if and only if |s| < 1 [9], we may conclude spaces H r,s (ΣT ) and H 2 the following result. Theorem 3.1. Let 0 < s < 1. The operator α

α

e −s,− 2 s (ΣT ) → H e 1−s, 2 (1−s) (ΣT ) V :H is continuous.

(3.7)

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3.3. The jump relations As usual we define the jump of the traces across the boundary as [γ(u)] = γ(u+ ) − γ(u− ), where γ is the spatial trace operator and u+ = u|Ωc (u− = u|Ω ) is a function which is defined in the exterior (interior) of the domain Ω. Respectively, the jump of the normal derivative across the boundary is defined as [γ1 (u)] = [γ(∂n u)] = γ(∂n u+ ) − γ(∂n u− ). For the proof of jump relations we need some basic properties of fractional derivatives and Green’s formula in the case of fractional time–derivatives. Because the properties of fractional derivatives are crucial for the proof of Green’s formula, we consider them first. α In the sequel ∂tα := c D0+ denotes the left Caputo-derivative on time interval (0, T ), and the right Caputo-derivative on the interval (0, T ) is denoted by c DTα − . They are defined by the formulas Z t ϕ0 (s) 1 c α ds, D0+ ϕ(t) = Γ(1 − α) 0 (t − s)α Z T 1 ϕ0 (s) c α DT − ϕ(t) = − ds. Γ(1 − α) t (s − t)α The right and left Riemann-Liouville derivatives on the interval (0, T ) are defined by setting Z d t ϕ(s) 1 α D0+ ϕ(t) = ds, Γ(1 − α) dt 0 (t − s)α Z 1 d T ϕ(s) α DT − ϕ(t) = − ds, Γ(1 − α) dt t (s − t)α respectively. Note that for sufficiently smooth functions ϕ for which ϕ(0) = 0 the left Caputo and Riemann-Liouville derivatives coincide ([12], Formula (2.165)), i.e. c

α α D0+ ϕ(t) = D0+ ϕ(t).

(3.8)

Integration by parts gives the following relation between the left Caputo and the right Riemann-Liouville derivative: Z T Z T α ∂t ϕ(t)ψ(t) dt = ϕ(t)DTα − ψ(t) dt (3.9) 0

0

for ϕ ∈ C 1 ([0, T ]) with ϕ(0) = 0 and ψ ∈ C 1 ([0, T ]). The time reversal operator on the interval (0, T ) is defined by setting κT ϕ(t) = ϕ(T − t). Applying the time reversal operator to the left Riemann–Liouville derivative we have α DTα − (κT ϕ)(t) = κT D0+ ϕ(t). (3.10)

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Combining (3.8), (3.9) and (3.10) we have Z T Z ∂tα ϕ(t)(κT ψ)(t) dt = 0

IEOT

T

(κT ϕ)(t)∂tα ψ(t) dt

(3.11)

0

for ϕ ∈ C 1 ([0, T ]) and ψ ∈ C 1 ([0, T ]) with ψ(0) = 0. Let us next consider the Green’s formula for the fractional diffusion equation. By the Green’s formula with respect to the space variable we have the relation Z Z Z Z α ∇u · ∇v dx dt + ∂t u v dx dt − ∂n u v dsΓ dt = (∂tα − ∆)u v dx dt. QT

QT

ΣT

QT

In the previous identity we have assumed that functions are at least in C 1 (QT ). On the other hand, using the time reversal on the function v(x, t) we get Z Z Z (∂tα − ∆)u κT v dx dt = ∇u · ∇κT v dx dt + ∂tα u κT v dx dt QT QT QT Z (3.12) − ∂n u κT v dsΓ dt. ΣT

Changing in the previous identity the role of u and v we get Z Z Z α ∂tα v κT u dx dt ∇κT u · ∇v dx dt + (∂t − ∆)v κT u dx dt = QT QT QT Z − ∂n v κT u dsΓ dt.

(3.13)

ΣT

Assuming that v has zero initial condition and using (3.11) in (3.13) we have Z Z Z (∂tα − ∆)v κT u dx dt = ∇u · ∇κT v dx dt + ∂tα uκT v dx dt QT QT QT Z (3.14) − ∂n vκT u dsΓ dt. ΣT

Cancelling equation (3.12) from equation (3.14) we finally obtain the Green formula for the fractional diffusion equation: Z Z {(∂tα − ∆)u κT v − κT u (∂tα − ∆)v} dx dt = (u∂n κT v − ∂n u κT v) dsΓ dt QT

ΣT

= hγ(u), γ1 (κT v)i − hγ1 (u), γ(κT v)i. Now we are able to state and prove the jump relations for the single layer potential of the fractional diffusion operator: 1

α

Theorem 3.2. For every ψ ∈ H − 2 ,− 4 (ΣT ) there hold the jump relations: [γ(Sψ)] = 0, [γ1 (Sψ)] = −ψ. Proof. For the function ψ let us denote u = Sψ. By the assumption on ψ and e 1, α2 (BR × (0, T )), the properties of the trace map we have the inclusion u ∈ H where the radius of the ball BR is so large that Ω ⊂ BR . Hence by the trace

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theorem ([9], Theorem 4.2 of Chapter 1, and [10], Theorem 2.1 of Chapter 4) γ(u|QT ) = γ(u|QcT ) which proves the continuity of the trace across the boundary. Using the representation formula (3.3) we have (∂tα − ∆)u = γ 0 (ψ) in the distributional sense in Rn × (0, T ). Choosing φ ∈ C0∞ (BR × (0, T )) we get hψ, γ(φ)i = hγ 0 (ψ), φi = h(∂tα − ∆)u, φi =

hu, (DTα −

− ∆)φi.

(3.15) (3.16)

Making the time reversal and using the properties of the Caputo fractional derivatives, DTα − κT φ = κT ∂tα φ, we obtain from the previous equation Z hψ, γ(κT φ)i = (∂tα − ∆)φ κT u dx dt. (3.17) BR ×(0,T )

Using the Green formula for the fractional diffusion operator with respect to the sets QT and QcT we get (remember that on QT ∪ QcT we have (∂tα − ∆)u = 0) Z (∂tα − ∆)φ κT u dx dt = hγ1 (u), γ(κT φ)i − hγ(u), γ1 (κT φ)i (3.18) Q Z T (∂tα − ∆)φ κT u dx dt = −hγ1 (u), γ(κT φ)i + hγ(u), γ1 (κT φ)i. (3.19) QcT

The jump of the traces for u is [γ(u)] = 0 by the first part of the theorem. Since the test function φ is smooth it’s traces are continuous across the boundary, i.e. [γ(κT φ)] = [γ1 (κT φ)] = 0. Adding equations (3.18) and (3.19) together and using the previous trace properties of u and κT φ we obtain Z (∂tα − ∆)φ κT u dx dt = −h[γ1 (u)], γ(κT φ)i. (3.20) BR ×(0,T )

Combining the equations (3.18) and (3.20) we finally obtain hψ, γ(κT φ)i = −h[γ1 (u)], γ(κT φ)i ∀φ ∈ C0∞ (BR × (0, T )), which proves the second statement.

(3.21)

3.4. The coercivity of the single layer potential For the proof of coercivity we use the standard technique by proving G˚ arding’s inequality and positivity for the single layer potential (see the proof of Theorem 3.11 in [2]). To begin with we apply the Green formula to the function u = Sψ, where 1 α ψ ∈ H − 2 ,− 4 (ΣT ). By the Gauss divergence formula we have h∇u, ∇viQT + h∂tα (u), viQT = hγ1 (u), viΣT + h(∂tα − ∆)u, viQT , for some appropriate test function v. Since u is the solution of the homogeneous fractional diffusion equation we obtain by setting v = u: h∇u, ∇uiQT + h∂tα (u), uiQT = hγ1 (u), uiΣT .

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Since the Caputo derivative is positive semidefinite, we’ll get Z hγ1 (u|QT ), γ(u|QT )iΣT ≥ |∇u|2 dx dt. QT

QcR

QTc

Respectively, on the domain = ∩ BR we’ll have Z Z −hγ1 (u|QcR ), γ(u|QcR )i = |∇u|2 dx dt + h∂tα u, uiQcR − QcR

u∂n u ds∂BR dt.

∂BR ×(0,T )

By the jump relations we have hψ, V ψi = hγ1 (u|QT ), γ(u|QT )i − hγ1 (u|QcR ), γ(u|QcR )i. Hence we obtain Z

Z

2

hψ, V ψi ≥

|∇u| dx dt − QT ∪QcR

u∂n u ds∂BR dt. ∂BR ×(0,T )

Since the fundamental solution E(x, t) is smooth on the boundary ∂BR [8], α 1 α 1 there exists a compact operator T1 : H − 2 ,− 4 (ΣT ) → H 2 , 4 (ΣT ) such that Z u∂n u ds∂BR dt = hψ, T1 ψiΣT . ∂BR ×(0,T ) α

e 1, 2 (QT ) ,→ L2 (QT ) is a compact embedding, On the other hand, since H 1 1 α α there exists a compact operator T2 : H − 2 ,− 4 (ΣT ) → H 2 , 4 (ΣT ) such that Z |∇u|2 dx dt = kuk2H 1,0 (QT ) + kuk2H 1,0 (Qc ) − hψ, T2 ψi. R

QT ∪QcR

We need the following lemma. e 1, α2 (QT ), 0 < α < 1, and H e 1,0 (QT ) are equivalent on Lemma 3.3. The norms of H the subspace of functions satisfying the homogeneous fractional diffusion equation. This lemma can be proved similarly as Lemma 2.15 in [2] by defining the space V(QT ) consisting of those functions u ∈ L2 (I; H 1 (Ω)) for which there holds ∂tα u ∈ L2 (I; H −1 (Ω)). After that we use proper interpolation results to conclude the claim. Utilizing the norm equivalence we obtain hψ, (V + T1 + T2 )ψi ≥ C(kuk2H 1, α2 (Q

T)

+ kuk2H 1, α2 (Qc ) ).

(3.22)

R

By Theorem 3.2 we have kψk

1

α

H − 2 ,− 4 (ΣT )

= kγ1 (u|QT ) − γ1 (u|QcR )k

1

α

H − 2 ,− 4 (ΣT )

.

Combining this with the inequality (3.22) and trace theorem and denoting T := T1 + T2 , we finally get G˚ arding’s inequality: hψ, (V + T )ψi ≥ Ckψk2

1

α

H − 2 ,− 4 (ΣT )

.

Now we consider the positivity of the single layer operator.

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α

e − 2 ,− 4 (ΣT ) we have Lemma 3.4. For all σ ∈ H Re(V σ, σ) > 0, if σ 6= 0. Proof. By the standard density argument it is enough to show the positivity for smooth functions σ(x, t) for which the initial condition σ(x, 0) = 0 is valid. Let us define the potential φ = Sσ which is the solution of the homogeneous equation: (∂tα − ∆)φ(x, t) = 0 ∀(x, t) ∈ QT ∪ QcT . For a fixed t > 0 we get by the Green formula Z Z Z Z 0 = (∂tα φ − ∆φ) · φ dx = ∂tα φ · φ dx + |∇φ|2 dx − ∂n φ− · φ dsΓ , Ω Γ ZΩ Z Z ZΩ α α 2 (∂t φ − ∆φ) · φ dx = ∂t φ · φ dx + |∇φ| dx + ∂n φ+ · φ dsΓ . 0= Ω

c

Ω

c

Ω

c

Γ

Adding the identities together we obtain Z Z −[γ1 φ]φ dsΓ = (∂tα φ · φ + |∇φ|2 ) dx. Γ

Ω∪Ω

c

Note that in the right hand side we have used the continuity of the traces of the single layer potential proved in Theorem 3.2. Integrating the previous identity with respect to the time variable over the interval [0, T ] yields us Z Z (∂tα φ · φ + |∇φ|2 ) dx dt. [γ1 φ]φ dsΓ dt = − QT ∪QcT

ΣT

Using the jump relations once again we obtain Z Z σV σ dsΓ dt = (∂tα Sσ · Sσ + |∇Sσ|2 ) dx dt. QT ∪QcT

ΣT 1

1

Since the operators J and D commute when acting on functions with zero initial conditions, and they possess the semigroup property J α J β = J α+β , we obtain by using the positive semidefiniteness of the operator J α : Z T Z T Z T ∂tα φ · φ dt = J 1−α D1 φ dt = J 1−α D1 φ · J 1 D1 φ dt 0

0

Z

0 T

J 1−α D1 φ · J α J 1−α D1 φ dt

= 0

Z = 0

T

J |

1−α

1

α

D φ ·J J {z } | g

1−α

1

D φ dt = {z } g

Z

T

J α g · g dt ≥ 0.

0

Hence the single layer operator is at least positive semidefinite: Z Z V σ · σ dsΓ dt ≥ |∇Sσ|2 dx dt ≥ 0. ΣT

QT ∪QcT

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If there exists a boundary distribution such that Z σV σ dsΓ dt = 0 ΣT

we obtain that ∇Sσ = 0 for all x ∈ QT ∪ QcT and 0 < t < T . Thus we have ∂tα φ − ∆φ = ∂tα φ = 0. Now for every fixed x ∈ QT ∪ QcT we have ∂tα φ = J 1−α D1 φ = 0. Since the Abel integral equation J 1−α ϕ = ψ is uniquely solvable we have that D1 φ = 0, and thus φ(x, t) = C. By the zero initial condition we finally obtain that the constant φ(x, t) = C = 0 proving the positivity of the single layer operator. As in [6] or in [2] we obtain the strong coercivity of the single layer operator, and we thus are able to state our main theorem: e 12 , α4 (ΣT ) is an e − 21 ,− α4 (ΣT ) → H Theorem 3.5. The single layer operator V : H isomorphism. Furthermore, it is coercive, i.e. there exists a positive constant c such that Re(V σ, σ) ≥ ckσk2 − 1 ,− α H

e − 12 ,− α4

for all σ ∈ H

2

4

(ΣT )

(ΣT ).

e 12 , α4 (ΣT ) the fractional diffusion equation admits a Corollary 3.6. For every g ∈ H α e 1, 2 (QT ) which is given by the single layer potential: unique solution Φ(x, t) ∈ H Φ(x, t) = Sσ(x, t), e − 21 ,− α4

where σ ∈ H

(ΣT ) is the unique solution of the boundary integral equation V σ = g.

References [1] M. Abramowitz, I.A. Stegun (eds.), Handbook of mathematical functions and with formulas, graphs and mathematical tables, U.S. Government Printing Office, Washington DC, 1971. [2] M. Costabel, Boundary integral operators for the heat equation, Integr. Equ. Oper. Theory, 13(4), (1992), 498–552. [3] M. Costabel, Time-dependent problems with the boundary integral equation method. In: E. Stein, R. de Borst and T.J.R. Hughes (eds.), Encyclopedia of Computational Mechanics. John Wiley & Sons, Ltd. 2004. [4] M. Costabel, J. Saranen, Parabolic boundary integral operators, symbolic representations and basic properties, Integr. Equ. Oper. Theory, 40, (2001), 185–211. [5] I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series and products, Academic Press, New York, 1996. [6] G.C. Hsiao, J. Saranen, Coercivity of the single layer heat operator, Report 89–2, Center for Mathematics and Waves, Newark, Delaware, 1989.

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[7] G.C. Hsiao, J. Saranen, Boundary Integral Solution of the Two-dimensional Heat Equation, Math. Meth. Appl. Sci. 16 (1993), 87–114. [8] A.A. Kilbas, M. Saigo, H-transforms: Theory and Applications, CRC Press, LLC, 2004. [9] J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications I, Springer, Berlin, 1972. [10] J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications II, Springer, Berlin, 1972. [11] F. Oberhettinger, Fourier expansions, Academic Press, New York, London, 1973. [12] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, California, 1999. [13] A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series, vol. 3. More special functions, Overseas Publishers Association, Amsterdam, 1990. J. Kemppainen and K. Ruotsalainen University of Oulu Mathematics Division P.O. Box 4500 FI-90014 Oulu Finland e-mail: [email protected] [email protected] Submitted: August 1, 2008.

Integr. equ. oper. theory 64 (2009), 251–260 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020251-10, published online June 5, 2009 DOI 10.1007/s00020-009-1690-1

Integral Equations and Operator Theory

Bloch-Bergman Pullbacks with Logarithmic Weights E. G. Kwon Abstract. We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight. For 0 < p < ∞, 1 < α < ∞, let Ap,log α denote the space of holomorphic functions F in the unit disc D for which −α Z e dxdy p |F (z)| log < ∞, 2 1 − |z| 1 − |z|2 D α and let Ap,log denote the class of holomorphic self maps f of D for which σ p −α Z 1 dxdy e log log < ∞. 1 − |f (z)|2 1 − |z|2 1 − |z|2 D

Then for the Bloch pullback operator Cf , the following are equivalent: (1) Cf maps Bloch space B boundedly into A2p,log α α (2) f ∈ Ap,log p−1 0 2 1−α Z σ |f (z)| 1 e (3) log (1 − |z|) log dxdy < ∞. 1 − |f (z)|2 1 − |z|2 1 − |z| D Mathematics Subject Classification (2000). Primary 47B33; Secondary 30D45, 32A37. Keywords. Bloch space, weighted Bergman space, composition operators.

1. Introduction Let D = {z : |z| < 1} be the open unit disc of the complex plane. For functions f on D and for 0 < p < ∞, 0 ≤ r < 1, Mp (r, f ) and kf kp are defined respectively as usual by p1 Z 2π 1 iθ p Mp (r, f ) = |f (re )| dθ and kf kp = sup Mp (r, f ). 2π 0 r This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-313-C00026).

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The classical Hardy space H p = H p (D) is defined to consist of those f holomorphic in D for which kf kp < ∞, and the Yamashita hyperbolic Hardy class Hσp is defined as the set of those holomorphic self maps f of D for which kσ(f )kp < ∞, where σ(z) denotes the hyperbolic distance of z and 0 in D i.e. σ(z) =

1 1 + |z| log . 2 1 − |z|

We set, following Yamashita, λ(f ) = log

1 1 − |f |2

and f ] =

|f 0 | 1 − |f |2

for holomorphic self maps f of D. Note that f ∈ Hσp if and only if kλ(f )kp < ∞ and that f ] is derivative of f in the hyperbolic version. For −1 < α < ∞ and 0 < p < ∞, the weighted Bergman space Ap,α is defined to consist of those holomorphic functions f in D for which Z p1 kf kAp,α := |f (z)|p (1 − |z|2 )α dxdy < ∞. D

Ap,α is a Banach space or a Frechet space depending on the size of p. We refer to [D, G] for H p and Ap,α , and to [Y] for Hσp . As a hyperbolic counterpart, in this paper we are going to consider the hyperbolic Bergman class. We let Ap,α σ , −1 < α < ∞, 0 < p < ∞, be the set of those holomorphic self maps f of D for which Z p [λ(f )(z)] (1 − |z|2 )α dxdy < ∞. D

But, compared to the hyperbolic Hardy class the hyperbolic Bergman class Ap,α σ was less considered in the literature mainly because it simply turns out to be Ap,α σ = {f : f holomorphic self map of D}. We pass to the case of logarithmic weights. For 1 < α < ∞ and 0 < p < ∞, α , respectively the euclidian and let us define in this paper Ap,log α and Ap,log σ hyperbolic Bergman class of logarithmic weights, consisting of those holomorphic functions F of D and holomorphic self maps f of D for which (Z ) p1 −α e dxdy p kF kAp,log α := |F (z)| log < ∞ 1 − |z|2 1 − |z|2 D and (Z kf kAp,log α := σ

p

[λ(f )(z)] D

e log 1 − |z|2

−α

dxdy 1 − |z|2

) p1 < ∞.

Ap,log α also is a Banach space when 1 ≤ p < ∞ or a Frechet space when 0 < p < 1.

Vol. 64 (2009)

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253

Note that H p and Hσp may be regarded as the limiting class of Ap,log α and α respectively as α → 1. Note also that Ap,log is meaningless if α < 1 σ because then Z 1 dr α = ∞. 0 (1 − r) log e 1−r Aσp,log α

The subjective of this paper is the positive subharmonic Bergman function α class Ap,log . Our aim is to use the space in characterizing the Bloch pullback σ problems; see [RU, K4, K5] for the problems. Our results are described in Section 2 and are proven in the remaining sections. Main result of this paper, Theorem 2.3 α α in Section 2, says that Ap,log characterizes the Bloch-A2p,log pullback problem. σ σ

2. Statements of Results We begin with the Euclidian space Ap,log α and are going to establish an equivalent condition involving derivatives for the membership of the space. Among studies on Ap,α , we pay attention to the well-known equivalence Z f ∈ Ap,α ⇐⇒ |f (z)|p−2 |f 0 (z)|2 (1 − |z|2 )α+2 dxdy < ∞. (2.1) D

The question on characterizing general weight functions substituting (1 − |z|2 )α in (2.1) that maintain the equivalence was considered quite extensively. See, for example, [K1, K6]. We establish the following which verifies an example of the weight that the equivalence fails. Theorem 2.1. Let 0 < p < ∞ and 1 < α < ∞, and f be holomorphic on D. Then 1−α Z e dxdy. |f (z)|p−2 |f 0 (z)|2 (1 − |z|) log kf kpAp,log α ≈ |f (0)|p + 1 − |z| D α , we have the following parallel result. Also for the hyperbolic class Ap,log σ

Theorem 2.2. Let 0 < p < ∞ and 1 < α < ∞, and f : D −→ D be holomorphic. Then 1−α Z 2 e p−1 ] kf kp p,log α ≈ |f (0)|p + [λ(f )(z)] f (z) (1 − |z|) log dxdy. Aσ 1 − |z| D We next pay attention to the composition operators. We note that Hσp first occurred in the literature as a hyperbolic counterpart of H p , but later it is found out to have a close connection with the behavior of composition operators. We note the following equivalence (see [K2, K4]) on holomorphic self maps f : g ◦ f ∈ H 2p for all g ∈ B ⇐⇒ f ∈ Hσp ,

(2.2)

where B denotes the space of Bloch functions which is defined to consist of holomorphic functions g on D for which sup (1 − |z|2 )|g 0 (z)| < ∞. z∈D

(2.3)

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It is a general phenomenon that the composition operator maps the Bloch space boundedly into a (Euclidian) holomorphic function space if the inducing holomorphic self map belongs to the corresponding hyperbolic function class; see, for example, [K3]. Parallel to the general phenomena, we conjecture the following α g ◦ f ∈ A2p,log α for all g ∈ B ⇐⇒ f ∈ Ap,log . σ

(2.4)

Actually two cases are true : the limiting case α = 1 of (2.4) is nothing but (2.2) and, in [K5], the case p = 1 of (2.4) was verified. We prove (2.4) in general in this paper. The Bloch space B is a Banach space if the norm kgkB of g ∈ B is defined to be the sum of |g(0)| and the left side of (2.3). If we let Cf denote the composition operator defined by Cf g = g ◦ f , then Theorem 2.2 and (2.4) give, by the closed graph theorem, Theorem 2.3. For 0 < p < ∞, 1 < α < ∞, and for holomorphic self map f of D, the following are equivalent. (1) Cf maps Bloch space B boundedly into A2p,log α . α . (2) f ∈ Ap,log σ Z 2 p−1 ] (3) [λ(f )(z)] f (z) (1 − |z|) log D

e 1 − |z|

1−α dxdy < ∞.

Finally, we see that the compactness of Cf is equivalent to the boundedness in our case. Recall that a linear operator T from a Banach space X into another Banach space Y is compact if, for each bounded subset B of X, T (B) is relatively compact in Y . Theorem 2.4. Let f be a holomorphic self map of D and 1 < α < ∞, 1 ≤ p < ∞. Then Cf : B → Ap,log α is bounded if and only if it is compact.

3. Proofs of Theorem 2.1 and Theorem 2.2 Proof of Theorem 2.1. By the Hardy-Stein identity we have Z 2π Z p2 r iθ p dθ p |f (re )| = |f (0)| + |f (z)|p−2 |f 0 (z)|2 log dA(z), 2π 2 |z| 0 rD so that Z |f (z)|p α dA(z) kf kpAp,log α = e D (1 − |z|) log 1−|z| Z Z 1 r r dr α |f (z)|p−2 |f 0 (z)|2 log dA(z) ≈ |f (0)|p + |z| rD 0 (1 − r) log e 1−r   Z Z 1 r r log |z| α dr dA(z). = |f (0)|p + |f (z)|p−2 |f 0 (z)|2  D |z| (1 − r) log e 1−r

Vol. 64 (2009)

Bloch-Bergman Pullbacks with Logarithmic Weights

Let us consider the inner integral Z 1 r |z| (1 − r) log

e 1−r

α log

255

r dr. |z|

Integration by parts gives " 1−α # Z 1 d 1 e r log dr r log dr 1 − α 1 − r |z| |z| 1−α Z 1 1 e r log log + 1 dr. = α − 1 |z| 1−r |z| If

1 2

≤ |z| ≤ 1, then the inner integral is comparable to 1−α Z 1 e dr. Iα−1 (z) = log 1−r |z|

But, since α > 1, Z

1

Iα−1 (z) =

log |z|

and simultaneously Z 1 Iα−1 (z) = log |z|

e 1−r

e 1−r

1−α

dr ≤

log

e 1 − |z|

1−α (1 − |z|)

1−α dr

1−α d e (1 − r) dr =− log 1−r dr |z| " #1 1−α −α Z 1 e e = − log (1 − r) + (1 − α) log dr 1−r 1−r |z| |z| 1−α e = log (1 − |z|) − (α − 1) Iα (z) 1 − |z| 1−α e ≥ log (1 − |z|) − (α − 1) Iα−1 (z). 1 − |z| Z

1

Thus, 1 α

log

e 1 − |z|

1−α

(1 − |z|) ≤ Iα−1 (z) ≤

log

e 1 − |z|

1−α (1 − |z|).

Noting that all singularities on {|z| < 21 } are integrable, we have 1−α Z e p p p−2 0 2 kf kAp,log α ≈ |f (0)| + |f (z)| |f (z)| (1 − |z|) log dA(z). 1 − |z| D

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Proof of Theorem 2.2. The idea and the process of the proof is quite similar to Theorem 2.1. Green’s Theorem with a limiting process gives that Z 2π Z r iθ p dθ p [λ(f )(re )] ≈ [λ(f )(0)] + ∆[λ(f )(z)]p log dA(z); 2π |z| 0 rD see [K4]. Hence, Z kf kp p,log α = Aσ

[λ(f )(z)]p α dA(z) e D (1 − |z|) log 1−|z| Z 1 Z r dr r α ≈ [λ(f )(0)]p + dA(z) ∆[λ(f )(z)]p log |z| e 0 (1 − r) log rD 1−r   Z Z 1 r r log |z| α dr dA(z). = [λ(f )(0)]p + ∆[λ(f )(z)]p  D |z| (1 − r) log e 1−r

The remaining process of dealing the inner integral is identical to that of proof of Theorem 2.1. The result then follows by noting that 2 ∆[λ(f )(z)]p ≈ [λ(f )(z)]p−1 f ] (z) ; see [K4].

4. Proof of Theorem 2.3 By means of Theorem 2.2, we are left to show (1) ⇐⇒ (2). (1) =⇒ (2) : Suppose Cf : B −→ A2p,log α is bounded. Then kg ◦ f kA2p,log α ≤ CkgkB for a constant C independent of g. By the subharmonicity of |g ◦ f |2p , M2p (g ◦ f )2p is increasing function of r, so that we obtain Z 1 Z π dr dθ α |g ◦ f (reiθ )|2p ≤ Ckgk2p (4.1) B 2π e 0 (1 − r) log −π 1−r for all g ∈ B. Let us take for each non-dyadic t ∈ [0, 1] the function gt (z) =

∞ X

k

γk (t)z 2 ,

z ∈ D,

0

where γk is the Rademacher function; γk (t) = sign sin(2k+1 πt),

t ∈ [0, 1].

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Then it follows that kgt kB = |gt (0)| + sup(1 − |z|2 )|gt0 (z)| z∈B

≤ 2 sup(1 − |z|) z∈B

∞ X

2k |z|2

k

−1

0

≤ 4 sup(1 − |z|) z∈B

∞ X

|z|k ≤ 4

0

simply because 1 + 2|z|1 + 22 |z|3 + 23 |z|7 + 24 |z|15 + · · · ≤ 2 + 2|z| + 2|z|2 + 2|z|3 + 2|z|4 + · · · . So by (4.1) Z 0

1

dr (1 − r) log

Z e 1−r

α

π

|gt ◦ f (reiθ )|2p

−π

dθ ≤ C 2π

for some C independent of gt . Integrating (4.2) with respect to t and applying Fubini’s Theorem, Z 1 Z Z π dr dθ 1 α |gt ◦ f (reiθ )|2p dt ≤ C. 0 (1 − r) log e −π 2π 0 1−r

(4.2)

(4.3)

But by Khinchin’s Inequality ([Z]), the inner integral of the left hand side of (4.3) is 2p Z Z ∞ Z π Z π dθ 1 dθ 1 X iθ 2k iθ 2p γk (t)f (re ) dt |gt ◦ f (re )| dt = −π 2π 0 −π 2π 0 0 ! p Z π X ∞ k 2 1 dθ (4.4) ≥ f (reiθ )2 C −π 2π 0 p Z 1 π dθ 1 ≥ , log C −π 1 − |f (reiθ )|2 2π where we used the inequality ∞

∞

X |f (reiθ )2 |k X k 1 log = = |f (reiθ )2 |2 . iθ 2 1 − |f (re )| k 0 0 Here C in (4.4) is another constant independent of gt and f . Plugging (4.4) into (4.3), we finally obtain p Z π Z 1 dr 1 dθ α log < ∞. iθ 2 1 − |f (re )| 2π 0 (1 − r) log e −π 1−r α (2) =⇒ (1) : Suppose f ∈ Ap,log . By the Closed Graph Theorem, it is σ 2p,log α sufficient to show that g ◦ f ∈ A for all g ∈ B. By a routine argument, we may assume f (0) = 0.

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We will make use of the following two equivalences. One is the well-known euclidian g-function equivalence Z

π

1

Z

0

iθ

2

p

(1 − r)|F (re )| dr −π

dθ ≈ kF k2p 2p

(4.5)

0

valid for all holomorphic F on D with F (0) = 0. The other one is the hyperbolic g-function equivalence ([K3]) Z

π

−π

Z

1

2 (1 − r) f # (reiθ ) dr

p

dθ ≈ kλ(f )kpp

(4.6)

0

valid for all holomorphic self maps f on D with f (0) = 0. Let g ∈ B. Then by (4.5), 1

Z π dr α |g ◦ f (reiθ ) − g(0)|2p dθ e 0 (1 − r) log −π 1−r p Z 1 Z π Z 1 dr 0 iθ 2 α ≤C (1 − ρ)|(g ◦ fr ) (ρe )| dr dθ. 0 (1 − r) log e −π 0 1−r

Z

(4.7)

Since |(g ◦ fr )0 (ρeiθ )| = |g 0 ◦ fr (ρeiθ )| · |fr0 (ρeiθ )| ≤ kgkB fr# (ρeiθ ), by (4.7) and (4.6) we have Z π dr α |g ◦ f (reiθ )|2p dθ − C|g(0)|2p e −π 0 (1 − r) log 1−r p Z 1 Z π Z 1 # iθ 2 dr 2p α ≤ CkgkB (1 − ρ) fr (ρe ) dρ dθ 0 (1 − r) log e −π 0 1−r p Z 1 Z π 1 1 2p α lim ≤ CkgkB log dθ dr ρ→1 −π 1 − |fr (ρeiθ )|2 0 (1 − r) log e 1−r p Z π Z 1 dr 1 = Ckgk2p log dθ, α B 1 − |f (reiθ )|2 0 (1 − r) log e −π 1−r

Z

1

where we used the subharmonicity of log

1 1−|f |2

p

in the last equality.

The last integral is finite by the hypothesis so that g ◦ f ∈ Ap,log α , which completes the proof.

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5. Proof of Theorem 2.4 To prove Theorem 2.4, we note that Cf is compact if and only if, for each bounded sequence gn in B, the sequence Cf gn contains a subsequence converging to some limit in Ap,log α (see [TL]). Since every compact linear operator is continuous, one direction is obvious. Suppose that Cf : B → Ap,log α is bounded. Then g ◦ f ∈ Ap,log α for all g ∈ B. Let gn ∈ B be such that kgn kB ≤ 1. We are going to show that Cf gn has a convergent subsequence in Ap,log α . Since 1 1 + |z| |gn (z)| ≤ |gn (0)| + kgn kB log , z ∈ D, (5.1) 2 1 − |z| {gn } forms a normal family, so that there is a subsequence of {gn } that converges on compact subsets of D to an analytic function g. g belong to B with kgkB ≤ 1 because 2

2

|g 0 (z)|(1 − |z| ) = lim |gn0 (z)|(1 − |z| ) ≤ 1. n→∞

Thus, by (5.1) and Minkowski’s Inequality, 1/p e −α 1 dxdy log lim |(gn − g) ◦ f (z)| n→∞ 1 − |z| 1 − |z| D ≤ lim kgn − gkB n→∞ Z 1/p p 1 e −α 1 1 + |f (z)| × log 1 + log dxdy 1 − |z| 2 1 − |f (z)| D 1 − |z| ≤ Cp lim kgn − gkB n→∞ # "Z 1/p p e −α 1 1 + |f (z)| 1 log log dxdy +1 . × 1 − |z| 2 1 − |f (z)| D 1 − |z| Z

p

p

Since Cf is bounded, the last quantity is finite by Theorem 2.3. So |(gn − g) ◦ f (z)| −α 1 e is dominated by an L1 function 1−|z| log 1−|z| p

kgn − gkpB

1 e −α 1 1 + |f (z)| log 1 + log . 1 − |z| 1 − |z| 2 1 − |f (z)|

Now by the hypothesis we have Cf g ∈ Ap,log α , and by the Dominated Convergence Theorem we obtain Z e −α 1 p log dxdy = 0. lim |(gn − g) ◦ f (z)| n→∞ D 1 − |z| 1 − |z| Therefore Cf gn → Cf g in Ap,log α .

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References [D] [G] [K1] [K2] [K3] [K4] [K5] [K6] [RU] [TL] [Y] [Z]

P. L. Duren, The theory of H p spaces, Academic Press, New York, 1970. J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981. E. G. Kwon, A characterization of Bloch space and Besov space, J. Math. Anal. Appl. 324 (2006), 1429–1437. , Composition of Blochs with bounded analytic functions, Proceedings of American Mathematical Society 124 (1996), 1473–1480. , Hyperbolic g-function and Bloch pullback operators, J. Math. Anal. Appl. 309 (2005), 626–637. , Hyperbolic mean growth of bounded holomorphic functions in the ball, Trans. Amer. Math. Soc. 355 (2003), 1269–1294. , On analytic functions of Bergman BMO in the ball, Can. Math. Bull. 42(1) (1999), 97–103. , Quantities equivalent to the norm of a weighted Bergman space, J. Math. Anal. Appl. 338 (2008), 758–770. W. Ramey and D. Ullrich, Bounded mean oscillations of Bloch pullbacks, Math. Ann. 291 (1991), 591–606. A. E. Taylor and D. C. Lay, Introduction to functional analysis, John Wiley and Sons, Inc., New York, 1980. S. Yamashita, Hyperbolic Hardy class and hyperbolically Dirichlet finite functions, Hokkaido Math. J. Special Issue 10 (1981), 709–722. A. Zygmund, Trigonometric series, Cambridge Univ. Press, London, 1959.

E. G. Kwon Department of Mathematics Education Andong National University Andong 760-749 Korea e-mail: [email protected] Submitted: January 7, 2009. Revised: April 2, 2009.

Integr. equ. oper. theory 64 (2009), 261–271 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020261-11, published online June 5, 2009 DOI 10.1007/s00020-009-1689-7

Integral Equations and Operator Theory

Lie Derivations of Reflexive Algebras Fangyan Lu and Benhong Liu Abstract. A Lie derivation is called standard if it is a sum of a derivation and a linear map with image in the center vanishing on commutators. In this paper we show that Lie derivations of a reflexive algebra AlgL on a Banach space are standard if L is a nest, or has the non-trivial smallest element, or has the non-trivial greatest element. Mathematics Subject Classification (2000). Primary 47L35; Secondary 17B40, 17B60. Keywords. Lie derivations, derivations, reflexive algebras, nest algebras.

1. Introduction and Preliminaries Let A be an associative algebra and M be an A-bimodule. By Z(M, A) we denote the center of M relative to A, that is, Z(M, A) = {M ∈ M : AM = M A for all A ∈ A}. A linear map δ : A → M is called a derivation if δ(AB) = δ(A)B + Aδ(B) for all A, B ∈ A, and a Lie derivation if δ([A, B]) = [δ(A), B] + [A, δ(B)] for all A, B, ∈ A, where [A, B] = AB − BA is the usual Lie product. We say that a Lie derivation δ is standard if it can be decomposed as δ = d + τ , where d is a derivation from A into M and τ is a linear map from A into the center Z(M, A) of M relative to A vanishing on each commutator. The classical problem, which has been studied for many years, is to find conditions on A under which each Lie derivation is standard or standard-like. This problem has been investigated for prime rings in [2, 5, 14, 17, 18], for C*-algebras and for more general semisimple Banach algebras in [1, 9, 15, 16, 19], for triangular algebras in [3, 6]. In the present note, we pursue this line of investigation for reflexive algebras. Throughout, all algebras and vector spaces will be over F, where F is either the real field R or the complex field C. Given a Banach space X with topological dual X ∗ , by B(X) we mean the algebra of all bounded linear operators on X. The terms operator on X and subspace of X will mean ‘bounded linear map of X This work was supported by NNSFC (No. 10771154) and PNSFJ (No. BK2007049).

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into itself’ and ‘norm closed linear manifold of X’, respectively. For A ∈ B(X), denote by A∗ the adjoint of A. For any non-empty subset L ⊆ X, L⊥ denotes its annihilator, that is, L⊥ = {f ∈ X ∗ : f (x) = 0 for all x ∈ L}. A family L of subspaces of X is a subspace lattice if it contains (0) and X, and is complete in the sense that it is closed under the formation of arbitrary closed linear spans (denoted by ∨) and intersections (denoted by ∧). A nest is a totally ordered subspace lattice. Given a subspace lattice L on X, the associated subspace lattice algebra AlgL is the set of operators on X leaving every subspace in L invariant, that is, AlgL = {A ∈ B(X) : Ax ∈ E for every x ∈ E and for every E ∈ L}. Dually, if A is a subalgebra of B(X), by LatA we denote the lattice of subspaces of X that are left invariant by each operator in A. An algebra A is reflexive if A = AlgLatA, and a lattice L is reflexive if L = LatAlgL. Clearly, every reflexive algebra is of the form AlgL for some subspace lattice L and vice versa. In this paper, we are mainly interested in a certain tractable class of reflexive algebras, namely those which are rich in rank one operators. Let x ∈ X and f ∈ X ∗ be non-zero. The rank one operator x ⊗ f is defined by y 7→ f (y)x for y ∈ X. If L is a subspace lattice of X and E ∈ L, we define E− = ∨{F ∈ L : F 6⊇ E}, E 6= 0, E+ = ∧{F ∈ L : F 6⊆ E}, E 6= X. It is well known that x ⊗ f belongs to AlgL if and only if there exists an element ⊥ ⊥ means (E− )⊥ . . Here and subsequently, E− E ∈ L such that x ∈ E and f ∈ E− It turns out that the problem of characterizing Lie derivations of algebras on Banach spaces is much more difficult than the problem of describing Lie derivations of algebras on Hilbert spaces. Suppose that each derivation d of a subalgebra A of B(X) is spatial, i.e, there exists an operator T in B(X) such that d(A) = T A−AT for A ∈ A. Then it is easily seen that d(A) ker(A) is contained in the range of A for each A in A. Hence if a Lie derivation δ of A is standard, then there exits a center-valued map τ such that (δ(A) − τ (A)) ker(A) is contained in the range of A for each A ∈ A. The main idea in this paper is to show that this is true in some cases. The following proposition makes it possible to prove this only for the set of rank one operators. Proposition 1.1. Let E and F be non-zero subspaces of X and X ∗ , respectively. Let Φ : E × F → B(X) be a bilinear map such that Φ(x, f ) ker(f ) ⊆ Fx for all x ∈ E and f ∈ F . Then there exist two linear maps T : E → X and S : F → X ∗ such that Φ(x, f ) = T x ⊗ f + x ⊗ Sf for all x ∈ E and f ∈ F . ⊥ Proof. For any non-zero vectors x ∈ X and f ∈ X− , since Φ(x, f ) ker(f ) ⊆ Fx , there is a continuous linear functional hx,f on ker(f ) such that, for each z ∈ ker(f ),

Φ(x, f )z = hx,f (z)x.

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˜ x,f be a Let zf be in X such that f (zf ) = 1. Then X = Fzf ⊕ ker(f ). Let h ˜ ˜ continuous extension of hx,f to X. Then hx,f − hx,f (zf )f is also an extension of hx,f which vanishes at zf . Still by hx,f we denote such a special extension. Define a linear map Tf : E → X by Tf x = Φ(x, f )zf for x ∈ E. Then for all λ ∈ F and all z ∈ ker(f ), we have that Φ(x, f )(λzf + z) = λTf x + hx,f (z)x = λTf x + hx,f (λzf + z)x. So for all x ∈ E and f ∈ F , Φ(x, f ) = Tf x ⊗ f + x ⊗ hx,f .

(1.1)

We claim that hx,f depends only on f . To see this, fix a non-zero functional f in F , and let x1 and x2 be non-zero vectors in E. First suppose that x1 and x2 are linearly independent. For all z ∈ ker(f ), by (1.1) we have that Tf (x1 + x2 ) + hx1 +x2 ,f (z)(x1 + x2 ) = Φ(x1 + x2 , f )(zf + z) = (Φ(x1 , f ) + Φ(x2 , f ))(zf + z) = Tf x1 + hx1 ,f (z)x1 + Tf x2 + hx2 ,f (z)x2 , from which we get (hx1 +x2 ,f (z) − hx1 ,f (z))x1 = (hx2 ,f (z) − hx1 +x2 ,f (z))x2 . So hx1 ,f = hx1 +x2 ,f = hx2 ,f . Now suppose that x1 and x2 are linearly dependent, say x2 = λx1 . Then Tf x2 ⊗ f + x2 ⊗ hx2 ,f = Φ(x2 ⊗ f ) = λΦ(x1 ⊗ f ) = λ(Tf x1 ⊗ f + x1 ⊗ hx1 ,f ) = Tf x2 ⊗ f + x2 ⊗ hx1 ,f . So hx1 ,f = hx2 ,f , establishing the claim. Therefore, for each f ∈ F there exists a unique functional hf in X ∗ which vanishes at zf such that Φ(x, f ) = Tf x ⊗ f + x ⊗ hf

(1.2)

holds for all x ∈ E. We now claim that if f1 and f2 in F are linearly independent then the difference Tf1 − Tf2 is a scalar multiple of the identity IE on E. By the independency of f1 and f2 , we have ker(f1 ) 6⊆ ker(f2 ) and ker(f2 ) 6⊆ ker(f1 ). Accordingly, there exist two vectors x1 and x2 such that fi (xj ) = δij . By (1.2), for all x ∈ E, Tf1 x ⊗ f1 + x ⊗ hf1 + Tf2 x ⊗ f2 + x ⊗ hf2 = Φ(x, f1 + f2 ) = Tf1 +f2 x ⊗ (f1 + f2 ) + x ⊗ hf1 +f2 . Applying this equation to x1 − x2 , we get a scalar λi such that (Tf1 − Tf2 )x = λi x for all x ∈ E, proving the claim. Now fix a non-zero functional f0 ∈ F and set T = Tf0 . Let f be in F . If f is linearly dependent of f0 , say f = µf f0 , then by (1.2) we have, for all x ∈ E, that Φ(x, f ) = µf Φ(x, f0 ) = µf (Tf0 x ⊗ f0 + x ⊗ hf0 ) = T x ⊗ f + x ⊗ (µf hf0 );

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if f is linearly independent of f0 , then Tf = Tf0 + λf IE for some λf ∈ F. Thus, by (1.2) we have, for all x ∈ E, that Φ(x, f ) = Tf x ⊗ f + x ⊗ hf = (Tf0 + λf IE )x ⊗ f + x ⊗ hf = T x ⊗ f + x ⊗ (λf f + hf ). So there exists a unique functional Sf from F into X ∗ such that Φ(x, f ) = T x ⊗ f + x ⊗ Sf . It is easy to see that the map S : F → X ∗ is well-defined and linear. The proof is complete.

2. Lie derivations of reflexive algebras In this section, we shall study Lie derivations of reflexive algebras AlgL for which L has the non-trivial smallest element or the non-trivial greatest element. Theorem 2.1. Let X be a Banach space with dimension greater than 1. Let L be a subspace lattice of X with X− 6= X. Let δ : AlgL → B(X) be a Lie derivation. Then δ is standard. Proof. If X− = (0), then AlgL = B(X). The result follows from [14]. If (0) < X− < X and X is 2-dimensional, then δ is a Lie derivation from the two-by-two upper triangular matrix algebra into the two-by-two matrix algebra. The result follows from [3, 4, 6]. In the sequel, we assume that X is of dimension greater than 2 and (0) < X− < X. We first establish two claims. ⊥ Claim 1. There is a map ψ : X×X− → F such that (δ(x⊗f )−ψ(x, f )I) ker(f ) ⊆ Fx ⊥ for all x ∈ X and f ∈ X− . Moreover, if f (x) = 0 then ψ(x, f ) = 0. ⊥ . First suppose that f (x) 6= 0. For z ∈ ker(f ), we Let x be in X and f in X− have f (x)δ(z ⊗ f ) = δ([z ⊗ f, x ⊗ f ]) = [δ(z ⊗ f ), x ⊗ f ] + [z ⊗ f, δ(x ⊗ f )] = δ(z ⊗ f )x ⊗ f − (x ⊗ f )δ(z ⊗ f ) + (z ⊗ f )δ(x ⊗ f ) − δ(x ⊗ f )z ⊗ f. Applying this equation to x, we get f (x)δ(x ⊗ f )z = f (δ(x ⊗ f )x)z − f (δ(z ⊗ f )x)x 1 for all z ∈ ker(f ). Let ψ(x, f ) = f (x) f (δ(x ⊗ f )x), as required. Now suppose that f (x) = 0. Since ker(f ) is of dimension at least 2, we can take x1 from ker(f ) which is linearly independent of x. Take y1 from X such that

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f (y1 ) = 1. Let y2 = y1 + x1 . Then y1 , y2 , x are linearly independent. For i = 1, 2, a computation gives δ(x ⊗ f ) = δ([x ⊗ f, yi ⊗ f ]) = δ(x ⊗ f )yi ⊗ f − (yi ⊗ f )δ(x ⊗ f ) + (x ⊗ f )δ(yi ⊗ f ) − δ(yi ⊗ f )x ⊗ f. Let z be in ker(f ). Applying the above equation to z we get δ(x ⊗ f )z = λi yi + µi x for some λi , µi ∈ F, i = 1, 2. Thus λ1 y1 + µ1 x = λ2 y2 + µ2 x. Hence λ1 = λ2 = 0 since y1 , y2 , x are linearly independent. Consequently, δ(x ⊗ f ) ker(f ) ⊆ Fx. Claim 2. The map ψ obtained in Claim 1 is bilinear. The homogeneity is obvious. ⊥ Let f be in X− . Let x1 and x2 be in X. If both x1 and x2 are in ker(f ), then ψ(x1 , f ) = ψ(x2 , f ) = ψ(x1 + x2 , f ) = 0. So ψ(x1 + x2 , f ) = ψ(x1 , f ) + ψ(x2 , f ). If one of x1 and x2 is not in ker(f ), then span{x1 , x2 } ∩ ker(f ) is of dimension at most one. So we can take y from ker(f ) such that y ∈ / span{x1 , x2 }. Then we have δ(x1 ⊗ f )y = ψ(x1 , f )y + µ1 x1 , δ(x2 ⊗ f )y = ψ(x2 , f )y + µ2 x2 and δ((x1 + x2 ) ⊗ f )y = ψ(x1 + x2 , f )y + µ(x1 + x2 ) for some µ, µ1 , µ2 ∈ F. Comparing those equations and noting δ((x1 + x2 ) ⊗ f ) = δ(x1 , f ) + δ(x2 , f ), we get (ψ(x1 + x2 , f ) − ψ(x1 , f ) − ψ(x2 , f ))y = µ1 x1 + µ2 x2 − µ(x1 + x2 ). Since y ∈ / span{x1 , x2 }, it follows that ψ(x1 + x2 , f ) − ψ(x1 , f ) − ψ(x2 , f ) = 0. So ψ is additive in the first variable. ⊥ . To show that ψ is additive in the second variable, we let f1 and f2 be in X− Let x be in X. If x ∈ ker(f1 ) ∩ ker(f2 ), then ψ(x, f1 ) = ψ(x, f2 ) = ψ(x, f1 + f2 ). So ψ(x, f1 + f2 ) = ψ(x, f1 ) + ψ(x, f2 ). If x ∈ / ker(f1 ) ∩ ker(f2 ), then we can take z ∈ X− which is in k(f1 ) ∩ ker(f2 ) and is linearly independent of x. By Claim 1 we have δ(x ⊗ f1 )z = ψ(x, f1 )z + λ1 x, δ(x ⊗ f2 )z = ψ(x, f2 )z + λ2 x and δ(x ⊗ (f1 + f2 ))z = ψ(x, f1 + f2 )z + λx for some λ, λ1 , λ2 ∈ F. Comparing those equations and noting that δ(x ⊗ (f1 + f2 )) = δ(x ⊗ f1 ) + δ(x ⊗ f2 ), we get (ψ(x, f1 + f2 ) − ψ(x, f1 ) − ψ(x, f2 ))z = (λ1 + λ2 − λ)x. Since z and x are linearly independent, it follows that ψ(x, f1 + f2 ) = ψ(x, f1 ) + ψ(x, f2 ), establishing the claim.

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⊥ Now, for x ∈ X and f ∈ X− , let Φ(x, f ) = δ(x ⊗ f ) − ψ(x, f )I. By the two claims above, Φ(x, f ) is bilinear and Φ(x, f ) ker(f ) ⊆ Fx. Hence by Proposition 1.1,

Φ(x, f ) = T x ⊗ f + x ⊗ Sf, ⊥ where maps T : X → X and S : X− → X ∗ are linear. ⊥ Fix a non-zero functional f in X− . Let A be in AlgL. For x ∈ X,

δ([A, x ⊗ f ]) = δ(Ax ⊗ f − x ⊗ A∗ f ) = T Ax ⊗ f + Ax ⊗ Sf + ψ(Ax, f )I − T x ⊗ A∗ f − x ⊗ SA∗ f − ψ(x, A∗ f )I, on the other hand, δ([A, x ⊗ f ]) = [δ(A), x ⊗ f ] + [A, δ(x ⊗ f )] = δ(A)x ⊗ f − x ⊗ δ(A)∗ f + AT x ⊗ f + Ax ⊗ Sf − T x ⊗ A∗ f − x ⊗ A∗ Sf. So (δ(A) + AT − T A)x ⊗ f = x ⊗ (δ(A)∗ + A∗ S − SA∗ )f + (ψ(Ax, f ) − ψ(x, A∗ f ))I. Let z be in ker(f ) which is linearly independent of x. Applying both sides to z, we get λx + (ψ(Ax, f ) − ψ(x, A∗ f ))z = 0 for some λ ∈ F and hence ψ(Ax, f ) − ψ(x, A∗ f ) = 0. So (δ(A) + AT − T A)x ⊗ f = x ⊗ (δ(A)∗ + A∗ S − SA∗ )f for all x ∈ X. Applying this to a vector u with f (u) = 1, we get δ(A) + AT − T A = τ (A) for some τ (A) ∈ FI. Now for A ∈ AlgL, we define d(A) = δ(A) − τ (A). Then d(A) = AT − T A. Using this, it is easy to verify that d is a derivation. Hence τ is a linear map into FI vanishing on commutators. Adapting the ideas in the above proof, we give a kind of a dual. Note that if L is a subspace lattice of a reflexive Banach space, then L⊥ = {L⊥ : L ∈ L} is a subspace lattice. In nonreflexive spaces, the set L⊥ of subspaces of X ∗ fails, usually, to be a lattice. But even in the case of reflexive spaces, the elements (L− )⊥ and (L⊥ )− of L⊥ bear no relation: Examples show that they can be incomparable. So the following theorem does not seem to follow from the previous one. Theorem 2.2. Let L be a subspace lattice of a Banach space X with dimension at least 2 and suppose that (0)+ 6= (0). Let δ : AlgL → B(X) be a Lie derivation. Then δ is standard. Proof. (sketch) We can assume that (0) < (0)+ < X and X has dimension at least 3. Let x 7→ x ˆ be the canonical map from X into X ∗∗ . With x ∈ X, we fix a functional ϕx in X ∗ such that ϕx (x) = 1. Then X ∗ = Fϕx ⊕ker(ˆ x). Hence there is a x) ⊆ Ff bilinear map ψ from (0)+ ×X ∗ to B(X) such that (δ(x⊗f )−ψ(x, f )I)∗ ker(ˆ

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for all (x, f ) ∈ (0)+ ×X ∗ . Let Φ(f, x ˆ) = (δ(x⊗f )−ψ(x, f )I)∗ . Then Φ is a bilinear d+ into B(X ∗ ). Hence there exist linear maps T : X ∗ → X ∗ and map from X ∗ × (0) d+ → X ∗∗ such that S : (0) Φ(f, x ˆ) = T f ⊗ x ˆ + f ⊗ Sx ˆ ∗

for all (x, f ) ∈ (0)+ × X . Hence with A ∈ AlgL, we have that \ + A∗∗ S x c (δ(A)∗ + A∗ T − T A∗ )f ⊗ x ˆ = f ⊗ (δ(A)x ˆ − S Ax) for all (x, f ) ∈ (0)+ × X ∗ . So δ(A)∗ + A∗ T − T A∗ = τ (A) for some τ (A) ∈ FI. Clearly, τ is a linear map. Define d(A) = δ(A) − τ (A) for A ∈ AlgL. Then d is linear. Moreover, for A, B ∈ L, d(AB)∗ = T (AB)∗ − (AB)∗ T = B ∗ d(A)∗ + d(B)∗ A∗ = (d(A)B + Ad(B))∗ . Consequently, d(AB) = d(A)B +Ad(B) for all A, B ∈ A. Namely, d is a derivation. Now δ = d + τ . From this, it is easy to see that τ vanishes on commutators, completing the proof.

3. Lie derivations of nest algebras We now turn to the nest algebra case. To prove the main result we need some lemmas. Lemma 3.1. Let E be an infinite-dimensional subspace of a Banach space X. Let f be in X ∗ . Then the subspace ker(f ) ∩ E is infinite-dimensional. Proof. If f (E) = {0}, then the lemma is obviously true. Now suppose that f (x) = 1 for some x ∈ E. Let P = x ⊗ f . Then ker(f ) = (I − P )X and hence ker(f ) ∩ E = (I − P )E. Since E is infinite-dimensional, so is (I − P )E, proving the lemma. An alternative proof is as follows. Let g = f |E . Then ker(f ) ∩ E = ker(g) and hence it is infinite-dimensional since E is infinite-dimensional. Lemma 3.2. Let E and F be infinite-dimensional subspaces of X and X ∗ respectively. Let φ and ψ be two maps from E × F into B(X). Suppose that φ is bilinear and (φ(x, f ) − ψ(x, f )) ker(f ) ⊆ Fx for all x ∈ E and f ∈ F . Then ψ is bilinear. Proof. Let f be a functional in F . Let x1 and x2 be in E and α1 and α2 be in F. Choose y in ker(f ) such that y ∈ / span{x1 , x2 }. By the assumption, we have φ(x1 , f )y = ψ(x1 , f )y + µ1 x1 , φ(x2 , f )y = ψ(x2 , f )y + µ2 x2 and φ(α1 x1 + α2 x2 , f )y = ψ(α1 x1 + α2 x2 , f )y + µ(α1 x1 + α2 x2 ) for some µ, µ1 , µ2 ∈ F. Comparing those equations and noting φ(α1 x1 +α2 x2 , f ) = α1 φ(x1 , f ) + α2 φ(x2 , f ), we get (ψ(α1 x1 + α2 x2 , f ) − α1 ψ(x1 , f ) − α2 ψ(x2 , f ))y = µ1 x1 + µ2 x2 − µ(α1 x1 + α2 x2 ).

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Since y ∈ / span{x1 , x2 }, it follows that ψ(α1 x1 + α2 x2 , f ) − α1 ψ(x1 , f ) − α2 ψ(x2 , f ) = 0. So ψ is linear in the first variable. To show that ψ is linear in the second variable, we let f1 and f2 be in F and β1 and β2 be in F. By Lemma 3.1, ker(f1 ) ∩ ker(f2 ) is infinite dimensional. Let x be in E. Choose z from ker(f1 ) ∩ ker(f2 ) such that z is linearly independent of x. Note that z ∈ ker(f1 + f2 ). By the assumption, we have φ(x, f1 )z = ψ(x, f1 )z + λ1 x, φ(x, f2 )z = ψ(x, f2 )z + λ2 x and φ(x, β1 f1 + β2 f2 )z = ψ(x, β1 f1 + β2 f2 )z + λx for some λ, λ1 , λ2 ∈ F. Comparing those equations and noting that φ(x, β1 f1 + β2 f2 ) = β1 φ(x, f1 ) + β2 δ(x, f2 ), we get (ψ(x, β1 f1 + β2 f2 ) − β1 ψ(x, f1 ) − β2 ψ(x, f2 ))z = (λ1 + λ2 − λ)x. Since z and x are linearly independent, it follows that ψ(x, β1 f1 + β2 f2 ) = β1 ψ(x, f1 ) + β2 ψ(x, f2 ), completing the proof.

The following was proved in the Hilbert space case. The method here is very different from one in [13] because of the lack of invariant projections. Theorem 3.3. Let N be a nest on a Banach space X. Let δ : AlgN → B(X) be a Lie derivation. Then δ is standard. Proof. By Theorems 2.1 and 2.2, we can assume that (0)+ = (0) and X− = X. Let N 0 = N \ {(0), X}. Then (0) = ∧{N : N ∈ N 0 } and X = ∨{N : N ∈ N 0 }. We first prove two claims. Claim 1. Let f be in X ∗ . Then ker(f ) = ∨{ker(f ) ∩ N : N ∈ N 0 }. If f = 0, then the claim is obviously true. Now suppose that f 6= 0. Since ∨{N : N ∈ N 0 } = X, there must be an M in N 0 such that f (z) = 1 for some z ∈ M . Let x be in ker(f ). Since ∨{N : N ∈ N 0 } = X again, there is a net Nk in N 0 and a net yk with each yk ∈ Nk such that limk yk = x. Since X = Fz + ker(f ), we can write yk = λk z +xk with λk ∈ F and xk ∈ ker(f ). Then xk ∈ M ∨Nk ∈ N 0 . Now lim λk = lim f (yk ) = f (x) = 0. k

k

So lim xk = lim(yk − λk z) = lim yk = x. k

k

k

Thus we have shown that ker(f ) ⊆ ∨{ker(f ) ∩ N : N ∈ N 0 }. But the inverse inclusion is clear, establishing the claim 1. Claim 2. There is a map ψ from the set of all rank one operators in AlgN into F such that (δ(x ⊗ f ) − ψ(x, f )I) ker(f ) ⊆ Fx for all x ⊗ f ∈ AlgN .

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Let x ⊗ f in AlgN be non-zero. Let N be in N 0 such that f |N 6= 0. It follows from x ⊗ f ∈ AlgN that x ∈ N . Choose a non-zero functional g ∈ N ⊥ . Obviously, g is linearly independent of f . Accordingly, there is a vector z in X such that f (z) = 0 and g(z) = 1. For y ∈ ker(f ) ∩ N , we have that 0 = δ([x ⊗ f, y ⊗ g]) = δ(x ⊗ f )y ⊗ g − y ⊗ gδ(x ⊗ f ) + x ⊗ f δ(y ⊗ g) − δ(y ⊗ g)x ⊗ f. Applying this to z, we get, for each y ∈ ker(f ) ∩ N , that δ(x ⊗ f )y = µ(x, f, y, N )x + ψ(x, f, N )y

(3.1)

for some µ(x, f, y, N ), ψ(x, f, N ) ∈ F. We first show that the quantity ψ(x, f, N ) in Eq. (3.1) is unique. To do this, suppose that the scalar λ satisfies that for each y ∈ ker(f ) ∩ N there is a scalar µ(y) such that δ(x ⊗ f )y = µ(y)x + λy. (3.2) Since (0)+ = (0), N is infinite-dimensional and so is ker(f ) ∩ N . Thus we can choose y0 ∈ ker(f ) ∩ N which is linearly independent of x. By Eqs. (3.1) and (3.2), we have that (ψ(x, f, N ) − λ)y0 = (µ(y0 ) − µ(x, f, y0 , N ))x. By the linear independence of y0 and x, we get that λ = ψ(x, f, N ). We next show that ψ(x, f, N ) = ψ(x, f, M ) for all N, M ∈ N 0 with f |N 6= 0 and f |M 6= 0. In fact, without loss of gernerality, we can assume that N ≤ M . Then for each y ∈ ker(f ) ∩ N , since ker(f ) ∩ N ⊆ ker(f ) ∩ M , we have δ(x ⊗ f )y = µ(x, f, y, M )x + ψ(x, f, M )y. By the uniqueness of the quantity ψ(x, f, N ), ψ(x, f, M ) = ψ(x, f, N ). Now by ψ(x, f ) denote the common values ψ(x, f, N ). Then (δ(x ⊗ f ) − ψ(x, f )I)y ∈ Fx for all N ∈ N 0 with f |N 6= 0 and all y ∈ ker(f ) ∩ N . However, by the claim 1 span{ker(f ) ∩ N : N ∈ N 0 , f |N 6= 0} is dense in ker(f ). It follows that (δ(x ⊗ f ) − ψ(x, f )I) ker(f ) ⊆ Fx, establishing the claim 2. Therefore, for N ∈ N 0 , (δ(x ⊗ f ) − ψ(x, f )I) ker(f ) ⊆ Fx ⊥ . N−

By Lemma 3.2 and Proposition 1.1, there are two linear for all x ∈ N and f ∈ ⊥ maps TN : N → X and SN : N− → X ∗ such that δ(x ⊗ f ) − ψ(x, f )I = TN x ⊗ f + x ⊗ SN f

(3.3)

⊥ for all x ∈ N and f ∈ N− . Then for N, M ∈ N 0 , we have

T N x ⊗ f + x ⊗ SN f = T M x ⊗ f + x ⊗ SM f for all x ∈ N ∩ M and f ∈ (N ∨ M )⊥ − . So the restriction of TN − TM to N ∩ M is a scalar multiple of the identity on N ∩ M .

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⊥ Let A be in AlgN . Let N be in N 0 . Then for x ∈ N and f ∈ N− , by δ([A, x ⊗ f ]) = [δ(A), x ⊗ f ] + [A, δ(x ⊗ f )] and Eq.(3.3), we have

(δ(A)+ATN −TN A)x⊗f = x⊗(δ(A)∗ +A∗ SN −SN A∗ )f −(ψ(x, A∗ f )−ψ(Ax, f ))I. Since I is infinite-rank, it follows that ψ(x, A∗ f ) − ψ(Ax, f ) = 0. So (δ(A) + ATN − TN A)x ⊗ f = x ⊗ (δ(A)∗ + A∗ SN − SN A∗ )f. Hence there is a scalar τN (A) such that (δ(A) + ATN − TN A)x = τN (A)x

(3.4)

for all x ∈ N . Fix an N0 in N 0 . Set τ (A) = τN0 (A). For N ∈ N 0 and x ∈ N ∩ N0 , by Eq. (3.4) we have (A(TN − TN0 ) − (TN − TN0 )A)x = (τN (A) − τ (A))x. Since we have shown that the restriction of TN − TN0 to N ∩ N0 is a scalar multiple of the identity on N ∩ N0 , it follows that τN (A) = τ (A) for all N ∈ N 0 . Moreover, τ is linear. Now let d = δ − τ . Then for N ∈ N and x ∈ N , d(A)x = (TN A − ATN ) and hence d(AB)x = (d(A)B + Ad(B))x. Since ∪{N : N ∈ N 0 } is dense in X, it follows that d(AB) = d(A)B + Ad(B). Namely d is a derivation, completing the proof.

References [1] J. Alaminos, M. Mathieu and A. R. Villena, Symmetric amenability and Lie derivations. Math. Proc. Cambridge Philos. Soc. 137(2004), 433–439. [2] K. I. Beidar and M. A. Chebotar, On Lie derivations of Lie ideals of prime rings. Israel J. Math. 123(2001), 131–148. [3] D. Benkovi˘c, Lie derivations on triangular matrices. Linear Multilinear Algebra 55(2007), 619–626. [4] D. Benkovi˘c and D. Eremita, Commuting traces and commutativity preserving maps on triangular algebras. J. Algebra 280(2004), 797–824. [5] M. Bre˘sar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings. Trans. Amer. Math. Soc. 335(1993), 525–546. [6] W. Cheung, Lie derivations of triangular algebras. Linear Multilinear Algebra 51(2003), 299–310. [7] K. Davidson, Nest algebras. Pitman Res. Notes, Math. Ser. 191, Longman Sci. Tech., New York, 1988. [8] P. R. Halmos, A Hilbert Space Problem Book. 2nd Edition, Springer-Verlag, New York/Heidelberg/Berlin, 1982. [9] B. E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations. Math. Proc. Cambridge Philos. Soc. 120(1996), 455–473. [10] M. S. Lambrou, Approximants, commutants and double commutants in normed algebras. J. London Math. Soc. 25(1982), 499–513.

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[11] M. S. Lambrou, Automatic continuity and implementation of homomorphisms. (manuscript). [12] W. E. Longstaff, Strongly reflexive lattices. J. London Math. Soc. 11(1975), 491–498. [13] F. Lu, Lie triple derivations on nest algebras. Math. Nachr. 280(2007), 882–887. [14] W. S. Martindale, Lie derivations of primitive rings. Michigan J. Math. 11(1964), 183–187. [15] M. Mathieu and A. R. Villena, The structure of Lie derivations on C*-algebras. J. Funct. Anal. 202(2003), 504–525. [16] C. R. Miers, Lie derivations of von Neumann algebras. Duke Math. J. 40(1973), 403–409. [17] G. A. Swain, Lie derivations of the skew elements of prime rings with involution. J. Algebra 184(1996), 679–704. [18] G. A. Swain and P. S. Blau, Lie derivations in prime rings with involution. Canad. Math. Bull. 42(1999), 401–411. [19] R. A. Villena, Lie derivations on Banach algebras. J. Algebra 226(2000), 390–409. Fangyan Lu and Benhong Liu Department of Mathematics Suzhou University Suzhou 215006 People’s Republic of China e-mail: [email protected] honglb− [email protected] Submitted: November 17, 2008. Revised: January 14, 2009.

Integr. equ. oper. theory 64 (2009), 273–299 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/020273-27, published online June 2, 2009 DOI 10.1007/s00020-009-1686-x

Integral Equations and Operator Theory

A Schur Analysis of the Minimal Weak Unitary Dilations of a Contraction Operator and the Relaxed Commutant Lifting Theorem S.A.M. Marcantognini and M.D. Mor´an Abstract. A Schur-type analysis of the minimal weak unitary dilations of a given contraction operator is obtained from the Arov-Grossman functional model. The result is combined with the coupling method to give a description of the interpolants in the Relaxed Commutant Lifting Theorem. Mathematics Subject Classification (2000). Primary 47A20; Secondary 47A57, 47A56. Keywords. Weak unitary dilations of contractions, Schur analysis, interpolants, Relaxed Commutant Lifting Theorem, Arov-Grossman model, coupling.

1. Introduction The Arov-Grossman model [3, 4] gives a labeling of the minimal unitary extensions of a given Hilbert space isometry by means of operator valued Schur functions. In this setting if V is an isometry defined in a closed linear subspace D of a separable Hilbert space H (all Hilbert spaces are supposed to be separable,) by a minimal e⊇H unitary extension of V we mean a unitary operator U on a Hilbert space H e such that U |D = V (the extension property) and H is the minimal Hilbert space containing all the orbits of H under U and U −1 (the minimality condition.) If V : D ⊆ H → H is an isometry then its defect subspaces are N := H D and M := H V (D). These spaces are relevant in the Arov-Grossman model since the values of the Schur functions in the description provided by the model are bounded linear operators from N into M. Roughly speaking, if ϑ is an L(N , M)valued Schur function on the open unit disk D of the complex plane, then there exists a model unitary operator Uϑ on a Hilbert space Fϑ such that Uϑ is a minimal unitary extension of V . Conversely, if U is a minimal unitary extension of V acting e then ϑ(z) := PM U (1 − zP e U )−1 |N (z ∈ D) is an L(N , M)-valued Schur on H H H

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e function whose associated model Uϑ ∈ L(Fϑ ) is undistinguishable from U ∈ L(H) e via an isometric isomorphism τ : H → Fϑ such that τ |H = 1 and τ U = Uϑ τ . Above and hereafter 1 denotes the identity operator on the corresponding Hilbert e PE is the orthogonal space and, for any closed subspace E of a Hilbert space E, e projection from E onto E. The Arov-Grossman model is a useful tool in problem solving. Its use as a theoretical device in finding solutions and as a practical method for constructing them relies on its combination with the coupling method. The coupling method is an operator theoretic technique that, when applicable to a given problem, provides a Hilbert space operator containing all the information within the data of the problem. In several interesting cases, for instance, in the framework of the classical Commutant Lifting Theorem, the coupling operator is an isometry V on a Hilbert space H and it happens that the solutions of the problem are in oneto-one correspondence, up to isometric isomorphism, with some (or all) minimal unitary extensions of V . Therefore, in this setting, the combination of the coupling method and the Arov-Grossman model gives a parameterization of the solutions of the problem, yielding, as by-product, solvability and non-uniqueness criteria. In some other problems, associated with more complex situations, for instance, in dealing with the Relaxed Commutant Lifting Theorem, the coupling operator is no longer an isometry but just a contraction. Nonetheless some unitary operators related to the coupling contraction, a subclass of its minimal weak unitary dilations (to be defined in Section 2,) still happen to correspond to the solutions we are looking for. If X is an isometry then a weak unitary dilation of X is no different than a unitary extension of X. Even though a more general situation is determined when X is only a contraction operator, a description of its minimal weak unitary dilations can still be obtained from the Arov-Grossman model. We present such a description in Section 2 while in Section 3 we use it to describe all the interpolants in the Relaxed Commutant Lifting Theorem. The description is given by a map from certain set of Schur functions (the parameters) onto the set of all interpolants. The map, however, does not establish a one-to-one correspondence between the parameter and the interpolant, as it may happen that different parameters provide the same interpolant. We give a necessary and sufficient condition for two parameters to yield the same interpolant. The special case of the classical Commutant Lifting Theorem is also discussed in this framework. For this case we get that the map provides a proper parameterization. We complement this note with an Appendix containing a sketch of the proof of the Arov-Grossman model as we state it and use it in Section 2. The relaxation of the Commutant Lifting Theorem was introduced by C. Foias, A.E. Frazho and M.A. Kaashoek in [9]. Descriptions of the interpolants in the relaxed version of the Commutant Lifting Theorem were provided by A.E. Frazho, S. ter Host and M.A. Kaashoek in [10] and [11], and by W.S. Li and D. Timotin in [12]. The coupling method was used in [10] in combination with system theory techniques and in [12] in conjunction with a choice sequence approach.

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The results we obtained differ from the ones given by those authors, in particular, because ours are proved by merging the coupling method with the Arov-Grossman functional model. In [11] the use of an elementary harmonic majorant argument results in a more refined and more explicit description of the interpolants than the one presented by the same authors in the previous paper. The coupling method was used to study commutant lifting problems by R. Arocena [1, 2]. In essence the same treatment was independently outlined by M. Cotlar and C. Sadosky in [6] (see also [8, Section VII.8] and [7, Section 5].) A proper parameterization of the interpolants in the classical Commutant Lifting Theorem was obtained in [13] by methods which are similar to the ones we used here.

2. The minimal weak unitary dilations of a contraction Let X : B ⊆ A → A be a Hilbert space contraction. A unitary operator W on a Hilbert space Ae ⊇ A is a weak unitary dilation of X if PA W |B = X. A weak unitary dilation W of X on Ae is said to be minimal if Ae is the minimal Hilbert space containing all the orbits of A under W and W −1 . Two minimal weak unitary e and W 0 ∈ L(Ae0 ), are regarded as identical whenever dilations of X, say W ∈ L(A) there is an isometric isomorphism τ : Ae0 → Ae such that τ |A = 1 and τ W 0 = W τ . In the sequel we denote by W U D(X) the set of the undistinguishable minimal weak unitary dilations of X. We use the term “weak” dilation for contrast as the term dilation has been used in the literature for “strong” or “power” dilation, meaning that D ∈ L(D) is a dilation of C ∈ L(C) if C ⊆ D and PC Dn |C = C n for all n = 0, 1, . . .. When V : D ⊆ H → H is an isometry, W U D(V ) is the set of the undistinguishable minimal unitary extensions of V . We write U(V ) for W U D(V ) in this case. 1 If X : B ⊆ A → A is a contraction, then DX := (1 − X ∗ X) 2 and DX ∗ := 1 (1−XX ∗ ) 2 are the defect operators of X and X ∗ , respectively. The corresponding defect spaces are DX := DX (B) and DX ∗ := DX ∗ (A). Clearly, the operator V1 defined in B by X A V1 := :B→ (2.1) DX DX is an isometry on the Hilbert space H1 := A ⊕ DX .

(2.2)

e is a weak unitary dilation of X, then W admits a 2 × 2 block If W ∈ L(A) matrix representation in the form B A X DX ∗ Z W = : e → e , Y DX W22 A B A A

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e e where Y : DX → A A is an isometry, Z : A B → DX ∗ ⊆ A is a co-isometry, that is, Z ∗ : DX ∗ → Ae B is an isometry, and W22 : Ae B → Ae A is a contraction such that Y ∗ W22 = −X ∗ Z, ∗ ZW22 = −XY ∗ , ∗ W22 W22 (1 − Z ∗ Z) = 1 − Z ∗ Z,

and ∗ W22 W22 (1 − Y Y ∗ ) = 1 − Y Y ∗ .

In particular, VY := W |B is an isometry on HY := A ⊕ Y (DX ): X A VY = :B→ ; Y DX Y (DX ) and if W is minimal, that is, W ∈ W U D(X), then W ∈ U(VY ). On the other hand, if V1 is the isometry in (2.1)-(2.2) (in which case V1 = VY with Y the identity operator on DX ) then U(V1 ) ⊆ W U D(X). These simple observations lead to contrast W U D(X) with U(V1 ) by comparX DX ∗ Z ing U(V1 ) to U(VY ) for each W = in W U D(X). To accomplish Y DX W22 the task we appeal to the Arov-Grossman model. In the sequel, if N and M are two Hilbert spaces, then S(N , M) stands for the L(N , M)-Schur class, so that ϑ ∈ S(N , M) if and only if ϑ : D → L(N , M) is an analytic function such that sup kϑ(z)k ≤ 1. z∈D

For any (separable) Hilbert space E we denote by L2 (E) the class of functions f on the unit circle T of the complex plane with values in E, measurable (strongly or weakly, which comes to be the same due to the separability of E) and such that Z 2π 1 2 kf (eit )k2 dt < ∞. kf k := 2π 0 With the pointwise linear operations and the scalar product Z 2π 1 hf, giL2 (E) := hf (eit ), g(eit )iE dt (f, g ∈ L2 (E)) 2π 0 L2 (E) becomes a (separable) Hilbert space under the interpretation that two functions in L2 (E) are considered identical if they coincide almost everywhere. MoreL∞ over, L2 (E) = n=−∞ Gn (E), where, for each integer number n, Gn (E) is the subspace of those functions f ∈ L2 (E) such that f (eit ) = eint x for some x ∈ E. elements of H 2 (E) are all the analytic functions u : D → E, u(z) = P∞ The n n=0 z un , z ∈ D and {un } ⊆ E, such that kuk2 :=

∞ X n=0

kun k2 < ∞.

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We recall that H 2 (E) is a Hilbert space with the pointwise linear operations and the scalar product ! ∞ ∞ ∞ X X X n n 2 hu, viH 2 (E) := hun , vn iE u(z) = z un , v(z) = z vn ∈ H (E) . n=0

n=0

n=0

As a consequence of Fatou’s Theorem, the radial limit limr↑1 u(reit ) exists almost everywhere. The application that maps each u(z) ∈ H 2 (E) into its radial limit provides an embedding of H 2 (E) into L2 (E) preserving the Hilbert space structures. Via the Poisson be shown that the application maps H 2 (E) L∞integral, it can 2 ontoL the subspace n=0 Gn (E) of L (E). Therefore we may consider that H 2 (E) ∞ and n=0 Gn (E) amount to the same Hilbert space. If ϑ ∈ S(N , M) then limr↑1 ϑ(reit ) exists almost everywhere as a strong limit of operators and determines a contraction operator in L(N , M). With each ϑ ∈ S(N , M) we associate a contraction operator from L2 (N ) into L2 (M) defined by f (eit ) 7→ ϑ(eit )f (eit ) (f (eit ) ∈ L2 (N )) and a contraction operator from H 2 (N ) into H 2 (M) defined by u(z) 7→ ϑ(z)u(z) u(z) ∈ H 2 (N ) and z ∈ D . L∞ Due to identification of H 2 (N ) (and H 2 (M)) with the subspace n=0 Gn (N ) Lthe ∞ (and n=0 Gn (M), respectively) the latter operator may be consider as a restriction of the former one. We denote both of them by ϑ. When N = M = E and ϑ(z) ≡ z (z times the identity operator on E) the associated operator is the (forward) shift S. Given ϑ ∈ S(N , M) we can likewise consider the operator ∆(eit ) = Dϑ(eit ) almost everywhere. The basic reference for vector and operator valued analytic functions is [14]. We refer the reader to the detailed exposition given therein. As a matter of notation, if C, D are Hilbert spaces and E = C ⊕ D, we will c write the elements of E either as sums c ⊕ d or as columns . d Theorem 2.1. (Arov-Grossman [3, 4]) Let V : D ⊆ H → H be an isometry with defect subspaces N and M. Given ϑ ∈ S(N , M), set Eϑ := H 2 (M) ⊕ ∆L2 (N ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (N )}⊥ , where ∆(ζ) := Dϑ(ζ) ,

|ζ| = 1.

Define Fϑ := H ⊕ Eϑ and Uϑ : Fϑ → Fϑ by     h V PD h + ϑ(0)PN h + φ(0)  S ∗ (φ + ϑPN h) Uϑ  φ  :=  ψ S ∗ (ψ + ∆PN h)

h ∈ H,

φ ∈ Eϑ ψ

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where S is the shift on either H 2 (M) or L2 (N ), depending on context. Then: (i) Uϑ ∈ L(Fϑ ) is a minimal unitary extension of V such that PM Uϑ (1 − zPEϑ Uϑ )−1 |N = ϑ(z) for all z ∈ D. e the function (ii) For any minimal unitary extension U of V on H, U )−1 |N ϑ(z) := PM U (1 − zPH H e

(z ∈ D)

belongs to S(N , M). e 0 ) is a minimal unitary extension of V undistinguishable from (iii) U 0 ∈ L(H e under an isometric isomorphism τ : H e0 → H e such that τ |H = 1 U ∈ L(H), and τ U 0 = U τ , if and only if PM U 0 (1 − zPHe0 H U 0 )−1 |N = PM U (1 − zPH H U )−1 |N e for all z ∈ D. Therefore, the map ϑ 7→ Uϑ ∈ L(Fϑ ) establishes a bijective correspondence between S(N , M) and U(V ) (up to isometric isomorphisms as far as U(V ) is concerned.) As we have already remarked, the model in Theorem 2.1 is due to D.Z. Arov and L.Z. Grossman (see [3, 4].) For the sake of completeness we include a sketch of the proof in the Appendix. In what follows we freely apply the model to analyze U(VY ) and the archetypical representative U(V1 ). The defect subspaces of V1 are N1 = (A B) ⊕ DX

and M1 = {x ∈ H1 : X ∗ PA x + DX PDX x = 0}.

On the other hand, for a given W ∈ W U D(X), the defect subspaces of the associated isometry VY are NY = (A B)⊕Y (DX )

and MY = {x ∈ HY : X ∗ PA x+DX Y ∗ PY (DX ) x = 0}.

So, if γY : HY → H1 is defined to be the identity operator on A and Y ∗ on Y (DX ), then γY is a unitary operator such that γY VY = V1 = V1 γY |B , γY (NY ) = N1 and γY (MY ) = M1 . Let ρ be the function in the Schur class S(NY , MY ) corresponding with the given W ∈ W U D(X), when W is viewed as an element of U(VY ) through the ArovGrossman model. Let σ be the isometric isomorphism from Ae onto Fρ = HY ⊕ Eρ satisfying σ|HY = 1 and σW = Uρ σ. Here Eρ = H 2 (MY ) ⊕ EL2 (NY ) ∩ {(ρχ, Eχ) : χ ∈ H 2 (NY )}⊥ , where E(ζ) := Dρ(ζ) ,

|ζ| = 1,

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and     hY VY PB hY + ρ(0)PNY hY + φ(0)  S ∗ (φ + ρPNY hY ) Uρ  φ  :=  ψ S ∗ (ψ + EPNY hY )

hY ∈ H Y ,

φ ∈ Eρ . ψ

A function ϑ in the Schur class S(N1 , M1 ) is obtained from ρ and γY by setting ϑ(z) := γY ρ(z)γY∗ |N1 (z ∈ D). Let Uϑ ∈ L(Fϑ ) be the element in W U D(X) given by the minimal unitary extension of V1 associated with ϑ in the ArovGrossman model, so that Fϑ = H1 ⊕ Eϑ , Eϑ := H 2 (M1 ) ⊕ ∆L2 (N1 ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (N1 )}⊥ , where ∆(ζ) := Dϑ(ζ) ,

|ζ| = 1,

and     h1 V1 PB h1 + ϑ(0)PN1 h1 + φ(0)  S ∗ (φ + ϑPN1 h1 ) Uϑ  φ  :=  ∗ ψ S (ψ + ∆PN1 h1 )

φ h1 ∈ H 1 , ∈ Eϑ . ψ

Extend γY to a unitary operator from Fρ onto Fϑ by setting     hY γY hY φ     γY φ = γY φ , hY ∈ HY , ∈ Eρ , ψ ψ γY ψ and let τ be the unitary operator from Ae onto Fϑ given by τ := γY σ. e and Uϑ ∈ L(Fϑ ) are undistinguishable We claim that the given W ∈ L(A) under τ as elements in W U D(X). Note that τ |A = 1, since the restrictions to A of both σ and γY equal 1. Hence we only need to show the intertwinnig relation τ W = Uϑ τ . e For all e a ∈ A, τWe a = γY σW e a = γY Uρ σe a.   hY a then Whence, if  φ  := σe ψ     VY PB hY + ρ(0)PNY hY + φ(0) hY  S ∗ (φ + ρPNY hY ) τWe a = γY Uρ  φ  = γY  ψ S ∗ (ψ + EPNY hY )

  V1 γY PB hY + ϑ(0)γY PNY hY + γY φ(0) , S ∗ (γY φ + ϑγY PNY hY ) = S ∗ (γY ψ + γY EPNY hY )

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  V1 PB γY hY + ϑ(0)PN1 γY hY + γY φ(0)  S ∗ (γY φ + ϑPN1 γY hY ) τWe a = Uϑ τe a= ∗ S (γY ψ + ∆PN1 γY hY )

whenever γY PB |HY = PB γY |HY ,

(2.3)

γY PNY |HY = PN1 γY |HY ,

(2.4)

γY E = ∆γY |NY .

(2.5)

The condition (2.3) is satisfied since, for all hY ∈ HY , hγY hY , BiH1 = hγY hY , γY (B)iH1 = hhY , BiHY = hPB hY , BiHY = hγY PB hY , BiH1 . Similarly, the condition (2.4) is equivalent to hγY hY , N1 iH1 = hγY hY , γY (NY )iH1 = hhY , NY iHY = hPNY hY , NY iHY = hγY PNY hY , N1 iH1 , for all hY ∈ HY . As for (2.5), note that, for any polynomial p, p(1 − ϑ∗ ϑ)γY |NY = γY p(1 − ρ∗ ρ), so that ∆γY |NY = γY E. The above arguments show that the map S(N1 , M1 ) → W U D(X) ϑ 7→ Uθ ∈ L(Fϑ ) e belonging to W U D(X), is surjective, in the sense that, given any W ∈ L(A) e is isometrically there exists ϑ in the Schur class S(N1 , M1 ) such that W ∈ L(A) isomorphic to Uϑ ∈ L(Fϑ ). Furthermore, as we will see next, the map is also one-to-one. Theorem 2.2. (Labeling of WUD(X)) Given a contraction X : B ⊆ A → A, set N1 := (A B) ⊕ DX

and

M1 := {x ∈ A ⊕ DX : X ∗ PA x + DX PDX x = 0}.

Given ϑ ∈ S(N1 , M1 ), set Eϑ := H 2 (M1 ) ⊕ ∆L2 (N1 ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (N1 )}⊥ , where ∆(ζ) := Dϑ(ζ) ,

|ζ| = 1.

Define Fϑ := A ⊕ DX ⊕ Eϑ and Uϑ : Fϑ → Fϑ by     h XPB h + DX PB h + ϑ(0)PN1 h + φ(0)  S ∗ (φ + ϑPN1 h) Uϑ  φ  :=  ∗ ψ S (ψ + ∆PN1 h)

φ h ∈ A ⊕ DX , ∈ Eϑ ψ

where S is the shift on either H 2 (M1 ) or L2 (N1 ), depending on context. Then the map ϑ 7→ Uθ ∈ L(Fϑ )

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establishes a bijective correspondence between S(N1 , M1 ) and W U D(X) (up to isometric isomorphisms as far as W U D(X) is concerned.) Proof. Only the injectivity is to be shown as we have already seen that for each ϑ ∈ S(N1 , M1 ) the unitary operator Uϑ ∈ L(Fϑ ) belongs to W U D(X) and any e is undistinguishable from some Uϑ ∈ L(Fϑ ). Consider two S(N1 , M1 )W ∈ L(A) functions α, β and let Uα ∈ L(Fα ), Uβ ∈ L(Fβ ) be the corresponding unitary operators. Assume there exists an isometric isomorphism σ : Fα → Fβ such that σ|A = 1 and σUα = Uβ σ. Then, for all a ∈ A and all integer number n, hUαn a, AiFα = hσUαn a, σAiFβ = hUβn a, AiFβ . Therefore PA (1 − zUα )−1 |A = PA (1 − zUβ )−1 |A ,

z ∈ D.

(2.6)

Conversely, if (2.6) holds then hUαn a, a0 iFα = hUβn a, a0 iFβ for all a, a0 ∈ A and all natural number n. Hence the operator defined on the linear n n span of {Uαn (A)}∞ n=−∞ and mapping Uα a into Uβ a (a ∈ A, n = 0, ±1, ±2, . . .) can be extended by continuity to an isometric isomorphism σ : Fα → Fβ such that σ|A = 1 and σUα = Uβ σ. These arguments show that Uα ∈ L(Fα ) and Uβ ∈ L(Fβ ) can be regarded as undistinguishable elements of W U D(X) if and only if (2.6) is satisfied. Next we show that if Uϑ ∈ L(Fϑ ) is the unitary operator corresponding with an arbitrary ϑ ∈ S(N1 , M1 ) then, for all z ∈ D, PA (1 − zUϑ )−1 |A n h io−1 −1 = 1 − z XPB + PA ϑ(z) (1 − zPDX ϑ(z)) (PA B + zDX PB ) .

(2.7)

For a given a ∈ A, define a1 (= a1 (z)) (z ∈ D) by a1 := PA (1 − zUϑ )−1 a. φ The vector a1 is related to a as above if and only if there exist b ∈ B and ∈ Eϑ ψ φ φ b = b(z), = (z) (z ∈ D) such that ψ ψ   a1 + DX b , φ (1 − zUϑ )−1 a =  ψ

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that is, such that       a a1 + DX b a1 + DX b 0 =   − zUϑ   φ φ 0 ψ ψ     a1 + DX b XPB a1 + DX PB a1 + ϑ(0)PN1 (a1 + DX b) + φ(0) −z . φ S ∗ (φ + ϑPN1 (a1 + DX b)) = ψ S ∗ (ψ + ∆PN1 (a1 + DX b)) Equivalently, a1 := PA (1 − zUϑ )−1 a φ if and only if there exist b ∈ B and ∈ Eϑ such that ψ a = a1 − z {XPB a1 + PA [ϑ(0)(PA B a1 + DX b) + φ(0)]} ,

(2.8)

0 = DX b − z {DX PB a1 + PDX [ϑ(0)(PA B a1 + DX b) + φ(0)]} ,

(2.9)

∗

0 = φ − zS [φ + ϑ(PA B a1 + DX b)] ,

(2.10)

∗

0 = ψ − zS [ψ + ∆(PA B a1 + DX b)] .

(2.11) ∗ −1

∗

Set ω := PA B a1 + DX b ∈ N1 . Then (2.10) says that φ = (1 − zS ) zS ϑω. It follows that φ(0) = (ϑ(z) − ϑ(0))ω. Replacing φ(0) by (ϑ(z) − ϑ(0))ω in (2.9) we get that (2.9) is equivalent to 0 = DX b − z(DX PB a1 + PDX ϑ(0)ω + PDX φ(0)) = DX b − z(DX PB a1 + PDX ϑ(z)ω) = (1 − zPDX ϑ(z))DX b − z(DX PB a1 + PDX ϑ(z)PA B a1 ), that is, DX b = z(1 − zPDX ϑ(z))−1 (DX PB a1 + PDX ϑ(z)PA B a1 ). Hence ω = PA B a1 + DX b = (1−zPDX ϑ(z))−1 [(1−zPDX ϑ(z))PA B a1 +z(DX PB a1 +PDX PA B ϑ(z)a1 )] = (1−zPDX ϑ(z))−1 (PA B a1 +zDX PB a1 ) and (2.8) can be rewritten as a = a1 − z(XPB a1 + PA ϑ(0)ω + PA φ(0)) = a1 − z(XPB a1 + PA ϑ(z)ω) = a1 − z XPB + PA ϑ(z)(1 − zPDX ϑ(z))−1 (PA B + zDX PB ) a1 . Therefore a1 := PA (1 − zUϑ )−1 a if and only if n h io−1 −1 a1 = 1 − z XPB + PA ϑ(z) (1 − zPDX ϑ(z)) (PA B + zDX PB ) a.

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This shows (2.7). In particular, (2.6) holds if and only if n h io−1 −1 1 − z XPB + PA α(z) (1 − zPDX α(z)) (PA B + zDX PB ) n h io−1 −1 = 1 − z XPB + PA β(z) (1 − zPDX β(z)) (PA B + zDX PB ) for all z ∈ D. Therefore Uα ∈ L(Fα ) and Uβ ∈ L(Fβ ) can be regarded as undistinguishable elements of W U D(X) if and only if, for all z ∈ D, PA α(z)(1 − zPDX α(z))−1 PA B = PA β(z)(1 − zPDX β(z))−1 PA B , PA α(z)(1 − zPDX α(z))−1 DX PB = PA β(z)(1 − zPDX β(z))−1 DX PB that is, PA α(z)(1 − zPDX α(z))−1 |N1 = PA β(z)(1 − zPDX β(z))−1 |N1 . Note that, for all z ∈ D, PA α(z)(1 − zPDX α(z))−1 − β(z)(1 − zPDX β(z))−1 |N1 = PA (1 − zα(z)PDX )−1 (α(z) − β(z))(1 − zPDX β(z))−1 |N1 . For all u ∈ N1 and all z ∈ D, (1 − zPDX β(z))u ∈ N1

and (1 − zα(z)PDX )−1 (α(z) − β(z))u ∈ M1 .

So PA (1 − zα(z)PDX )−1 (α(z) − β(z))(1 − zPDX β(z))−1 |N1 = 0,

z∈D

(2.12)

whenever, for all u ∈ N1 , the M1 -vector v(z) := (1 − zα(z)PDX )−1 (α(z) − β(z))u verifies PA v(z) = 0. Since X ∗ PA v(z)+DX PDX v(z) = 0, it follows that PA v(z) = 0 only when v(z) = 0. Thus Uα ∈ L(Fα ) and Uβ ∈ L(Fβ ) are undistinguishable as elements of W U D(X), meaning that (2.12) holds, if and only if α ≡ β. This shows that the map S(N1 , M1 ) → W U D(X) ϑ 7→ Uθ ∈ L(Fϑ ) is one-to-one. The proof of the theorem is complete.

3. The Relaxed Commutant Lifting Theorem We are given five Hilbert space operators • a contraction C : E → H, • a contraction T : H → H with minimal isometric dilation VT : K → K, • two contractions R, Q : E0 → E,

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so that T CR = CQ and R∗ R ≤ Q∗ Q. The Relaxed Commutant Lifting Theorem states that there exists a contraction D:E →K satisfying the relations PH D = C

and VT DR = DQ.

An operator D with the above properties is said to be an interpolant for {C, T, VT , R, Q}. The problem we address is to give a parametric description of the set of interpolants for {C, T, VT , R, Q}. We recall that VT ∈ L(K) is an isometry such that K ⊇ H, PH VTn |H = T n for all n = 0, 1, 2, . . . and K is the least Hilbert space containing {VTn (H)}∞ n=0 . The isometric dilation VT is essentially unique, in the sense that any other isometry V 0 ∈ L(K0 ) with the same properties is undistinguishable from VT ∈ L(K) under an isometric isomorphism τ : K0 → K such that τ |H = 1 and τ V 0 = VT τ . The space H is invariant under VT∗ and T ∗ = VT∗ |H . In particular, VT is an isometric lifting of T , meaning that T PH = PH VT . A minimal unitary dilation of T is a unitary operator UT acting on a larger Hilbert space G ⊇ H satisfying PH UTn |H = T n and PH UT−n |H = T ∗n for all n = 0, 1, 2, . . . and such that G is the least Hilbert space containing {UTn (H)}∞ n=−∞ . The latter condition determines UT up to isometric isomorphism, thus one can call it “the” minimal unitary dilation of T . If UT ∈ L(G) is the minimal unitary dilation of T then the subspace containing all the orbits of H under UT is invariant for UT , contains H and the restriction of UT to it is a minimal isometric dilation of T . Therefore, when speaking on the minimal isometric dilation VT ∈ L(K) of T we always consider that K ⊆ G and VT = UT |K . Likewise if WT : F → F is the minimal isometric dilation of T ∗ then F ⊆ G and WT = UT∗ |F . For a complete account concerning isometric and unitary dilations of contractions we refer to [14]. Consider the product space E × F with the sesquilinear hermitian form 0 0 0 e e 1 C ∗ PH e e e e , 0 := , 0 , 0 ∈E ×F f f C 1 f f f f C E⊕F so that 0 e e = he, e0 iE + hf, Ce0 iF + hCe, f 0 iF + hf, f 0 iF , 0 f f C 0 e e for all , 0 ∈ E × F. f f It can be easily seen that AC := DC ⊕ F is the Hilbert space which is obtained from (E × F, h·, ·iC ) by modding out by isotropic vectors and completing. The Hilbert space AC is the coupling space associated with C, when C is viewed as a linear operator from E into F ⊇ H. It readily follows:

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Lemma 3.1. If σC : E × F → AC is the linear map given by e e σC := DC e ⊕ (Ce + f ) ∈E ×F f f then σC is an isometric map from (E × F, h·, ·iC ) into AC with dense range and such that e 0 σC = DC e and σC =f −Ce f for all e ∈ E and all f ∈ F. Also, BC := DC Q(E0 )⊕F is a closed subspace of AC which equals the closure of σC (Q(E0 ) × F). Define σC

Qe0 Re0 7→ σC f WT f

Qe0 ∈ Q(E0 ) × F . f

(3.1)

Then, for all e0 ∈ E0 and f ∈ F,

2

σC Re0 = kRe0 k2 + 2RehCRe0 , WT f iF + kWT f k2

WT f = kRe0 k2 + 2RehT CRe0 , f iF + kf k2 = kRe0 k2 + 2RehCQe0 , f iF + kf k2 since it is assumed that T CR = CQ. As R∗ R ≤ Q∗ Q it follows that, for all e0 ∈ E0 and f ∈ F,

2 2

σC Re0 ≤ kQe0 k2 + 2RehCQe0 , f iF + kf k2 = σC Qe0 .

WT f f Therefore (3.1) gives rise to a contraction XC : BC ⊆ AC → AC satisfying XC σC

Qe0 Re0 = σC f WT f

for all e0 ∈ E0 and f ∈ F. Furthermore, in the particular case that R∗ R = Q∗ Q, XC is an isometry on AC with domain BC and range XC (BC ) = {DC Re0 ⊕ (CRe0 + WT f ) : e0 ∈ E0 , f ∈ F}. The contraction XC : BC ⊆ AC → AC is the coupling contraction underlying the lifting data set {C, T, VT , R, Q}. 1 Write D0 := (Q∗ Q − R∗ R) 2 and D0 = D0 (E0 ). Notice that

2 2

XC σC Qe0 = σC Qe0 − kD0 e0 k2

f f

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for all e0 ∈ E0 and f ∈ F. Hence the defect operator DXC of XC is determined by D0 via the relation

Qe0

DX σC Qe0 = kD0 e0 k ∈ Q(E0 ) × F .

C f f We now consider the set W U D(XC ) as described in Theorem 2.2. Here, instead of N1 and M1 , we deal with NC := (AC BC ) ⊕ DXC and MC := {x ∈ AC ⊕ DXC : XC∗ PAC x + DXC PDXC x = 0}, so that for each ϑ ∈ S(NC , MC ) Eϑ := H 2 (MC ) ⊕ ∆L2 (NC ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (NC )}⊥ , where ∆(ζ) := Dϑ(ζ) , |ζ| = 1, Fϑ := AC ⊕ DXC ⊕ Eϑ and Uϑ : Fϑ → Fϑ is given by     XC PBC h+DXC PBC h+ϑ(0)PNC h+φ(0) h φ ∗     S (φ+ϑPNC h) Uϑ φ := h ∈ AC ⊕DXC , ∈ Eϑ . ψ ψ S ∗ (ψ+∆PNC h) Given ϑ ∈ S(NC , MC ), define ϕϑ VTn h := Uϑ−n h,

h ∈ H, n = 0, 1, 2, . . .

and extend ϕϑ by linearity to the linear span of {VTn (H)}∞ n=0 . Note that, for all h ∈ H and all n = 0, 1, 2, . . ., Uϑn h = WTn h. Therefore, for all h, h0 ∈ H and all n = 0, 1, 2, . . ., hϕϑ VTn h, ϕϑ h0 iFϑ = hUϑ−n h, h0 iFϑ = hh, Uϑn h0 iFϑ = hh, WTn h0 iFϑ = hh, WTn h0 iF = hh, T ∗n h0 iH = hT n h, h0 iH = hVTn h, h0 iK . It follows that ϕϑ is an isometric map from the linear span of {VTn (H)}∞ n=0 into Fϑ . Thus its extension by continuity to all of K, say Φϑ , is a unitary operator from K onto the least closed subspace of Fϑ containing all the subspaces Uϑ−n (H), n = 0, 1, 2, . . .. Moreover, Φϑ |H = 1 and Φϑ VT = Uϑ−1 Φϑ . Now define D : E → K by hDe, kiK :=

e σC , Φϑ k 0 F

(e ∈ E, k ∈ K). ϑ

(3.2)

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For all e ∈ E and all k ∈ K,

e

|hDe, kiK | ≤

σC 0 kΦϑ kk = kekkkk. Therefore kDk ≤ 1. For all e ∈ E and all h ∈ H, e e hDe, hiK := σC , Φϑ h = σC ,h = hCe, hiH . 0 0 F F ϑ

ϑ

Hence PH D = C. Let e0 ∈ E0 and h ∈ H be given. Then hVT DRe0 , hiK = hDRe0 , T ∗ hiK = hCRe0 , T ∗ hiH = hT CRe0 , hiH = hCQe0 , hiH = hDQe0 , hiK and Re0 σC ,h 0 A C Qe0 = Uϑ σC ,h 0 Fϑ Qe0 = σC , Φϑ V T h 0 F

hVT DRe0 , VT hiK = hDRe0 , hiK = hCRe0 , hiH = Qe0 XC σC ,h 0 A C Qe0 = σC , Uϑ−1 h 0 F =

ϑ

ϑ

= hDQe0 , VT hiK , so that hVT DRe0 , (VT − T )hiK = hDQe0 , (VT − T )hiK . As L := (VT − T )(H) is known to be the wandering subspace for VT , in the sense that VTn (L) ⊥ L for all n ∈ N (N for the set of natural numbers) and K = H ⊕ L ⊕ VT (L) ⊕ VT2 (L) ⊕ · · · , it is clear that VT DR = DQ as far as hVT DRe0 , VTn (VT − T )hiK = hDQe0 , VTn (VT − T )hiK for any given e0 ∈ E0 , h ∈ H and all n ∈ N. Note that Φϑ VTn (VT − T )h = Uϑ−n−1 h − Uϑ−n T h = Uϑ−n−1 (1 − WT T )h and Qe0 Re0 Qe0 Uϑ σC = σC + DXC σC . 0 0 0

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Whence Qe0 σC , Φϑ VTn (VT − T )h 0 Fϑ Qe0 −n−1 = σC , Uϑ (1 − WT T )h 0 F ϑ Qe0 = Uϑ σC , Uϑ−n (1 − WT T )h 0 Fϑ Re0 Qe0 −n = σC + DXC σC , Uϑ (1 − WT T )h 0 0 F

hDQe0 , VTn (VT − T )hiK =

ϑ

and hVT DRe0 , VTn (VT − T )hiK = hDRe0 , VTn−1 (VT − T )hiK Re0 = σC , Φϑ VTn−1 (VT − T )h 0 F ϑ Re0 = σC , Uϑ−n (1 − WT T )h . 0 F ϑ

Thus VT DR = DQ if and only if, for all e0 ∈ E0 , h ∈ H and n ∈ N, Qe0 Uϑn DXC σC , (1 − WT T )h = 0. 0 F

(3.3)

ϑ

Therefore, ϑ ∈ S(NC , MC ) gives rise to an interpolant D for {C, T, VT , R, Q} if and only if the corresponding element in W U D(XC ), Uϑ ∈ L(Fϑ ), satisfies

(1 − zUϑ )−1 Uϑ u, (1 − WT T )h F = 0, z ∈ D, u ∈ DXC and h ∈ H. (3.4) ϑ

Set J := {(1 − WT T )h : h ∈ H}. Then J = ker(WT∗ ), the null space of WT∗ , and (3.4) is equivalent to PJ (1 − zUϑ )−1 Uϑ |DXC = 0,

z ∈ D.

(3.5)

In a similar way as we established (2.7) in the proof of Theorem 2.2, we can see that, for all z ∈ D, PJ (1 − zUϑ )−1 Uϑ |DX= PJ (1 − zXC PBC − zDXC PBC − zϑ(z)PNC )−1 ϑ(z)|DXC C =PJ (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)|DXC . So, for (3.5) to hold, we must grant that PJ (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)|DXC = 0. Given x ∈ NC , define y(= y(z)) (z ∈ D) by y := ϑ(z)x. Since y ∈ MC , then XC∗ PAC y + DXC PDXC y = 0. Hence, for all f ∈ F, 0 = hXC∗ PAC y + DXC PDXC y, f iAC = hy, XC f iAC = hy, WT f iAC = hWT∗ PF y, f iF ,

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so that PF y ∈ J . On the other hand, for all e ∈ DC , PBC e = PDC Q(E0 ) e while, for all e0 ∈ E0 , Qe0 Re0 Re0 XC DC Qe0 = XC σC = σC = σC −CQe0 −WT CQe0 −WT T CRe0 = DC Re0 ⊕ (1 − WT T )CRe0 . Whence PF XC PBC e ∈ J for all e ∈ DC as well. Thus, for any given u ∈ DXC and all z ∈ D, PF(1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u = PF ϑ(z)u + zPF ϑ(z)PNC (1 − zXC PBC − zϑ(z)PNC)−1 ϑ(z)u + zPF XC PBC PDC (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u + zWT PF (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u = v(u, z) + w(u, z), where v(u, z) := PF ϑ(z)u + zPF ϑ(z)PNC (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u + zPF XC PBC PDC (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u ∈ J and w(u, z) := zWT PF (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u ⊥ J . Consequently, for any given u ∈ DXC and all z ∈ D, PJ (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u = v(u, z). We get that v(u, z) = (1 − zWT )PF (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u. Since 1 − zWT is an invertible operator for each z ∈ D, we can conclude that v(u, z) = 0 for all u ∈ DXC and all z ∈ D whenever PF (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)u = 0. We point out that the above discussion gives a proof of the following: Proposition 3.2. The following statements are equivalent: (a) The operator D associated with a give ϑ ∈ S(NC , MC ) by means of (3.2) satisfies VT DR = DQ and, hence, is an interpolant for {C, T, VT , R, Q}. (b) PJ (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)|DXC = 0. (c) PF (1 − zXC PBC − zϑ(z)PNC )−1 ϑ(z)|DXC = 0. The proposition leads to conclude that the operator D associated with a given ϑ ∈ S(NC , MC ) by means of (3.2) satisfies VT DR = DQ and, hence, is an interpolant for {C, T, VT , R, Q} if, in particular, ϑ(z)|DXC = 0 for each z ∈ D. Before embarking in the problem of analyzing the map (3.2), let us establish a closed formula for the direct connection between ϑ and D.

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Hereafter it could be helpful to recall that NC and MC can be interpreted as the defect subspaces of the isometry VC on HC := AC ⊕ DXC defined in BC by XC AC VC := : BC → DXC DXC and that each Uϑ ∈ L(Fϑ ) belongs not only to W U D(XC ) but also to U(VC ). In particular, under this viewpoint, we could cope with a more compact notation. P∞ It is clear that D = C + n=0 PVTn (L) D, with L := (VT − T )(H) the wandering subspace for VT . Therefore D is determined by the sequence of operators {PVTn (L) D}∞ n=0 (cf. [5].) With each D we associate the (formal) power series SD (z) :=

∞ X

z n Sc D (n)

(z ∈ D)

n=0

where ∗n Sc D (n) := VT PVTn (L) D

(n = 0, 1, 2, . . .).

For all e ∈ E and all h ∈ H, hSD (z)e, (VT − T )hiK =

e PJ (1 − zUϑ )−1 Uϑ σC , (1 − WT T )h . 0 F ϑ

e Set F e := σC (e ∈ E) so that F : E → AC is the embedding of E into AC . 0 Define G(1 − WT T )h := (VT − T )h (h ∈ H) to get a unitary operator G : J → L. Then SD (z) = GPJ (1 − zUϑ )−1 Uϑ F, z ∈ D. In computing PJ (1 − zUϑ )−1 Uϑ |HC it is relevant to take into account that ⊆ BC and V n+1 (J ) ⊥ J for all n = 0, 1, 2, . . .. Then a straightforward computation we omit in the present discussion gives the following:

VCn (J )

Lemma 3.3. For all z ∈ D, PJ (1 − zUϑ )−1 Uϑ |HC = a(z) + b(z)ϑ(z)(1 − c(z)ϑ(z))−1 d(z), where −1 a(z) := PJ X C → J, C PBC (1 − z(1 − PJ )VC PBC ) : H−1 b(z) := PJ 1 + zXC PBC (1 − z(1 − PJ )VC PBC ) (1 − PJ ) |MC : MC → J , c(z) := zPNC (1 − z(1 − PJ )VC PBC )−1 (1 − PJ )|MC : MC → NC , d(z) := PNC (1 − z(1 − PJ )VC PBC )−1 : HC → NC .

Therefore ˜ SD (z) = a ˜(z) + ˜b(z)ϑ(z)(1 − c˜(z)ϑ(z))−1 d(z), ˜ := d(z)F and a(z), b(z), where a ˜(z) := Ga(z)F , ˜b(z) := Gb(z), c˜(z) := c(z), d(z) 2 c(z), d(z) as in the lemma. So if Γ : H (L) → L ⊕ VT (L) ⊕ VT2 (L) ⊕ · · · is the

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unitary operator given by ! ∞ ∞ X X n z xn := Γ VTn xn n=0

then

∞ X

x(z) =

n=0

291

∞ X

! n

2

z xn ∈ H (L) ,

n=0

˜ PVTn (L) D := Γ a ˜(z) + ˜b(z)ϑ(z)(1 − c˜(z)ϑ(z))−1 d(z) .

n=0

Theorem 3.4. (Description of the interpolants in the Relaxed Commutant Lifting Theorem) Consider the lifting data set {C, T, VT , R, Q}. Let WT ∈ L(F) be the minimal isometric dilation of T ∗ . Let AC = DC ⊕ F be the coupling Hilbert space associated with C, when C is viewed as a linear operator from E into F ⊇ H, and let XC : BC ⊆ AC → AC be the coupling contraction underlying the lifting data set {C, T, VT , R, Q}. Write L for the wandering subspace of VT and Γ : H 2 (L) → L ⊕ VT (L) ⊕ VT2 (L) ⊕ · · · for the unitary operator given by ! ! ∞ ∞ ∞ X X X n n n 2 Γ z xn := VT xn x(z) = z xn ∈ H (L) . n=0

n=0

n=0

Set F e := DC e ⊕ Ce (e ∈ E) so that F : E → AC is the embedding of E into AC . Define G(1 − WT T )h := (VT − T )h (h ∈ H) to get a unitary operator G : J → L, where J := (1 − WT T )(H) is the null space of WT∗ . Set NC := (AC BC ) ⊕ DXC and MC := {x ∈ AC ⊕ DXC : XC∗ PAC x + DXC PDXC x = 0}. Given ϑ ∈ S(NC , MC ) such that ϑ(z)|DXC = 0 for each z ∈ D, put     C H ∞   : E →  L D :=  VTn (L) −1 ˜ ˜ Γ a ˜(z) + b(z)ϑ(z)(1 − c˜(z)ϑ(z)) d(z) n=0

(3.6)

where a ˜(z) := GPJ XC PBC (1−z(1−PJ )(XC PBC +DXC PBC ))−1 F : E → L, ˜b(z) := GPJ [1 + zXC PB C ×(1−z(1−PJ )(XC PBC +DXC PBC ))−1 (1−PJ ) |MC : MC → L, c˜(z) := zPNC (1−z(1−PJ )(XC PBC +DXC PBC ))−1 (1−PJ )|MC : MC → NC , ˜ := PN (1−z(1−PJ )(XC PB +DX PB ))−1 F : E → NC . d(z) C

C

C

C

Then D is an interpolant for {C, T, VT , R, Q}. Moreover, in this way all interpolants for {C, T, VT , R, Q} are obtained. A Schur class function ϑ ∈ S(NC , MC ) satisfying ϑ(z)|DXC = 0 for each z ∈ D always exists. For instance, one can take ϑ ≡ 0. This choice for ϑ yields the central interpolant.

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Proof. We have already seen that D is an interpolant for {C, T, VT , R, Q}. Now, given an interpolant D, so that D : E → K is a contraction satisfying PH D = C

and VT DR = DQ,

let us show that D can be obtained from a function ϑ ∈ S(NC , MC ) as in (3.6). Let UT : G → G be the minimal unitary dilation of T (recall that K ⊆ G and VT = UT |K .) View D as a linear operator from E into G and let AD := DD ⊕ G be the coupling space associated with D. Since PH D = C and PK |F = PK |H (indeed G = F ⊕ L ⊕ VT (L) ⊕ VT2 (L) ⊕ · · · ) it can be seen that kDD e ⊕ (De + f )k = kDC e ⊕ (Ce + f )k for all e ∈ E and all f ∈ F. If the product space E × G is endowed with the sesquilinear hermitian form 0 0 0 e e 1 D∗ PH e e e e , 0 := , 0 , ∈E ×G g g D 1 g g g g D E⊕G and e σD := DD e ⊕ (De + g) g

e ∈E ×G g

then σD is an isometry from (E × G, h·, ·iD ) into AD with dense range and such that e 0 σD = DD e and σD =g −De g for all e ∈ E and all g ∈ G. Moreover, if σC is like in Lemma 3.1 then

σD e = σC e

f f for all e ∈ E and all f ∈ F. Therefore, e e ρσC := σD f f defines an isometry from AC into AD . Set BD := DQ(E0 ) × G and define Qe0 Re0 σD 7→ σD g UT∗ g

e ∈E ×F f

Qe0 ∈ Q(E0 ) × G . g

(3.7)

As VT DR = DQ, VT = UT |K and R∗ R ≤ Q∗ Q, it follows that (3.7) gives rise to a contraction XD : BD ⊆ AD → AD . Since WT = UT∗ |F , it holds that ρXC = XD ρ|BC . As before for XC , the defect operator DXD of XD is determined by D0 := 1 ∗ (Q Q − R∗ R) 2 via the identity

Qe0

DX σD Qe0 = kD0 e0 k ∈ Q(E0 ) × G .

D g g

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Write HD := AD ⊕ DXD and define XD AD VD := : BD → . DXD DXD Extend ρ from AC to all of HC by setting ρDXC := DXD ρ|BC . Then ρ : HC → HD is an isometry such that ρVC = VD ρ|BC . If ND := (AD BD ) ⊕ DXD and ∗ MD := {x ∈ HD : XD PAD x + DXD PDXD x = 0}

are the defect subspaces of VD then fix ν ∈ S(ND , MD ) and let Uν ∈ L(Fν ) be the corresponding minimal weak unitary dilation of XD . For the sake of simplicity we consider ν ≡ 0 and call U0 the corresponding unitary operator on F0 . Notice that, for all h ∈ H and all n ∈ N, U0n UTn h = VDn UTn h = h, whence, for all e ∈ E, hDe, VTn hiK =

e e σD = ρσC . , U0−n h , U0−n ρh 0 0 F F 0

0

So, if U = ∨n∈Z U0n ρ(AC ) (the least Hilbert space containing U0n ρ(AC ) for all n ∈ Z) and U = U0 |U then U = ∨n∈Z U0n ρ(HC ) and, for all e ∈ E, h ∈ H and n ∈ N, e hDe, VTn hiK = U n ρσC , ρh . (3.8) 0 U If V = VD |ρBC then V is an isometry acting on ρHC with defect subspaces N = ρNC and M = ρMC and U is a minimal unitary extension of V . Set ϑU (z) = PρMC U (1 − zPU ρHC U )−1 |ρNC and ϑ(z) = ρ∗ ϑU (z)ρ|NC . Then ϑ ∈ S(NC , MC ) and the associated unitary operator Uϑ ∈ L(Fϑ ) is a minimal weak unitary dilation of XC . It can be computed ρ∗ PρHC U (1 − zU )−1 ρ|HC to yield −1

ρ∗ PρHC U (1−zU )−1 ρ|HC = (VC PBC +ϑ(z)PNC ) [1−z(VC PBC +ϑ(z)PNC )]

|HC

= PHC Uϑ (1 − zUϑ )−1 |HC . From here and (3.8) it follows that, for all e ∈ E, h ∈ H and n ∈ N, e hDe, VTn hiK = σC , Uϑ−n h . 0 F ϑ

Therefore D is given by ϑ as in (3.2), hence, by formula (3.6). It remains to see that ϑ(z)|DXC = 0 for each z ∈ D. Notice that ϑ can also be written as ϑ(z) = PMC ρ∗ PρHC (1 − zVD PBD ρBC PU ρHC )−1 VD PBD ρHC ρ|NC .

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As DXC ⊥ BD it readily follows that ϑ is constrained as required. This completes the proof. The setting for the Commutant Lifting Theorem appears when R is the identity on E (thus E0 = E) and Q is an isometry on E. In this case, R∗ R = Q∗ Q = 1 and the underlying contraction XC is an isometry. As a matter of fact, the formula (3.6) is known to yield a proper parameterization of the interpolants (see [13]). That is, in the Commutant Lifting Theorem, the map ϑ 7→ D, which is defined for all ϑ ∈ S(NC , MC ), is one-to-one and onto. The latter is not true in general. It may happen that different parameters ϑ’s in S(NC , MC ), constrained to satisfy ϑ(z)|DXC = 0 for each z ∈ D, provide the same interpolant D via (3.6). As for a concrete example, consider C = 0, T an isometry and R = Q such that ker(Q∗ ) 6= {0}. In this particular case, the coupling Hilbert space is E ⊕ F and the coupling contraction is indeed an isometry. Its domain and both defect subspaces are given by Q(E0 )⊕F and ker(Q∗ ), respectively. Since T is isometric, VT = T . Hence, there is only one interpolant D for the data set {0, T, T, Q, Q}, namely D = C = 0. Also, as T is isometric, WT∗ = T . Thus J = {0}. Therefore, a ˜, ˜b ≡ 0 in formula (3.6) and any ϑ ∈ S(ker(Q∗ ), ker(Q∗ )) gives D. Theorem 3.5. With the notation of Theorem 3.4, set S := {ϑ ∈ S(NC , MC ) : ϑ(z)|DXC = 0 for each z ∈ D}. Let D be the interpolant obtained from a given α ∈ S via (3.6). Then β ∈ S is mapped to D via (3.6) if and only if there exist analytic functions µ : D → L(AC BC , DC ) and ν : D → L(AC BC , DXC ) such that, for each z ∈ D, β(z) = α(z)(1 − zPAC BC µ(z)) + (1 − zXC PBC )µ(z) + ν(z). In particular, β produces the central interpolant if and only if PF (1 − zXC PBC )−1 β(z) ≡ 0. Proof. It is clear that α, β ∈ S provide the same interpolant D via (3.6) if and only if ˜b(z)α(z)(1 − c˜(z)α(z))−1 d(z) ˜ ≡ ˜b(z)β(z)(1 − c˜(z)β(z))−1 d(z), ˜ that is, if and only if PJ (1 − zXC PBC − zPAC α(z)PAC BC )−1 F ≡ PJ (1 − zXC PBC − zPAC β(z)PAC BC )−1 F. Since PJ (1 − zXC PBC − zPAC θ(z)PAC BC )−1 Ce = PJ Ce for all e ∈ E, we get that the above condition is equivalent to PJ (1 − zXC PBC − zPAC α(z)PAC BC )−1 |DC ≡ PJ (1 − zXC PBC − zPAC β(z)PAC BC )−1 |DC

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or PJ (1 − zXC PBC − zα(z)PAC BC )−1 (α(z)−β(z)) ≡ 0. The same reasoning as in the proof of Proposition 3.2 yields that PJ (1 − zXC PBC − zα(z)PAC BC )−1 |MC = PF (1 − zXC PBC PDC − zα(z)PAC BC )−1 |MC . Therefore two functions α, β ∈ S are mapped to the same D via (3.6) if and only if PF (1 − zXC PBC PDC − zα(z)PAC BC )−1 (α(z)−β(z)) ≡ 0. (3.9) For (3.9) to hold it is necessary and sufficient that there exist analytic functions µ : D → L(AC BC , DC ) and ν : D → L(AC BC , DXC ) such that, for each z ∈ D, (1 − zXC PBC PDC − zα(z)PAC BC )−1 (α(z)−β(z)) = −µ(z) − ν(z). A straightforward computation shows that the above equation can be rewritten as β(z) = α(z)(1 − zPAC BC µ(z)) + (1 − zXC PBC )µ(z) + ν(z). When α ≡ 0 (in which case the corresponding D is the central interpolant) we get that β ∈ S is mapped to the central interpolant by (3.6) if and only if there exist analytic functions µ : D → L(AC BC , DC ) and ν : D → L(AC BC , DXC ) such that, for each z ∈ D, β(z) = (1 − zXC PBC )µ(z) + ν(z). As the above equation holds if and only if (1 − zXC PBC )−1 β(z) = µ(z) + ν(z), the last statement in the theorem is proved and the proof of the theorem is complete itself. Note that if µ : D → L(AC BC , DC ) and ν : D → L(AC BC , DXC ) verify β(z) ≡ α(z)(1 − zPAC BC µ(z)) + (1 − zXC PBC )µ(z) + ν(z) for given α, β ∈ S, then, for each z ∈ D, kα(z)(1 − zPAC BC µ(z)) + (1 − zXC PBC )µ(z) + ν(z)k ≤ 1 and XC∗ (1 − zXC PBC )µ(z) + DXC ν(z) = 0. The classical Commutant Lifting Theorem is included as a particular case of the following corollary. Corollary 3.6. Assume that R∗ R = Q∗ Q and DC R(E0 ) = DC . Then the map in (3.6) establishes a one-to-one correspondence between ϑ ∈ S(NC , MC ) and the interpolant D for {C, T, VT , R, Q}, with NC and MC being given by NC = DC DC Q(E0 )

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and MC = DC ⊕ J {DC Re0 ⊕ (1 − WT T )CRe0 : e0 ∈ E0 }. Proof. From the assumption that R∗ R = Q∗ Q it readily follows that XC is an isometry. In particular NC = AC BC = DC DC Q(E0 ). As for the defect subspace MC , we get that MC = ker(XC∗ ) = DC ⊕ J {DC Re0 ⊕ (1 − WT T )CRe0 : e0 ∈ E0 }. If α is any given function in S(NC , MC ) and D is the corresponding interpolant, then β ∈ S(NC , MC ) yields the same D via (3.6) if and only if there exists an analytic function µ : D → L(AC BC , DC ) such that, for each z ∈ D, β(z) = α(z)(1 − zPAC BC µ(z)) + (1 − zXC PBC )µ(z). Note that, for each z ∈ D, 0 = XC∗ (1 − zXC PBC )µ(z) = XC∗ µ(z) − zPBC µ(z). In particular, XC∗ µ(0) = 0. As DC R(E0 ) = DC , it follows that µ(0) = 0. Hence, µ(z) ≡ zµ1 (z) for some analytic function µ1 : D → L(AC BC , DC ). The same argument as before yields µ1 (0) = 0. By iteration we get µ ≡ 0 and β ≡ α. Other sufficient conditions under which Theorem 3.4 provides a proper parameterization are simple to obtain. To find necessary and sufficient conditions for this to happen remains an open problem.

4. Appendix We herein include a sketch of the proof of Theorem 2.1. P∞ (i) Given ϑ ∈ S(N , M), ϑ(z) = k=0 z k ϑk , z ∈ D, let Uϑ be the linear operator defined in Fϑ := H ⊕ Eϑ by     h V PD h + ϑ(0)PN h + φ(0) φ ∗     S (φ + ϑPN h) Uϑ φ := h ∈ H, ∈ Eϑ . ψ ψ S ∗ (ψ + ∆PN h) Recall that Eϑ := H 2 (M) ⊕ ∆L2 (N ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (N )}⊥ , with ∆(ζ) := Dϑ(ζ) , 2

|ζ| = 1, 2

and that S is the shift on either H (M) or L (N ), depending on context.

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φ ∈ Eϑ , we ψ get that S ∗ (φ + ϑPN h) ∈ H 2 (M), S ∗ (ψ + ∆PN h) ∈ ∆L2 (N ) and that, for all u ∈ H 2 (N ), ∗ S (φ + ϑPN h) ϑu , = hϑPN h, ϑSui + h∆PN h, ∆Sui = hPN h, Sui = 0. S ∗ (ψ + ∆PN h) ∆u As Sϑ = ϑS|H 2 (N ) and S∆ = ∆S, given any h ∈ H and

Therefore Uϑ ∈ L(Fϑ ). φ It can be seen that Uϑ is unitary. In point of fact, if h ∈ H and ∈ Eϑ , ψ  ∗   h h then  φ∗  = Uϑ∗  φ  is given by ψ∗ ψ h∗ = V ∗ PV (D) h + ϑ(0)∗ PM h + PN (ϑ∗ Sφ + ∆Sψ), φ∗ = PM h + Sφ − ϑ[ϑ(0)∗ PM h + PN ((ϑ∗ Sφ + ∆Sψ)], ψ ∗ = Sψ − ∆[ϑ(0)∗ PM h + PN ((ϑ∗ Sφ + ∆Sψ). Also, it can be seen that Fϑ is the least Hilbert space containing Uϑn (H) for all n = 0, ±1, ±2, . . .. From this and since Uϑ |D = V it comes that Uϑ ∈ L(Fϑ ) is a minimal unitary extension of V . It remains to show that PM Uϑ (1 − zPEϑ Uϑ )−1 |N = ϑ(z) for all z ∈ D. This follows from the relations   0 k (PEϑ Uϑ ) h =  S ∗k ϑh  , S ∗k ∆h

h ∈ N , k ∈ N,

and k

PM Uϑ (PEϑ Uϑ ) h = ϑk h,

h ∈ N , k = 0, 1, 2, . . . .

e be a minimal unitary extension of V . Define (ii) Let U ∈ L(H) ϑ(z) := PM U (1 − zPH H U )−1 |N e

(z ∈ D).

Set U11 := PM U |N , U12 := PM U |H H , U21 = PH H U |N , U22 := PH H U |H H . e e e e Then ϑ(z) = U11 + zU12 (1 − zU22 )−1 U21 ,

z ∈ D,

where (Ujk )j,k=1,2 is the 2×2 block matrix representation of the isometric operator e H) onto M ⊕ (H e H). Hence ϑ is the characteristic U |N ⊕(H H) from N ⊕ (H e n o e H; U | function of the unitary colligation N , M, H with state space e N ⊕(H H)

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e H H, input space N and output space M. It is known from the theory of operator colligations that it is the case that ϑ ∈ S(N , M). e 0 ) be a minimal unitary extension of V undistinguishable from (iii) Let U 0 ∈ L(H e e0 → H e such that τ |H = 1 and U ∈ L(H), under an isometric isomorphism τ : H 0 τU = Uτ. Then τ PH H U = PHe0 H U 0 τ . So, for all x ∈ N , y ∈ M and k = 0, 1, 2, . . ., e k k 0 0 x, y . PM U PH H U x, y = PM U PHe0 H U e H

H

Hence U )−1 |N = PM U 0 (1 − zPHe0 H U 0 )−1 |N PM U (1 − zPH H e for all z ∈ D. As for the converse, write ϑ(z) := PM U (1 − zPH H U )−1 |N = PM U 0 (1 − zPHe0 H U 0 )−1 |N e

(z ∈ D)

and note that, for all z ∈ D, −1

PH U (1 − zU )−1 |H = (V PD + ϑ(z)PN ) [1 − z(V PD + ϑ(z)PN )] = PH U 0 (1 − zU 0 )−1 |H . Therefore hU n h, HiHe = hU 0n h, HiHe0

for all h ∈ H and n ∈ N. Hence the operator defined on the linear span of n 0n {U n (H)}∞ n=−∞ and mapping U h into U h (h ∈ H, n = 0, ±1, ±2, . . .) can be exe→H e 0 such that τ |H = 1 and τ U 0 = U τ . tended to an isometric isomorphism τ : H

References [1] R. Arocena, Generalized Toeplitz kernels and dilations of intertwining operators, Integral Equations and Operator Theory, 6(1983), 759–778. [2] ,Unitary extensions of isometries and contractive intertwining dilations, in: The Gohberg Anniversary Collection II, Operator Theory: Advances and Applications 41, Birkh¨ auser-Verlag Basel, 1989, pp. 13–23. [3] D.Z. Arov and L.Z. Grossman, Scattering matrices in the theory of dilations of isometric operators, Soviet Math. Dokl., 27(1983), No. 3, 518–522. [4] , Scattering matrices in the theory of unitary extension of isometric operators, Math. Nachr., 157(1992), 105–123. [5] Gr. Arsene, Z. Ceau¸sescu and C. Foia¸s, On intertwining dilations. VIII. J. Operator Theory 4(1980), No. 1, 55–91. [6] M. Cotlar and C. Sadosky, Transference of metrics induced by unitary couplings, a Sarason theorem for the bidimensional torus and a Sz.-Nagy-Foia¸s theorem for two pairs of dilations, J. Funct. Anal., 111(1993), 473–488.

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[7] C. Foia¸s, On the extension of intertwining operators, in: Harmonic Analysis and Operator Theory, A Conference in Honor of Mischa Cotlar, January 3–8, 1994, Caracas, Venezuela, Contemporary Mathematics 189, American Mathematical Society, Providence, Rhode Island, 1995, pp. 227–234. [8] C. Foia¸s and A.E Frazho, The Commutant Lifting Approach to Interpolation Problems, Operator Theory: Advances and Applications 44, Birkh¨ auser-Verlag, Basel, 1990. [9] C. Foia¸s, A.E. Frazho and M.A. Kaashoek, Relaxation of metric constrained interpolation and a new lifting theorem, Integral Equations and Operator Theory, 42(2002), 253–310. [10] A.E. Frazho, S. ter Horst and M.A. Kaashoek, Coupling and relaxed commutant lifting, Integral Equations and Operator Theory, 54(2006), 33–67. [11] , All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged), 72(2006), No. 1-2, 299–318. [12] W.S. Li and D. Timotin, The relaxed intertwining lifting in the coupling approach, Integral Equations and Operator Theory, 54(2006), 97–111. [13] M.D. Mor´ an, On intertwining dilations, J. Math. Anal. Appl., 141(1989), No. 1, 219–234. [14] B. Sz.-Nagy and C. Foia¸s, Harmonic analysis of operators on Hilbert space, NorthHolland Publishing Co., Amsterdam-London, 1970. S.A.M. Marcantognini Departamento de Matem´ aticas Instituto Venezolano de Investigaciones Cient´ıficas Apartado Postal 21827 Caracas 1020A Venezuela e-mail: [email protected] M.D. Mor´ an Escuela de Matem´ aticas Facultad de Ciencias Universidad Central de Venezuela Apartado Postal 20513 Caracas 1020A Venezuela e-mail: [email protected] Submitted: 05 October 2008.

Integr. equ. oper. theory 64 (2009), 301–323 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030301-23, published online June 26, 2009 DOI 10.1007/s00020-009-1696-8

Integral Equations and Operator Theory

Factorisation of Non-negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators S. Albeverio, R. Hryniv and Ya. Mykytyuk Abstract. We study the problem of factorisation of non-negative Fredholm operators acting in the Hilbert space L2 (0, 1) and its relation to the inverse spectral problem for Bessel operators. In particular, we derive an algorithm of reconstructing the singular potential of the Bessel operator from its spectrum and the sequence of norming constants. Mathematics Subject Classification (2000). Primary 47A68; Secondary 34A55, 34B24, 34B30, 47E05. Keywords. Factorisation in operator algebras, non-negative operators, Bessel operators, inverse problems.

1. Introduction In the Hilbert space L2 (0, 1), we consider an operator S = I + F , where F is of the Hilbert–Schmidt class. It is well known [11,20] that if S is positive, then it can uniquely be factorised in the form S = (I + K)−1 (I + K ∗ )−1 ,

(1.1)

where K is a Volterra integral operator of the Hilbert–Schmidt class with uppertriangular kernel k (i.e., k(x, t) = 0 for a.e. (x, t) satisfying 0 ≤ t ≤ x ≤ 1). The kernel k can be found from the equation Z 1 k(x, t) + f (x, t) + k(x, s)f (s, t) ds = 0, 0 ≤ x < t ≤ 1, (1.2) x

where f is the kernel of F . In the special case where f has the form f (x, t) = φ(2 − x − t) ± φ(|x − t|)

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for some function φ ∈ L2 (0, 2), equation (1.2) is called the Gelfand–Levitan– Marchenko (GLM) equation, and if f (x, t) = φ(|x − t|), then (1.2) is the Krein equation. These equations naturally arise while solving the inverse spectral problems for the Sturm–Liouville operators d2 +q dx2 on the interval (0, 1) subject to suitable boundary conditions; then F is constructed from the spectral data for T , and I + K is the transformation operator between the unperturbed (q = 0) and perturbed (q 6= 0) Sturm–Liouville operators, see Subsection 4.2. Moreover, under suitable regularity assumptions on the potential q the kernel k is continuous in the domain {(x, t) | 0 ≤ x ≤ t ≤ 1}, and the potential q is related to k via T =−

dk(x, x) . dx On the other hand, there are Sturm–Liouville operators with singular potentials, for which the above reconstruction procedure is impossible because the corresponding operator I +F is only non-negative and has a non-trivial null-space. One such an example is given by the so called Bessel operators arising as follows. It is well known (see, e.g., Example 4 of Appendix to X.1 of [18]) that a radial Schr¨ odinger operator −∆+q(|x|) considered in the unit ball of R3 with q supported on (0, 1) decomposes into the direct sum of the Bessel operators Tm corresponding to the angular momenta m ∈ Z+ , i.e., the operators generated in L2 (0, 1) by the differential expressions d2 m(m + 1) − 2+ +q (1.3) dx x2 and suitable boundary conditions. If one tries to follow the same classical approach to solve the inverse spectral problem for the Bessel operators Tm with m > 0 as for the Sturm–Liouville case m = 0, then one immediately encounters the problem that the operator S constructed via the spectral data for Tm is non-negative and has a non-trivial null-space. Therefore the representation (1.1) is clearly impossible, and the very existence of the transformation operators is questionable. However, such an operator S (i.e., non-negative and with finite-dimensional null-space) might still be factorisable as q(x) = −2

∗ S = S+ S+ ,

(1.4)

where S+ is an upper-triangular operator, see definitions in Section 2. This question is studied in detail in Section 3, and the main result there (Theorem 3.12) states that such a representation is always possible and, moreover, it is even unique under some extra conditions imposed on S and S+ . In Section 4 we study properties of the factor S+ in the case where S is constructed from the spectral data for some Bessel operator Tm , m > 0, as explained

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above. We prove in Theorem 4.2 that then the factor S+ is the transformation operator between Tm and the unperturbed Sturm–Liouville operators and might be used to reconstruct the potential q, as in the regular case of Sturm–Liouville operators. This leads to a new algorithm of solving the inverse spectral problems for Bessel operators with momenta m ∈ N, which is described in Section 4. It should be mentioned that the inverse spectral problem for Bessel operators Tm was earlier studied in several papers; see, e.g., [1,4,5,7,21,24]. For m ≥ − 14 , uniqueness of solution was established in [5, 24] and the spectra of Tm where characterised in [4]. In [21] the author gave a complete solution of the inverse spectral problem for m ∈ N and q ∈ L2 (0, 1) using the method of [17]. Another approach based on the double commutation method was suggested in [7] and further developed in [1] to treat the case m ∈ N and a wide class of singular potentials. Our treatment of the inverse spectral problem for Bessel operators is close to the classical one as suggested by Gelfand and Levitan [8] and Marchenko [14] in that it uses the transformation operators and the related integral equations. Although the reconstruction method of this paper generalises in a straightforward manner to the class of Bessel operators with real-valued potentials in L1 (0, 1) or to those treated in [1], potentials q ∈ L2 (0, 1) produce spectral data that are easy to characterise, and we decided to sacrifice generality for the sake of simplicity. We remark that, to the best of our knowledge, no complete solution to the inverse spectral problem for Bessel operators with non-integer m has been given so far. In particular, the problems arising after decomposition of the radial Laplace operator in even dimensions lead to Tm with half-integer m and have not been solved; in odd dimensions greater than 3 the analysis remains unchanged. The main obstacle for applying the method of this paper to solve the inverse spectral problem for the Bessel operators Tm with non-integer m is not a non-trivial null-space of the corresponding operator I + F (e.g., this operator is positive if m ∈ [− 21 , 12 )), but rather the fact that F is then no longer compact. Unfortunately, there is no general theory of factorization of positive operators I + F with non-compact F . We hope that for particular operators F related to the Bessel operators Tm such a theory can be developed. This would make it possible to extend the method of reconstructing the Bessel operators Tm to arbitrary angular momenta m ≥ − 12 . Such a project will be discussed elsewhere. Although we study the the factorization problem for operators acting in L2 (0, 1), a generalization to the spaces L2 (0, a) with any a > 0 or a = ∞ is straightforward. We restricted ourselves to the unit interval since we apply the results obtained to the inverse spectral problems for Bessel operators defined on finite intervals. Inverse problems for Schr¨odinger operators on the semi-axis generated by the differential expressions (1.3) with angular momenta m ∈ N have been treated in the framework of the scattering theory [6, Ch. 14,15] and operate with different objects.

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2. Factorisation in operator algebras 2.1. Definitions In the Hilbert space H := L2 (0, 1), we introduce the chain of orthoprojectors P (t), t ∈ [0, 1], given by P (t)f (x) := χt (x)f (x), where χt is the characteristic function of the interval [0, t], and set Q(t) := I −P (t). A bounded operator S+ (resp. S− ) is called an upper-triangular (resp. lowertriangular ) operator if, for all t ∈ [0, 1], Q(t)S+ P (t) = 0

(resp.,

P (t)S− Q(t) = 0).

All bounded upper-triangular (resp. lower-triangular) operators constitute a closed subalgebra B + (resp. B − ) in the Banach algebra B = B(H) of all bounded linear operators in H. It is clear that the involution A 7→ A∗ maps B + (resp. B − ) into B − (resp. B + ). Definition 2.1. Let A be a subalgebra of B. We say that an operator S ∈ A admits factorisation, or is factorisable in A if there exist S+ ∈ A ∩ B + and S− ∈ A ∩ B − such that S = S+ S− . Clearly, this notion is an infinite-dimensional generalisation of the Gauss method of inverting a square matrix S using its LU -decomposition, i.e., using the representation of S as the product of lower- and upper-triangular matrices. M. Krein [12, 13] was seemingly the first to consider the factorisation problem in an infinite-dimensional algebra—namely, in the algebra of operators of the form zI + K, where z ∈ C, I is the identity operator, and K is an integral operator with continuous kernel. A more general algebra A∞ = {zI + B | B ∈ B∞ }, with B∞ denoting the ideal of all compact operators was studied in detail in the book by I. Gokhberg and M. Krein [11], and the case of a Banach space H was considered in [3]. L. Sakhnovich in [19, 20] investigated factorisation in the group Binv of all invertible operators; he required that the factors S+ and S− in S = S+ S− be in addition invertible in B + and B − and called such a factorisation the special factorisation. We observe that also in [3,11–13] only invertible operators were considered. 2.2. The necessary condition for special factorisability In all cases above, there is a simple necessary condition for special factorisability. Indeed, assume that an operator S admits a factorisation S = S+ S− , in which S+ and S− are invertible in B + and B − respectively. We set, for every t ∈ [0, 1], S+ (t) := P (t) + Q(t)S+ ∈ B + , S− (t) := P (t) + S− Q(t) ∈ B − , S(t) := P (t) + Q(t)SQ(t).

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It is easy to see that S(t) = S+ (t)S− (t) and that the operators S+ (t) and S− (t) are boundedly invertible, with −1 −1 S+ (t) = P (t) + Q(t)S+ , −1 −1 S− (t) = P (t) + S− Q(t). Hence if an invertible operator S ∈ Binv admits a special factorisation, then the following condition holds: (I) for every t ∈ [0, 1], the operator S(t) is boundedly invertible in the algebra B. For a generic algebra A condition (I) is by no means sufficient in order that an operator S ∈ A be factorisable in A . However, (I) is sufficient, e.g., in the algebras Ap := CI + Bp := {zI + B | z ∈ C, B ∈ Bp }, where Bp is the Schatten–von Neumann ideal, cf. [11, IV.5]. In particular, we have the following statement: Proposition 2.2. Assume that F is a Hilbert–Schmidt operator such that S := I +F is positive. Then S admits a special factorisation in the algebra A2 . Some other non-classical operator algebras in which (I) is sufficient for factorisability were given in [15,16], where also methods for constructing such algebras were presented. 2.3. Uniqueness Assume that S ∈ A admits two special factorisations in the algebra A , S = S+ S− = S˜+ S˜− . By definition, the factors S+ and S˜+ are invertible in A ∩ B + , and S− and S˜− are invertible in A ∩ B − ; therefore the operator −1 −1 S˜+ S+ = S˜− S−

belongs to the subalgebra D := B + ∩ B − of diagonal operators. Every diagonal operator D is in fact the operator of multiplication by a function d ∈ L∞ (0, 1). Every diagonal operator D that is invertible in A causes therefore non-uniqueness of factorisation in A via S = S+ S− = (S+ D)(D−1 S− ). Conversely, if the only diagonal operators in A that are invertible in A are the scalar operators (i.e., the operators zI for z ∈ C \ {0}), then every S ∈ A can admit at most one special factorisation in A modulo scalar factors.

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These scalar factors can actually be fixed by requiring that the factors S± should belong to the set BI of operators having diagonal equal to the identity operator I. By definition, an operator A ∈ B has diagonal D if the limit n(τ )

s-lim diam(τ )→0

X

∆j P A∆j P

(2.1)

j=1

exists and is equal to D; here τ = {0 =: t0 < t1 < · · · < tn(τ ) := 1} is a partition of the interval [0, 1], diam(τ ) := max{tj − tj−1 | j = 1, . . . , n(τ )} is its diameter, ∆j P := P (tj ) − P (tj−1 ), and the limit is taken over all possible partitions in the strong operator topology. Clearly, if D is the diagonal of an operator A, then D belongs to D. Also, every operator D in D coincides with its own diagonal. We shall need later the following auxiliary results. Lemma 2.3. (i) Every compact operator has diagonal equal to zero. (ii) Assume that operators A and B in B + have diagonals DA and DB respectively; then AB has diagonal DA DB . Proof. Part (i) follows from a stronger result established in [11, Lemma I.5.1]. If A and B belong to B + and τ is an arbitrary partition of [0, 1], then ∆j P A = ∆j P A(I − Ptj−1 ),

B∆j P = P (tj )B∆j P.

Therefore, n(τ )

X

n(τ )

∆j P AB∆j P =

j=1

X

n(τ )

∆j P A∆j P B∆j P =

j=1

X j=1

and (ii) follows.

n(τ )

∆j P A∆j P

X

∆k P B∆k P,

k=1

Corollary 2.4. Under the assumptions of Proposition 2.2, the operator I + F has a unique factorisation in A2 with factors S± belonging to the sets BI± := B ± ∩ BI ; ∗ . moreover, S− = S+ Proof. In view of the above lemma elements of A2 that belong to BI have the form I + K with K ∈ B2 . Moreover, if I + K ∈ A2 ∩ BI± and I + K is invertible in B, then the inverse (I + K)−1 =: I + K1 also belongs to A2 ∩ BI± . The inclusion K1 ∈ B2 follows from the equality K1 = −K(I + K1 ), and that K1 ∈ B ± can be proved directly by studying the integral equation Z 1 k(x, t) + k1 (x, t) + k(x, s)k1 (s, t) ds = 0 0

for the corresponding kernels. Assume now that a positive operator I+F with a Hilbert–Schmidt operator F is factorisable in A2 as S+ S− , where S+ and S− belong to BI+ and BI− respectively. Then S± have the form I + K± with some K± ∈ B2 . Clearly, S− has a trivial nullspace and thus is a bijection; hence S+ has a trivial null-space and is a bijection as well. Therefore every factorisation of I + F in A2 with factors having diagonal

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equal to I is a special one. Since the only diagonal operator belonging to A2 ∩ BI is the identity operator, the arguments at the beginning of this subsection imply that such a factorisation is unique. Since I + F = (S− )∗ (S+ )∗ is also a factorisation of I + F with the specified properties, it follows that S− = (S+ )∗ as claimed.

3. Factorisation of non-negative Fredholm operators In this section, we consider the factorisation problem for non-invertible Fredholm operators, which, to the best of our knowledge, has not been studied yet. We denote by F the algebra of all Fredholm operators in B and set F ± := F ∩ B ± , FI := F ∩ BI , and FI± := F ± ∩ FI . We shall start with the simplest situation of factorisation of orthogonal projectors with one-dimensional null-space. 3.1. Factorisation of orthogonal projectors with one-dimensional null-space Let φ be a function in L2 (0, 1) of norm 1 whose support contains 0. We set Z x a(x) := |φ(t)|2 dt 0

and introduce an operator Vφ in L2 (0, 1) via Z

1

(Vφ y)(x) := y(x) − φ(x) x

φ(t)y(t) dt. a(t)

A variant of the following lemma has appeared in [10]. We give the proof here for the sake of completeness. Lemma 3.1. The operator Vφ is isometric and Vφ Vφ∗ = I − Pφ , where Pφ := (·, φ)φ is the orthogonal projector onto φ. Proof. Take arbitrary functions f and g in L2 (0, 1) whose support does not contain the origin. Direct calculations give Z 1 Z 1 Z 1 φ(t)g(t) (Vφ f, Vφ g) = f (t)g(t) dt − dxf (x)φ(x) dt a(t) 0 0 x Z 1 Z 1 φ(t)f (t) − dxφ(x)g(x) dt a(t) 0 x Z 1 Z 1 Z 1 φ(t)g(t) φ(t)f (t) 2 dt dt. + dx|φ(x)| a(t) a(t) x 0 x

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Observing that |φ(x)|2 = a0 (x) and integrating by parts in the last integral, we see that the last three integrals sum up to zero, thus showing that Vφ is isometric. The equality Vφ∗ y = 0 implies that Z a0 (x) x φ(x)y(x) = φ(t)y(t) dt, a(x) 0 i.e., by integrating, that Z

x

φ(t)y(t) dt = ca(x) 0

for some constant c, and finally, upon differentiating, that y = cφ. It thus follows that ker Vφ∗ is spanned by φ. Since Vφ Vφ∗ is the orthogonal projector onto (ker Vφ∗ )⊥ , we get Vφ Vφ∗ = I − Pφ as claimed. Clearly, the factorisation of I −Pφ as Vφ Vφ∗ is a very special one, and it would be desirable to understand in what sense it is unique. It follows from Lemma 3.6 below (take B = I therein) that Vφ has diagonal equal to I and thus belongs to FI+ . However, there are many different operators V ∈ FI+ such that I − Pφ = V V ∗ , as the following example demonstrates. Example 3.2. Let B be any co-isometry in L2 (0, 1) belonging to FI+ ; then V := Vφ B is in FI+ by Lemma 2.3(ii) and V V ∗ = Vφ BB ∗ Vφ∗ = Vφ Vφ∗ = I − Pφ . Such a co-isometry B can be constructed e.g. as B = RV ∗ R, where R is the reflection operator, (Rf )(x) = f (1 − x), and V is any isometry in FI+ (see Remark 3.11). Every operator V constructed as suggested in the above example has a nontrivial null-space, and one could hope that Vφ is singled out by requiring that the factor V in the equality I − Pφ = V V ∗ should be injective. However, this is also not true, since there are unitary operators in BI+ different from the identity operator I. We construct one such an operator in the example below; it should be clear how to modify and/or iterate this construction to get many other examples. It seems that the problem of characterising all unitary operators in BI+ is quite difficult; we shall not touch upon it here as it goes beyond our main aims. Example 3.3. We construct here a non-trivial unitary operator U in H = L2 (0, 1) belonging to BI+ . The main idea of the construction is to take an isometry U1 and an co-isometry U2 in BI+ , make them act on L2 (0, 1) and L2 (1, 2) respectively, take their direct sum in L2 (0, 2), then perturb the sum by a partial isometry that sends vectors in the null-space of U2 in L2 (1, 2) into the orthogonal complement of the range of U1 in L2 (0, 1), and finally make the resulting operator act in L2 (0, 1) by an appropriate scaling. To this end we take arbitrary unit vectors φ1 and φ2 of L2 (0, 1) whose support R2 contain the point x = 0, set U1 := Vφ1 , ψ(x) := φ2 (2 − x) and b(x) := x |ψ(t)|2 dt

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for x ∈ (1, 2), and define (U2 f2 )(x) := f2 (x) −

ψ(x) b(x)

Z

2

ψ(t)f2 (t) dt x

for x ∈ (1, 2) and f2 ∈ L2 (1, 2). Then the operator f1 U1 f1 φ1 U := + (f2 , ψ)L2 (1,2) f2 U2 f2 0 −1 is unitary in L2 (0, 1) √ ⊕ L2 (1, 2) = L2 (0, 2), and it remains to set U := W UW , where (W f )(x) := 2f (2x) is a unitary dilation of L2 (0, 2) into L2 (0, 1). It is clear that U has diagonal I and belongs to B + .

To point out yet another important property of the operator Vφ , we introduce the following definition. Definition 3.4. We say that a bounded operator T is almost Hilbert–Schmidt if, for every t ∈ (0, 1), the compression Q(t)T Q(t) L2 (t,1) of T onto the subspace L2 (t, 1) is a Hilbert–Schmidt operator therein. Clearly, the operator Vφ − I is almost Hilbert–Schmidt in the above sense. The main result of this subsection now reads as follows: Theorem 3.5. Let φ be an arbitrary function in L2 (0, 1) of unit norm whose support contains x = 0. (a) If V satisfies the relation V V ∗ = I − Pφ and V − I is an almost Hilbert– Schmidt operator, then V = Vφ . (b) If V is an injective operator in FI+ such that V V ∗ = I − Pφ , then V = Vφ U , where U is some unitary operator in BI+ . Our proof will rely on several auxiliary results, which we establish first. Lemma 3.6. Assume that B ∈ B and that φ is a unit vector in L2 (0, 1) whose support contains 0. Then the operator A := Vφ B − B has diagonal equal to zero. Proof. For B ∈ B and f ∈ H, we set )

n(τ

X ∆j P B∆j P f , D(B, f ) := lim sup diam(τ )→0 j=1

with notations explained after the displayed formula (2.1). It is easily seen that for every B and C in B and every f ∈ H, one gets D(B, f ) ≤ kBkkf k, D(B + C, f ) ≤ D(B, f ) + D(C, f ), D(P (t)B, f ) = D(B, P (t)f ),

(3.1)

t ∈ [0, 1].

Fix an arbitrary t ∈ (0, 1) and observe that the operator Q(t)(Vφ − I) is compact. Therefore for such t compact is also the operator At := Q(t)A,

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so that At has diagonal zero and D(At , f ) = 0 for every f ∈ H. Using properties (3.1), we conclude that D(A, f ) ≤ D(P (t)A, f ) = D(A, P (t)f ) ≤ kAkkP (t)f k. Since t ∈ (0, 1) is arbitrary, it follows that D(A, f ) = 0, i.e., that A has diagonal zero. Corollary 3.7. If, under the above assumptions, one of the operators Vφ B or B belongs to BI , then so does the other one. Lemma 3.8. Assume that B ∈ B and that φ is a unit vector in L2 (0, 1) whose support contains 0. If Vφ B belongs to B + , then B belongs to B + , too. Proof. Since Vφ B ∈ B + by assumption, we have for every t ∈ (0, 1) Q(t)Vφ BP (t) = 0. Using the fact that Vφ belongs to B + , we rewrite the above equality as Q(t)Vφ Q(t) Q(t)BP (t) = 0.

(3.2)

The operator Q(t)Vφ Q(t) L2 (t,1) , being equal to the identity operator plus a Volterra one, is invertible in L2 (t, 1); hence (3.2) yields Q(t)BP (t) = 0 for every t ∈ (0, 1). Thus B ∈ B + as claimed.

Proof of Theorem 3.5. (a) Assume that V = I + K with an almost Hilbert– Schmidt operator K ∈ BI+ is such that I − Pφ = V V ∗ . Applying Q(t), t ∈ (0, 1), from both sides, using the fact that K ∈ B + and K ∗ ∈ B − , and setting K(t) := Q(t)KQ(t), we get Q(t) − Q(t)Pφ Q(t) = Q(t) + K(t) Q(t) + K ∗ (t) . (3.3) We restrict this equality onto L2 (t, 1); since Q(t) acts there as the identity operator, (3.3) can be regarded as a factorisation in the algebra of operators on L2 (t, 1), viz. I − Q(t)Pφ Q(t) = I + K(t) I + K ∗ (t) . L2 (t,1)

L2 (t,1)

L2 (t,1)

The assumption 0 ∈ supp φ implies that kQ(t)φk < kφk, and thus the operator I −Q(t)Pφ Q(t) is positive in L2 (t, 1). Applying Corollary 2.4, we conclude that the operator K(t) is uniquely determined by φ and thus coincides with the operator Q(t)(Vφ −I)Q(t). Since t ∈ (0, 1) was arbitrary, this yields the equality K = Vφ −I, i.e., V = Vφ as claimed. (b) Assume that V verifies the assumptions of the theorem. Then the equality kV ∗ f k = k(I − Pφ )f k shows that V ∗ is a partial isometry and ker V ∗ is spanned by φ. Therefore, ran V = H ker V ∗ = ran(I − Pφ ),

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so that V V ∗ V = (I − Pφ )V = V , i.e., V (V ∗ V − I) = 0. Since V is injective, it follows that V ∗ V = I and thus V is an isometry. We conclude that the operator U := Vφ∗ V is unitary and that Vφ U = (I − Pφ )V = V . By Lemma 3.8 and Corollary 3.7, the latter equality implies that U ∈ BI+ as claimed. Remark 3.9. The assumption that the support of the function φ contains zero was made only to simplify the constructions and can easily be dropped. Indeed, let φ be of unit norm in L2 (0, 1) and inf supp φ = c > 0. We then interpret the operator Vφ as acting via ( f (x) if x ≤ c, R1 (Vφ f )(x) := if x > c. f (x) + φ(x) x φ(t)y(t)/a(t) dt Thus Vφ is the direct sum of the identity operator in L2 (0, c) and an uppertriangular √ operator in L2 (c, 1) that is unitarily equivalent to the operator Vψ with ψ(x) := 1 − c φ c + (1 − c)x of unit norm in L2 (0, 1) whose support contains 0. It is straightforward to verify that under suitable modifications all the results of this subsection remain valid. Therefore we do not need to have 0 ∈ supp φ, although this assumption will be satisfied for concrete applications to Bessel operators treated in Section 4. 3.2. Factorisation of orthogonal projectors with finite-dimensional null-space We now extend the results of the previous subsection to Fredholm orthogonal projectors as follows: Theorem 3.10. Assume that P is an orthoprojector in L2 (0, 1) of finite rank such that, for every t ∈ (0, 1), ran P ∩ Q(t)L2 (0, 1) = {0}. Then there exists a unique operator VP in FI+ such that VP − I is almost Hilbert–Schmidt and VP VP∗ = I − P. The operator VP is isometric; moreover, every injective operator V in FI+ satisfying V V ∗ = I − P is isometric and equals VP U , where U is a unitary operator in BI+ . Proof. The proof of existence and uniqueness of VP with the stated properties is by induction on rank P . Theorem 3.5 handles the case where rank P = 1. We assume next that, for some n ∈ N, the theorem is already proved whenever rank P ≤ n and let P be an orthoprojector of rank n + 1. Fix an element φ ∈ ran P of unit norm whose support contains 0 and set Q := Vφ∗ P Vφ ; existence of such a φ is guaranteed by the assumption on ran P . Since Q = Q∗ and, by Lemma 3.1, Q2 = Vφ∗ P (I − Pφ )P Vφ = Q, we conclude that Q is an orthoprojector on ran Q. Taking into account that Vφ is injective and ran Vφ = L2 (0, 1) φ, we see that Q is of rank n. We set VP := Vφ VQ ,

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where VQ is a unique operator in FI+ such that VQ − I is almost Hilbert–Schmidt and VQ VQ∗ = I − Q, whose existence is guaranteed by the hypothesis of induction. The operator VP is isometric and belongs to F + . Since VQ has diagonal equal to I, so does VP by Corollary 3.7, i.e., VP ∈ FI+ . Next we see that VP VP∗ = Vφ VQ VQ∗ Vφ∗ = Vφ (I − Q)Vφ∗ = Vφ Vφ∗ (I − P )Vφ Vφ∗ = (I − Pφ )(I − P )(I − Pφ ) = I − P, as required. Finally, VP − I = (Vφ − I)VQ + (VQ − I) is an almost Hilbert–Schmidt operator. To show uniqueness of VP with the specified properties, we assume that V V ∗ = I − P for some operator V ∈ FI+ such that V − I is almost Hilbert– Schmidt. Since, for every t ∈ (0, 1), the operator I − Q(t)P Q(t) is positive in L2 (t, 1), we conclude as in the proof of Theorem 3.5 that Q(t)V Q(t) is uniquely determined by P and thus coincides with Q(t)VP Q(t). As t ∈ (0, 1) was arbitrary, we conclude that V = VP . Finally, assuming that there is an injection V in FI+ such that V V ∗ = I − P and putting VeQ := Vφ∗ V , we see that VeQ VeQ∗ = Vφ∗ V V ∗ Vφ = Vφ∗ (I − P )Vφ = I − Q. As in the proof of Theorem 3.5 we can show that V ∗ is a partial isometry and thus V is an isometry. Since ran V = ran(I − P ), VeQ is injective, and the equalities Vφ VeQ = (I − Pφ )V = V and Lemma 3.8 imply then that VeQ belongs to FI+ . We showed above that Q is an orthoprojector of rank n, so that by the assumption of induction we have VeQ = VQ U for some unitary U ∈ BI+ resulting in V = Vφ VeQ = V φ V Q U = VP U . Remark 3.11. It follows from the proof that the operator VP has the form VP = Vφ1 Vφ2 · · · Vφn

(3.4)

for a suitable (not unique!) choice of vectors φ1 , φ2 , . . . , φn of unit length; here n is the rank of the projector P . Indeed, take an orthonormal basis ψ1 , . . . , ψn of the range of P and set recursively φ1 := ψ1 , φ2 := Vφ∗1 ψ2 , . . . , φn := Vφ∗n−1· · · Vφ∗2 Vφ∗1 ψn . That φk are of unit norm can be deduced from the fact that Vφ∗ is a partial isometry between H φ and H. It is clear now that the null-space of the operator Vφ∗n Vφ∗n−1 · · · Vφ∗2 Vφ∗1

(3.5)

is of dimension n and is spanned by the vectors ψ1 , ψ2 , . . . , ψn , hence it coincides with the range of P . Thus the operator of (3.5) coincides with VP∗ , and VP takes the form (3.4).

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3.3. Factorisation of non-negative Fredholm operators We combine here the above partial results to treat a general case of non-negative Fredholm operator I + F with F ∈ B2 . Theorem 3.12. Assume that F is a self-adjoint operator of Hilbert–Schmidt class such that I + F is non-negative but I + Q(t)F Q(t) is positive for every t ∈ (0, 1), and denote by P the orthoprojector on ker(I + F ). Then I + F admits a unique factorisation of the form I + F = (VP + K+ )(VP + K+ )∗ ,

(3.6)

in which VP is the isometric operator introduced in Theorem 3.10 and K+ ∈ B + is a Hilbert–Schmidt operator. Proof. The assumptions of the theorem imply that the support of every nontrivial function in ker(I + F ) contains zero and thus ran P ∩ Q(t)L2 (0, 1) = {0} for every t ∈ (0, 1). By Theorem 3.10, we have I − P = VP VP∗ , so that I + F = VP VP∗ (I + F )VP VP∗ . It is easily seen that the operator Se = VP∗ (I + F )VP = I + VP∗ F VP is strictly positive and thus by Corollary 2.4 it admits a unique factorisation as e + )(I + K e + )∗ Se = (I + K e + ∈ B + ∩ B2 . Now we conclude that with K e + )(I + K e + )∗ VP∗ =: (VP + K+ )(VP + K+ )∗ , I + F = VP (I + K

(3.7)

e + is a Hilbert–Schmidt operator in B + as required. where K+ := VP K Uniqueness is proved as in the previous subsections, by considering the induced factorisation in the operator algebra over L2 (t, 1), t ∈ (0, 1) of the positive operator Q(t) + Q(t)F Q(t). There is another way to show that factorisation of I + F of the form (3.6) is unique if we assume in addition that the null-space of the factor VP + K+ is trivial. Then the range of (VP + K+ )∗ is the whole space H, and thus ran(VP + K+ ) = ran(I − P ). Therefore we have VP + K+ = VP VP∗ (VP + K+ ) = VP (I + VP∗ K+ ). In view of Remark 3.11, repeated application of Lemma 3.8 shows that, for a bounded ope+ erator B, the inclusion VP B ∈ B + yields B ∈ B + . Therefore VP∗ K+ =: K + belongs to B ∩ B2 , and thus every factorisation of the form (3.6) can be recast e + = V ∗ K+ ∈ B + ∩ B2 . It follows that the operator in the form (3.7) with K P ∗ Se := I + VP F VP is factorised as e + )(I + K e + )∗ Se = (I + K

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e + is unique. Therefore K+ = VP K e+ in A2 , and thus by Corollary 2.4 the operator K is also uniquely determined by F .

4. Application to the inverse spectral problem for Bessel operators In this section, we shall consider the factorisation problem for the operator I + F constructed from the spectral data for a Bessel operator and study the properties of the corresponding factors. 4.1. Special property of Vφ A careful examination of the transformation Vφ studied in Subsection 3.1 reveals that it is inverse to the well known double commutation transformation used in the spectral analysis of the Sturm–Liouville and Dirac operators [1, 2, 9, 10, 23]. This suggests that the factorisation problem discussed in the previous section is intimately related to Sturm–Liouville and Dirac differential operators and might be of much use in the spectral analysis of the latter. We demonstrate this for the Sturm–Liouville case. Assume that q0 ∈ L2 (0, 1) is real-valued and that a real-valued function φ0 of unit norm in L2 (0, 1) obeys the terminal condition φ0 (1) = 0 and is a solution of the differential equation −y 00 + q0 y = λ20 y with a real λ0 . We set q1 := q0 − 2(log a)00 , (4.1) Rx 2 where a(x) := 0 |φ0 (t)| dt, and, for every λ ∈ C, denote by u(·, λ) a solution of the equation −u00 + q1 u = λ2 u satisfying the terminal conditions u(1) = 0 and u0 (1) = 1. Lemma 4.1. Under the above assumptions, for every λ ∈ C the function Z 1 φ0 (t)u(t, λ) dt v(·, λ) := Vφ0 u(·, λ) = u(x, λ) − φ0 (x) a(t) x

(4.2)

verifies the relations v(1) = 0 and v 0 (1) = 1 and solves the equation −y 00 + q0 y = λ2 y. Proof. We start by observing that the function φ0 /a is collinear to u(·, λ0 ). Indeed, direct calculations give 0 2 φ 00 φ00 a0 φ00 a0 φ0 a0 φ0 0 = 0 −2 − + , a a a a a a a a which in view of the relations 2a0 φ00 = a00 φ0 and φ000 = (q0 − λ20 )φ0 implies that φ 00 φ φ0 0 0 = q0 − 2(log a)00 − λ20 = (q1 − λ2 ) . a a a

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It remains to observe that φ0 /a satisfies along with φ0 the Dirichlet condition at x = 1. Next, it is clear that v(·, λ) obeys the stated conditions at x = 1. Differentiating (4.2), we get Z 1 φ0 (t)u(t, λ) a0 (x) 0 0 0 v (x, λ) = u (x, λ) − φ0 (x) dt + u(x, λ), a(t) a(x) x and then, using the fact that u(·, λ) and φ0 satisfy the corresponding differential equations, we find that Z 1 φ0 (t)u(t, λ) 00 00 00 dt v (x, λ) = u (x, λ) − φ0 (x) a(t) x a0 (x) 0 1 a00 (x) a0 (x) 0 + u(x, λ) + u(x, λ) + u (x, λ) 2 a(x) a(x) a(x) Z 1 φ0 (t)u(t, λ) dt = q1 (x) − λ2 u(x, λ) − q0 (x) − λ20 φ0 (x) a(t) x a0 (x) 0 1 a00 (x) a0 (x) 0 + u(x, λ) + u(x, λ) + u (x, λ) 2 a(x) a(x) a(x) Z 1 φ0 (t)u(t, λ) dt = (q0 − λ2 )v(x, λ) + (λ20 − λ2 )φ0 (x) a(t) x a0 (x) 0 1 a00 (x) a0 (x) 0 + u(x, λ) − u(x, λ) + u (x, λ). 2 a(x) a(x) a(x) Recalling that φ0 /a is collinear to u(·, λ0 ) and using the Lagrange identity, we derive the relation Z 1 φ (x) 0 φ0 (x) 0 φ0 (t)u(t, λ) 0 (λ20 − λ2 ) dt = u(x, λ) − u (x, λ) a(t) a(x) a(x) x which, on account of the Riccati identity (a0 /a)0 = a00 /a − (a0 /a)2 , shows that the last four summands above cancel out. Therefore v 00 (x, λ) = (q0 − λ2 )v(x, λ). 4.2. Solution of the classical Sturm–Liouville inverse spectral problem We recall first in some more detail the classical method of reconstruction of a Sturm–Liouville operator d2 S(θ, q) := − 2 + q dx subject to the boundary conditions cos θ y(0) − sin θ y 0 (0) = y(1) = 0 from its spectral data, the sequences of eigenvalues and norming constants. We assume that the potential q is real-valued and belongs to L2 (0, 1). For every λ ∈ C, we denote by y(·, λ) a solution to the equation −y 00 + qy = 2 λ y subject to the terminal conditions y(1) = 0 and y 0 (1) = λ. Then y(·, λ) has

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the representation Z

1

k(x, t) sin λ(t − 1) dt,

y(x, λ) = sin λ(x − 1) +

(4.3)

x

where k is the kernel of the corresponding transformation operator I + K. The kernel k has the property that, for every fixed x ∈ [0, 1), the function k(x, ·) belongs to the Sobolev space W21 (x, 1). The eigenvalues λ2n of the operator S(0, q) are then squared zeros of the characteristic function Z 1 ∆(λ) := sin λ + k(0, t) sin λ(1 − t) dt 0

and have the asymptotics ˜n, λ2n = π 2 n2 + A + λ (4.4) ˜ with A := 0 q(t) dt and an `2 -sequence (λn ). For the non-Dirichlet boundary condition at x = 0 (θ > 0) the leading term of the above asymptotics becomes π 2 (n − 1/2)2 . For the eigenvalue λ2n , we denote the norming constant αn as the squared L2 -norm of the eigenfunction y(·, λn ), i.e., Z 1 αn := |y(x, λn )|2 dx. (4.5) R1

0

Using the properties of the operator K, it is easy to prove that the αn have the form 1 α ˜n αn = + , (4.6) 2 n where the numbers α ˜ n form a sequence from `2 . Now the resolution of the identity for the operator S(θ, q) reads I = s-lim

N X

N →∞

αn−1 ·, yn yn ,

n=1

with yn := y(·, λn ). Setting sn (x) := sin λn (x − 1) and using the representation (4.3), we conclude that N h X i I = (I + K) s-lim αn−1 ·, sn sn (I + K ∗ ). N →∞

n=1

The operator in the square brackets is uniformly positive in L2 (0, 1) and has the form I + F , where F is a Hilbert–Schmidt operator with kernel f (x, y) = ϕ(2 − x − y) − ϕ(|x − y|), where ϕ(s) :=

i 1 Xh 2 cos πks − αk−1 cos λk s 2 k∈N

(4.7) (4.8)

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is a function from W21 (0, 2). Setting I + L := (I + K)−1 , we get I + F = (I + L)(I + L∗ ); since I + L belongs to BI+ , by Corollary 2.4 this is a unique factorisation of I + F with factors in BI+ . Finally, we recall that the potential q of the operator S(θ, q) can be reconstructed through dl(x, x) , (4.9) dx where l is the kernel of L; it is proved in the classical inverse theory that the function l(x, x) belongs to W21 (0, 1) and thus the above formula yields a function from L2 (0, 1) as required. Summarising, we arrive at the following reconstruction algorithm: first, given the spectral data, we construct the function ϕ and the operator F , then factorise the operator I + F , call the factor I + L, and finally use the kernel l of L to determine q via (4.9). It is worth noting that any two sequences of real numbers λ2n and αn such that λ2n strictly increase and satisfy the asymptotics of (4.4) and αn are positive and satisfy (4.6), are sequences of eigenvalues and norming constants of a unique Sturm–Liouville operator S(0, q) with q ∈ L2 (0, 1). The method of finding this q is precisely the same as above. q(x) = 2

4.3. Reconstruction in the Bessel case Now assume that m ∈ N and q ∈ L2 (0, 1) is real-valued and consider the Bessel operator T (m; q) given by the differential expression d m d m m(m + 1) d2 − + +q =− +q t(m; q) := − 2 + 2 dx x dx x dx x subject to the Dirichlet boundary condition at x = 1. The differential expression t(m; q) is well defined on the set of functions y that together with their quasiderivatives y [1] := y+(m/x)y are absolutely continuous on [ε, 1] for every ε ∈ (0, 1). It is well known [1, 4, 5, 7, 21] that being considered on the domain dom T (m; q) := {y ∈ dom t(m; q) ∩ L2 (0, 1) | t(m; q)y ∈ L2 (0, 1), y(1) = 0} the operator T (m; q) becomes self-adjoint, bounded below, and has a discrete spectrum. As earlier, for a nonzero λ ∈ C, we denote by y(·, λ) a solution of the equation m(m + 1) −y 00 + y + q(x)y = λ2 y (4.10) x2 satisfying the terminal conditions y(1) = 0 and y 0 (1) = λ. The function y(x, λ) either vanishes at x = 0 or has there a pole. In the former case λ2 is an eigenvalue of T (m; q) (and thus is real) and y(·, λ) is a corresponding eigenfunction. We enumerate the eigenvalues λ21 < λ22 < . . . in increasing order and recall [4, 5, 7, 21] the asymptotic relation 2 ˜n, λ2n = π 2 n − m +A+λ (4.11) 2

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˜ n ). Without loss of generality we assume that with A ∈ R and an `2 -sequence (λ λ21 > 0 as otherwise we can shift the spectrum by adding a suitable constant to q. Slightly modifying the arguments of [1, Lemma 2.2], we can show that the eigenfunctions y(·, λn ) have the form y(x, λn ) = xm+1 un (x),

(4.12)

W22 (0, 1)

where un is a function from that does not vanish at x = 0. Next, we introduce the norming constants αn corresponding to λ2n via (4.5); then [1, 7] the αn have the same asymptotics as in the case m = 0. Keeping the notations of the previous subsection, we introduce the function ϕ of (4.8), the kernel f of (4.7), and the corresponding integral operator F . Then I + F = s-lim

N →∞

N X

αn−1 ·, sn sn

n=1

is non-negative but in view of the asymptotic behaviour of λn it has a non-trivial null-space of dimension [(m + 1)/2], [a] denoting the integral part of a number a. Since the system {sn }∞ n=1 forms a Riesz basis of its closed linear span and is complete in L2 (ε, 1) for every ε > 0, the support of every function in ker(I + F ) contains zero. Therefore by Theorem 3.12 there is a unique operator S+ := VP +K+ such that ∗ I + F = S+ S + ; here P is the orthogonal projector on the null-space of I +F , VP is the isometric operator introduced in Theorem 3.10, and K+ is a Hilbert–Schmidt operator in B + . Since the operator VP − I is almost Hilbert–Schmidt in the sense of Definition 3.4, it follows that S+ has the form I + L with an almost Hilbert–Schmidt operator L. The properties of the kernel l of the operator L are given in the following theorem. Theorem 4.2. Under the above assumptions, (I + L)y(x, λ) = sin λ(x − 1), i.e., I +L transforms solutions of the equation (4.10) into those for the unperturbed Sturm–Liouville equation −y 00 = λ2 y. Moreover, the kernel l is continuous if x > 0, the function l(x, x) belongs to W21 (ε, 1) for every ε > 0, and dl(x, x) m(m + 1) + q(x) = 2 . 2 x dx Proof. The proof is by induction on m ∈ N; moreover, we have to consider separately the cases of even and odd m. Even m: base of induction. We start with the case m = 2 and fix arbitrary positive λ20 and α0 such that λ20 < λ21 . By the results of Subsection 4.2 the sets {λ2n }n≥0 and {αn }n≥0 are sets of eigenvalues and norming constants for a unique Dirichlet Sturm–Liouville operator T (0, q0 ) := −

d2 + q0 , dx2

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with some q0 ∈ L2 (0, 1). Put F0 := F + α0−1 (·, s0 )s0 with s0 (x) := sin λ0 (x − 1). Then I + F0 = (I + L0 )(I + L∗0 ), where I + L0 ∈ BI+ , the kernel l0 of L0 is continuous on Ω, and dl0 (x, x) . dx The operator I + L0 is the transformation operator between T (0, q0 ) and T (0, 0), i.e., if v(·, λ) is the solution of the equation `(0, q0 )v = λ2 v subject to the terminal conditions v(1) = 0 and v 0 (1) = λ, then (I + L0 )v(x, λ) = sin λ(x − 1). Next we find that I + F = (I + L0 ) I − (·, φ0 )φ0 (I + L∗0 ), q0 (x) = 2

where φ0 := α−1/2 (I + L0 )−1 s0 is an eigenfunction of the operator T (0, q0 ) of norm 1 corresponding to the eigenvalue λ20 . Therefore I − (·, φ0 )φ0 = Vφ0 Vφ∗0 (see Subsection 3.1), and, setting I +L := (I +L0 )Vφ0 , we get the required factorisation I + F = (I + L)(I + L∗ ) of I + F . The kernel l of L equals l(x, t) = l0 (x, t) − with a(x) := over,

Rx 0

φ0 (x)φ0 (t) − a(t)

Z

t

l0 (x, s) x

φ0 (s)φ0 (t) ds a(t)

|φ0 (t)|2 dt and thus it has the stated smoothness properties; more-

dl(x, x) = q0 (x) − 2 (log a)00 (x) =: q1 (x). (4.13) dx Applying Lemma 4.1, we conclude that the operator I +L transforms the solutions of the equation −y 00 + q1 y = λ2 y (4.14) satisfying the terminal conditions y(1) = 0 and y 0 (1) = λ into such solutions sin λ(x − 1) for zero potential (i.e., for q1 ≡ 0). Next we observe that the function a has the form a(x) = x3 b(x) for some function b ∈ W22 (0, 1) that is positive on [0, 1]; this follows from the behaviour of the eigenfunction φ0 at the origin and the properties of the Hardy operators, see details in [1, App. A]. Therefore, 2(log a)00 = −2 · 3/x2 + 2(log b)00 ; it follows that the function q1 has the form 2·3 q1 (x) = 2 + q˜(x) x with q˜ := q0 (x) − 2(log b)00 ∈ L2 (0, 1), and it remains to show that q˜ = q. To this end we recall that the operator Vφ0 maps isometrically L2 (0, 1) onto its range L2 (0, 1) φ0 . Thus the pre-images of the eigenfunctions v(·, λn ), n ≥ 1, of the 1/2 operator T (0, q0 ) have norm αn in L2 (0, 1), satisfy the Dirichlet condition at x = 1, and solve (4.14) with λ = λn . Therefore the operator T (2, q˜) has eigenvalues λ2n , n ≥ 1, and the corresponding norming constants are equal to αn . The direct 2

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spectral analysis of Bessel operators (in particular, the known asymptotics of their eigenvalues) suggests that T (2, q˜) has no other eigenvalues. By the Borg–Levinson uniqueness result [1, 4, 5], we get q˜ = q, and the case m = 2 is done. Even m: Induction step. Assume that we have already proved the theorem for all even m less than 2k and consider the case m = 2k. We again augment the spectral data {(λ2n )n∈N , (αn )n∈N } of the operator T (2k, q) by the pair (λ20 , α0 ) with λ20 < λ21 and α0 > 0. The augmented data {(λ2n )n≥0 , (αn )n≥0 } are the spectral data for a unique operator T0 := T (2k − 2, q0 ) with some q0 ∈ L2 (0, 1). We denote by F0 the operator constructed for T0 and observe that, by the induction hypothesis, I + F0 = (I + L0 )(I + L∗0 ) for some L0 ∈ BI+ . Denote by φ0 an eigenfunction of the operator T (2k − 2, q0 ) of norm 1 corresponding to the eigenvalue λ20 and satisfying the relation φ00 (1) > 0; −1/2 then (I + L0 )φ0 = α0 s0 with s0 (x) := sin λ0 (x − 1), and thus we get I + F = I + F0 − α0−1 (·, s0 )s0 = (I + L0 ) I − (·, φ0 )φ0 (I + L∗0 ). Again the required factorisation follows with I + L := (I + L0 )Vφ0 ; moreover, by the inductive assumption, the kernel l of L satisfies 2

dl0 (x, x) d a0 (x) (2k − 2)(2k − 1) dl(x, x) =2 −2 = y + q1 dx dx dx a(x) x2

with q1 := q0 − 2(log a)00 . By Lemma 4.1 and the induction assumption, I + L transforms solutions of the equation (2k − 2)(2k − 1) y + q 1 y = λ2 y x2 into sin λ(x−1). The representation (4.12) with m = 2k−2 and the properties of the Hardy operators [1, App. A] imply that a(x) = x4k−1 b(x) with some b ∈ W22 (0, 1) that is positive on (0, 1); therefore, −y 00 +

(2k − 2)(2k − 1) 2k(2k + 1) + q1 = + q˜, 2 x x2 with q˜ := q0 − 2(log b)00 ∈ L2 (0, 1). The equality q˜ = q is justified as above, using the Borg–Levinson uniqueness theorem. The proof by induction for even m is complete. Odd m. Augmenting the spectral data for the operator T (1, q), we get spectral data for a unique Sturm–Liouville operator Th (0, q0 ) with a real-valued potential q0 ∈ L2 (0, 1) and subject to the Robin–Dirichlet boundary conditions y 0 (0) − hy(0) = y(1) = 0 for some h ∈ R. Repeating the arguments used in the case m = 2, we find an operator I + L that factorises the operator I + F constructed for T (1, q) and maps solutions of equation (4.14) with q1 given by (4.13) into sin λ(x − 1). However, now φ0 does not vanish at x = 0, and thus a has a simple zero at x = 0, which results in 2 q0 − 2(log a)00 = 2 + q˜ x

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321

with some q˜ ∈ L2 . We then justify that q˜ = q in the usual way. Finally, the induction step proceeds then as for the case of even m. 4.4. Reconstruction algorithm ∞ Assume that we are given two sequences (λ2n )∞ n=1 and (αn )n=1 of real numbers satisfying the following conditions: ˜ n ) ∈ `2 ; (i) the sequence (λ2n ) is increasing and λ2n obey (4.11) with A ∈ R and (λ (ii) the αn are positive and obey (4.6) with (˜ αn ) ∈ `2 . The results of [7,21] imply that there is a unique real-valued q ∈ L2 (0, 1) such that λ2n are all eigenvalues and αn the corresponding norming constants for the Bessel operator T (m; q). To find this q, we perform the following steps: (1) construct the integral operator F as explained in the previous subsection; (2) uniquely factorise I + F as (I + L)(I + L∗ ) with an almost Hilbert–Schmidt integral operator L ∈ FI+ with kernel l; (3) set dl(x, x) m(m + 1) − . q(x) := 2 dx x2 Theorem 4.2 implies that the function q obtained on the third step gives the required potential. We conclude with remark that similar results also hold for the Dirac operators with singular potentials appearing in the angular momentum decomposition of the radial Dirac operators in the unit ball of R3 , cf. [2,22]. All constructions and proofs can be carried over by analogy with those presented here using the results of [2,22]. Acknowledgements. The authors thank the anonymous referee for stimulating criticism and express their gratitude to Deutsche Forschungsgemeinschaft, DFG, for financial support of the project 436 UKR 113/84. The research of the second author was partially supported by the Alexander von Humboldt Foundation, which is gratefully acknowledged. The second and the third authors thank the Institute for Applied Mathematics of Bonn University for the warm hospitality.

References [1] S. Albeverio, R. Hryniv, and Ya. Mykytyuk, Inverse spectral problems for Bessel operators, J. Differential Equations 241 (2007), no. 1, 130–159. [2] S. Albeverio, R. Hryniv, and Ya. Mykytyuk, Reconstruction of radial Dirac operators, J. Math. Phys. 48 (2007), no. 4, 043501 (14 p.). [3] M. A. Barkar0 and I. Ts. Gohberg, Factorization of operators in a Banach space, Mat. Issled. 1 (1966), no. 2, 98–129 (in Russian). [4] R. Carlson, Inverse spectral theory for some singular Sturm–Liouville problems, J. Diff. Equat. 106 (1993), 121–140. [5] R. Carlson, A Borg–Levinson theorem for Bessel operators, Pacific J. Math. 177 (1997), 1–26.

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[6] L. D. Faddeev, The inverse problem in the quantum theory of scattering, Uspekhi Mat. Nauk 14 (1959), 57–119 (in Russian); Engl. transl. in J. Math. Phys. 4 (1963), 72–104. [7] M. G. Gasymov, Determination of a Sturm–Liouville equation with a singularity by two spectra, Dokl. Akad. Nauk SSSR 161 (1965), 274–276 (in Russian); Engl. transl. in Soviet Math. Dokl. 6 (1965), 396–399. [8] I. M. Gelfand and B. M. Levitan, On determination of a differential equation by its spectral function, Izv. AN USSR, Ser. Mat. 15 (1951), no. 4, 309–360 (in Russian); Engl. transl. in Amer. Math. Soc. Transl. (2) 1 (1955), 253–304. [9] F. Gesztesy, A complete spectral characterization of the double commutation method, J. Funct. Anal. 117 (1993), 401–446. [10] F. Gesztesy and G. Teschl, On the double commutation method, Proc. Amer. Math. Soc. 124 (1996), 1831–1840. [11] I. Gohberg and M. Krein, Theory of Volterra Operators in Hilbert Space and its Applications, Nauka Publ., Moscow, 1967 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Math. Monographs, 24, Amer. Math. Soc., Providence, RI, 1970. [12] M. G. Kre˘ın, On integral equations generating differential equations of 2nd order, Dokl. Akad. Nauk SSSR (N.S.) 97 (1954), no. 1, 21–24 (in Russian). [13] M. G. Kre˘ın, On a new method of solution of linear integral equations of first and second kinds, Dokl. Akad. Nauk SSSR (N.S.) 100 (1955), no. 3, 413–416 (in Russian). [14] V. A. Marchenko, Some questions of the theory of one-dimensional linear differential operators of the second order. I, Trudy Moskov. Mat. Obˇsˇc. 1 (1952), 327–420 (in Russian); Engl. transl. in Amer. Math. Soc. Transl. (2) 101 (1973), 1–104. [15] Ya. V. Mykytyuk, Factorization of Fredholm operators, Mat. Stud. 20 (2003), no. 2, 185–199 (in Ukrainian). [16] Ya. V. Mykytyuk, Factorization of Fredholm operators in operator algebras, Mat. Stud. 21 (2004), no. 1, 87–97 (in Ukrainian). [17] J. P¨ oschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Orlando, Florida, 1987 (Pure and Applied Math., Vol. 130). [18] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness, Academic Press, New York-London, 1975. [19] L. A. Sakhnovich, Factorization of operators in L2 (a, b), Funkt. Anal. Prilozh. 13 (1979), 40–45 (in Russian). [20] L. A. Sakhnovich, Factorization of operators, theory and applications, Ukrain. Mat. Zh. 46 (1994), no. 3, 293–304 (in Russian); Engl. transl. in Ukrainian Math. J. 46 (1994), no. 3, 304–317. [21] F. Serier, The inverse spectral problem for radial Schr¨ odinger operators on [0, 1], J. Differential Equations 235 (2007), no. 1, 101–126. [22] F. Serier, Inverse spectral problem for singular Ablowitz–Kaup–Newell–Segur operators on [0, 1], Inverse Problems 22 (2006), no. 4, 1457–1484. [23] G. Teschl, Deforming the point spectra of one-dimensional Dirac operators, Proc. Amer. Math. Soc. 126 (1998), 2873–2881. [24] L. A. Zhornitskaya and V. S. Serov, Inverse eigenvalue problems for a singular Sturm– Liouville operator on [0, 1], Inverse Problems 10 (1994), no. 4, 975–987.

Vol. 64 (2009)

Factorisation of non-negative Fredholm operators

S. Albeverio Institut f¨ ur Angewandte Mathematik Universit¨ at Bonn Wegelerstr. 6 D–53115 Bonn Germany; IZKS and SFB 611, Bonn, Germany; BiBoS, Bielefeld, Germany; CERFIM, Locarno, Switzerland and Accademia di Architettura, Mendrisio, Switzerland e-mail: [email protected] R. Hryniv Institute for Applied Problems of Mechanics and Mathematics 3b Naukova st. 79601 Lviv Ukraine and Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine e-mail: [email protected] Ya. Mykytyuk Lviv National University 1 Universytetska st. 79602 Lviv Ukraine e-mail: [email protected] Submitted: May 28, 2008. Revised: May 8, 2009.

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Integr. equ. oper. theory 64 (2009), 325–355 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030325-31, published online June 26, 2009 DOI 10.1007/s00020-009-1692-z

Integral Equations and Operator Theory

On Positive Linear Volterra-Stieltjes Differential Systems P. H. Anh Ngoc, S. Murakami, T. Naito, J. Son Shin and Y. Nagabuchi Abstract. We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integrodifferential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new. Mathematics Subject Classification (2000). Primary 45J05; Secondary 34K20, 93D09. Keywords. Linear Volterra-Stieltjes system, positive system, Perron-Frobenius theorem, stability and robust stability.

1. Introduction Roughly speaking, a dynamical system is called positive if for any nonnegative initial condition, the corresponding solution of the system is also nonnegative. In particular, a dynamical system with state space Rn is positive if any trajectory of the system starting at an initial state in the positive orthant Rn+ remains forever in Rn+ . Positive dynamical systems play an important role in the modeling of dynamical phenomena whose variables are restricted to be nonnegative, see [3], [22]. They The first author is supported by the Alexander von Humboldt Foundation.

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are often encountered in applications, for example, networks of reservoirs, industrial processes involving chemical reactors, heat exchangers, distillation columns, storage systems, hierarchical systems, compartmental systems used for modeling transport and accumulation phenomena of substances, etc. In the book “Introduction to Dynamic Systems: Theory, Models and Applications” [22], David G. Luenberger wrote that “the theory of positive systems is deep and elegant and yet pleasantly consistent with intuition ... . It is for positive systems, therefore, that dynamic systems theory assumes one of its most potent forms”. The mathematical theory of positive systems is based on the theory of nonnegative matrices founded by Perron and Frobenius. As references we mention [3], [6], [22]. In the last few years, problems of positive systems have attracted a lot of attention from researchers, see e.g. [2], [6], [10]–[11], [15]–[16], [36]–[35], [41]–[43]. In the literature, there are some criteria for familiar positive linear systems such as positive linear time-invariant differential (difference) systems, positive linear time delay systems of retarded type. For example, it is well-known that a linear time-delay differential system of the form x(t) ˙ = A0 x(t) + A1 x(t − h), t ≥ 0, is positive if and only if A0 is a Metzler matrix and A1 is a nonnegative matrix and a linear discrete time system of the form x(k + 1) = A0 x(k) + A1 x(k − h), k ∈ N, is positive if and only if A0 , A1 are nonnegative matrices, see e.g. [28], [29], [43]. Recently, we developed an advanced theory of positive systems for some new classes of linear systems such as: positive linear functional (difference) differential system, see e.g. [32]–[33], [35]; positive linear Volterra integro-differential system [26], [36] and positive linear Volterra integral system [27]. More precisely, we first introduced various notions of positive system for these classes of systems. Then, we offered explicit criteria for them in terms of positivity of system matrices. Furthermore, we gave some extensions of the classical Perron-Frobenius theorems which are important tools for analyzing stability and robust stability of positive systems. Finally, we obtained new criteria for asymptotic stability of the above classes of positive systems. For example, in recent paper [35], we showed that a linear functional differential system of the form Z

0

x(t) ˙ = Ax(t) +

d[η(θ)]x(t + θ),

x(t) ∈ Rn ,

t ≥ 0,

(1)

−h

is positive (it means that its solution semi-group is positive) if and only if A is a Metzler matrix and η(·) is an increasing matrix function on [−h, 0]. Then a positive system of the form (1) is exponentially stable if and only if the spectral abscissa of the matrix A + η(0) is strictly less than zero. Moreover, stability radius problems of positive linear functional differential systems (1) under multi-perturbations and affine perturbations have been studied in [32] where some explicit formulae for these stability radii were given. Some similar results for positive linear Volterra integro-differential systems [26], [36] and for positive linear Volterra integral systems [27] have just been given in only recent time.

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In the present paper, we continue to introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give characterizations of these positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are presented. Finally, we study stability and robust stability of positive linear Volterra-Stieltjes differential systems. The results of this paper include ones established earlier for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [32]–[35] as particular cases. The organization of the paper is as follows. In the next section, we summarize some notations and preliminary results which will be used in what follows. In Section 3, we first introduce the notion of positive linear Volterra-Stieltjes differential systems and then offer an explicit criterion for these positive systems. As direct consequences, we get back the criteria for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [35]. In Section 4, a new Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems is given, which includes those for positive linear Volterra integro-differential systems [36], positive linear functional differential systems [33] and for positive quasi-polynomial matrices [34] as particular cases. In Section 5, we offer an explicit criterion for uniformly asymptotic stability of positive linear Volterra-Stieltjes differential systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive systems under structured perturbations coincide and can be computed by an explicit formula. Some examples are given to illustrate the obtained results. Section 6 gives a brief summary of the obtained results of this paper and the last section are appendices which include proofs of some technical lemmas utilized in the previous sections

2. Preliminaries In this section we shall define some notations and recall some well-known results which will be used in the subsequent sections. Let K = C or R where C and R denote the sets of all complex and all real numbers, respectively. For an integer l, q ≥ 1, Kl denotes the l−dimensional vector space over K, (Kl )∗ is its dual and Kl×q stands for the set of all l × q-matrices with entries in K. Inequalities between real matrices or vectors will be understood componentwise, i.e. for two real matrices A = (aij ) and B = (bij ) in Rl×q , we write A ≥ B if and only if aij ≥ bij for i = 1, · · · , l, j = 1, · · · , q. In particular, if aij > bij for i = 1, · · · , l, j = 1, · · · , q, then we write A B instead of A ≥ B. We denote by Rl×q + the set of all nonnegative matrices A ≥ 0. Similar notations are adopted for vectors. For x ∈ Kn and P ∈ Kl×q we define |x| = (|xi |) and |P | = (|pij |). For any matrix A ∈ Kn×n the spectral abscissa of A is denoted by µ(A) = max{<λ; λ ∈ σ(A)}, where σ(A) := {s ∈ C; det(sIn − A) = 0} is the spectrum of A. A matrix A ∈ Rn×n is called a Metzler matrix if all off-diagonal elements of A are nonnegative.

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A norm k · k on Kn is said to be monotonic if kxk ≤ kyk whenever |x| ≤ |y|, x, y ∈ Kn . Every p-norm on Kn , 1 ≤ p ≤ ∞, is monotonic. Throughout the paper, if otherwise not stated, the norm of a matrix P ∈ Kl×q is understood as its operator norm associated with a given pair of monotonic vector norms on Kl and Kq , that is kP k = max{kP yk; kyk = 1}. We note that the operator norm is in general not monotonic norm on Kl×q even if Kl , Kq are provided with monotonic norms. However, such monotonicity holds for nonnegative matrices. Moreover, we have (see, e.g. [42]) P ∈ Kl×q , Q ∈ Rl×q + , |P | ≤ Q

⇒

kP k ≤ k |P | k ≤ kQk.

The following theorem summarizes some existing results on properties of Metzler matrices which will be used in what follows. Theorem 2.1. [42] Suppose that A ∈ Rn×n is a Metzler matrix. Then: (i) (Perron-Frobenius) µ(A) is an eigenvalue of A and there exists a nonnegative eigenvector x ≥ 0, x 6= 0 such that Ax = µ(A)x. (ii) Given α ∈ R, there exists a nonzero vector x ≥ 0 such that Ax ≥ αx if and only if µ(A) ≥ α. (iii) (tIn − A)−1 exists and is nonnegative if and only if t > µ(A). n×n (iv) Given B ∈ Rn×n . Then + , C ∈C |C| ≤ B

=⇒

µ(A + C) ≤ µ(A + B).

Let Km×n be endowed with the norm k · k and C([α, β], Km×n ) be the Banach space of all continuous functions on [α, β] with values in Km×n normed by the maximum norm kφk = maxθ∈[α,β] kφ(θ)k. For φ ∈ C([α, β], Rm×n ), the notation φ(·) ≥ 0 (or simply φ ≥ 0) means that φ(θ) ≥ 0 for every θ ∈ [α, β]. To make the presentation self-contained we present here some basic facts on vector functions of bounded variation and relative knowledge. Let J be an interval of R. A matrix function η(·) : J → Rl×q is called a increasing matrix function if η(θ2 ) ≥ η(θ1 ) for θ1 , θ2 ∈ J, θ1 < θ2 . A matrix function η(·) : [α, β] → Km×n is said to be of bounded variation if X Var(η; α, β) := sup kη(θk ) − η(θk−1 )k < +∞, P [α,β] k

where the supremum is taken over the set of all finite partitions of the interval [α, β]. The set BV ([α, β], Km×n ) of all matrix functions η(·) of bounded variation on [α, β] satisfying η(α) = 0 is a Banach space endowed with the norm kηk = Var(η; α, β). Let η(·) : R+ → Kl×q be given. For T > 0, we define Vη (T ) := sup

N X

kη(θk ) − η(θk−1 )k,

k=1

where the supremum is taken over all N and over all choices of θk such that 0 ≤ θ1 < θ2 < ... < θN = T. In general, 0 ≤ Vη (T1 ) ≤ Vη (T2 ) ≤ ∞, 0 < T1 < T2 .

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If the limit limT →+∞ Vη (T ) exists and is finite then we say that η(·) is of bounded variation on R+ and Vη := limT →+∞ Vη (T ) is called the total variation of η(·). R +∞ One often writes 0 |dη(s)| instead of Vη . Given η(·) ∈ BV ([α, β], Km×n ) then for any γ ∈ C([α, β], K) and φ ∈ C([α, β], Kn ), Rβ Rβ the integrals α γ(θ)d[η(θ)] and α d[η(θ)]φ(θ) exist and are defined respectively as Pp Pp the limits of S1 (P ) := k=1 γ(ζk )(η(θk ) − η(θk−1 )) and S2 (P ) := k=1 (η(θk ) − η(θk−1 ))φ(ζk ) as d(P ) := maxk |θk − θk−1 | → 0, where P = {θ1 = α ≤ θ2 ≤ · · · ≤ θp = β} is any finite partition of the interval [α, β] and ζk ∈ [θk−1 , θk ]. It is immediate from the definition that

Z β

γ(θ)d[η(θ)]

≤ max |γ(θ)| kηk,

Z

α β

α

θ∈[α,β]

d[η(θ)]φ(θ)

≤ max kφ(θ)k kηk. θ∈[α,β]

Let L : C([α, β], Kn ) → Kn be a linear bounded operator. Then, by the Riesz representation theorem, there exists a unique matrix function η(·) ∈ BV ([α, β], Kn×n ) which is continuous from the right (or briefly c.f.r.) on (α, β) such that Z β Lφ = d[η(θ)]φ(θ), ∀φ ∈ C([α, β], Kn ). α

In particular, an operator L : C([α, β], Rn ) → Rn is called positive if Lφ ≥ 0, for every φ ∈ C([α, β], Rn ), φ ≥ 0. Finally, the following spaces will be used frequently in the subsequent sections N BV ([α, β], Kl×q ) := {η ∈ BV ([α, β], Kl×q ), η(α) = 0, η is c.f.r. on [α, β]}; l×q N BV (R+ , K ) := δ(·) : R+ → Kl×q / δ(·) is c.f.r. on R+ , δ(0) = 0, Z +∞ and kδk := |dδ(s)| < +∞ . 0

3. Positive linear Volterra-Stieltjes differential systems Consider a linear Volterra-Stieltjes differential system of the form Z t x(t) ˙ = Ax(t) + d[B(s)]x(t − s), for a.a. t ∈ R+ ,

(2)

0

where A ∈ Rn×n is a given matrix and B(·) : R+ → Rn×n is a given matrix function of locally bounded variation on R+ . Furthermore, we always assume that B(·) is normalized to be right-continuous on R+ and vanishes at 0. From the theory of integro-differential systems, it is well-known that there exists a unique locally absolutely continuous matrix function R(·) : R+ → Rn×n

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such that ˙ R(t) = AR(t) +

t

Z

d[B(s)]R(t − s),

a.a. t ∈ R+ ,

R(0) = In .

(3)

0

Then R(·) is called the resolvent of (2). Moreover, for given f ∈ L1loc (R+ , Rn ), the following nonhomogeneous system Z t x(t) ˙ = Ax(t) + d[B(s)]x(t − s) + f (t), a.a. t ∈ R+ , (4) 0

has a unique locally absolutely continuous solution x(·) satisfying the initial condition x(0) = x0 ∈ Rn and it is given by the variation of constants formula Z t x(t) = R(t)x0 + R(t − s)f (s)ds, t ∈ R+ , (5) 0

see e.g. [8, p. 81]. Definition 3.1. Let σ ∈ R+ and φ ∈ C([0, σ], Rn ). A vector function x(·) : R+ → Rn is called a solution of (2) through (σ, φ) if x(·) is absolutely continuous on any compact subinterval of [σ, +∞) and satisfies (2) for almost all t ∈ [σ, +∞) and x(t) = φ(t), ∀t ∈ [0, σ]. We denote it by x(· ; σ, φ). Remark 3.2. By the fact mentioned above on the solution of the nonhomogeneous system (4) and the variation of constants formula (5), it is easy to check that for a fixed σ ∈ R+ and a given φ ∈ C([0, σ], Rn ), there exists a unique solution of (2) through (σ, φ) and it is given by Z u+σ Z t x(t+σ; σ, φ) = R(t)φ(σ)+ R(t−u) d[B(s)]φ(u+σ−s) du, t ∈ R+ . (6) 0

In the above, it is understood that

u

R u+σ u

d[B(s)]φ(u + σ − s) = 0 when σ = 0.

Definition 3.3. We say that (2) is positive, if for every σ ≥ 0 and every φ ∈ C([0, σ], Rn ), φ ≥ 0, the corresponding solution x(· ; σ, φ) is also nonnegative, that is x(t; σ, φ) ≥ 0, ∀t ≥ σ. Remark 3.4. Roughly speaking, (2) is positive if for any “input” φ ∈ C([0, σ], Rn ) being nonnegative, the corresponding “output” x(· ; σ, φ), is also nonnegative. Furthermore, we will see below that the notion of positive linear Volterra-Stieltjes differential system is an extension of that of linear functional differential systems. Proposition 3.5. If A ∈ Rn×n is a Metzler matrix and B(·) is an increasing matrix function on R+ then for every x0 ∈ Rn+ , the solution x(· ; 0, x0 ) of (2) is nonnegative. In particular, we have R(·) ≥ 0. Proof. Let x0 ∈ Rn+ , and x(t) := x(t; 0, x0 ), t ≥ 0. Then x(·) satisfies Z u Z t At A(t−u) x(t) = e x0 + e d[B(s)]x(u − s) du, t ≥ 0. 0

0

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Set f (t) := eAt x0 , t ≥ 0. Fix σ > 0 and consider the operator T defined by C([0, σ], Rn ) −→ C([0, σ], Rn ) Z u Z t d[B(s)]φ(u − s) du, eA(t−u) φ 7→ T φ(t) := f (t) + T :

t ∈ [0, σ].

0

0

By induction, it is easy to show that for φ1 , φ2 ∈ C([0, σ], Rn ) and k ∈ N, we have kT k φ2 (t) − T k φ1 (t)k ≤

M k tk kφ2 − φ1 k, ∀t ∈ [0, σ], k!

where M := M1 M2 and M1 := maxs∈[0,σ] keAs k, M2 := Var(B; 0, σ). This implies that T k is a contraction for k ∈ N sufficiently large. Fix k0 ∈ N sufficiently large, by the contraction mapping principal, there exists a unique solution of the equation x = T x in C([0, σ], Rn ). Moreover, it is well-known that the sequence (T mk0 φ0 )m∈N , with an arbitrary φ0 ∈ C([0, σ], Rn ) converges to this solution in the space C([0, σ], Rn ). Choose φ0 ∈ C([0, σ], Rn ), φ0 ≥ 0. Since A ∈ Rn×n is a At Metzler matrix, it follows R u that e ≥ 0, ∀t ≥ 0. Moreover, since B(·) is increasing, it is easy to see that 0 d[B(s)]φ(u − s) ≥ 0 for every φ ∈ C([0, u], Rn ), φ ≥ 0. Thus, T mk0 φ0 ≥ 0, ∀m ∈ N and we deduce that x(t) := x(t; 0, x0 ) ≥ 0, ∀t ∈ [0, σ]. Since σ > 0 is arbitrary, we conclude that x(t) ≥ 0, ∀t ≥ 0. To prove a criterion for positive linear Volterra-Stieltjes differential systems, we need the two following auxiliary lemmas whose proofs are given in Appendices A and B, respectively. Lemma 3.6. Assume that B(·) ∈ N BV (R+ , Rn×n ), that is Z +∞ |dB(t)| < +∞.

(7)

0

Then the resolvent R(·) of (2) is of exponential order. Lemma 3.7. Let σ > 0 and suppose the linear operator L is defined by Z σ n n L : C([0, σ], R ) → R , φ 7→ Lφ = d[η(θ)]φ(θ), 0 n×n

where η ∈ N BV ([0, σ], R ). Then L is a positive operator if and only if η is an increasing matrix function. Let h : [0, +∞) → R. Then the Laplace transform of h is formally defined to be Z +∞ ˆ h(z) := e−zt h(t)dt. 0

R +∞

ˆ If β ∈ R and 0 e |h(t)|dt < +∞, then h(z) exists for z ∈ C, β}. If ˆ := (dˆij ). D(t) = (dij (t)) is a matrix function, then we define D −βt

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In the remainder of this section, we assume that (7) holds true. Then, the Laplace-Stieltjes transform of B(·) is defined by Z +∞ ˜ B(z) := e−zs dB(s). 0

which is well-defined for every z ∈ C,
0

R u+σ Since B(·) is increasing and φ ≥ 0, it follows that u d[B(s)]φ(u + σ − s) ≥ 0, ∀u ≥ 0. By this fact and R(·) ≥ 0, (8) implies that x(t + σ) ≥ 0, ∀t ≥ 0. (The “only if” part) Assume that (2) is positive. Let σ > 0 and let φ ∈ C([0, σ], Rn ) satisfy φ(σ) = 0. Set x(t) := x(t; σ, φ), t ≥ σ. Suppose the sequence (tk )k , tk ∈ [σ, σ + 1/k] satisfies Z tk x(t ˙ k ) = Ax(tk ) + d[B(s)]x(tk − s). (9) 0

Then, we have

Z tk

Z σ

d[B(s)]x(t − s) − d[B(s)]φ(σ − s) k

0 0

Z tk

Z σ

+ d[B(s)]x(t − s) d[B(s)](x(t − s) − x(σ − s)) ≤ k k

0

σ

≤ Var(B; 0, σ) sup kx(tk − s) − x(σ − s)k + Var(B; σ, tk )L, s∈[0,σ]

for some positive number L. As x(·) is uniformly continuous on the interval [0, σ+1] and B(·) is continuous from the right at σ, it is easy to see that Z tk Z σ d[B(s)]x(tk − s) → d[B(s)]φ(σ − s) as k → +∞. 0

0

It follows that

Z k→+∞

σ

d[B(s)]φ(σ − s).

lim x(t ˙ k) =

(10)

0

Let B(·) = (bij (·)). Using this fact, we prove that bij (·) is increasing on R+ for every i, j ∈ {1, 2, ..., n}. To do so, let ψ ∈ C([0, σ], R), ψ ≥ 0, ψ(σ) = 0. Fix i0 ∈ {1, 2, ..., n} and let us define φ := (φ1 , ..., φn )T ∈ C([0, σ], Rn ), where φi (·) = 0 if i 6= i0 otherwise φi = ψ. Set x(t) := x(t; σ, φ) = (x1 (t), x2 (t), ..., xn (t))T ≥ 0, ∀t ≥ σ. Fix k ∈ N. Since x1 (·) is absolutely continuous on [σ, σ + 1/k] and

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x1 (·) ≥ 0, x1 (σ) = 0, there exists tk ∈ [σ, σ + 1/k] such that x(·) satisfies (2) at tk and x˙ 1 (tk ) ≥ 0. In particular, R σ(9) holds true. Therefore, we get (10). In particular, we have limk→+∞ x˙ 1 (tk ) = 0 ψ(σ − s)d[b1i0 (s)] ≥ 0. Thus, the linear functional defined by Z σ L : C([0, σ], R) → R, ψ 7→ Lψ := ψ(σ − s)d[b1i0 (s)], 0

is positive. Taking into account Lemma 3.7 and Remark 7.1, we conclude that b1i0 (·) is increasing on [0, σ). Since σ > 0 is arbitrary, b1i0 (·) is increasing on R+ . By a similar way, we can show that bij (·) is increasing on R+ for any i, j ∈ {1, 2, ..., n}. We now show that A is a Metzler matrix. By Lemma 3.6, R(·) is of exponential order. Taking the Laplace transforms on both sides of (3), we get [sIn − A − ˜ ˆ B(s)] R(s) = R(0) = In , for s ∈ R sufficiently large. By Proposition 3.5, R(t) ≥ −1 ˆ ˜ 0, ∀t ≥ 0. It follows that R(s) = sIn − A − B(s) ≥ 0, for s ∈ R sufficiently large. Let A = (aij ) and assume on the contrary that ai0 j0 < 0 for some i0 6= j0 . ˜ It follows from the assumption (7) that B(s) → 0, as s → +∞. Therefore, we can represent −1 −1 ˜ ˜ sIn − A − B(s) = s−1 In − s−1 A + B(s) +∞ X k ˜ ˜ = s−1 In + s−2 A + B(s) + s−(k+1) A + B(s) , k=2

for s > 0 sufficiently large. We thus get, +∞ k X ˜ ˜ s−(k−1) A + B(s) sIn + A + B(s) + ≥0

(11)

k=2

k P+∞ ˜ for s > 0 sufficiently large. Note that lims→+∞ k=2 s−(k−1) A + B(s) = 0. Then, it follows from (11) that the entry bi0 j0 of the matrix on the left-hand side of (11) is negative for s > 0 sufficiently large. This is a contradiction. Hence, A must be a Metzler matrix. Corollary 3.9. Let A ∈ Rn×n be a given matrix and let C(·) : R+ → Rn×n be a given continuous matrix function. Assume that Z +∞ kC(s)kds < +∞. (12) 0

Then, a linear Volterra integro-differential system of convolution type Z t x(t) ˙ = Ax(t) + C(t − s)x(s)ds, t ∈ R+ ,

(13)

0

is positive if and only if A ∈ Rn×n is a Metzler matrix and C(t) ∈ Rn×n + , ∀t ∈ R+ . Proof. First, we note that (13) can be rewritten in the form (2) with B(t) = Rt C(s)ds, t ≥ 0. Furthermore, (12) implies that (7) holds true. The conclusion of 0 the corollary now follows directly from Theorem 3.8.

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Next, we consider a linear functional differential system of the form Z h d[η(s)]x(t − s), t ≥ h, x(t) ˙ = Ax(t) +

IEOT

(14)

0

where A ∈ Rn×n and η(·) ∈ N BV ([0, h], Rn×n ) are given. It is well-known that for an initial function φ ∈ C([0, h], Rn ), the linear functional differential system (14) has a unique solution x(·; φ) satisfying the initial condition x(t) = φ(t), t ∈ [0, h]. (15) Definition 3.10. The system (14) is called positive if for every nonnegative initial function φ ∈ C([0, h], Rn ), the unique solution of (14)–(15) x(· ; φ), is also nonnegative. Corollary 3.11. The system (14) is positive if and only if A ∈ Rn×n is a Metzler matrix and η(·) is an increasing matrix function on [0, h]. Proof. Let A ∈ Rn×n be a Metzler matrix and η(·) be an increasing matrix function on [0, h]. We now consider a linear Volterra-Stieltjes differential system of the form (2) where the matrix function B(·) is defined by ( η(s) if s ∈ [0, h), B(s) := η(h) if s ∈ [h, +∞). It is easy to see that (2) now coincides with (14) on the interval [h, +∞). Since B(·) is increasing on R+ and A ∈ Rn×n is a Metzler matrix, it follows that (2) is positive. Therefore, the solution x(· ; φ) is nonnegative whenever φ(·) is nonnegative. Conversely, if (14) is positive then by a similar argument as in the last part of the proof of Theorem 3.8, we can show that A ∈ Rn×n is a Metzler matrix and η(·) is an increasing matrix function on [0, h]. Remark 3.12. Corollary 3.9 and Corollary 3.11 are the main results of [36] and of [35], respectively, which have been given by ourselves in very recent time.

4. Perron-Frobenius type theorems for positive linear systems It is well-known that Perron-Frobenius type theorems are principal tools for analysis of stability and robust stability of positive linear time-invariant systems. To our knowledge, there is a large number of extensions of the classical Perron-Frobenius theorems, see e.g. [1], [7], [18], [30], [33], [34], [40] and the references therein. Recall that a linear time-invariant differential system of the form x(t) ˙ = Ax(t), t ≥ 0 is positive if and only if the system matrix A ∈ Rn×n is a Metzler matrix. Therefore, the classical Perron-Frobenius theorem (Theorem 2.1) can be seen as the Perron-Frobenius theorem for the class of these positive systems. From this dynamic point of view, we recently presented a series of extensions of the classical Perron-Frobenius theorem such as: Perron-Frobenius theorem for positive linear higher order difference systems [30], Perron-Frobenius theorem for positive linear

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time-delay systems [34], Perron-Frobenius theorem for positive linear functional differential systems [33], Perron-Frobenius theorem for positive linear Volterra systems [27], [36]. In this section, we first give a new Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems of the form (2). Then, as direct consequences, we get back some variations of the classical Perron-Frobenius theorem which have just been mentioned in the above. Let us define a formal formula Z +∞ ˜ ∆(z) := zIn − A − B(z) = zIn − A − e−zt dB(t). (16) 0

Set +∞

Z µ A, B(·) := sup
−
e

|dB(t)| < +∞, det ∆(z) = 0 ,

0

where by convention, µ A, B(·) := −∞ if

Z z∈C:

+∞

e−
= ∅.

0

Then, µ(A, B(·)) is called spectral abscissa of the Volterra system (2). It is important to note that if B(t) = 0, ∀t ≥ 0 then µ(A, B(·)) coincides with the spectral abscissa µ(A) of the matrix A. Theorem 4.1. Suppose A ∈ Rn×n is a Metzler matrix and B(·) is an increasing R +∞ matrix function on R+ . Let β := inf γ ∈ R : 0 e−γt |dB(t)| < +∞ and assume that β < +∞, α ∈ (β, +∞). If µ∗ := µ A, B(·) > −∞ then (i) (Perron-Frobenius theorem for positive linear Volterra-Stieltjes differential systems) µ∗ is a root of the characteristic equation, that is det ∆(µ∗ ) = 0. Moreover, there exists a vector x ≥ 0, x 6= 0 such that Z +∞ −µ∗ t A+ e dB(t) x = µ∗ x, 0

R +∞ (ii) there exists a nonzero vector x ≥ 0 such that A + 0 e−αt dB(t) x ≥ αx if and only if µ∗ ≥ α, −1 R +∞ (iii) the matrix ∆(α)−1 = αIn −A− 0 e−αt dB(t) exists and is nonnegative if and only if α > µ∗ . Proof. First, we show that µ∗ < +∞. To do this, we first assume that β > −∞ and det ∆(s) = 0 for some s ∈ C, <s ≥ β + 1. This implies that <s ≤ µ(A + R +∞ −st e dB(t)). Since B(·) is increasing on R+ , it follows that 0 Z +∞ Z +∞ Z +∞ −st −<st e dB(t) ≤ e dB(t) ≤ e−(β+1)t dB(t). 0

0

0

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Since A ∈ Rn×n is a Metzler matrix, by Theorem 2.1 (iv), Z +∞ Z +∞ −st −(β+1)t µ A+ e dB(t) ≤ µ A + e dB(t) . 0

0

Note that if det ∆(s) = 0 for some s ∈ C then Z +∞ e−(β+1)t dB(t) . <s ≤ max β + 1, µ A + 0

R +∞

−(β+1)t

Thus, µ∗ ≤ max{β + 1, µ(A + 0 e dB(t))}. In case of β = −∞, the proof is similar to the above. Next, we prove that Z +∞ e−µ∗ t dB(t) . (17) µ∗ ≤ µ A + 0

In fact, by the assumption µ∗ > −∞, there exists s0 ∈ C such that det ∆(s0 ) = 0, β ≤ <s0 ≤ µ∗ . If <s0 = µ∗ then using Theorem 2.1 (iv) again, we get (17). If β ≤ <s0 < µ∗ then there exists a sequence (sk )k such that det ∆(sk ) = 0, β < <sk < µ∗ , ∀k ∈ N and <sk → µ∗ as k → +∞. Then by Theorem 2.1 (iv), we get Z +∞ <sk ≤ µ A + e−<sk t dB(t) . (18) 0

Letting k → +∞ in (18), we also get (17). We now consider the continuous real function given by Z +∞ f (θ) := θ − µ A + e−θt dB(t) , 0

R +∞

−βt

where θ ∈ [β, +∞) if 0 e |dB(t)| < +∞, otherwise θ ∈ (β, +∞). By (17), f (µ∗ ) ≤ 0. Assume that f (µ∗ ) < 0. Since, clearly, limθ→+∞ f (θ) = +∞, we derive R +∞ that f (θ0 ) = 0, for some θ0 > µ∗ . This gives θ0 = µ A + 0 e−θ0 t dB(t) . By R +∞ −θ t Theorem 2.1(i), it implies that det(θ0 In − A − 0 e 0 dB(t)) = 0. However, this conflicts with the definition of µ∗ . Thus f (µ∗ ) = 0, or equivalently, µ∗ = R +∞ µ A0 + 0 e−µ∗ t dB(t) . Then, (i) now follows from Theorem 2.1(i). Moreover, Z +∞ Z +∞ µ A+ e−θ2 t dB(t) ≤ µ A + e−θ1 t dB(t) , β < θ1 ≤ θ2 , 0

0

by Theorem 2.1(iv). Therefore, f is strictly increasing on (β, +∞). Moreover, f (µ∗ ) = 0. Now, it is easy to see that (ii), (iii) follows from Theorem 2.1(ii), Theorem 2.1(iii), respectively. The following theorems are seen as Perron-Frobenius theorems for positive linear Volterra integro-differential systems and for positive linear functional differential systems, respectively. The proofs of them are straightforward from Theorem 4.1.

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Theorem 4.2. Suppose A ∈ Rn×n is a Metzler matrix and C(·) : R+ → Rn×n is a + continuous matrix function. Let Z +∞ β := inf γ ∈ R : e−γt kC(t)kdt < +∞ 0

and Z µ0 := sup −∞ then

+∞ −γt

e

kC(t)kdt < +∞, det ∆0 (z) = 0 ,

0 R +∞ −zt e C(t)dt. 0

Assume that β < +∞, α ∈ (β, +∞).

(i) (Perron-Frobenius theorem for positive linear Volterra integro-differential systems) µ0 is a root of the characteristic equation, that is det ∆0 (µ0 ) = 0. Moreover, there exists a vector x ≥ 0, x 6= 0 such that Z +∞ e−µ0 t C(t)dt x = µ0 x, A+ 0

R +∞ (ii) there exists a nonzero vector x ≥ 0 such that A + 0 e−αt C(t)dt x ≥ αx if and only if µ0 ≥ α, −1 R +∞ (iii) the matrix ∆0 (α)−1 = αIn − A − 0 e−αt C(t)dt exists and is nonnegative if and only if α > µ0 . Theorem 4.3. Let the linear functional differential equation (14) be positive. Set Rh µ1 := sup{
(ii) Given α ∈ R, there exists a nonzero vector x ≥ 0 such that ∆1 (α)x ≤ 0 if and only if µ1 ≥ α. (iii) ∆1 (t)−1 exists and is nonnegative if and only if t > µ1 . We now consider a quasi-polynomial matrix of the form Q(z) := zIn − A0 − e−h1 z A1 − ... − e−hm z Am ,

z ∈ C,

(19)

where Ai ∈ Rn×n , i ∈ {0, 1, 2, ..., m} are given matrices and 0 < h1 < h1 < ... < hm are given positive numbers. Let us denote µ Q(·) := sup
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a linear time delay system of the form x(t) ˙ = A0 x(t) + A1 x(t − h1 ) + ... + Am x(t − hm ),

t ≥ 0.

(20)

Then the quasi-polynomial matrix (19) is said to be positive if (20) is positive. That is, for any nonnegative initial function φ ∈ C([−hm , 0], Rn ), the unique solution of (20) satisfying the initial condition x(t) = φ(t), t ∈ [−hm , 0, ] is also nonnegative. Furthermore, it is shown that the quasi-polynomial matrix (19) is positive if and only if A0 ∈ Rn×n is a Metzler matrix and Ai ∈ Rn×n , i ∈ {1, 2, ..., m} are nonnegative matrices, see e.g. [35]. The following result is seen as a Perron-Frobenius theorem for positive quasipolynomial matrices of the form (19). Theorem 4.4. Let the quasi-polynomial matrix (19) be positive and α ∈ R be given. Then (i) (Perron-Frobenius theorem for positive quasi-polynomial matrices) µq := µ Q(·) is a root of the characteristic equation, that is det Q(µq ) = 0. Moreover, there exists a vector xq ≥ 0, xq 6= 0 such that Q(µq )xq = 0, (ii) there exists a nonzero vector x ≥ 0 such that Q(α)x ≤ 0 if and only if µq ≥ α. (iii) the matrix Q(α)−1 exists and is nonnegative if and only if α > µq , Proof. Let us consider the step function B(·) : R+   0 if      if A1 B(t) := A1 + A2 if   . . . if    A + A + · · · + A if 1 2 m

→ Rn×n defined by t ∈ [0, h1 ), t ∈ [h1 , h2 ), t ∈ [h2 , h3 ), ..., t ∈ [hm , +∞).

Then, the matrix function ∆(·) defined by (16) now becomes the polynomial matrix Q(·). Finally, Theorem 4.4 directly follows from Theorem 4.1. This completes the proof. Remark 4.5. Theorems 4.2, 4.3, 4.4 have been found recently by ourselves, see [36], [33] and [34].

5. Stability and robust stability of positive linear Volterra-Stieltjes differential systems 5.1. An explicit criterion for uniformly asymptotic stability of positive linear Volterra-Stieltjes differential systems In this subsection, we offer a new and novel criterion for uniformly asymptotic stability of positive linear Volterra-Stieltjes differential systems of the form (2). First, we give some definitions on stability of Volterra-Stieltjes differential systems.

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Definition 5.1. The zero solution of (2) is said to be uniformly stable (US) if for each > 0, there exists δ() > 0 such that φ ∈ C([0, σ], Rn ), kφk < δ ⇒ kx(t, σ, φ)k < ,

∀t ≥ σ.

Definition 5.2. The zero solution of (2) is said to be uniformly asymptotically stable (UAS) if it is US and if there exists δ0 > 0 such that ∀ > 0, ∃T () > 0 : φ ∈ C([0, σ], Rn ), kφk < δ0 ⇒ kx(t, σ, φ)k < ,

∀t ≥ σ + T ().

If the zero solution of (2) is US (UAS) then we say that (2) is US (UAS), respectively. The following theorem is a generalization of the Grossman and Miller’s result [9] to Volterra-Stieltjes differential systems of the form (2). Theorem 5.3. Let (7) hold true. Then the following statements are equivalent R +∞ (i) det ∆(z) 6= 0, ∀z ∈ C,
v

Hence we get Z v+σ kx(t + σ; σ, φ)k ≤ kR(t)kkφk + kφk kR(t − v)k |dB(s)| dv 0 v Z +∞ ≤N 1+ |dB(s)| kφk, t ≥ 0, Z

t

0

which shows that (2) is US. Likewise, we get Z Z t kx(t + σ; σ, φ)k ≤ kR(t)kkφk + kφk kR(t − v)k 0

Rt

+∞

|dB(s)| dv.

v

R +∞ Observe that R(·) ∈ L1 (R+ , Rn×n ) and 0 kR(t − v)k( v |dB(s)|)dv → 0 as R +∞ t → +∞ because of v |dB(s)| → 0 as v → +∞. This yields that (2) is UAS. Next we will prove that (iii) implies (i). Assume that (2) is UAS. Then, there exists δ0 > 0 such that for any > 0, ∃ T () > 0 : kx(t, 0, x0 )k = kR(t)x0 k < , ∀t > T (), whenever kx0 k < δ0 . It follows that limt→+∞ kR(t)k = 0. Given > 0 with < 1/2. Since (2) is UAS and limt→+∞ kR(t)k = 0, there exist constants K and T1 () > 0 such that kx(t + σ, σ, φ) − R(t)φ(σ)k ≤ K if t ≥ 0,

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and kx(t + σ, σ, φ) − R(t)φ(σ)k ≤ if t ≥ T1 (), whenever φ ∈ C([0, σ], Rn ) with kφk ≤ 1. Thus,

Z t ( Z τ +σ K, ∀ t ≥ 0,

d[B(s)]φ(τ + σ − s) dτ R(t − τ )

≤

, ∀ t ≥ T1 (), τ 0 or σ

Z t Z

R(t − τ )

0

0

( K, ∀ t ≥ 0,

du [B(u + τ )]φ(σ − u) dτ

≤ , ∀ t ≥ T1 (),

whenever φ ∈ C([0, σ], Rn ) with kφk ≤ 1. Notice that the function u ∈ [0, σ] 7→ Rt R(t − τ )B(u + τ )dτ is right continuous, and moreover it is of bounded variation 0 R +∞ Rt kR(s)kds 0 |dB(τ )|. Applying an whose total variation does not exceed 0 un-symmetric Fubini’s theorem (see [4] or [12, Th. 2.2.11]), we get Z σ Z t Z σ Z t R(t−τ ) du [B(u+τ )]φ(σ−u) dτ = du R(t−τ )B(u+τ )dτ φ(σ−u) 0

0

and hence

Z

0

σ

t

Z du

0

0

0

( K, ∀ t ≥ 0

≤ R(t − τ )B(u + τ )dτ φ(σ − u)

, ∀ t ≥ T1 (),

whenever φ ∈ C([0, σ], Rn ) with kφk ≤ 1. Therefore it follows that ( Z σ Z t K, ∀ t ≥ 0, du ≤ R(t − τ )B(u + τ )dτ , ∀ t ≥ T1 (), 0 0 for any σ > 0. Now assume that (i) is not true. Then there is a z0 ∈ C with
t

and hence we have Z

t

y(t) = R(t)x0 + 0

Z R(t − s)

+∞ z0 (s−τ )

e

d[B(τ )]x0 ds.

s

Note that

Z

s

s+u z0 (s−τ )

e

Z

d[B(τ )]x0

≤

0

+∞

|dB(τ )| < +∞

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for any u ≥ 0. Applying Lebesgue’s convergence theorem, we get Z +∞ Z t ez0 (s−τ ) d[B(τ )]x0 ds R(t − s) s 0 Z s+σ Z t ez0 (s−τ ) d[B(τ )]x0 ds, R(t − s) = lim σ→+∞

s

0

in turn, by an un-symmetric Fubini’s theorem the last term in the above becomes Z t Z σ −z0 u e du R(t − s)B(u + s)ds x0 . lim σ→+∞

0

0

Therefore, if t ≥ T1 () then kR(t)k < and

Z t Z σ

−z0 u e R(t − s)B(u + s)ds x0 R(t)x + lim d 1 ≤ ky(t)k = 0 u

σ→+∞ 0 0 Z σ Z t d u R(t − s)B(u + s)ds < 2 < 1, ≤ kR(t)k + lim sup σ→+∞

0

0

which is a contradiction. Thus (i) must hold true.

The following theorem offers an explicit criterion for uniformly asymptotic stability of positive linear Volterra-Stieltjes differential systems. Theorem 5.4. Suppose that (7) holds true and (2) is positive. Then, (2) is UAS if R +∞ and only if µ(A + 0 dB(t)) < 0. R +∞ Proof. Assume that µ(A + 0 dB(t)) < 0. For an arbitrary z ∈ C,
0

By Theorem 2.1 (iv), Z µ A+

+∞ −zt

e

0

Z dB(t) ≤ µ A +

0

+∞

dB(t) < 0,

0

R +∞

−zt

which implies that z 6∈ σ(A + 0 e dB(t)). That is, det ∆(z) = 6 0. By Theorem 5.3, (2) is UAS. Conversely, suppose (2) is UAS. Then, det ∆(z) 6= 0 ∀z ∈ C,
Clearly, f is continuous and limθ→+∞ f (θ) = +∞. We show that f (0) > 0. Seeking a contradiction, assume that f (0) ≤ 0. Then there is λ1 ≥ 0 such R +∞ that f (λ1 ) = 0. That is, λ1 = µ(A + 0 e−λ1 t dB(t)). Consequently, we have R +∞ −λ t λ1 ∈ σ(A + 0 e 1 dB(t)), by Theorem 2.1 (i). Thus, det ∆(λ1 ) = 0. Hence, R +∞ µ(A + 0 dB(t)) = −f (0) < 0, as required.

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The following is immediate from Theorem 5.4. Corollary 5.5. Let A ∈ Rn×n be a Metzler matrix and let C(·) : R+ → Rn×n be a + R +∞ continuous matrix function. Assume that 0 kC(s)kds < +∞. Then, a positive linear Volterra integro-differential system of the form (13) is UAS if and only if R +∞ µ(A + 0 C(t)dt) < 0. Before stating the next corollary, we recall the notion of exponential stability of the linear functional differential systems (14). Definition 5.6. The linear functional differential system (14) is said to be exponentially stable if there exists M ≥ 1 and α > 0 such that for any initial function φ ∈ C([0, h], Rn ) the solution of (14)–(15) x(· ; φ), satisfies kx(t; φ)k ≤ M e−αt kφk,

∀t ≥ 0.

Furthermore, it is well-known that (14) is exponentially stable if and only if its characteristic equation has no zeros in the closed right half complex plane, that is Z h −zt det zIn − A − e dη(t) 6= 0, ∀z ∈ C,
see e.g. [5]. Corollary 5.7. Let the linear functional differential system (14) be positive. Then it is exponentially stable if and only if µ(A + η(h)) < 0. Proof. The proof is immediate from Theorem 5.4 and omitted here.

We illustrate the obtained results by two examples. Example 5.8. Consider a positive linear Volterra integro-differential system in R2 given by Z t x(t) ˙ = Ax(t) + B(t − τ )x(τ )dτ, x(t) ∈ R2 , t ≥ 0, (21) 0

where A=

a 1 0 a

;

B(t) =

e−pt

0

1 (t+1)2

e−pt

,

t ≥ 0,

(22)

and a, p ∈ R, p > 0 are parameters. Since p > 0, the condition (12) is satisfied. By Corollary 5.5, (21) is UAS if and only if Z +∞ a + 1/p 1 µ A+ B(t)dt = µ = a + 1 + 1/p < 0. 1 a + 1/p 0 Taking into account that p > 0, it is easy to see that this is equivalent to a < −1, p > −1/(a + 1).

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Example 5.9. Consider the scalar linear functional differential equation given by Z x(t) ˙ = −x(t) +

1

e−s x(t − s)ds t ≥ 0, x(t) ∈ R.

(23)

0

The equation (23) can be rewritten in the form (14) with η(s) = 1 − e−s , s ∈ [0, 1]. Clearly, η(·) is increasing on [0, 1] and η(0) = 0. Moreover, −1 + η(1) = −e−1 . By Corollary 5.7, (23) is exponentially stable. 5.2. Stability radius of positive linear Volterra-Stieltjes differential systems Motivated by many applications in control engineering, problems of robust stability of dynamical systems have attracted a lot of attention from researchers during the last twenty years. In the study of these problems, the notion of stability radius was proved to be an effective tool. By definition, the stability radius of an asymptotically stable linear differential system x(t) ˙ = Ax(t), t ≥ 0 is the maximal r > 0 for which all the systems of the form x(t) ˙ = (A + D∆E)x(t), k∆k < r, are asymptotically stable. Here, ∆ is unknown disturbance matrix, D and E are given matrices defining the structure of the perturbations. Depending upon whether complex or real disturbances ∆ are considered this maximal r is called complex or real stability radius respectively. The basic problem in the study of robustness of stability of the system is to characterize and compute these radii in terms of given matrices A, D, E. It is important to note that these two stability radii are in general distinct. The analysis and computation of the complex stability radius for systems under structured perturbations has been done first in [13] and extended later in many subsequent papers (see [14] for a survey up till 1990) while the computation of the real stability radius, being a much more difficult problem, has been solved quite recently with a complicated solution, see e.g. [38]. The situation is much simpler for the class of positive systems. It has been shown in [15], [41] that if A is a Metzler matrix (i.e. x(t) ˙ = Ax(t), t ≥ 0 is a positive system) and D, E are nonnegative matrices, then the complex and the real stability radii coincide and can be computed directly by a simple formula. These results have been extended recently to many various classes of positive systems such as positive linear time-delay differential systems, see e.g. [29], [31], [43], positive linear discrete time-delay systems, see e.g. [16], [28] and positive linear functional differential systems, see e.g. [32], [44]. Although there have been many works dedicated to studying the stability radius problems of linear dynamical systems, however, the problem of computing of stability radii of linear Volterra-Stieltjes differential systems has not been studied yet in the literature. In this section, we deal with the problem of computing stability radii of positive linear Volterra-Stieltjes differential system (2) under structured perturbations.

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We now assume that (7) holds true and (2) is UAS. Consider perturbed systems of the following form Z t d[B + D1 δ(s)E]x(t − s), (24) x(t) ˙ = (A + D0 ∆E)x(t) + 0 n×li

where Di ∈ R (i ∈ I := {0, 1}), E ∈ Rq×n are given matrices determining the structure of perturbations and ∆ ∈ Kl0 ×q , δ(·) ∈ N BV (R+ , Kl1 ×q ), (K = R, C), are unknown disturbances. We shall measure the size of each perturbation (∆, δ(·)) by the norm Z +∞ k(∆, δ(·))k := k∆k + |dδ(s)|. 0

The main problem here is to find the maximal r > 0 for which the perturbed systems (24) remain UAS whenever k(∆, δ(·))k < r. In this case, we say that A, B(·) are subjected to structured perturbations of the form A

A + D0 ∆E;

B(·)

B(·) + D1 δ(·)E.

(25)

Let σ(A + D0 ∆E, B(·) + D1 δ(·)E) be the set of all roots of the characteristic equation of a perturbed system of the form (24). That is, σ(A + D0 ∆E, B(·) + D1 δ(·)E) Z := z ∈ C : det zIn − (A + D0 ∆E) −

+∞

e−zs d[B(s) + D1 δ(s)E] = 0 .

0

Recall that, by Theorem 5.3, a perturbed system of the form (24) is UAS if and only if σ(A + D0 ∆E, B(·) + D1 δ(·)E) ⊂ C− := {z ∈ C :
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From the above definition, it is easy to see that 0 < rC ≤ rR ≤ r+ ≤ +∞.

(26)

To get characterizations of the stability radii for the class of positive systems, we need two following technical lemmas whose proofs are given in Appendices C and D, respectively. q×n Lemma 5.11. Suppose (2) is positive, UAS and D ∈ Rn×l + , E ∈ R+ . Then, −1 −1 Z +∞ Z +∞ −zs zIn − A − |x|, ∀x ∈ Cn , dB(s) x ≤ − A − e dB(s) 0

0

for every z ∈ C,
−1 Z +∞ Z

−zs

E − A − = e dB(s) D max E zI − A − n

z∈C,
0

0

+∞

−1

dB(s) D

.

Lemma 5.12. Suppose (2) is positive, UAS and Di ∈ ∈ I, E ∈ Rq×n + . If R +∞ −1 maxi∈I kE(−A − 0 dB(s)) Di k 6= 0, then there exists a nonnegative perturbation (∆0 , δ1 (·)) ∈ D+ such that 1 k(∆0 , δ1 (·))k = (27) R +∞ maxi∈I kE(−A − 0 dB(s))−1 Di k i Rn×l ,i +

and 0 ∈ σ A + D0 ∆0 E, B(·) + D1 δ1 (·)E .

(28)

The following theorem shows that for the class of positive Volterra-Stieltjes differential systems, the complex, real and positive stability radius coincide and they can be computed by an explicit formula. n×li Theorem 5.13. Suppose (2) is positive, UAS and E ∈ Rq×n , i ∈ I. + , D i ∈ R+ Then 1 rC = rR = r+ = . R +∞ maxi∈I kE(−A − 0 dB(s))−1 Di k

Proof. Suppose that rC < +∞, as otherwise, there is nothing to show. Let (∆, δ(·)) ∈ DC be a destabilizing complex disturbance. By Theorem 5.3, σ(A + D0 ∆E, B(·) + D1 δ(·)E) 6⊂ C− . Then, there exist z ∈ C,
Since (2) is UAS, it follows that −1 Z +∞ Z zIn − A − e−zs dB(s) D0 ∆E + D1 0

+∞

e−zs d[D1 δ(s) E] x = x.

0

Therefore, Ex 6= 0 and we thus get −1 Z +∞ Z −zs E zIn − A − e dB(s) D0 ∆E + D1 0

0

+∞ −zs

e

dδ(s) E x = Ex.

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Taking norms, we deduce that

−1 Z +∞

−zs

E zIn − A − D0 e dB(s)

k∆kkExk

0

Z

+ E zIn − A −

+∞ −zs

e

−1

Z

D1 dB(s)

+∞ −zs

e

0

0

dδ(s)

kExk ≥ kExk.

Hence, Z

max E zIn − A − i∈I

+∞ −zs

e

−1 Z

dB(s) Di k∆k +

+∞

|dδ(t)| ≥ 1.

0

0

Using Lemma 5.11, we get Z

max E − A − i∈I

+∞

0

−1

Di dB(s)

k(∆, δ(·))k ≥ 1.

This is equivalent to k(∆, δ(·))k ≥ maxi∈I

1 −1 . R +∞ kE − A − 0 dB(s) Di k

Since this inequality holds true for an arbitrary destabilizing complex perturbation, we conclude that rC ≥

maxi∈I

1 −1 . R +∞ kE − A − 0 dB(s) Di k

Taking into account the inequalities (26), it remains to show that r+ ≤

maxi∈I

1 −1 . R +∞ kE − A − 0 dB(s) Di k

However, this is immediate from Lemma 5.12.

The following is immediate from Theorem 5.13. Corollary 5.14. Suppose that (12) holds and the linear Volterra integro-differential n×li system (13) is positive and UAS and E ∈ Rq×n , (i ∈ I = {0, 1}). Then + , Di ∈ R+ a perturbed system of the form Z t x(t) ˙ = A + D0 ∆E x(t) + C(t − s) + D1 δ(t − s)E x(s)ds, t ∈ R+ , 0

where ∆ ∈ R

l0 ×q

Z k∆k +

, δ(·) ∈ L ([0, +∞), Rl1 ×q ) ∩ C([0, +∞), Rl1 ×q ), is still UAS if 1

+∞

kδ(s)kds < 0

maxi∈I kE − A −

1 R +∞ 0

C(s)ds

−1

. Di k

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Corollary 5.15. Suppose the linear functional differential system (14) is positive n×li , (i ∈ I = {0, 1}). Then a and exponentially stable. Let E ∈ Rq×n + , Di ∈ R+ perturbed system of the form Z h x(t) ˙ = A + D0 ∆E x(t) + d[η(s) + D1 δ(s)E]x(t − s)ds, t ≥ h, 0

where ∆ ∈ R

l0 ×q

, δ(·) ∈ N BV ([0, h], Rl1 ×q ), is still exponentially stable if

k∆k + Var(δ; 0, h) <

1

−1 . maxi∈I kE − A − η(h) Di k

We conclude the paper by two examples. Example 5.16. Consider again the positive linear Volterra differential system defined by (21)-(22) with a = −2, p = 2. By Theorem 5.4, the system (21) is UAS. We now consider a perturbed system of the form Z t x(t) ˙ = A∆ x(t) + Bδ (τ )x(t − τ )dτ, x(t) ∈ R2 , t ≥ 0 (29) 0

where A∆ := Bδ (t) :=

−2 + 2a1 0

e−2t + b1 e−t

1 + 2a2 −2

0 e−2t + b2 e−t

1 (t+1)2

, ,

t ≥ 0,

where a1 , a2 , b1 , b2 ∈ R, are unknown parameters. It is important to note that we can rewrite A∆ and Bδ (·) in the following form A∆ = A + D0 ∆E; Bδ (·) = B(·) + D1 δ(·)E where D0 =

2 0

,

D1 =

0 1

and

E = I2 ,

and δ(t) = (b1 e−t , b2 e−t ),

∆ = (a1 , a2 );

t ≥ 0.

2

Assume that R is endowed with 2−norm, by Corollary 5.14, the system (29) is still UAS if √ Z +∞ q q q q 5 13 2 2 2 2 2 2 2 2 . a1 + a2 + δ1 (t) + δ2 (t) dt = a1 + a2 + b1 + b2 < 52 0 Example 5.17. Consider a positive linear functional differential equation Z 1 x(t) ˙ = −x(t) + e−s x(t − s)ds t ≥ 0, x(t) ∈ R. 0

(30)

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Equation (30) can be rewritten in the form of (14), with η(s) = 1 − e−s , s ∈ [0, 1]. By Theorem 5.4, it is easy to see that (30) is exponentially stable. Assume the system (30) is perturbed as follows Z 1 x(t) ˙ = − 1 + δ0 x(t) + e−s + 2006∆1 (s) + 2007∆2 (s) x(t − s)ds, (31) 0

where δ0 ∈ R is an unknown parameter scalar and ∆1 (θ), ∆2 (θ) are unknown integrable functions on [0, 1]. This perturbed system can be rewritten in the form Z 1 x(t) ˙ = − 1 + δ0 x(t) + d[η(θ) + 2006δ1 (s) + 2007δ2 (s)]x(t − s), 0

where Z

s

Z

s

∆2 (τ )dτ, s ∈ [0, 1].

∆1 (τ )dτ, δ2 (s) =

δ1 (s) =

0

0

By Corollary 5.15, we conclude that the perturbed system (31) is exponentially stable for all δ0 ∈ R, ∆1 (·), ∆2 (·) ∈ L1 ([0, 1], R) satisfying Z 1 Z 1 1 . |δ0 | + |∆1 (θ)|dθ + |∆2 (θ)|dθ < 2007e 0 0

6. Conclusion A general class of positive linear systems (namely, positive linear Volterra-Stieltjes differential systems) is introduced and then an explicit criterion for systems of this class is given. A new Perron-Frobenius type theorem for positive linear VolterraStieltjes differential equations is presented. Furthermore, an explicit criterion for uniformly asymptotic stability of positive systems is provided. Finally, it is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula.

7. Appendices 7.1. Appendix A: Proof of Lemma 3.6 It is sufficient to show that, for every x0 ∈ Rn , the solution x(t) := x(· ; 0, x0 ) of (2) is of exponential order. In fact, x(·) satisfies Z t Z tZ τ x(t) = x0 + Ax(τ )dτ + d[B(s)]x(τ − s) dτ, t ≥ 0. 0

0

0

Taking norms to two sides of the above equality, we deduce that Z t Z t kx(t)k ≤ kx0 k + kAkkx(τ )kdτ + Var(B(·); 0, τ ) max kx(s)kdτ, 0

0

s∈[0,τ ]

t ≥ 0.

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Hence, Z kx(t)k ≤ kx0 k +

t

Z

t

kAkkx(τ )kdτ + M

max kx(s)kdτ,

t ≥ 0,

0 s∈[0,τ ]

0

R +∞ where M := 0 |dB(s)|. For a fixed t ≥ 0, we set xt := maxs∈[0,t] kx(s)k. Then, it is easy to see that the real function defined by t 7→ xt , is continuous on R+ . Moreover, the last inequality implies that Z t Z t kx(t)k ≤ kx0 k + kAkxτ dτ + M xτ dτ, t ≥ 0. 0

0

Therefore

t

Z xt ≤ kx0 k +

t ≥ 0.

(kAk + M )xτ dτ, 0

By Gronwall’s inequality, we get kx(t)k ≤ xt ≤ kx0 ke(M +kAk)t ,

t ≥ 0.

This completes the proof. 7.2. Appendix B: Proof of Lemma 3.7 Suppose η is an increasing matrix function on [0, σ]. By the definition of RiemannStieltjes integrals, we have Lφ =

lim d(P )→0

p X

(η(θk ) − η(θk−1 ))φ(ζk ) ≥ 0,

k=1

for every φ ∈ C([−h, 0], Rn ), φ ≥ 0. It means that L is positive. Conversely, let η(·) = (ηij (·)) and let L be positive. We show that ηij (·) : [0, σ] → R is an increasing scalar function for every i, j ∈ {1, 2, ..., n}. Since L is positive, it is easy to see that the functional Z σ Lij : C([0, σ], R) −→ R; φ 7→ Lij φ := φ(θ)dηij (θ), 0

is also positive for every i, j ∈ {1, 2, ..., n}. Fix θ1 , θ2 ∈ [0, σ), θ1 < θ2 , k ∈ N, sufficiently large and consider the continuous function φk defined by   if θ ∈ [0, θ1 ],  0    if θ ∈ (θ1 , θ1 + 1/k], kθ − kθ1 (32) φk (θ) := 1 if θ ∈ (θ1 + 1/k, θ2 ],   −kθ + kθ2 + 1 if θ ∈ (θ2 , θ2 + 1/k],    0 if θ ∈ (θ2 + 1/k, σ]. Since φk is continuous on [0, σ], it follows from a standard property of RiemannStieltjes integrals that Z θ1 Z θ1 +1/k Z θ2 Z σ Z θ2 +1/k Z σ φk (θ)dηij (θ) = + + + + φk (θ)dηij (θ), 0

0

θ1

θ1 +1/k

θ2

θ2 +1/k

350

Anh Ngoc, Murakami, Naito, Son Shin and Nagabuchi

see e.g. [39]. This gives Z Z θ1 +1/k φk (θ)dηij (θ) + ηij (θ2 ) − ηij (θ1 + 1/k) +

IEOT

θ2 +1/k

φk (θ)dηij (θ) ≥ 0,

θ2

θ1

for every k ∈ N sufficiently large. Taking into account that ηij is continuous from the right at θ1 , θ2 and letting k → ∞, we get ηij (θ2 ) ≥ ηij (θ1 ) for every θ1 , θ2 ∈ [0, σ). In case of 0 < θ1 < θ2 = σ, by a similar way, we also get ηij (θ2 ) ≥ ηij (θ1 ). This completes the proof. Remark 7.1. Taking into account the formula of the function φk given by (32), by the same argument used in the proof of the above lemma, one can show that the matrix function η(·) is increasing on subinterval [0, σ), provided Lφ ≥ 0, for every φ ∈ C([0, σ], Rn ), φ ≥ 0, φ(σ) = 0. 7.3. Appendix C: Proof of Lemma 5.11 (i) Since (2) is positive, it follows that A is a Metzler matrix and B(·) is increasing on R+ . For every z ∈ C,
0

0

R +∞

On the other hand, because (2) is UAS, we get µ A+ 0 dB(s) < 0, by Theorem R +∞ −zs 5.4. Therefore, µ A + 0 e dB(s) < 0, for every z ∈ C,
0

for any x ∈ Cn , see e.g. [25], [37]. Since A is a Metzler matrix, there exists a real number α0 > 0 such that (A + α0 In ) ≥ 0. As (A + α0 In ) ≥ 0 and B(·) is increasing on R+ , it follows that R +∞ −zs R +∞ −zs eα0 θ eθ(A+ 0 e dB(s)) = eθ((A+α0 In )+ 0 e dB(s)) ≤ eθ((α0 In +A)+

R +∞ 0

dB(s))

This implies that R θ(A+R +∞ e−zs dB(s)) ≤ eθ(A+ 0+∞ dB(s)) , e 0 Taking (33), (34) into account, we get −1 Z Z +∞ −zs zIn − A − e dB(s)ds x ≤ 0

R +∞

= eα0 θ eθ(A+

0

dB(s))

θ ≥ 0, z ∈ C,
+∞

R +∞

eθ(A+

0

dB(s))

dθ|x|

0

=

Z −A− 0

+∞

−1 dB(s) |x|,

.

(34)

Vol. 64 (2009) On Positive Linear Volterra-Stieltjes Differential Systems

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for every z ∈ C,
0

for every z ∈ C,
−1 −1 Z +∞ Z +∞

−zs

E zIn − A − D dB(s) D ≤ E − A − e dB(s)dt

,

0

0

for every z ∈ C,
−1 R +∞ q×li By Theorem 2.1 (iii), we get E(−A − 0 B(t)dt Di ∈ R+ , i ∈ I. Assume that

−1 −1 Z +∞ Z +∞

dB(s) Di0 max kE −A− dB(s) Di k = E −A−

, i0 ∈ I. i∈I

0

Then

Z

E −A−

0

0

−1

dB(s) Di0

= max

+∞

li

u∈R+0 , kuk=1

Z

E −A−

0

−1

dB(s) Di0 u

,

+∞

l R+i0

such that ku0 k = 1 and see e.g. [17]. Therefore, we can choose u0 ∈

−1 −1 Z +∞ Z +∞

E − A −

dB(s) Di0 u0 = E − A − dB(s) Di0

. 0

0

R +∞

−1

By E(−A − 0 dB(s)) Di0 u0 ≥ 0, there exists, by a theorem of Krein and Rutman [19], a positive linear form y ∗ ∈ (Cq )∗ of dual norm ky ∗ k = 1 such that

−1 −1 Z +∞ Z +∞

E − A − dB(s) D u y∗ E − A − dB(s) Di0 u0 = i0 0 .

0

0

Define

Z

∆ := E − A −

0

+∞

−1 −1

li0 ×q ∗ dB(s) Di0

u 0 y ∈ R+ .

R +∞ It is easy to see that k∆k = kE(−A − 0 dB(s))−1 Di0 k−1 . Set x0 := (−A − R +∞ dB(s))−1 Di0 u0 . This implies that ∆Ex0 = u0 . Therefore, x0 6= 0 and x0 = 0 R +∞ (−A − 0 dB(s))−1 Di0 ∆Ex0 . It follows that Z +∞ Z +∞ (A + Di0 ∆E) + dB(s) x0 = A + d[B(s) + Di0 δ(s)E] x0 = 0, 0

0

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Rs where δ(s) := 0 e−τ ∆dτ = (1 − e−s )∆, s ≥ 0. We consider two separate cases as follows: - If i0 = 0, then we set ∆0 = ∆ and δ1 (·) = 0. Then it is easy to see that (∆0 , δ1 (·)) ∈ D+ satisfies (27)-(28). - If i0 = 1, then we set ∆0 = 0 and δ1 (·) = δ(·). Then, (∆0 , δ1 (·)) ∈ D+ satisfies (27)-(28). This completes the proof.

References [1] A. Aeyels and P. D. Leenheer, Extension of the Perron-Frobenius theorem to homogeneous systems, SIAM Journal on Control and Optimization, Vol. 41, no. 2, pp. 563–582, 2002. [2] L. Benvenuti and L. Farina, Eigenvalue regions for positive systems, Systems Control Lett. Vol. 51, no. 3–4, pp. 325–330, 2004. [3] A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Sciences, Acad. Press, New York, 1979. [4] R. H. Cameron and W. T. Martin, An unsymmetric Fubini theorem, Bull. Amer. Math. Soc. Vol. 47, pp. 121–125, 1941. [5] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations, Functional-, Complex- and Nonlinear Analysis, Springer-Verlag, New-york, 1995. [6] L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, John Wiley and Sons, New York, 2000. [7] S. Gaubert and J. Gunawardena, The Perron-Frobenius theorem for homogeneous, monotone functions, Trans. Amer. Math. Soc. Vol. 356, no. 12, pp. 4931–4950, 2004. [8] G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge Univeristy Press, 1990. [9] G. S. Grossman and R. K. Miller, Nonlinear Volterra integro-differential systems with L1 −kernels, Journal of Differential Equations Vol. 13, pp. 551–566, 1973. [10] W. M. Haddad and V. Chellaboina, Stability and dissipativity theory for nonnegative and compartmental dynamical systems with time delay, Advances in time-delay systems, pp. 421–435, Lect. Notes Comput. Sci. Eng., 38 (2004), Springer, Berlin. [11] W. M. Haddad and V. Chellaboina, Stability and dissipativity theory for nonnegative dynamical systems: a unified analysis framework for biological and physiological systems, Nonlinear Anal. Real World Appl. Vol. 6, no. 1, pp. 35–65, 2005. [12] Y. Hino, S. Murakami and T. Naito, Functional-differential equations with infinite delay, Lecture Notes in Mathematics, 1473. Springer-Verlag, Berlin, 1991. [13] D. Hinrichsen and A. J. Pritchard, Stability radius for structured perturbations and the algebraic Riccati equation, Systems & Control Letters, Vol. 8, no. 2, pp. 105–113, 1986. [14] D. Hinrichsen and A. J. Pritchard, Real and complex stability radii: a survey, in D. Hinrichsen and B. M˚ artensson (Eds), Control of Uncertain Systems, volume 6 of Progress in System and Control Theory, Basel. Birkh¨ auser (1990), 119–162.

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[15] D. Hinrichsen and N. K. Son, µ-analysis and robust stability of positive linear systems, Appl. Math. and Comp. Sci. Vol. 8, no. 2, pp. 253–268, 1998. [16] D. Hinrichsen N. K. Son and P. H. A. Ngoc, Stability radii of positive higher order difference systems, Systems & Control Letters, Vol. 49, no. 5, pp. 377–388, 2003. [17] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1993. [18] F. E. Kloeden and A. M. Rubinov, A generalization of Perron-Frobenius theorem, Nonlinear Analysis, Vol. 41, no. 1–2, pp. 97–115, 2000. [19] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl. Vol. 26, pp. 199–325, 1950. [20] V. Lakshmikantham and M. Rama Mohana Rao, Theory of integro-differential equations. Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, Lausanne, 1995 [21] De Leenheer and D. Patrick Aeyels, Stabilization of positive systems with first integrals, Automatica, Vol. 38, no. 9, pp. 1583–1589, 2002. [22] D. G. Luenberger, Introduction to Dynamic Systems, Theory, Models and Applications, J. Wiley, New York, 1979. [23] R. K. Miller, Asymptotic stability properties of Volterra integro-differential systems, Journal of Differential Equations, Vol. 10, pp. 485–506, 1971. [24] R. K. Miller, Structure of solutions of unstable linear Volterra integro-differential equations, Journal of Differential Equations, Vol. 15, pp. 129–157, 1974. [25] R. Nagel (Ed.), One-Parameter Semigroups of Positive Operators, Springer-Verlag, Berlin, 1986. [26] T. Naito, J. S. Shin, S. Murakami, P. H. A. Ngoc, Characterizations of positive linear Volterra integro-differential systems, Integral Equations and Operator Theory, Vol. 58, no.2, pp. 255–272, 2007. [27] T. Naito, J. S. Shin, S. Murakami, P. H. A. Ngoc, Characterizations of positive linear Volterra integral equations with nonnegative kernels, Journal of Mathematical Analysis and Applications, Vol. 335, no.1, pp. 298–313, 2007. [28] P. H. A. Ngoc and N. K. Son, Stability radii of positive linear difference equations under affine parameter perturbations, Applied Mathematics and Computation, Vol. 134, no. 2-3, pp. 577–594, 2003. [29] P. H. A. Ngoc and N. K. Son, Stability radii of linear systems under multiperturbations, Numer. Funct. Anal. Optim., Vol. 25, no. 3-4, pp. 221–238, 2004. [30] P. H. A. Ngoc, B. S. Lee and N. K. Son, Perron Frobenius theorem for positive polynomial matrices, Vietnam Journal of Mathematics, Vol. 32, no.4, pp. 475–481, 2004. [31] P. H. A. Ngoc, Strong stability radii of positive linear time-delay systems, International Journal of Robust and Nonlinear Control, Vol. 15, no. 10, pp. 459–472, 2005. [32] P. H. A. Ngoc and N. K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM Journal on Control and Optimization, Vol. 43, no. 6, pp. 2278–2295, 2005.

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[33] P. H. A. Ngoc and B. S. Lee, A characterization of spectral abscissa and PerronFrobenius theorem of postive linear functional differntial equations, IMA Journal of Mathematical Control and Information, Vol. 23, no. 3, pp. 259–268, 2006. [34] P. H. A. Ngoc, A Perron-Frobenius theorem for a class of positive quasi-polynomial matrices, Applied Mathematic Letters, Vol. 19, no. 8, pp. 747–751, 2006. [35] P. H. A. Ngoc, T. Naito and J. S. Shin, Characterizations of postive linear functional differential equations, Funkcialaj Ekvacioj, Vol. 50, no. 1, pp. 1–17, 2007. [36] P. H. A. Ngoc, T. Naito, J. S. Shin and S. Murakami, On stability and robust stability of positive linear Volterra equations, SIAM Journal on Control and Optimization, Vol. 47, no. 2, pp. 975–996, 2008. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. [38] L. Qiu et al., A formula for computation of the real stability radius, Automatica, Vol. 31, no. 5, pp. 879–890, 1995. [39] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987. [40] S. M. Rump, Theorems of Perron-Frobenius type for matrices without sign restrictions, Linear Algebra and its Applications, Vol. 266, pp. 1–42, 1997. [41] B. Shafai, J. Chen and M. Kothandaraman, Explicit formulas for stability radii of nonnegative and Metzlerian matrices, IEEE Trans. Autom. Control, Vol. 42, no. 2, pp. 265–269, 1997. [42] N. K. Son and D. Hinrichsen, Robust stability of positive continuous time systems, Numer. Funct. Anal. Optim., Vol. 17, no. 5-6, pp. 649–659, 1996. [43] N. K. Son and P. H. A. Ngoc, Robust stability of positive linear time delay systems under affine parameter perturbations, Acta Mathematica Vietnamica, Vol. 24, no. 3, pp. 353–372, 1999. [44] N. K. Son and P. H. A. Ngoc, Robust stability of linear functional differential equations, Advanced Studies in Contemporary Mathematics, Vol. 3, no. 2, pp. 43–59, 2001. P. H. Anh Ngoc Institute of Mathematics Technical University Ilmenau Weimarer Straße 25 98693 Ilmenau Germany e-mail: [email protected] S. Murakami and Y. Nagabuchi Department of Applied Mathematics Okayama University of Science Ridaicho Okayama 700-0005 Japan e-mail: [email protected] [email protected]

Vol. 64 (2009) On Positive Linear Volterra-Stieltjes Differential Systems T. Naito and J. Son Shin Department of Mathematics University of Electro-Communication Chofu Tokyo 182-8585 Japan e-mail: [email protected] [email protected] Submitted: October 23, 2008. Revised: November 28, 2008.

355

Integr. equ. oper. theory 64 (2009), 357–379 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030357-23, published online July 3, 2009 DOI 10.1007/s00020-009-1698-6

Integral Equations and Operator Theory

Theorem of Completeness for a Dirac-Type Operator with Generalized λ-Depending Boundary Conditions Seppo Hassi and Leonid Oridoroga Abstract. A completeness theorem is proved involving a system of integrodifferential equations with some λ-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established. Mathematics Subject Classification (2000). Primary 34L40; Secondary 47E05. Keywords. Dirac-type operator, λ-depending boundary condition, completeness theorem, Riesz basis.

1. Introduction It is well known [13, Chap. 1, §3] that the system of eigenfunctions and associated functions (SEAF) of the Sturm-Liouville problem −y 00 + q(x)y = λ2 y, 0

(1.1)

0

y (0) − h0 y(0) = y (1) − h1 y(1) = 0, (1.2) is complete in L2 [0, 1] for arbitrary complex valued potential q ∈ L1 [0, 1] and h0 , h1 ∈ C. A similar result is also known for arbitrary nondegenerate boundary conditions (see [13, Chap. 1, §3]). A completeness result for a boundary value problem of arbitrary order differential equations of the form y (n) +

n−2 X

qj (x)y = λn y,

(1.3)

j=0

with separated boundary conditions, has been announced by M.V. Keldysh [9] and was first proved by A.A. Shkalikov [19]. The research was supported by the Academy of Finland (project 129092).

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In [12] M.M. Malamud and one of the authors have generalized the above mentioned results from [13, Chap. 1, §3] to the case of first order systems with arbitrary boundary conditions (not depending on a spectral parameter). A more specific system, involving λ-polynomial boundary conditions, treated in [22] is using a linearization method and asymptotic estimates of the Green’s function, cf. [14, p. 388]; see also other references in [22]. In [20] and [21] the completeness results for the problem (1.1), (1.2) have been generalized to the case of nonlinear λ-depending boundary conditions of the form ( P11 (λ)y(0) + P12 (λ)y 0 (0) = 0, (1.4) 2 P21 (λ)y 2 ( 21 ) + P22 (λ)y( 21 )y 0 ( 12 ) + P23 (λ)y 0 ( 21 ) = 0 and of the form  2 2 P10 (λ)y 2 (0) + P11 (λ)y 0 (0) + P12 (λ)y 2 ( 13 ) + P13 (λ)y 0 ( 13 )      + P14 (λ)y(0)y 0 (0) + P15 (λ)y(0)y( 13 ) + P16 (λ)y(0)y 0 ( 31 )      + P17 (λ)y 0 (0)y( 1 ) + P18 (λ)y 0 (0)y 0 ( 1 ) + P19 (λ)y( 1 )y 0 ( 1 ) = 0, 3 3 3 3 2 2   P20 (λ)y 2 (0) + P21 (λ)y 0 (0) + P22 (λ)y 2 ( 31 ) + P23 (λ)y 0 ( 13 )      + P24 (λ)y(0)y 0 (0) + P25 (λ)y(0)y( 13 ) + P26 (λ)y(0)y 0 ( 31 )    + P27 (λ)y 0 (0)y( 13 ) + P28 (λ)y 0 (0)y 0 ( 31 ) + P29 (λ)y( 31 )y 0 ( 13 ) = 0,

(1.5)

where Pij (λ) are polynomials. Moreover, in [17] analogous results were obtained for a system with a pair of separated λ-depending boundary conditions similar to the conditions (1.4) and (1.5). In the recent papers [23], [24] a related problem concerning the Riesz basis property of the SEAF for a first-order system of the form (1.6) given below with separated boundary conditions, not depending on a spectral parameter, has been established. In the present paper completeness and Riesz basis property of the SEAF are considered for Dirac-type systems with certain λ-depending boundary conditions. Naturally the results cover the case of Dirac operators, which have been more extensively studied in the literature. In particular, we wish to mention the recent studies on spectral decompositions of 1D periodic Dirac operators and related convergence results by B. Mityagin and P. Djakov; see [15] and [2], [3, Sec. 4]; see also Remark 3.8. The paper is organized as follows. In Section 2 we prove some completeness results for the first order systems of certain integro-differential equations involving general linear or quadratic λ-depending boundary conditions. More precisely, consider in L2 [0, 1] ⊕ L2 [0, 1] a boundary value problem for the first order system of ordinary integro-differential equations of the form Z x 1 0 By + Q(x)y + M (x, t)y(t) dt = λy (1.6) i 0

Vol. 64 (2009)

Completeness Theorems for a Dirac-Type Operator

359

with B=

a−1 0

0

b−1

, Q(x) =

0 q1 (x) M11 (x, t) M12 (x, t) , M (x, t) = , q2 (x) 0 M21 (x, t) M22 (x, t)

where it is assumed that a < 0 < b, and that qj ∈ L∞ [0, 1],

Mij ∈ L∞ (Ω),

Ω = {0 ≤ t ≤ x ≤ 1},

i, j = 1, 2.

(1.7)

In addition, y stands for y(x) = (y1 (x), y2 (x))> . If −a = b = 1 and M (x, t) ≡ 0, then the system (1.6) reduces to a Dirac system (see [10]). Therefore it is natural to call the general system (1.6) a Dirac-type system. Two types of λ-depending boundary conditions will be treated. Namely: (i) arbitrary linear conditions of the form ( P11 (λ)y1 (0) + P12 (λ)y2 (0) + P13 (λ)y1 (1) + P14 (λ)y2 (1) = 0, P21 (λ)y1 (0) + P22 (λ)y2 (0) + P23 (λ)y1 (1) + P24 (λ)y2 (1) = 0, (ii) arbitrary quadratic conditions of the form  2 2 2 1 2 1   P10 (λ)y1 (0) + P11 (λ)y2 (0) + P12 (λ)y1 ( 2 ) + P13 (λ)y2 ( 2 )    + P14 (λ)y1 (0)y2 (0) + P15 (λ)y1 (0)y1 ( 21 ) + P16 (λ)y1 (0)y2 ( 12 )      + P17 (λ)y2 (0)y1 ( 1 ) + P18 (λ)y2 (0)y2 ( 1 ) + P19 (λ)y1 ( 1 )y2 ( 1 ) = 0, 2 2 2 2  P20 (λ)y12 (0) + P21 (λ)y22 (0) + P22 (λ)y12 ( 21 ) + P23 (λ)y22 ( 12 )      + P24 (λ)y1 (0)y2 (0) + P25 (λ)y1 (0)y1 ( 1 ) + P26 (λ)y1 (0)y2 ( 1 )   2 2   + P27 (λ)y2 (0)y1 ( 12 ) + P28 (λ)y2 (0)y2 ( 21 ) + P29 (λ)y1 ( 21 )y2 ( 12 ) = 0,

(1.8)

(1.9)

where Pij (λ) are polynomials. In what follows it is tacitly assumed that the two boundary conditions in (1.8) (as well as in (1.9)) are linearly independent from each other. In Section 3 some general sufficient conditions for polynomials Pij are established in order that the SEAF of the problem (1.6) with separated λ-depending boundary conditions forms a Riesz basis. Here instead of (1.7) the following stronger smoothness assumptions on Q(x) and M (x, t) are used Q ∈ C 1 [0, 1]⊗C2×2 ,

M ∈ C 1 (Ω)⊗C2×2 ,

Ω = {0 ≤ t ≤ x ≤ 1}.

Some of the main results of this paper has been announced without proofs in [6], [7], and have been published as a preprint [8].

2. Theorems on completeness of SEAF In this section some sufficient conditions for the completeness of the SEAF of the problems (1.6), (1.8) and (1.6), (1.9) in L2 [0, 1] ⊕ L2 [0, 1] are established. The starting point is to estimate the growth of the solution of the Cauchy problem for the system (1.6) with special initial conditions.

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Let ϕα (x; λ) =

ϕα1 (x; λ) ϕα2 (x; λ)

and

ψα (x; λ) =

ψα1 (x; λ) ψα2 (x; λ)

(2.1)

be the solutions of the Cauchy problem for the system (1.6) with the initial conditions ϕα1 (α; λ) = ψα2 (α; λ) = 1

and

ϕα2 (α; λ) = ψα1 (α; λ) = 0,

(2.2)

where α ∈ [0, 1]. The next lemma gives some estimates for the growth of ϕ0j (x; λ) and ψ0j (x; λ), j = 1, 2. Lemma 2.1. Let the solutions ϕ0 (x; λ) and ψ0 (x; λ) for the system (1.6) satisfying (1.7) be defined by (2.1), (2.2) with α = 0. Then ϕ0j (x; λ) and ψ0j (x; λ), j = 0, 1, satisfy the following estimates w.r.t. λ, uniformly in x (as λ → ∞): 1 )) exp(aλix), ϕ01 (x; λ) = (1 + O( =λ 1 ψ01 (x; λ) = O( =λ ) exp(aλix),

1 ϕ02 (x; λ) = O( =λ ) exp(aλix),

1 ψ02 (x; λ) = O( =λ ) exp(aλix),

(2.3)

when λ ∈ C+ , and 1 ) exp(bλix), ϕ01 (x; λ) = O( =λ

1 ϕ02 (x; λ) = O( =λ ) exp(bλix),

1 ψ01 (x; λ) = O( =λ ) exp(bλix),

1 ψ02 (x; λ) = (1 + O( =λ )) exp(bλix),

(2.4)

when λ ∈ C− . Proof. We prove the first of the estimates in (2.3). All the other estimates can be proved similarly. Due to the assumptions (1.7) the system (1.6) admits a triangular transformation operator; see [11, Thm. 1.1]. This means that the solution ϕ0 admits a representation iaλx Z x iaλt e e ϕ0 (x; λ) = + K(x, t) ibλt dt, (2.5) 0 e 0 K11 (x, t) K12 (x, t) where K(x, t) := ∈ L∞ (Ω) ⊗ C2×2 . In particular, K21 (x, t) K22 (x, t) Z x Z x ϕ01 (x; λ) = eiaλx + K11 (x, t)eiaλt dt + K12 (x, t)eibλt dt. 0

0

Rx

If =λ > 0, then |eibλt | < 1 and 0 K12 (x, t)eibλt dt = O(1) since a < Rx 0 < b and K12 (x, t) is bounded, and then, in particular, 0 K12 (x, t)eibλt dt = 1 O( =λ ) exp(aλix). Moreover, since K11 (x, t) is bounded, one has Z x Z x iaλx Z x e iaλt iaλt −a(=λ)t . K11 (x, t)e dt = O |e | dt = O e dt = O =λ 0 0 0

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Completeness Theorems for a Dirac-Type Operator

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Therefore, for =λ > 0 one gets iaλx

ϕ01 (x; λ) = e

iaλx e 1 +O = 1+O eiaλx . =λ =λ

When the functions Q(x) and M (x, t) admit some smoothness properties, the above estimates can be strengthened. The stronger assumptions stated in the next lemma are used in Section 3 when establishing Riesz basis properties for the root functions. Lemma 2.2. Let ϕ0 (x; λ) and ψ0 (x; λ) be as in Lemma 2.1 and, in addition, assume that Q(x) ∈ C 1 [0, 1] ⊗ C2×2 and M (x, t) ∈ C 1 (Ω)⊗C2×2 . Then the functions ϕj (x; λ) and ψj (x; λ), j = 0, 1, satisfy the following estimates w.r.t. λ, uniformly in x (as λ → ∞): ϕ01 (x; λ) = exp(aλix) + λ1 O(exp(aλix)), ϕ02 (x; λ) = λ1 O(exp(aλix)), ψ01 (x; λ) = λ1 O(exp(aλix)), ψ02 (x; λ) = λ1 O(exp(aλix)); ϕ11 (x; λ) = λ1 O(exp(bλi(x − 1))),

(2.6)

ϕ12 (x; λ) = λ1 O(exp(bλi(x − 1))), ψ11 (x; λ) = λ1 O(exp(bλi(x − 1))), ψ12 (x; λ) = exp(bλi(x − 1)) + λ1 O(exp(bλi(x − 1))), when λ ∈ C+ , and ϕ01 (x; λ) = λ1 O(exp(bλix)), ϕ02 (x; λ) = λ1 O(exp(bλix)), ψ01 (x; λ) = λ1 O(exp(bλix)), ψ02 (x; λ) = exp(bλix) + λ1 O(exp(bλix)); ϕ11 (x; λ) = exp(aλi(x − 1)) + λ1 O(exp(aλi(x − 1))),

(2.7)

ϕ12 (x; λ) = λ1 O(exp(aλi(x − 1))), ψ11 (x; λ) = λ1 O(exp(aλi(x − 1))), ψ12 (x; λ) = λ1 O(exp(aλi(x − 1))), when λ ∈ C− . Proof. As in Lemma 2.1 we just prove the first of the estimates in (2.6). For this purpose the formula (2.5) in the proof of Lemma 2.1 will be used. In [11] (see step (iii) of the proof [11, Thm 1.1]) it was shown that if Q(x) ∈ C 1 [0, 1] ⊗ C2×2 and M (x, t) ∈ C 1 (Ω)⊗C2×2 then Kij (x, t) ∈ C 1 (Ω)⊗C2×2 , too. Now integration by

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parts yields Z x Z x x 1 ∂ K11 (x, t)eiaλt dt = K11 (x, t)eiaλt t=0 − K11 (x, t)eiaλt dt iaλ 0 0 ∂t −a(=λ)x iaλx e iaλx − 1 1 1 K11 (x, x)e − K11 (x, 0) , − O = = λO e iaλ iaλ =λ and Z

x

1 dt = ibλ

x

∂ ibλt K12 (x, t)e − K12 (x, t)e dt 0 0 ∂t 1 1 = O(1) = O eiaλx , ibλ λ since =λ > 0 and a < 0 < b. Therefore from the formula (2.5) one gets ibλt

x K12 (x, t)eibλt t=0

Z

ϕ01 (x; λ) = exp(aλix) + λ1 O(exp(aλix)). All the other estimates in Lemma 2.2 are proved in a similar manner.

In the next lemma some estimates for the growth of the Wronski determinant are presented. Lemma 2.3. Let y1 (x; λ) and y2 (x; λ) be two linearly independent solutions of the system (1.6) satisfying the assumptions (1.7). Then the Wronski determinant y (x; λ) y21 (x; λ) W (x; λ) = det 11 (2.8) y12 (x; λ) y22 (x; λ) admits the following estimate w.r.t. λ, uniformly in x (as λ → ∞): W (x; λ) = (1 + o(1)) exp((a + b)λi)W (0; λ).

(2.9)

Proof. It was proved in [12] that in C± the system (1.6) with (1.7) has two linearly independent solutions ε1 (x, λ) and ε2 (x, λ) satisfying the following estimates (1 + o(1))exp(aλix) o(1)exp(bλix) ε1 (x, λ) = , ε2 (x, λ) = (2.10) o(1)exp(aλix) (1 + o(1))exp(bλix) for λ ∈ C± uniformly in x. Since yj (x; λ) is a linear combination of ε1 (x, λ) and ε2 (x, λ), one has y11 (x; λ) y21 (x; λ) y12 (x; λ) y22 (x; λ) −1 ε11 (x; λ) ε21 (x; λ) ε11 (0; λ) ε21 (0; λ) y11 (0; λ) y21 (0; λ) = . ε12 (x; λ) ε22 (x; λ) ε12 (0; λ) ε22 (0; λ) y12 (0; λ) y22 (0; λ) (2.11) It follows that −1 ε11 (x; λ) ε21 (x; λ) ε11 (0; λ) ε21 (0; λ) W (x; λ) = det det W (0; λ) ε12 (x; λ) ε22 (x; λ) ε12 (0; λ) ε22 (0; λ)

= (1 + o(1)) exp((a + b)λi)W (0; λ)

(2.12)

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which gives the required estimate (2.9).

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Now some further definitions and notations are given. First, recall the following definition from [16, Chap. 1]. The point λ0 ∈ C is an eigenvalue of the operator Z x 1 M (x, t)y(t) dt − λy (2.13) L(y; λ) = By 0 + Q(x)y + i 0 subject to the boundary conditions U1 (y; λ) = 0, U2 (y; λ) = 0 if there exists a nonzero vector function y0 such that L(y0 ; λ0 ) = 0,

U1 (y0 ; λ0 ) = U2 (y0 ; λ0 ) = 0.

Furthermore, a sequence y0 , y1 , y2 , . . . , yn is called a chain of the eigenfunction y0 and the associated functions y1 , y2 , . . . , yn (corresponding to the eigenvalue λ0 ) if the following equalities hold for k = 0, . . . , n: k X 1 ∂j = 0, L(yk−j ; λ) j j! ∂λ λ=λ0 j=0 k X 1 ∂j = 0, U (y ; λ) 1 k−j j j! ∂λ λ=λ0 j=0

(2.14)

k X 1 ∂j = 0. U2 (yk−j ; λ) j j! ∂λ λ=λ0 j=0 Clearly, for any eigenvalue λ0 of L(y, λ) defined by (2.13), dim ker(L(y, λ0 )) ≤ 2 holds. If dim ker(L(y, λ0 )) = 2, then the length of a maximal chain of an eigenfunction and its associated functions depends on the choice of an eigenvector. It is easily seen that the length of a maximal chain takes at most two values. De(1) (2) note these lengths by p0 and p0 . If the length of all such chains are equal, (1) (2) then p0 = p0 is equal to the length of each maximal chain. Otherwise, there (1) (2) is a maximal chain whose length is equal to min{p0 , p0 }, and maximal chains (1) (2) whose lengths are equal to max{p0 , p0 }. If dim ker(L(y, λ0 )) = 1 then we put (1) (2) min{p0 , p0 } = 0. Finally, the function χ(λ) defined by Q11 (λ) Q12 (λ) χ(λ) := det , (2.15) Q21 (λ) Q22 (λ) where Q11 (λ) = P11 (λ) + P13 (λ)ϕ01 (1; λ) + P14 (λ)ϕ02 (1; λ), Q12 (λ) = P12 (λ) + P13 (λ)ψ01 (1; λ) + P14 (λ)ψ02 (1; λ), Q21 (λ) = P21 (λ) + P23 (λ)ϕ01 (1; λ) + P24 (λ)ϕ02 (1; λ), Q22 (λ) = P22 (λ) + P23 (λ)ψ01 (1; λ) + P24 (λ)ψ02 (1; λ),

(2.16)

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is said to be the characteristic function of the problem (1.6), (1.8). The definition of χ(λ) in (2.15) is motivated by the next result. Proposition 2.4. The number λ0 ∈ C is an eigenvalue of the operator associated with the problem (1.6)–(1.8) if and only if χ(λ0 ) = 0. Moreover, the functions ω1 (x; λ0 ) and ω2 (x; λ0 ) given by ω1 (x; λ) = Q12 (λ)ϕ0 (x; λ) − Q11 (λ)ψ0 (x; λ)

(2.17)

ω2 (x; λ) = Q22 (λ)ϕ0 (x; λ) − Q21 (λ)ψ0 (x; λ)

(2.18)

and are eigenfunctions corresponding to the eigenvalue λ0 , or, one has ωj (x; λ0 ) ≡ 0. Moreover, all the eigenfunctions and associated functions corresponding to the eigenvalue λ0 are the nonzero functions of the form 1 ∂k , where 0 ≤ k < p0 , j = 1, 2. (2.19) ω (x; λ) j k k! ∂λ λ=λ0 Proof. It follows from (2.2) with α = 0 that for all λ ∈ C the function ω1 (x; λ) = (P12 (λ) + P13 (λ)ψ01 (1; λ) + P14 (λ)ψ02 (1; λ))ϕ0 (x; λ) − (P11 (λ) + P13 (λ)ϕ01 (1; λ) + P14 (λ)ϕ02 (1; λ))ψ0 (x; λ)

(2.20)

is a solution of the first equation in (1.8). Moreover, since P21 (λ)ω11 (0, λ) + P22 (λ)ω12 (0, λ) + P23 (λ)ω11 (1, λ) + P24 (λ)ω12 (1, λ) = −χ(λ), (2.21) ω1 (x; λ0 ) is a solution of the second equation in (1.8), if λ0 is a root of χ(λ). Similarly, for all λ ∈ C the function ω2 (x; λ) = (P22 (λ) + P23 (λ)ψ01 (1; λ) + P24 (λ)ψ02 (1; λ))ϕ0 (x; λ) − (P21 (λ) + P23 (λ)ϕ01 (1; λ) + P24 (λ)ϕ02 (1; λ))ψ0 (x; λ).

(2.22)

is a solution of the second equation in (1.8) and since P11 (λ)ω21 (0, λ) + P12 (λ)ω22 (0, λ) + P13 (λ)ω21 (1, λ) + P14 (λ)ω22 (1, λ) = χ(λ), (2.23) ω2 (x; λ0 ) is a solution of the first equation in (1.8) too, if λ0 is a root of χ(λ). Observe, that ω1 (x; λ) and ω2 (x; λ) are linearly independent for all λ 6= λ0 . If ωj (x; λ0 ), j = 1, 2, is nontrivial, then it is an eigenfunction corresponding to the eigenvalue λ0 . Moreover, if λ0 is a root of χ(λ) of the order p0 , then the functions ωj (x; λ) in (2.17), (2.17) give rise to at most p0 eigenfunctions and associated functions corresponding to λ0 . In fact, by differentiation w.r.t. λ it follows from (2.21), (2.23) that all nonzero functions given by 1 ∂k ω (x; λ) , where 0 ≤ k < p0 , j = 1, 2, (2.24) j k k! ∂λ λ=λ0

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are eigenfunctions and associated functions corresponding to the eigenvalue λ0 ; (1) (2) cf. (2.14). If dim ker(L(y, λ0 )) = 2 and p0 6= p0 and the maximal chains determined by the functions ω1 (x; λ) and ω2 (x; λ) via (2.24) both are of the length (1) (2) (2) (1) (2) max{p0 , p0 }, then we can put p0 = min{p0 , p0 } to get linearly independent chains. On the other hand, it follows from the definition (2.14) that all the eigenfunctions and the associate functions are obtained in this manner. The following two examples clarify the formulation of Proposition 2.4 and some of the definitions preceding it. Example. Consider the Dirac system, i.e. the system (1.6) with Q ≡ 0, M ≡ 0, a = −1, b = 1. The functions ϕ0 (x; λ) and ψ0 (x; λ) appearing in (2.1), (2.2) are given by −iλx e 0 ϕ0 (x; λ) = , ψ0 (x; λ) = iλx . 0 e Consider the boundary value problem for this system by adding the following boundary conditions ( y1 (0) + y2 (1) = 0 (2.25) y2 (0) + y1 (1) = 0. The characteristic determinant corresponding to the boundary condition (2.25) is now a zero function; 1 eiλ χ(λ) = det −iλ = 1 − 1 ≡ 0, e 1 though the boundary conditions are linearly independent. Since χ(λ) = 0, the eigenfunctions ω1 (x; λ) and ω2 (x; λ) in (2.17) and (2.18) are linearly dependent for each λ ∈ C. In fact, iλ(1−x) e ω1 (x; λ) = = eiλ ω2 (x; λ). −eiλx Example. Consider the Dirac system subject to the boundary conditions ( y1 (0) − y1 (1) = 0 y2 (0) − y2 (1) = 0. The characteristic determinant is now given by 1 − e−iλ 0 χ(λ) = det = −e−iλ (1 − eiλ )2 . 0 1 − eiλ The corresponding eigenvalues are λn = 2πn and their multiplicity is equal to two. Here the functions ω1 (x; λ) and ω2 (x; λ) in (2.17) and (2.18) associated to the 0 eigenvalues λn = 2πn satisfy ωj (x; λn ) ≡ 0. In fact, ω1 (x; λ) = −(1 − e−iλ ) eiλx , 0 and clearly ω1 (x; 2πn) ≡ 0 . At the same time, the corresponding derivative 0 ∂ ω1 (x; λ) = ∂λ −ie2πnix λ=2πn

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is an eigenfunction corresponding to the eigenvalue λn . −iλx Similarly, ω2 (x; λ) = (1−eiλ ) e 0 and ω2 (x; 2πn) ≡ 00 . The corresponding −2πnix ∂ derivative ∂λ ω2 (x; λ) λ=2πn = −ie 0 is a linearly independent eigenfunction associated to λn . The completeness result for the system (1.6) satisfying the assumptions (1.7) and involving the boundary conditions (1.8) can now be stated as follows. Theorem 2.5. Let Pij (i = 1, 2; j = 1, 2, 3, 4) be polynomials, let the rank of the polynomial matrix P11 (λ) P12 (λ) P13 (λ) P14 (λ) P (λ) = (2.26) P21 (λ) P22 (λ) P23 (λ) P24 (λ) be equal to 2 for all λ ∈ C, and let deg J14 = deg J32 ≥ max{deg J13 , deg J42 , M }, where M = max{ deg Pij : i ∈ {1, 2}; j ∈ {1, 2, 3, 4} } and P1i P1j Jij = det , i, j ∈ {1, 2, 3, 4}. P2i P2j

(2.27)

(2.28)

Then the SEAF of the problem (1.6)–(1.8) is complete in L2 [0, 1] ⊕ L2 [0, 1]. Moreover, let the set Φ, which consists of N := deg J14 − M eigenfunctions and associated functions, satisfy the following condition: If Φ contains either an eigenfunction or an associated function ωk corresponding to an eigenvalue λk , then it also contains all the associated functions of higher order corresponding to the same eigenvalue and the same function ωk . Then the SEAF of the problem (1.6)–(1.8) without the set Φ is also complete in the space L2 [0, 1] ⊕ L2 [0, 1]. Proof. Suppose that the SEAF of the problem (1.6), (1.8) without the set Φ is not complete in the space L2 [0, 1] ⊕ L2 [0, 1]. Then there exists a nonzero vector function f (x) = (f1 (x), f2 (x))> , which is orthogonal to the SEAF of the problem (1.6), (1.8) (possibly, excluding functions from the set Φ). Define Z 1 fj (λ) := hωj (x; λ), f (x)i = (2.29) F (ωj1 (x; λ)f1 (x) + ωj2 (x; λ)f2 (x)) dx. 0

fj (λ) is an entire function. If λs is an eigenvalue of multiplicity ps (= Clearly, F (1) (2) ps + ps ) and the set Φ contains neither an eigenfunction nor an associated function corresponding to λs , then it follows from Proposition 2.4 that λs is a root fj (λ) of order greater than or equal to ps . of F If Φ contains k eigenfunctions or associated functions corresponding to the fj (λ) of order greater than or equal to ps − k. eigenvalue λs , then λs is a root of F Let Φ 6= ∅ and denote by Λ the set of all eigenvalues of the problem (1.6), (1.8), such that the corresponding eigenfunctions or associated functions belong

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to the set Φ. For each λs ∈ Λ denote by qs the number of eigenfunctions and associated functions in Φ corresponding to λs . Define Y Π(λ) = (λ − λs )qs . (2.30) λs ∈Λ

Let λk be an eigenvalue of the problem (1.6), (1.8) of multiplicity pk . Then λk is a zero of the product Π(λ)Fej (λ) at least of order pk . Consequently, the functions fj (λ) Π(λ)F χ(λ) are entire. Next an estimate for these functions will be derived. One can rewrite χ(λ) as follows: Fj (λ) =

χ(λ) = J12 + J13 ψ01 (1; λ) + J14 ψ02 (1; λ) + J32 ϕ01 (1; λ) ϕ01 (1; λ) ψ01 (1; λ) + J42 ϕ02 (1; λ) + J34 det ϕ02 (1; λ) ψ02 (1; λ)

(2.31)

(2.32)

= J12 + J13 ψ01 (1; λ) + J14 ψ02 (1; λ) + J32 ϕ01 (1; λ) 1 + J42 ϕ02 (1; λ) + J34 1 + O =λ exp((a + b)λi). Then one obtains from (2.3), (2.4) in Lemma 2.1, and the assumption (2.27) the following estimates for χ(λ): 1 ))J32 exp(aλi), χ(λ) = (1 + O( =λ

λ ∈ C+ ;

1 O( =λ ))J14

−

χ(λ) = (1 + exp(bλi), λ ∈ C . Moreover, the definition of ωj (x; λ) (cf. (2.20) and (2.22)) implies that

(2.33) (2.34)

ωj (x; λ) = −Pj1 (λ)ψ0 (x; λ) + Pj2 (λ)ϕ0 (x; λ) 1 ) exp((a + b)λi)(−Pj3 (λ)ψ1 (x; λ) + Pj4 (λ)ϕ1 (x; λ)). + (1 + O =λ (2.35) If λ ∈ C+ , then (2.35) and the estimate (2.3) imply ωjk (x; λ) = (O(Pj1 (λ)) + O(Pj2 (λ))) exp(aλix) + (O(Pj3 (λ)) + O(Pj4 (λ))) exp(aλi) exp(bλix).

(2.36)

By using the Cauchy-Schwartz inequality one gets ! Z 1 exp(aλi) p , |fk (x) exp(aλix)| dx = O |=λ| 0 ! Z 1 1 |fk (x) exp(bλix)| dx = O p , |=λ| 0 since a < 0 < b. Consequently there exists a constant c1 > 0, such that, for =λ > c1 , ! ! exp(aλi) λM f Fj (λ) = O max |Pjk (λ)| p =O p eaλi . (2.37) k |=λ| |=λ|

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Similarly, there exists a constant c2 < 0, such that, for =λ < c2 , ! M λ fj (λ) = O p F ebλi . |=λ|

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(2.38)

From (2.33), (2.34), (2.37), (2.38), and the assumption (2.27) one obtains finally the estimate ! 1 , |=λ| > c. (2.39) Fj (λ) = O p |=λ| By applying the Phragmen-Lindel¨of theorem for a strip one concludes that Fj (λ) ≡ 0. Consequently, Fej (λ) ≡ 0, i.e. f (x) is orthogonal to ω1 (x; λ) and ω2 (x; λ) for all λ. Observe, that if χ(λ) 6= 0 then the functions ω1 (x; λ) and ω2 (x; λ) are linearly independent. Hence, the functions ω1 (x; λ) and ω2 (x; λ) form a fundamental system of solutions of the equation (1.6) if λ is not an eigenvalue. Since the set of eigenvalues coincides with the set of all roots of χ(λ), this set is discrete. This implies that f (x) is orthogonal to all solutions of the equation (1.6), so that f (x) ≡ 0. Therefore, there is no nontrivial function f (x) orthogonal to the SEAF of the problem (1.6)–(1.8) (maybe without the set Φ). Remark 2.6. The estimates in (2.37) and (2.38) can be sharpened as follows: ! Mj λ fj (λ) = O p eaλi if =λ > c1 , F |=λ| ! λMj f ebλi if =λ < c2 , Fj (λ) = O p |=λ| where Mj = max{ deg Pjk : k ∈ {1, 2, 3, 4} }, j = 1, 2. Next the completeness result for the system (1.6) satisfying the assumptions (1.7) and involving the quadratic boundary conditions (1.9) is given. Theorem 2.7. Let Pij (i = 1, 2; j = 0, 1, . . . , 9) be polynomials, let the rank of the matrix P10 (λ) P11 (λ) . . . P19 (λ) (2.40) P20 (λ) P21 (λ) . . . P29 (λ) be equal to 2 for all λ ∈ C, and let deg J03 = deg J12 = M, where

P1i Jij = det P2i

P1j , P2j

i, j = 0, 1, . . . , 9,

(2.41) (2.42)

and M = max{ deg Pij : i ∈ {1, 2}; j ∈ {0, 1, . . . , 9} }. Then the SEAF of the problem (1.6), (1.7), (1.9) is complete in L2 [0, 1] ⊕ L2 [0, 1]. Moreover, let the set Φ, which consists of M eigenfunctions and associated functions, satisfy the following condition:

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If Φ contains either an eigenfunction or an associated function ωk corresponding to an eigenvalue λk , then it also contains all the associated functions of higher order corresponding to the same eigenvalue and the same function ωk . Then the SEAF of the problem (1.6), (1.7), (1.9) without the set Φ is also complete in the space L2 [0, 1] ⊕ L2 [0, 1]. Proof. The proof of this theorem is similar to the proof of Theorem 2.5. As in Theorem 2.5 one considers the characteristic function and the functions ω1 (x; λ) and ω2 (x; λ), but in this case these functions will be defined by other formulas. Let ω(x; λ) be an arbitrary solution of the system (1.6) with (1.7). Then ω(x; λ) may be written in the form ω(x; λ) = Aϕ0 (x; λ) + Bψ0 (x; λ). It follows that the solution ω(x; λ) to (1.6) satisfies the boundary conditions (1.9) if and only if  P10 (λ)A2 + P11 (λ)B 2 + P12 (λ)(Aϕ01 ( 21 ; λ) + Bψ01 ( 12 ; λ))2      + P13 (λ)(Aϕ02 ( 12 ; λ) + Bψ02 ( 21 ; λ))2 + P14 (λ)AB      + P15 (λ)A(Aϕ01 ( 21 ; λ) + Bψ01 ( 21 ; λ)) + P16 (λ)A(Aϕ02 ( 12 ; λ) + Bψ02 ( 12 ; λ))      + P17 (λ)B(Aϕ01 ( 12 ; λ) + Bψ01 ( 12 ; λ)) + P18 (λ)B(Aϕ02 ( 21 ; λ) + Bψ02 ( 12 ; λ))      + P19 (λ)(Aϕ01 ( 1 ; λ) + Bψ01 ( 1 ; λ))(Aϕ02 ( 1 ; λ) + Bψ02 ( 1 ; λ)) = 0, 2 2 2 2  P20 (λ)A2 + P21 (λ)B 2 + P22 (λ)(Aϕ01 ( 21 ; λ) + Bψ01 ( 12 ; λ))2      + P23 (λ)(Aϕ02 ( 12 ; λ) + Bψ02 ( 21 ; λ))2 + P24 (λ)AB       + P25 (λ)A(Aϕ01 ( 21 ; λ) + Bψ01 ( 21 ; λ)) + P26 (λ)A(Aϕ02 ( 12 ; λ) + Bψ02 ( 12 ; λ))      + P27 (λ)B(Aϕ01 ( 12 ; λ) + Bψ01 ( 12 ; λ)) + P28 (λ)B(Aϕ02 ( 21 ; λ) + Bψ02 ( 12 ; λ))    + P29 (λ)(Aϕ01 ( 12 ; λ) + Bψ01 ( 21 ; λ))(Aϕ02 ( 12 ; λ) + Bψ02 ( 12 ; λ)) = 0. (2.43) The system (2.43) may be rewritten in the form ( Q11 (λ)A2 + Q12 (λ)AB + Q13 (λ)B 2 = 0, (2.44) Q21 (λ)A2 + Q22 (λ)AB + Q23 (λ)B 2 = 0, where Q11 (λ) = P10 (λ) + P12 (λ)ϕ201 ( 21 ; λ) + P13 (λ)ϕ202 ( 12 ; λ) + P15 (λ)ϕ01 ( 12 ; λ) + P16 (λ)ϕ02 ( 21 ; λ) + P19 (λ)ϕ01 ( 21 ; λ)ϕ02 ( 12 ; λ), Q12 (λ) = 2P12 (λ)ϕ01 ( 21 ; λ)ψ01 ( 12 ; λ) + 2P13 (λ)ϕ02 ( 21 ; λ)ψ02 ( 12 ; λ) + P14 (λ) + P15 (λ)ψ01 ( 12 ; λ) + P16 (λ)ψ02 ( 12 ; λ) + P17 (λ)ϕ01 ( 21 ; λ) + P18 (λ)ϕ02 ( 12 ; λ) + P19 (λ)(ϕ01 ( 21 ; λ)ψ02 ( 12 ; λ) + ψ01 ( 12 ; λ)ϕ02 ( 12 ; λ)),

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2 1 2 1 Q13 (λ) = P11 (λ) + P12 (λ)ψ01 ( 2 ; λ)) + P13 (λ)ψ02 ( 2 ; λ)

+ P17 (λ)ψ01 ( 12 ; λ)) + P18 (λ)ψ02 ( 12 ; λ) + P19 (λ)ψ01 ( 12 ; λ)ψ02 ( 12 ; λ), Q21 (λ = P20 (λ) + P22 (λ)ϕ201 ( 21 ; λ) + P23 (λ)ϕ202 ( 12 ; λ) + P25 (λ)ϕ01 ( 12 ; λ) + P26 (λ)ϕ02 ( 21 ; λ) + P29 (λ)ϕ01 ( 21 ; λ)ϕ02 ( 12 ; λ), Q22 (λ) = 2P22 (λ)ϕ01 ( 21 ; λ)ψ01 ( 12 ; λ) + 2P23 (λ)ϕ02 ( 21 ; λ)ψ02 ( 12 ; λ) + P24 (λ) + P25 (λ)ψ01 ( 12 ; λ) + P26 (λ)ψ02 ( 12 ; λ) + P27 (λ)ϕ01 ( 21 ; λ) + P28 (λ)ϕ02 ( 12 ; λ) + P29 (λ)(ϕ01 ( 21 ; λ)ψ02 ( 12 ; λ) + ψ01 ( 12 ; λ)ϕ02 ( 12 ; λ)), 2 1 2 1 Q23 (λ) = P21 (λ) + P22 (λ)ψ01 ( 2 ; λ)) + P23 (λ)ψ02 ( 2 ; λ)

+ P27 (λ)ψ01 ( 12 ; λ)) + P28 (λ)ψ02 ( 12 ; λ) + P29 (λ)ψ01 ( 12 ; λ)ψ02 ( 12 ; λ). It is well known (see, for example, [25, Chap. 5, §34]), that a system of two quadratic equations has a nonzero solution if and only if, the resultant is equal to 0. Therefore, λ0 is an eigenvalue of the problem (1.6), (1.9) if and only if χ(λ0 ) = 0, where   Q11 Q12 Q13 0  0 Q11 Q12 Q13  2  = D13 − D12 D23 (2.45) χ(λ) = det  Q21 Q22 Q23 0  0 Q21 Q22 Q23 with Q1i Q1j Dij = det . (2.46) Q2i Q2j Moreover, the multiplicity of λ0 as a zero of the function χ(λ) is equal to the number of eigenfunctions and associated functions corresponding to the eigenvalue λ0 . To see this, introduce the functions ω1 (x; λ) := D13 ϕ0 (x; λ) − D12 ψ0 (x; λ), (2.47) ω2 (x; λ) := D23 ϕ0 (x; λ) − D13 ψ0 (x; λ). The function ω1 (x; λ) satisfies the boundary conditions (1.9) if and only if ( 2 2 Γ1 (λ) := Q11 (λ)D13 (λ) − Q12 (λ)D13 (λ)D12 (λ) + Q13 (λ)D12 (λ) = 0 (2.48) 2 2 Γ2 (λ) := Q21 (λ)D13 (λ) − Q22 (λ)D13 (λ)D12 (λ) + Q23 (λ)D12 (λ) = 0. Observe that Γ1 (λ) = Q11 (λ)χ(λ)

and Γ2 (λ) = Q21 (λ)χ(λ).

Hence, if λ0 is an eigenvalue of multiplicity p0 , then ∂k ∂k Γ (x; λ) = 0 and Γ (x; λ) =0 1 2 k k ∂λ ∂λ λ=λ0 λ=λ0 Therefore, in this case, all nonzero functions ∂k ω (x; λ) with k < p0 , 1 k ∂λ λ=λ0

for all k < p0 .

(2.49)

(2.50)

(2.51)

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are eigenfunctions and associated functions, corresponding to the eigenvalue λ0 . Similarly all nonzero functions given by ∂k with k < p0 , (2.52) ω2 (x; λ) k ∂λ λ=λ0

are eigenfunctions and associated functions, corresponding to the eigenvalue λ0 , too. Now, introduce the Wronski determinant ϕ01 (x; λ) ψ01 (x; λ) W (x; λ) = det . ϕ02 (x; λ) ψ02 (x; λ) Then χ(λ) may be transformed to a polynomial of degree 4 with the arguments ϕ01 ( 21 ; λ), ψ01 ( 21 ; λ), ϕ02 ( 12 ; λ), ψ02 ( 12 ; λ), and W ( 12 , λ), having all the coefficients 2 of form Jij Jkl . In particular, the coefficient of ϕ40,1 ( 12 ; λ) is equal to J12 , and the 4 2 coefficient of ψ0,2 ( 21 ; λ) is equal to J03 . Therefore, from the condition (2.41), the estimates (2.3), (2.4), and the following estimate (cf. Lemma 2.3) 1 exp((a + b)λi), (2.53) W (x; λ) = 1 + O =λ one can derive the following estimates for the characteristic function χ(λ): 2 1 χ(λ) = (1 + O( =λ ))J12 exp(2aλi)

for λ ∈ C+ ;

(2.54)

and 2 1 χ(λ) = (1 + O( =λ ))J03 exp(2bλi) for λ ∈ C− . (2.55) Suppose that the SEAF of the problem (1.6), (1.9) without the set Φ is not complete in the space L2 [0, 1]⊕L2 [0, 1]. Then there exists a nonzero vector-function f1 (x) f (x) = (2.56) f2 (x)

which is orthogonal to the SEAF of problem (1.6)–(1.9) (possibly excluding the functions from the set Φ). Just as in the proof of Theorem 2.5 introduce the functions F˜j (λ) and Π(λ) by the formulae (2.29) and (2.30). Then, as before, the functions Π(λ)F˜j (λ) (2.57) Fj (λ) = χ(λ) are entire. Let g(x) = (c1 ψ01 ( 12 ; λ) + c2 ψ02 ( 21 ; λ))ϕ0 (x; λ) − (c1 ϕ01 ( 21 ; λ) + c2 ϕ02 ( 12 ; λ))ψ0 (x; λ), where c1 and c2 are arbitrary complex coefficients. From (2.53) one gets the estimate (−c1 + O λ1 ) exp((a + b)λi) 1 g( 2 ) = (c2 + O λ1 ) exp((a + b)λi) and this implies that the function g(x) satisfies the estimate 1 g(x) = 1 + O =λ exp( 12 (a + b)λi)(−c1 ψ 1 (x; λ) + c2 ϕ 1 (x; λ)), (2.58) 2

2

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where the functions ϕ 1 (x; λ) and ψ 1 (x; λ) are solutions of the Cauchy problem 2

2

for the system (1.6) with the initial conditions ϕ 1 ( 12 ; λ) = ψ 1 ( 12 ; λ) = 1 21

22

and

ϕ 1 ( 21 ; λ) = ψ 1 ( 12 ; λ) = 0. 22

21

(2.59)

As in Lemma 2.1 one can derive for these functions the following estimates: • if λ ∈ C+ and x > 21 , then 1 )) exp(aλi(x − 21 )), ϕ 1 (x; λ) = (1 + O( =λ 21

1 ϕ 1 (x; λ) = O( =λ ) exp(aλi(x − 21 )), 22

1 ψ 1 (x; λ) = O( =λ ) exp(aλi(x − 21 )),

(2.60)

21

1 ψ 1 (x; λ) = O( =λ ) exp(aλi(x − 21 )); 22

• if λ ∈ C+ and x < 12 , then 1 ϕ 1 (x; λ) = O( =λ ) exp(bλi(x − 21 )), 21

1 ϕ 1 (x; λ) = O( =λ ) exp(bλi(x − 21 )), 22

1 ψ 1 (x; λ) = O( =λ ) exp(bλi(x − 21 )),

(2.61)

21

1 ψ 1 (x; λ) = (1 + O( =λ )) exp(bλi(x − 12 )); 22

• if λ ∈ C− and x > 12 , then 1 ϕ 1 (x; λ) = O( =λ ) exp(bλi(x − 21 )), 21

1 ϕ 1 (x; λ) = O( =λ ) exp(bλi(x − 21 )), 22

1 ψ 1 (x; λ) = O( =λ ) exp(bλi(x − 21 )),

(2.62)

21

1 ψ 1 (x; λ) = (1 + O( =λ )) exp(bλi(x − 12 )); 22

• if λ ∈ C− and x < 12 , then 1 ϕ 1 (x; λ) = (1 + O( =λ )) exp(bλi(x − 12 )), 21

1 ) exp(bλi(x − 21 )), ϕ 1 (x; λ) = O( =λ 22

1 ψ 1 (x; λ) = O( =λ ) exp(bλi(x − 21 )),

(2.63)

21

1 ψ 1 (x; λ) = O( =λ ) exp(aλi(x − 12 )). 22

Using the formula (2.58) and the estimates (2.3), (2.4), (2.60)–(2.63) one gets the following estimates for the functions in (2.47) ωi (x; λ) = O(λM exp(iaλ(x + 1)))

if =λ > 0,

(2.64)

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and ωi (x; λ) = O(λM exp(ibλ(x + 1))) if =λ < 0. (2.65) From the estimates (2.54), (2.55), (2.64), (2.65) one finally gets the estimates ! 1 Fi (λ) = O p , |=λ| > C, (2.66) |=λ| where C > 0 is a constant. Then, by the Phragmen-Lindel¨of theorem for a strip, one again concludes that Fj (λ) ≡ 0, and therefore F˜j (λ) ≡ 0, i.e. f (x) is orthogonal to ω1 (x; λ) and ω2 (x; λ) for all λ. Observe that, if χ(λ) 6= 0, then the functions ω1 (x; λ) and ω2 (x; λ) are linearly independent. Therefore, for these values of λ, ω1 (x; λ) and ω2 (x; λ) form a fundamental system of solutions of the system (1.6). Now as in the proof of Theorem 2.5 one concludes that f (x) is orthogonal to all solutions of the system (1.6). Consequently, f (x) ≡ 0 and this completes the proof.

3. Riesz basis property of the SEAF In this section some sufficient conditions for the Riesz basis property of the SEAF of the system (1.6) with separated λ-depending boundary conditions will be established. First recall the definition of a Riesz basis. Definition 3.1. A system of vectors {ψn }∞ n=1 is called a Riesz basis in the Hilbert space H if there exists a bounded operator A with bounded inverse A−1 , such that the transformed system {Aψn }∞ n=1 forms an orthonormal basis in H. The following lemma is well known (see [5, Chap. VI, §3]). Lemma 3.2. Let the system of the vectors {ψn }∞ a Hilbert space n=1 P∞be complete in 2 H. Let {ϕn }∞ be a Riesz basis of H such that kψ − φ k < ∞. Then the n n n=1 n=1 system {ψn }∞ is a Riesz basis of H, too. n=1 Also the following lemma, which concerns the spectrum of the system (1.6) with separated λ-depending boundary conditions, will be needed. Lemma 3.3. Assume that Q(x) ∈ C 1 [0, 1] ⊗ C2×2 and M (x, t) ∈ C 1 (Ω)⊗C2×2 . Let P11 (λ) and P12 (λ) be relatively prime polynomials such that deg P11 = deg P12 = N0 and let P21 (λ) and P22 (λ) be relatively prime polynomials such that deg P21 = deg P22 = N1 . Let Cij be the leading coefficient of the polynomial Pij (λ) and denote C1 = C12 C21 and C2 = C11 C22 . Furthermore, assume that the set Λ contains N = N0 + N1 (arbitrary) eigenvalues of the problem (1.6) with separated λ-depending boundary conditions given by ( P11 (λ)y1 (0) + P12 (λ)y2 (0) = 0 (3.1) P21 (λ)y1 (1) + P22 (λ)y2 (1) = 0.

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Then it is possible to enumerate the remaining (all but N ) eigenvalues, so that i ln(C1 /C2 ) + 2πn 1 λn = +O , where n ∈ Z. (3.2) b−a |n| Proof. The characteristic function χ(λ) in (2.15) of the system (1.6) with the boundary conditions in (3.1) takes the form (compare (2.21), (2.23)) χ(λ) = P11 (λ)(P21 (λ)ψ01 (1; λ) + P22 (λ)ψ02 (1; λ)) − P12 (λ)(P21 (λ)ϕ01 (1; λ) + P22 (λ)ϕ02 (1; λ)), where the products of polynomials are all of degree N by assumptions. Introduce the function Y ˜ s )ps , (λ − λ Π(λ) = ˜ s ∈Λ λ

˜ s in the set Λ. Then the eigenvalues where ps is the multiplicity of the eigenvalue λ which do not belong to the set Λ are the roots of the entire function χ(λ) ˜ := −

χ(λ) . Π(λ)

It follows from Lemma 2.2 that the function χ(λ) ˜ satisfies the following estimate: χ(λ) ˜ = C1 exp(aλi) − C2 exp(bλi) +

1 O(max{exp(aλi), exp(bλi)}), λ

(3.3)

where C1 C2 6= 0. On the line = ln(C1 /C2 ) + (2n + 1)π , b−a which is determined by the equation arg(C1 exp(aλi)) = arg(−C2 exp(bλi)), one has |C1 exp(aλi) − C2 exp(bλi)| = |C1 exp(aλi)| + |C2 exp(bλi)|. (3.4) <λ =

From (3.3) and (3.4) one concludes that on this line, with |λ| large enough, |χ(λ) ˜ − (C1 exp(aλi) − C2 exp(bλi))| < |C1 exp(aλi) − C2 exp(bλi)|.

(3.5)

Therefore, it follows from Rouche’s theorem (see [18, Theorem 10.43; Exercise 24, p.229]) and the estimates (3.3) and (3.5) that for |n| large enough there exists precisely one root of χ(λ) ˜ in the strip (2n − 1)π < (b − a)<λ + = ln(C1 /C2 ) < (2n + 1)π,

(3.6)

and, furthermore, that there are 2|n| − 1 roots of the function χ(λ) ˜ in the strip −(2|n| − 1)π < (b − a)<λ + = ln(C1 /C2 ) < (2|n| − 1)π.

(3.7)

Thus, in particular, the roots of χ(λ) ˜ except for, possibly, a finite number of them, are simple. Moreover, the roots λn of χ(λ) ˜ can be ordered as a bilateral sequence, so that for |n| > n0 , λn belongs to the strip (3.6).

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Let λn,0 =

i ln(C1 /C2 ) + 2πn b−a

(3.8)

be the root of the function χ ˜0 (λ) = C1 exp(aλi) − C2 exp(bλi). Consider a disk D(n, ρ) with the radius ρ and the center λn,0 . For λ ∈ D(n, ρ) it follows from (3.3) that there exist K1 and K2 , such that |λ||χ(λ) ˜ −χ ˜0 (λ)| < K2 (| exp(a=λ)| + | exp(b=λ)|) < K1 .

(3.9)

Moreover, because =λ is independent of n, K1 is independent of n, too. On the other hand, because the derivative of the function χ ˜0 (λ) at λn,0 is nonzero, then, for ρ small enough there exists K3 , such that |χ ˜0 (λ)| > K3 |λ−λn,0 |. 1 and |λ − λn,0 | = ρ, then (3.9) implies that Therefore, if ρ ≥ KK3 |λ| |χ ˜0 (λ)| >

K1 | exp(a=λ)| + | exp(b=λ)| > K2 > |χ(λ) ˜ −χ ˜0 (λ)|. |λ| |λ|

(3.10)

Hence, again by Rouche’s theorem, in this disk the functions χ(λ) ˜ and χ ˜0 (λ) have the same number of roots, i.e., precisely one root. 2πn Therefore, |λn − λn,0 | < K3K|λ1 n | and since λn = b−a + O(1), one has K3K|λ1 n | = 1 ). Now, using the formula (3.8), the statement in (3.2) follows. O( |n| Lemmas 3.2 and 3.3 are used to prove the following theorem. Theorem 3.4. Assume that Q(x) ∈ C 1 [0, 1] ⊗ C2×2 and M (x, t) ∈ C 1 (Ω)⊗C2×2 . Let P11 (λ) and P12 (λ) be relatively prime polynomials such that deg P11 = deg P12 = N0 and let P21 (λ) and P22 (λ) be relatively prime polynomials such that deg P21 = deg P22 = N1 . Moreover, let Φ be a set, which consists of N = N0 + N1 eigenfunctions and associated functions of the problem (1.6), (3.1) and assume that the SEAF of this problem without the set Φ is complete in the space L2 [0, 1] ⊕ L2 [0, 1]. Then the SEAF of problem (1.6), (3.1) without the set Φ forms a Riesz basis in the space L2 [0, 1] ⊕ L2 [0, 1]. Proof. By Lemma 3.3, it is possible to enumerate the eigenvalues λn , corresponding to the eigenfunctions ωn (x) which are not contained in the set Φ, such that i ln(C1 /C2 ) + 2πn 1 λn = +O . b−a |n| Because ωn (x) satisfies the first of the conditions in (3.1), it may be written in the form ωn (x) = P12 (λn )ϕ0 (x; λn ) − P11 (λn )ψ0 (x; λn ) (up to a constant multiplier); cf. (2.17). Then, by Lemma 2.2, C12 exp(aλn ix) 1 (O(exp(aλn ix)) + O(exp(bλn ix))). (3.11) ωn (x) = + −C11 exp(bλn ix) λn

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Combining the estimates (3.2) in Lemma 3.3 with (3.11) one obtains /C2 )+2πn i ln(C /C ) + 2πn C12 exp(a i ln(C1b−a ix) 1 1 2 ωn (x) = + O exp a ix /C2 )+2πn n b−a ix) −C11 exp(b i ln(C1b−a i ln(C /C ) + 2πn 1 2 ix + O exp b b−a /C2 )+2πn 1 C12 exp(a i ln(C1b−a ix) . (3.12) = + O n −C11 exp(b i ln(C1 /C2 )+2πn ix) b−a

Now define the operator A : L2 [0, 1] ⊕ L2 [0, 1] → L2 [a, b] via ( 1 y1 ( x ), where a < x < 0 y1 A (x) = C121 a x y2 − C11 y2 ( b ), where 0 < x < b.

(3.13)

Then A and A−1 are bounded. Therefore, the system ωn (x) is a Riesz basis in the space L2 [0, 1] ⊕ L2 [0, 1] if and only if the system A(ωn ) is a Riesz basis in the space L2 [a, b]. From the estimate (3.12) and the definition of A in (3.13) one obtains 1 i ln(C /C ) + 2πn 1 2 ix + O . (3.14) A(ωn ) = exp b−a n It is obvious that the system i ln(C /C ) + 2πn 1 2 ix ω ˜ n = exp b−a is a Riesz basis in the space L2 [a, b] and that the norms of ω ˜ n are given by Z b <(ln(C /C )) 1 2 k˜ ωn k2 = x dx, n ∈ Z. exp −2 b − a a From the estimate (3.14) one concludes that ∞ X

kA(ωn ) − ω ˜ n k2 < ∞.

n=−∞

Therefore, by Lemma 3.2, A(ωn ) is a Riesz basis in L2 [a, b].

Theorem 3.4 is conditional in character since it assumes completeness of the root vectors of the problem (1.6), (3.1). Using Theorem 2.5 one can obtain an unconditional result under some additional assumptions on the set Φ and the polynomials Pjk (λ) in the boundary conditions (3.1). Theorem 3.5. Let Q(x) ∈ C 1 [0, 1] ⊗ C2×2 and M (x, t) ∈ C 1 (Ω)⊗C2×2 . Let deg P11 = deg P12 = 0, i.e., P11 6= 0 and P12 6= 0 are constants. Let P21 (λ) and P22 (λ) be relatively prime polynomials with deg P21 = deg P22 = N . Moreover, let Φ be a set, which consists of N eigenfunctions and associated functions, which satisfies the following condition:

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If Φ contains either an eigenfunction or an associated function uk corresponding to an eigenvalue λk , then it also contains all the associated functions of higher order corresponding to the same eigenvalue and the same function uk . Then the SEAF of the problem (1.6), (3.1) without the set Φ forms a Riesz basis in the space L2 [0, 1] ⊕ L2 [0, 1]. Proof. It is shown that the SEAF of the problem (1.6), (3.1) without the set Φ is complete in the space L2 [0, 1]⊕L2 [0, 1]. Let ωn (x) be the eigenfunctions associated to the eigenvalues λn as in the proof of Theorem 3.4, i.e., they corresponding to the function ω1 (x; λ) in Proposition 2.4. Now it suffices to consider orthogonality of f (x) only to ω1 (x; λ) in Theorem 2.5. The estimates in Remark 2.6 with j = 1 show that in the present case we can allow in Theorem 2.5 deg J14 − M1 = deg J14 = N eigenfunctions and associated functions in the set Ω therein. Hence, Theorem 2.5 implies that the SEAF of the problem (1.6), (3.1) without the set Φ is complete in the space L2 [0, 1] ⊕ L2 [0, 1]. Therefore, by Theorem 3.4 it is also a Riesz basis. By taking N = 0 in Theorem 3.5 we arrive at the following corollary for the system (1.6) with boundary conditions not depending on a spectral parameter. Corollary 3.6. Let Q(x) ∈ C 1 [0, 1] ⊗ C2×2 and M (x, t) ∈ C 1 (Ω)⊗C2×2 , and let h1 and h2 be nonzero numbers. Then the SEAF of problem (1.6) with the boundary conditions ( y1 (0) + h1 y2 (0) = 0, (3.15) y1 (1) + h2 y2 (1) = 0, forms a Riesz basis in the space L2 [0, 1] ⊕ L2 [0, 1]. Remark 3.7. For the special case of Dirac systems (b = −a = 1, M (x, t) ≡ 0) the result in Corollary 3.6 was proved earlier by I. Trooshin and M. Yamamoto [23] under the same assumption Q ∈ C 1 [0, 1] ⊗ C2×2 ; this smoothness assumption has been weakened to Q ∈ L2 [0, 1] ⊗ C2×2 in [24]. Also B. Mityagin has relaxed the smoothness assumptions on Q and established in [15, Theorem 8.8] an unconditional convergence result for the associated spectral decompositions involving Riesz projections for Dirac systems with periodic or anti-periodic boundary conditions. Remark 3.8. A referee has drawn our attention to the recent preprint [4] by P. Djakov and B. Mityagin which was submitted on arXiv.org after the submission of the present paper on arXiv.org and to IEOT. In [4] the Riesz basis property for Dirac systems has been established under the assumption Q ∈ L2 [0, 1] ⊗ C2×2 in the case of periodic, anti-periodic, and Dirichlet boundary conditions; see [4, Theorem 9]. Furthermore, in [4, Section 5.2] a similar statement is announced also for general regular boundary conditions. Their approach relies on the convergence P of the series |n|>N kPn − Pn0 k2 < ∞ (N sufficiently large) involving certain Riesz projections corresponding to the Dirac operator L with an L2 -potential and the free Dirac operator L0 (see [4, Theorem 3]) and then applies Bari-Markus Theorem (see e.g. [5]).

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Acknowledgment The authors would like to thank Mark Malamud for the fruitful discussions and useful remarks during the preparation of this paper.

References [1] N.I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert space, Moskow, “Nauka”, 1966. [2] P. Djakov, B. Mityagin, Instability zones of periodic 1D Schr¨ odinger and Dirac opero ators (Russian), Uspehi Mat. Nauk 61 N-4 (2006), 77–182 (English: Russian Math. o Surveys 61 N-4 (2006), 663–766). [3] P. Djakov, B. Mityagin, Bari-Markus property for Riesz projections of Hill operators with singular potentials, Manuscript, arXiv:0803.3170. [4] P. Djakov, B. Mityagin, Bari-Markus property for Riesz projections of 1D Periodic Dirac operators, Manuscript, arXiv:0901.0856. [5] I.C. Gohberg, M.G. Kre˘ın, Introduction to the theory of linear nonselfajoint operators, Moskva, “Nauka”, 1965. [6] S. Hassi, L.L. Oridoroga, Completeness theorems for Dirac-type operators with boundary conditions of general form depending on the spectral parameter, Math. Notes 74 o N-2 (2003), 316–320. [7] S. Hassi, L.L. Oridoroga, Completeness and Riesz basis property of systems of eigenfunctions and associated functions of Dirac-type operators with boundary conditions o depending on the spectral parameter, Math. Notes 79 N-4 (2006), 636–640. [8] S. Hassi, L.L. Oridoroga, Theorem of completeness for a Dirac-type operator with generalized λ-depending boundary conditions, Manuscript, arXiv:0808.0135v1. [9] M.V. Keldysh, On the characteristic values and characteristic functions of certain o classes of non-self-adjoint equations (Russian), Dokl. Akad. Nauk SSSR 77 N-1 (1951), 11–14. [10] B.M. Levitan, I.S. Sargsjan, Introduction to the spectral theory, Moscow, “Nauka”, 1970. [11] M.M. Malamud, Problems of the uniqueness in the inverse problems for the system of differential equations in bounded interval, Trans. Moscow Math. Soc. 60 (1999), 199–258. [12] M.M. Malamud, L.L. Oridoroga, Theorems of the completeness for the systems of o ordinary differential equations, Functional Analysis and Applications 34 N-3 (2000), 88–90. [13] V.A. Marchenko, Sturm-Liouville operators and their applications, Kyiv, “Naukowa dumka”, 1977. [14] R. Mennicken, M. M¨ oller, Non-self-adjoint boundary eigenvalue problems, NorthHolland Mathematics Studies, 192. North-Holland Publishing Co., Amsterdam, 2003. [15] B. Mityagin, Spectral expansions of one-dimensional periodic Dirac operators, Dyn. o Partial Differ. Equ. 1 N-2 (2004), 125–191. [16] M.A. Na˘ımark, Linear differential operators, Moscow, “Nauka”, 1968.

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[17] L.L. Oridoroga, Boundary value problems for 2 × 2 Dirac type systems with spectral parameter in boundary conditions, Methods of Functional Analysis and Topology 7 o N-1 (2001), 82–87. [18] W. Rudin, Real and complex analysis 3rd Edition, McGraw-Hill, 1986. [19] A.A. Shkalikov, The completeness of eigenfunctions and associated functions of an ordinary differential operator with irregular-spliting boundary conditions, Functional o Analysis and Applications 10 N-4 (1976), 69–80. [20] E.I. Tarapowa, Boundary-value problem of Sturm–Liouville equations with nonlinear boundary conditions. I, Theory of Functions, Functional Analysis and Applications 31 (1979), 157–160. [21] E.I. Tarapowa, Boundary-value problem of Sturm–Liouville equations with nonlinear boundary conditions. II, Theory of Functions, Functional Analysis and Applications 33 (1979), 82–87. [22] C. Tretter, Spectral problems for systems of differential equations y 0 + A0 y = λA1 y with λ-polynomial boundary conditions, Math. Nachr. 214 (2000), 129–172. [23] I. Trooshin, M. Yamamoto, Riesz basis of root vectors of a nonsymmetric system of first-order ordinary differential operators and application to inverse eigenvalue o problems, Appl. Anal. 80 N-1-2 (2001), 19–51. [24] I. Trooshin, M. Yamamoto, Spectral properties and an inverse eigenvalue problem for nonsymmetric systems of ordinary differential operators, J. Inverse Ill-Posed Probl. o 10 N-6 (2002), 643–658. [25] B.L. van der Waerden, Algebra, Frederick Ungar Publishing Co., New York, 1970. Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700 65101 Vaasa Finland e-mail: [email protected] Leonid Oridoroga Department of Mathematical Analysis Donetsk National University Universitetskaya str. 24 83055 Donetsk Ukraine e-mail: [email protected] Submitted: August 28, 2008. Revised: June 18, 2009.

Integr. equ. oper. theory 64 (2009), 381–398 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030381-18, published online June 26, 2009 DOI 10.1007/s00020-009-1695-9

Integral Equations and Operator Theory

Maximal Abelian von Neumann Algebras and Toeplitz Operators with Separately Radial Symbols Hansong Huang Abstract. This paper mainly concerns abelian von Neumann algebras generated by Toeplitz operators on weighted Bergman spaces. Recently a family of abelian w∗ -closed Toeplitz algebras has been obtained (see [5,6,7,8]). We show that this algebra is maximal abelian and is singly generated by a Toeplitz operator with a “common” symbol. A characterization for Toeplitz operators with radial symbols is obtained and generalized to the high dimensional case. We give several examples for abelian von Neumann algebras in the case of high dimensional weighted Bergman spaces, which are different from the one dimensional case. Mathematics Subject Classification (2000). Primary 47C15; Secondary 32A36. Keywords. Weighted Bergman spaces, abelian von Neumann algebra, Toeplitz operator, radial, separately radial.

1. Introduction Let D be the unit disk in C and dA the normalized area measure on it, i.e. 2 dA(z) = dxdy π (z = x+iy). Consider the weighted Bergman space Aα (D) (α > −1), consisting of all holomorphic functions which are square integrable on D with respect to the weighted measure dAα (z) = cα (1 − |z|2 )α dA(z), where cα satisfies R c (1 − |z|2 )α dA(z) = 1. Given a function a(z) ∈ L1 (D, dAα ), the Toeplitz operD α ator Taα with the symbol a is defined on A2α (D) as follows: Z Taα f (z) = K (α) (z, w)f (w)a(w)dAα (w), f ∈ A2α (D), D

where K (α) (z, ·) is the reproducing kernel at the point z ∈ D, i.e. 1 K α (z, w) = , w ∈ D. (1 − zw)2+α

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It is well known (for example, see [11]) that Toeplitz operators with radial symbols commute. Recently, S. Grudsky, A. Karapetyants, R. Quiroga-Barranco and N. Vasilevski have found a rich family of symbols which generate an abelian C ∗ algebra of Toeplitz operators. These abelian properties strongly depend on the geometry of the unit disk, the hyperbolic geometry, but less on the smoothness of symbols. Precisely, given a pencil of geodesics on D, consider the set A(D) of all (bounded) symbols constant on the corresponding cycles (see [2,8] for the definitions of pencil and cycles). The C ∗ -algebra or von Neumann algebra generated by Toeplitz operators with such symbols is abelian on each weighted Bergman space. For details, the reader can consult [5,6,7] and [20,21]. The inverse is true in an appropriate sense. In [8], they showed that under certain conditions, there are no other sets of symbols inducing abelian C ∗ -algebras of Toeplitz operators on each weighted Bergman space. The characterization for commutativity is surprising and deep. And very recently, a high dimensional approach to the unit ball Bn has been made by R. Quiroga-Barranco and N. Vasilevski, see [14] and [15]. There the commutativity property of C ∗ -algebras generated by Toeplitz operators and the geometric properties of these symbols of those operators are tightly related. These fruitful results are both interesting and instructive. It seems that the von Neumann algebra generated by Toeplitz operators related to a pencil (as mentioned above) is maximal abelian on each weighted Bergman space A2α (D). Actually, this is the case. Given a pencil of geodesics, the corresponding von Neumann algebra is maximal abelian. Moreover, we find that there is a single bounded symbol, whose corresponding Toeplitz operator generates this von Neumann algebra. And this symbol is independent of the parameter α of the space A2α (D). Moreover, it can be chosen to be smooth. A similar result is partially obtained in the high dimensional case. But there the term radial is substituted by separately radial, which plays an important role in our discussion. In what follows, when we say a symbol a is radial, we mean a is essentially radial. As far as the elliptic case is concerned, an interesting phenomenon is found, named “cycles determining cycles”. Given a Hilbert space H and two operators A and B on H, let [A, B] denote the commutator of A and B, i.e. AB − BA. Now assume that two Toeplitz operators commute on a weighted Bergman space: [Ta , Tb ] = 0 and a is radial and nontrivial, then b is radial. When a and b are bounded, this result is first proved on Bergman space as a corollary in [4]. Moreover, it can be translated to generic elliptic case via a M¨obius transformation and is reformulated as follows: suppose [Ta , Tb ] = 0 and neither a nor b is constant, and a is constant on cycles C corresponding to an elliptic pencil, then b is also constant on the same cycles C. Yet this result fails on the Fock space. When passing to high dimensional Bergman spaces, it fails either; and we find nontrivial examples for symbol class which induce abelian von Neumann algebra on each weighted Bergman space. These examples are different from one dimensional case. The author guesses that the cycles-determining-cycles phenomenon is likely to appear in the hyperbolic and parabolic cases.

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As for weighted Bergman spaces, if a Toeplitz operator is diagonal with respect to the standard orthogonal basis, then its symbol is radial. This result can nontrivially be passed to the high dimensional case. The paper is organized as follows: In Section 2 we show the maximal abelian property of the von Neumann algebra generated by Toeplitz operators with symbols which are constant on cycles corresponding to a given pencil. And we pick out the common symbol for each given pencil. In Section 3 we discuss with the elliptic case, which has been reduced to the radial case there. A cycles-determining-cycles phenomenon is found. Separately in Section 4 the radial case is treated in high dimensional weighted Bergman spaces. We give examples for abelian von Neumann algebras.

2. Maximal abelian von Neumann algebra For the definitions of pencil and cycles, the reader can consult [2] or [8]. Given a pencil P of geodesics, consider the set A(D) of L∞ -symbols which are constant on the corresponding cycles, then our main theorem in this section can be stated as follows: Theorem 2.1. On each weighted Bergman space A2α (D), let Tα (A(D)) be the Toeplitz algebra generated by Toeplitz operators with symbols in A(D). Then the w∗ -closure of Tα (A(D)) is maximal abelian. That is, any operator in the commutator of Tα (A(D)) lies in the w∗ -closure of the Toeplitz algebra Tα (A(D)). To prove Theorem 2.1, we need the following two lemmas. In fact, they are known to be consequences of a general result. For example, [3, p.55, Exercise 6] shows that if (X, Ω, µ) is a separable measure space and φ ∈ L∞ (X, µ), then the von Neumann algebra W ∗ (φ) generated by Mφ equals {Mg : g ∈ L∞ (X, µ)} if and only if there is a measurable subset Y of X having full measure on which φ is one-to-one. However, we include a proof independently. Lemma 2.2. Let X = R+ or R and f be a bounded, strictly increasing (or decreasing) function on X. Suppose T is a (bounded) operator acting on L2 (X) such that [T, Mf ] = T Mf − Mf T = 0. Then there is a g ∈ L∞ (X) such that T = Mg , where Mg denotes the multiplication operator by g on L2 (X). Proof. We will apply the spectral projection and deal with the case X = R only. The proof for the case X = R+ is similar. Let Eλ = E(f −1 (−∞, λ]) be the orthogonal projection from L2 (X) onto χf −1 (−∞,λ] L2 (X) . It is easily checked that Z Mf = f dEλ . X

It is well known that an operator T commutes with Mf if and only if it commutes with all spectral projections Eλ of Mf . Since f is strictly increasing (or decreasing),

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T commutes with all projections from L2 (X) onto χ(λ,µ] L2 (X) (λ < µ). Namely, T commutes with Mχ(λ,µ] . Thus T commutes with MPk ci χ(a ,b ] . A standard i=1 i i argument using the Lebesgue dominated convergence theorem shows that T commutes with each Mh with h ∈ L∞ (X). Hence the assertion follows at once from Theorem 4.1.2 in [1]. Applying the spectral projection or using functional calculus, we have the following lemma. Lemma 2.3. Let T be an operator on l2 = l2 (Z+ ) and S a diagonal operator with respect to the standard orthogonal basis of l2 . Write S = diag(λ1 , λ2 , λ3 , . . . ) with λi 6= λj provided i 6= j. If [T, S] = 0, then T is also a diagonal operator. In Lemma 2.2, the von Neumann algebra W ∗ (f ) generated by Mf is maximal abelian. The reasoning is as follows. By Lemma 2.2, it is not difficult to see that an operator T belongs to {Mf , Mf∗ }0 if and only if T is a multiplication operator. Thus the commutant W ∗ (f )0 of W ∗ (f ) equals {Mg : g ∈ L∞ (X)}, and then it is easy to see that W ∗ (f )00 = W ∗ (f )0 . By von Neumann’s bicommutant theorem, W ∗ (f )00 = W ∗ (f ), and hence W ∗ (f ) = W ∗ (f )0 , which implies that W ∗ (f ) is a maximal abelian von Neumann algebra. A similar argument shows that in Lemma 2.3 the von Neumann algebra generated by S is maximal abelian. Now we come to the proof of Theorem 2.1. Proof of Theorem 2.1. The theorem will be proved for three cases: elliptic, parabolic and hyperbolic pencils. 1. Elliptic case. In this case, it suffices to consider the radial case, which is also considered in [18]. To see this, notice that Uz is a unitary operator on the weighted Bergman space A2α (D) defined by ∀f ∈ A2α (D),

Uz f = (f ◦ ϕz )kz ,

where ϕz is the M¨ obius transformation that interchanges z and 0, and kz is the normalized reproducing kernel at the point z. Notice also that Uz intertwines Ta and Ta◦ϕz , i.e., Uz Ta = Ta◦ϕz Uz , which implies that studying the elliptic case can be reduced to studying the radial case. Assume without loss of generality that a is radial. The symbol a = a(r) defines a Toeplitz operator Ta on the weighted Bergman space A2α (D), which is unitarily isomorphic to the diagonal operator diag(λ(1), λ(2), λ(3), . . . ) on l2 (see [5]), where Z 1 √ λ(n) = B(n + 1, α + 1)−1 a( r)rn (1 − r)α dr. 0

Here B is the Beta function.

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To prove the theorem in this case, it suffices to show that the von Neumann algebra generated by some single operator Ta is maximal abelian. In particular, √ we take a( r) = r. Now B(n + 2, α + 1) B(n + 1, α + 1) n+1 α+1 = =1− . n+α+2 n+α+2

λ(n) =

Notice that α > −1, so α + 1 > 0. Thus λ(n) is strictly increasing. By Lemma 2.3, the corresponding diagonal operator diag(λ(1), λ(2), λ(3), . . . ) generates a maximal abelian w∗ -closed algebra (that is, a maximal abelian von Neumann algebra). The same is true for Ta . 2. Parabolic case. In this case, we write λ for α. Let A2λ (Π) denote the weighted Bergman space consisting of functions that are analytic in the upper half-plane λ and square integrable with respect to the measure (λ+1) π (2y) dxdy (z = x + iy). It is shown in [6] that a symbol a = a(y) on the upper half plane induces the Toeplitz operator Ta on A2λ (Π), which is unitarily isomorphic to the multiplication operator Mγλ on L2 (R+ ) with Z ∞ xλ+1 γλ (x) = a(t/2)tλ e−xt dt. Γ(λ + 1) 0 The study of Toeplitz algebra in parabolic case is equivalent to the study of the subalgebra of the multiplication algebra generated by multipliers of the above form. It is easy to check that Z ∞ Γ(λ + 1)γλ0 (x) = xλ a(t/2)tλ e−xt (λ + 1 − xt)dt. 0

Write I =

R∞ 0

a(t/2)tλ e−xt (λ + 1 − xt)dt. 1

Put h(t) = e− 1−t χ[0,1) (t) and set a(t) = h(2t)(t > 0). Then a(t/2) = h(t). Notice that h is a decreasing function in C 1 [0, 1] satisfying h(1) = 0. Integration by parts yields Z 1 I= h(t)tλ e−xt (λ + 1 − xt)dt 0

Z

1

h(t)tλ e−xt dt +

= (λ + 1) 0

Z = (λ + 1) =− 0

1

h(t)tλ tde−xt

0 1 λ −xt

h(t)t e

Z dt + 0 − 0 −

0

Z

Z

1

h0 (t)tλ+1 e−xt dt > 0.

0

1

[h0 (t)tλ+1 + (λ + 1)tλ h]e−xt dt

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Then it follows that γλ (x) is a strictly increasing function. Its boundedness follows from that of a. Thus by Lemma 2.2, Mγλ generates a maximal abelian von Neumann algebra in B(L2 (R+ )). Thus Ta generates a maximal abelian von Neumann algebra in B(A2λ (Π)). 3. Hyperbolic case. In this case, we write λ for α. The symbol a on L∞ (Π) can be written as a = a(θ). It is shown in [7] that the corresponding Toeplitz operator Ta on A2λ (Π) is unitarily isomorphic to the multiplier on L2 (R) by Z π γλ (ξ) = 2λ (λ + 1)ϑ2 (ξ) a(θ)e−2ξθ sinλ (θ)dθ, ξ ∈ R 0

−1 . where ϑ2 (ξ) = [2λ (λ + 1)]−1 0 e−2ξθ sinλ (θ)dθ Let us take a = χ[δ,π] , where δ ∈ (0, π). It will be shown that this a gives a strictly decreasing function γλ for each λ ∈ (−1, ∞). For any 4ξ > 0, R π −2(ξθ+4ξθ) λ e sin (θ)dθ γλ (ξ + 4ξ) = Rδπ −2(ξθ+4ξθ) sinλ (θ)dθ e 0 R −24ξθ 0 π −2(ξθ) e e sinλ (θ)dθ δ = Rπ e−2(ξθ+4ξθ) sinλ (θ)dθ 0 Rπ

where θ0 ∈ (δ, π). The last identity is derived from the mean-value theorem of integral. Notice that Z δ Z π Z π e−2(ξθ+4ξθ) sinλ (θ)dθ = + e−2(ξθ+4ξθ) sinλ (θ)dθ 0

0 −24ξθ 00

Z

δ −2(ξθ)

=e

> e−24ξθ

0

e Z

δ −24ξθ 0

λ

0 δ

Z

=e

π

sin (θ)dθ + e

e−2(ξθ) sinλ (θ)dθ + e−24ξθ

0 −24ξθ 0

Z

0

Z

e−2(ξθ) sinλ (θ)dθ

δ π

e−2(ξθ) sinλ (θ)dθ

δ π

e−2(ξθ) sinλ (θ)dθ,

0 e−24ξθ

0

Rπ

where θ00 ∈ (0, δ). Thus γλ (ξ + 4ξ) < e−24ξθ0 Rδπ = γλ (ξ), and then γλ is strictly 0 decreasing. If a is substituted by the finite linear span of χ[δ,π] with positive coefficients, the same result holds. The above a can be chosen to be smooth. For example, let a(θ) = θ. Since this a can be uniformly approximated by the finite linear span of {χ[δ,π] : 0 < δ < 1} with positive coefficients, then γλ is decreasing. Moreover, γλ is strictly decreasing. To see this, assume conversely that there exist ξ1 < ξ2 and a constant c such that γλ (ξ) = c, ξ ∈ (ξ1 , ξ2 ). That is, Z π e−2ξθ sinλ (θ)(a(θ) − c)dθ = 0, ξ ∈ (ξ1 , ξ2 ). 0

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Fix ξ = ξ0 ∈ (ξ1 , ξ2 ) and take the nth derivative at x0 of the above identity. Then we get Z π

θn e−2ξ0 θ sinλ (θ)(a(θ) − c)dθ,

n = 1, 2, . . . .

0

Since the span of {θn } is dense (with respect to maximum norm) in those continuous functions on [0, π], the above shows that e−2(ξ0 θ) sinλ (θ)(a(θ) − c)) = 0,

a.e. θ.

So a = c, e.e., which is a contradiction. Therefore γλ is strictly decreasing. Again, Lemma 2.2 guarantees the maximality of the von Neumann algebra generated by the multiplication operator Mγλ on L2 (R) and of the von Neumann algebra generated by the Toeplitz operator Ta on A2λ (Π). The proof of Theorem 2.1 is complete. From the above proof, we have the following consequence of Theorem 2.1. Corollary 2.4. The von Neumann algebra Tα (A(D)) can be generated by a single Ta , whose symbol a is smooth and independent of α ∈ (−1, +∞).

3. Toeplitz operators with radial symbols In this section, we will study the cycles-determining-cycles phenomenon. That is, if a Toeplitz operator commutes with another Toeplitz operator with nonconstant radial symbol, then the symbol of the first Toeplitz operator is also radial. In this section, for a fixed α, each symbol is assumed to be in L1 (D, dAα ). We call a symbol trivial if it is almost everywhere constant with respect to the area measure over D. When we say a symbol is radial, we mean that the symbol is essentially radial. When a and b are bounded, the following theorem appears as a corollary of Theorem 6 in [4]. However, the proof is completely different. Theorem 3.1. Given a weighted Bergman space A2α (D), if [Ta , Tb ] = 0 and a is radial and nontrivial, then b is radial. As an immediate consequence of Theorem 3.1, we have Corollary 3.2 (cycles-determining-cycles phenomenon). Given an elliptic pencil, let a be a nontrivial symbol which is constant on the corresponding cycles. Suppose b is such that [Ta , Tb ] = 0. Then b is constant on the same cycles. The cycles in Corollary 3.2 are exactly the image of all those circles in D with the center 0 under some M¨ obius map; equivalently, they are just those hyperbolic circles in D with a same center w0 ∈ D. However, Theorem 3.1 does not hold for the Fock space F 2 (C) consisting of square integrable entire functions over C with respect to the Gaussian measure 2 e−|z| dA(z). The following example shows this.

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Example. Consider the Fock space F 2 (C). For each pair of integers (m, t) with 0 ≤ m < t, there is a radial symbol whose associated Toeplitz operator is precisely the projection P onto the closed span of {z m+nt |n ∈ Z+ }. The existence is granted by Theorem 3.7 in [9]. Then any bounded operators Ta and Ta∗ commute with P , provided that a is of the form z t f (|z|). The following lemmas are the essential ingredients of Theorem 3.1. Lemma 3.3. Let f be a bounded holomorphic function defined on the half plane {z | Rez > a} (a is a real number), and let λ1 , λ2 , λ3 , . . . be all its zeros on the half real axis {x|x > a} (repeated according to multiplicity). Then for any δ > 0, X |λi |−1 < +∞. 06=λi >a+δ z−(a+δ) is bi-holomorphic from {z | Rez > a} Proof. The mapping w = w(z) = z−(a−δ) onto the unit disk. Let z = z(w) be its inverse. So f ◦z(w) is a bounded holomorphic function on the unit disk. w(λ1 ), w(λ2 ), w(λ3 ), . . . are contained in the zero-set of f , so they satisfy the Blaschke condition: X (1 − |w(λi )|) < +∞. P P −1 So < +∞. Basic λi >a+δ (1 − |w(λi )|) < +∞, i.e., 2δ λi >a+δ (λi − a + δ) arguments lead to the desired conclusion.

The following proposition is inspired by the Muntz-Szasz Theorem ([16, Theorem 15.26]), which is of independent interest. P 1 Proposition 3.4. Let Λ be a subset of Z+ satisfying n∈Λ n+1 = +∞. For a Toeplitz operator Ta on the weighted Bergman space A2α (D) (−1 < α < ∞), if Ta z n = λ(n)z n for all n ∈ Λ, then a is radial and Ta is diagonal with respect to {z n |n ∈ Z+ }. Moreover, if λ(n) ≡ c, then a = c, a.e. Proof. Assume Ta z n = λ(n)z n , ∀n ∈ Λ. Then for any n ∈ Λ and m 6= n, hTa z n , z m i = 0. So hTa z n , z n+k i = 0, where 0 < |k| ≤ n. By computations, for such n and k we have Z 1 Z 2π r2n+k {r(1 − r2 )α a(reiθ )e−ikθ dθ} dr = 0. 0

0

Set Z fk (z) =

1

Z r2z+k r(1 − r2 )α

0

2π

a(reiθ )e−ikθ dθ dr

0 2 α

R 2π

iθ

−ikθ

and let gk (r) = r(1 − r ) 0 a(re )e dθ. Notice that fk (z) is analytic in Ω = {z| 2Rez + k > 0}. Moreover fk can be continuously extended to the closure Ω of Ω; and by Fubini’s theorem, fk is bounded since gk lies in L1 [0, 1].

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Notice also that for all integers n ≥ |k|, n are zeros of fk . Then by Lemma 3.3, fk is constantly 0. In particular, for those z satisfying 2z+k = 0, 1, 2, . . . , fk (z) = 0. That is, Z 1 rm gk (r)dr = 0, ∀ m ∈ Z+ . 0

(The identity fk (−k/2) = 0 is deduced from the continuity property of fk ). Since the linear span of polynomials on the unit interval is dense in C[0, 1], gk = 0, a.e. Therefore for each integer k 6= 0, there is a Lebesgue null set Ek in [0, 1] such that Z 2π a(reiθ )e−ikθ dθ = 0, r ∈ / Ek . (3.1) 0 S For each r outside the null set Ek , (3.1) holds for all k ∈ Z − {0}. Since a is iθ integrable, / E0 . S there is a null set E0 such that a(re ) is integrable in θ for all r ∈ For r 6∈ k∈Z Ek , put Z 2π 1 a(reiθ )dθ, ar = 2π 0 and by (3.1), Z 2π (a(reiθ ) − ar )e−ikθ dθ, k ∈ Z. 0 S By the theory of Fourier analysis, a(reiθ ) − ar = 0, a.e. θ holds for r 6∈ k∈Z Ek . Thus a is radial. The proof of the second assertion is similar and is omitted here. Now we are ready to prove Theorem 3.1. Proof of Theorem 3.1. To prove the theorem, it suffices to deal with the case that a is real valued. In fact, we notice that Ta is diagonal. Write a = a1 + ia2 , where a1 and a2 are real valued. Since a is radial, a1 and a2 are radial. Without loss of generality, a1 is nontrivial. We will show [Ta1 , Tb ] = 0. Since Ta is diagonal, Ta is normal. It is well known that any operator that commutes with a normal operator N also commutes with its adjoint N ∗ (see [12, Proposition 4.4.12]). Consequently, if V commutes with Ta , then V commutes with Ta∗ or Ta¯ . It follows immediately that V commutes with Ta1 and Ta2 . Taking V = Tb , we have [Ta1 , Tb ] = 0. Thus it suffices to prove the theorem in the case that a is real valued. We will deal with the unweighted case first, and the proof is split into two steps for convenience. Step 1. The Toeplitz operator Ta is unitarily isomorphic to the the diagonal operator diag(λ(1), λ(2), λ(3), . . . ) on l2 where Z 1 −1 Z 1 √ λ(n) = rn (1 − r)α dr a( r)rn (1 − r)α dr. 0

0

We will show that there is a strictly increasing sequence {µn }: 0 < µ1 < µ2 < µ3 < . . .

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such that

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< +∞ and λ(j) 6= λ(k), ∀j < k

if there is an l such that j, k ∈ [µl , µl+1 ]. R1 R1 √ Write λ(z) = ( 0 rz (1 − r)α dr)−1 0 a( r)rz (1 − r)α dr with α = 0. Then Z 1 Z 1 √ √ λ0 (z) 1 a( r)rz dr. = a( r)rz ln rdr + z+1 z + 1 0 0 λ0 (z) is not constantly zero. Otherwise λ(z) is constant on {x ∈ R|x ≥ 1}. Therefore λ(n) = λ(2)(n ≥ 2), and by Proposition 3.4, a is trivial. This is a contradiction to the assumption. 0 (z) is a bounded holomorphic function on the right half plane, Notice that λz+1 and it has the same zero set as λ0 on {x ∈ R|x > 0}. By Lemma 3.3, we may assume without loss of generality that there are infinitely many zeros of λ0 on (0, +∞) which we may write as µ1 < µ2 < µ3 < . . . satisfying ∞ X 1 < +∞. µ i=1 i

Since λ(x) is real on R+ , λ(x) is strictly monotone in [µi , µi+1 ], which leads to our assertion as desired. Step 2. Since Tb commutes with TaP , Tb commutes with all spectral projections of Ta . Precisely, we can write Ta = λ(nj )Ej , where λ(nj ) are pairwise different and Ej are pairwise orthogonal projections. Denote the range of Ej by Hj . Let Fj denote the set {n ∈ Z+ |z n ∈ Hj }. Since Tb commutes with each Ej , hTb z n , z m i = 0,

where n ∈ Fj , m ∈ Fk with j 6= k.

(3.2)

Given an integer ∆ 6= 0, we will seek integers n with |∆| ≤ n, such that there are two integers j, k (j 6= k) (depending of both n and ∆) satisfying n ∈ Fj , m = n + ∆ ∈ Fk . Then by (3.2), we have Z 1 r

2n+4

Z r

0

2π iθ

−i4θ

b(re )e

dθ dr = 0.

(3.3)

0

P We will show that for each nonzero integer ∆, there is a set Λ such that n∈Λ +∞ and that each n ∈ Λ satisfies (3.3). If this has been shown, then the proof of Proposition 3.4 yields that Z 2π r b(reiθ )e−i4θ dθ = 0

1 n

=

0

for a.e. r and all nonzero integers 4. Then following the last paragraph of the proof of Proposition 3.4, b is radial. Thus it suffices to construct the set Λ.

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Without loss of generality, fix 4 > 0. As for those intervals of the form (µj , µj+1 ] whose lengths are ≤ 24, we choose all the integers in such intervals to compose Γ. And on each interval (µi , µi+1 ] whose length is > 24, we pick the integers mi , mi + 4, mi + 24, . . . , mi + ti 4, where ti is the largest integer satisfying mi + (ti + 1)4 ≤ µi+1 . These numbers make up Λ. Now we have a trivial identity: X1 X1 X 1 = + = +∞, n n n c n≥1

n∈Γ

n∈Γ

c

where Γ denotes the complement of Γ in Z+ − {0}. It is not difficult to see that there exist two positive constants c1 and c2 such that X 1 X1 ≤ c1 + c2 < +∞. n µi n∈Γ P So n∈Γc n1 = +∞. But a simple calculation shows that there are also constants M1 , M2 > 0 such that X 1 X 1 X 1 ≤ M1 + M2 4 , forcing = +∞. n n n c n∈Γ

n∈Λ

n∈Λ

Since λ(x) is strictly monotone in [µi , µi+1 ], we have, by the construction of Λ, that each n ∈ Λ satisfies (3.3). So the construction of Λ is complete and we are done in the unweighted case. For the weighted case, it is sufficient to discuss the step 1. Step 2 is similar. We regard γ as a real valued function on R+ , and then discussing with its derivation comes down to studying the function Z 1 √ Z 1 x α Dα (x) = a( r)r ln r(1 − r) dr rx (1 − r)α dr 0

−

Z

1

√ a( r)rx (1 − r)α dr

0

Z

0 1

rx ln r(1 − r)α dr .

0

This Dα can be naturally extended to a bounded holomorphic function on the half plane {z | Rez > 1}. The remaining is similar. Remark 3.5. In some sense, P the1 result of Proposition 3.4 is the best possible. Precisely, the condition n∈Λ n+1 = ∞ is sharp. In fact, by the proof of [16, Theorem 15.26], we P have the following: for any sequence of real numbers 0 < λ1 < λ2 < . . . satisfying n λ1n < ∞, there exists a bounded continuous function u on (0, 1], such that Z 1 F (z) = tz u(t)dt 0

is a bounded holomorphic function over {z ∈ C : Re z ≥ −1}, whose zero set equals {λn }. For example, consider the Bergman space A2 (D). Take a subset Λ = {λn } of

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P 1 Z+ satisfying n∈Λ n+1 < ∞ and put f (z) = u(|z|2 ). Then by simple calculations n we have Tf z = 0, n ∈ Λ. However, Tf 6= 0. In the high dimensional case, Theorem 3.1 fails. Theorem 1.2 in [19] shows that if f is a nontrivial bounded radial function over the unit ball Bn , and g is a square-integrable function over Bn , then [Tf , Tg ] = 0 holds on the Bergman space A2 (Bn ) if and only if g(eiθ z) = g(z) holds for a.e. (θ, z) ∈ R × Bn . In what follows, we shall discuss in detail the counterpart of the radial case in the high dimensional case.

4. High dimensional case In this section we study Toeplitz operators with radial symbols and separately radial symbols on high dimensional weighted Bergman spaces. Proposition 3.4 is generalized. Examples of abelian von Neumann algebras are given. And Theorem 2.1 is partially generalized. We adopt the notations in [17]. Bd will denote the open unit ball in Cd , z = (z1 , z2 , . . . , zd ) and z 0 = (z1 , z2 , . . . , zd−1 ). Let v be a normalized Lebesgue measure on Cd such that v(Bd ) = 1. If m2d is the ordinary Lebesgue measure d on R2d , then cd v = m2d where cd = πd! . Let β and γ (or β 0 , γ 0 ) be multi-indices referring to an ordered d-tuple (respectively (d−1)-tuple ) of nonnegative integers. Multi-indexes I and J are adopted in a symbol. The following abbreviated notations will be used: |β| = β1 + β2 + · · · + βd , Pd 2 β! = β1 !β2 ! . . . βd !, and z β = z1β1 z2β2 . . . zdβd . Let |z|2 = j=1 |zj | . A function on Bd is called separately radial if it depends only on |z1 |, |z2 |, . . . and |zd |. As in Section 3, when we say a symbol is separately radial, we mean the symbol is essentially separately radial. The following theorem can be regarded as an extension of Proposition 3.4, which was independently obtained by L. Trieu [18]. We include a proof for completeness. Actually the “if” part is an immediate consequence of Theorem 3.1 in [13], which states that a Teoplitz operator with radial like symbol is diagonal in the weighted Bergman space A2µ (Ω), consisting of analytic functions square integrable with respect to the weighted measure µdm (note that the term separately radial is synonymous with the term radial like in [13]). Here µ is a radial like, positive measurable function and Ω is a logarithmically convex complete Reinhardt domain centered at the origin. Theorem 4.1. Suppose that a ∈ L1 (Bd ). Then Ta is diagonal with respect to {z β | β ∈ Zd+ } on the Bergman space A2 (Bd ) if and only if a is separately radial. Proof. “If” part: It suffices to show that for all β 6= γ, hTa z β , z γ i = 0, i.e. haz β , z γ i = 0.

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Without loss of generality, assume that βd 6= γd . Since a is separately radial, Fubini’s theorem implies that Z Z 0 γ0 haz β , z γ i = d z 0 β z 0 dv(z 0 ) √ a(z 0 , |zd |)zdβd zd γd dv(zd ). 1−|z 0 |2 D

Bd−1

(Notice that in the above identity, the former v denotes the normalized Lebesgue measure such that v(Bd−1 ) = 1, and the latter v denotes the normalized Lebesgue measure such that v(D) = 1.) But for each fixed z 0 , the latter integral is killed by passing to polar coordinates. So haz β , z γ i = 0. “Only if” part: We shall first show that, for almost every z 0 , a(z 0 , zd ) is essentially constant on {zd ||zd | = c} for a.e. c ∈ (0, 1). To this end, consider β, γ with βd 6= γd . Thus haz β , z γ i = 0 by assumption. We deduce that Z Z 0 γ0 βd γd a(z 0 , zd )z 0β z 0 dv(z 0 ) = 0. zd zd dv(zd ) √ 1−|zd |2 Bd−1

D

R 0 γ0 Denote √1−|z |2 B a(z 0 , zd )z 0β z 0 dv(z 0 ) by I = I(zd ), which is in L1 (D) apd d−1 pealing to Fubini’s theorem. Thus applying Proposition 3.4 shows that I is a radial function in zd . R 2π Denote by a ˜ =a ˜(z 0 , |zd |) the integral 0 a(z 0 , zd eiθ )dθ/(2π) and put g = a−a ˜. We write g = gzd (z 0 ) now. It is evident that for fixed |zd |, Z 0 γ0 gzd (z 0 )z 0β z 0 dv(z 0 ) = 0 √ 1−|zd |2 Bd−1

holds for all β 0 , γ 0 . Then the Stone-Weierstrass theorem forces gzd = 0 a.e. z 0 . Recall that g = a − a ˜ and a ˜=a ˜(z 0 , |zd |), and we will have a = a ˜ by changing the values of a on a null set. So a = a(z 0 , |zd |), as desired. A similar argument shows that after changing the values of a on a null set carefully, we have a = a(z1 , . . . , zd−2 , |zd−1 |, |zd |). By induction, we have a = a(|z1 |, . . . , |zd−1 |, |zd |) after changing the values of a on a null set, which completes the proof. Using the idea in Proposition 3.4 and that in Theorem 4.1, we get a more general result as follows, which is probably contained in [18]. Proposition 4.2. Pick d subsets of Z+ , Λ1 , Λ2 , . . . , Λd such that X 1 = +∞, ∀1 ≤ j ≤ d. n+1 n∈Λj

If a ∈ L1 (Bd ) and Ta is diagonal with respect to {z β |β ∈ Λ1 × Λ2 × · · · × Λd } on A2 (Bd ), then a is separately radial. Notice that Proposition 4.2 still holds if A2 (Bd ) is replaced with A2 (Dd ).

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Before continuing, let us introduced two spaces of holomorphic functions. Denote by A2ρ (Bd ) the function space Z n o f ∈ Hol(Bd ) kf k2 = |f (z)|2 ρ(|z|)dv(z) < +∞ , Bd

which can be viewed as a special case of A2τ (Bd ) defined by Z o n |f (z)|2 τ (|z1 |, . . . , |zd |)dv(z) < +∞ f ∈ Hol(Bd ) kf k2 = Bd

where τ = τ (r1 , r2 , . . . , rd ) is a strictly positive continuous function. Both A2ρ (Bd ) and A2τ (Bd ) are Hilbert spaces with reproducing kernels. They are special cases of A2µ (Ω) in [13]. Denote by K τ (z, ·) (z ∈ Bd ) the reproducing kernel in A2τ (Bd ) satisfying hf, K τ (z, ·)i = f (z), ∀f ∈ A2τ (Bd ). Then define the Toeplitz operator Taτ with the symbol a ∈ L1 (Bd , τ dA(z)) on A2τ (Bd ) as follows: Z Taτ f (z) = a(w)f (w)K τ (z, w)τ (w)dA(w), f ∈ A2τ (Bd ). Bd 2 α In particular, R if we take ρ(r) = cα (1 − r ) where α ∈ (−1, +∞), and cα is such that cα Bd (1 − |z|2 )α dv(z) = 1, the weights cα (1 − r2 )α define the oneparameter family of weighted Bergman spaces. In this case, we write A2α (Bd ) for A2ρ (Bd ). Generally, we rewrite Ta for Taτ if there is no confusion. Theorem 4.1 wins on generality. Precisely, it remains valid on A2τ (Bd ). The proof is essentially the same. Our next objective is to show that all separately radial symbols induce a maximal abelian von Neumann algebra on each weighted Bergman space and that we can pick a “common” symbol which gives the generator as in Section 2.

Theorem 4.3. Denote by A(Bd ) the set of all bounded separately radial symbols over Bd . On each weighted Bergman space A2α (Bd ), let Tα (A(Bd )) be the Toeplitz algebra generated by Toeplitz operators with symbols in A(Bd ). Then the w∗ closure of Tα (A(Bd )) is maximal abelian. Moreover, we can pick a smooth symbol b (independent of α) in A(Bd ) so that Tb generates Tα (A(Bd )) for each α ∈ (−1, +∞). Proof. To prove the theorem, it suffices to show that there is a symbol b such that the von Neumann algebra generated by Tb is maximal abelian. On the weighted Bergman space A2α (Bd ), consider the d + 1 symbols a0 , a1 , . . . , ad , where aj = |zj |2 ( i = 1, 2, . . . , d) and a0 = a0 (r) is chosen to be the function in the elliptic case in Section 2 (for example a0 (r) = r2 ). All Taj are diagonal.

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Let Taj z β = λi (β)z β . By [17, 1.4.3], we have βj + 1 κ(|β|, α) for 1 ≤ j ≤ d, d + |β| R1 R1 where κ(|β|, α) = 0 r2d+1+2|β| (1−r2 )α dr/ 0 r2d−1+2|β| (1−r2 )α dr. We also have Z 1 Z 1 r2d−1+2|β| (1 − r2 )α dr. r2d−1+2|β| a0 (r)(1 − r2 )α dr/ λ0 (β) = λj (β) =

0

0

λ0 (β) is injective in |β| by our choice. That is, if |β| = 6 |γ|, then λ0 (β) 6= λ0 (γ). The imaginary unit i simplifies our construction of the common symbol. βj +1 Note that all d+|β| are rational numbers. Pick d rationally linear independent real Pd irrational numbers ε1 , . . . , εd and let b = ia0 + j=1 εj aj . Write Tb z β = λb (β)z β , Pd where λb (β) = iλ0 (β) + j=1 εj λj (β). λb (β) is injective in β. Namely, the λb (β) are pairwise different. In fact, if Pd λb (β) = λb (γ), we have λ0 (β) = λ0 (γ), forcing |β| = |γ|. So j=1 εj λj (β) = Pd j=1 εj λj (γ). That is, d X j=1

d

εj

X γj + 1 βj + 1 κ(|β|, α) = εj κ(|β|, α). d + |β| d + |β| j=1

Noting that κ is never zero, we have d X j=1

d

εj

X γj + 1 βj + 1 = εj , d + |β| j=1 d + |β|

forcing βj = γj , ∀j.

By Lemma 2.3, the usual unitary isomorphism between B(A2α (Bd )) and B(l2 ) shows that Tb generates a maximal abelian von Neumann algebra on A2α (Bd ). Remark 4.4. After the author gave the proof of Theorem 4.3, he found that a similar result has also been obtained by L. Trieu, see [18] and [19]. But there the smoothness for the symbol b is not required. Generally speaking, TzI1 z¯J1 does not commute with TzI2 z¯J2 . However, the following proposition will give abundant examples for abelian von Neumann algebras generated by Toeplitz operators. Note that a d-tuple index can also be regarded as a function on {1, 2, . . . , d}, so the support supp I of I makes sense. Proposition 4.5. Consider A2ρ (Bd ). If supp(I1 + J1 ) ∩ supp(I2 + J2 ) = φ and |I1 | = |J1 | = |I2 | = |J2 |, then TzI1 z¯J1 h1 (r) commutes with TzI2 z¯J2 h2 (r) , where qP d 2 r = j=1 |zj | . Moreover Tz I1 z¯J1 h1 (r) commutes with Toeplitz operators with radial symbols.

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Proof. To prove the proposition, we shall first compute the integral Z |z γ |2 b(|z|)ρ(|z|)dv(z) = I. Bd

R1 R By [17, 1.4.3], we have I = 2d 0 r2d−1+2|γ| b(r)ρ(r)dr S |ζ γ |2 dσ(ζ). Thus I = c1 (b, |γ|)γ!, where c1 (b, |γ|) denotes some constant depending on b and |γ|. It is easy to see that TzI z¯J h(r) z β = λ(β)z β+I−J (if one component of β +I −J is < 0, we understand z β+I−J to be 0). We must determine λ(β) now. Since hTzI z¯J h(r) z β , z β+I−J i = λ(β)hz β+I−J , z β+I−J i, we have Z |z

β+I 2

Z

| h(|z|)ρ(|z|)dv(z) = λ(β)

Bd

|z β+I−J |2 h(|z|)ρ(|z|)dv(z).

Bd

(β+I)! . It is remarkBy some manipulations, we get λ(β) = c2 (|β|, |I|, |J|) (β+I−J)! (β+I)! able that (β+I−J)! depends only on I, J and the components of β on supp J. This observation grantees our first conclusion. A similar argument shows that any Toeplitz operator with radial symbol is of P the form n≥0 c0n Pn , where Pn is the orthogonal projection onto the linear span of {z β | |β| = n}. Then our second conclusion follows and the proof is complete.

This proposition immediately gives one type of abelian von Neumann algebras. Example. Now take a = z I1 z¯J1 h1 (r) and b = z I2 z¯J2 h2 (r) where I1 , J1 , I2 , J2 satisfy the conditions in Proposition 4.5 and h1 , h2 are real valued. Let a ˜ = a+a ¯ and ˜b = b + ¯b. Proposition 4.5 shows that if we denote by A(Bd ) the function space spanned by {˜ a, ˜b} ∪ {radial functions over Bd }, then A(Bd ) induces an abelian von Neumann algebra on each A2ρ (Bd ). But it is notable that there is a symbol ˜b which is not separately radial. This implies that abelian C ∗ -algebras or von Neumann algebras on weighted Bergman spaces A2α (Bd ) in the high dimensional case are different from those in the one dimensional case. Recall that in the dimension-one case, no nontrivial examples were given except the standard three cases: elliptic, parabolic and hyperbolic. Let us see more examples. Example. On the weighted Bergman space A2α (Bd ) (d ≥ 4), let a = a(|z1 |2 + |z2 |2 ). Then for each pair of natural numbers m ≤ n, each orthogonal projection P onto the span of {z β | |β| = n, β1 + β2 = m} is in the commutant of Ta . Indeed, Ta can be written as the sum (in the strong topology sense) of multiples of the above pairwise orthogonal projections. Thus it is readily seen that Tb commutes with Ta provided b has one of the following forms: z12 z2 2 , z1 z2 , or z3 z4 . As in the above example, let now A(Bd ) be the linear span of {z3 z4 + z4 z3 } ∪ {b| b = b(|z1 |2 + |z2 |2 ) or b = b(r)}.

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Note also that Tz1 z2 and Tz2 z1 commute with Tb if b = b(|z1 |2 + |z2 |2 ), but Tz1 z2 does not commute with Tz2 z1 . This is different from the one-dimensional case: by Theorem 3.1, if Tb1 and Tb2 commute with some Ta where a is radial and nontrivial, then Tb1 commutes with Tb2 . The next example can be viewed as interesting, compared with the example below Proposition 4.5. Example. Let ˜bh = (z I z¯J + z J z¯I )h(r) (|I| = |J|). Then T˜bh commutes with each orthogonal projection Pn onto the linear span of {z β | |β| = n}. So T˜bh can be decomposed into diagonal blocks. Precisely, T˜bh is of the form Vh0 ⊕ Vh1 ⊕ · · · ⊕ Vhn ⊕ . . . , where Vhn is an operator on the linear span of {z β ||β| = n}. It is remarkable that for fixed I, J and n, the linear space {Vhn | h = h(r)} is at most one dimensional. Thus if there is another function h0 = h0 (r), then we have [T˜bh , T˜bh0 ] = 0. So in the example below Proposition 4.5, when we choose a and b, the restriction that h1 and h2 are real valued can be removed. Moreover, in that example, we can take A(Bd ) as the function space spanned by {˜ ah , ˜bh | h = h(r)} ∪ {radial functions over Bd }. Acknowledgment The author is deeply indebted to his advisor Professor Kunyu Guo for his advices and suggestions. He also takes the opportunity to express his appreciation to Professor N. Vasilevski for his emails helping a lot in writing this paper.

References [1] W. Arveson, A Short Course on Spectral Theory, GTM 209, Springer-Verlag, New York, 2001. [2] A. Beardon, The Geometry of Discrete Groups, Springer, Berlin, 1983. [3] J. Conway, A course in operator theory, Graduate Studies in Mathematics 21, AMS, Rhode Island, 2000. [4] Z. Cuckovic and N. Rao, Mellin Transform, monomial symbols, and commuting Toeplitz operators, J. Funct. Anal. 154(1998), 195–214. [5] S. Grudsky, A. Karapetyants and N. Vasilevski, Dynamics of properties of Toeplitz operators with radial symbols, Integral Equations and Operator Theory 20(2004), 217–253. [6] S. Grudsky, A. Karapetyants and N. Vasilevski, Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case, J. Operator Theory 52(2004), 185– 204. [7] S. Grudsky, A. Karapetyants and N. Vasilevski, Dynamics of properties of Toeplitz operators on the upper half-plane: Hyperbolic case, Bol. Soc. Mat. Mexicana 10(2004), 119–138.

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[8] S. Grudsky, R. Quiroga-Barranco and N. Vasilevski, Commutative C ∗ -algebras of Toeplitz operators and quantization on the unit disk, J. Funct. Anal. 234(2006), 1– 44. [9] S. Grudsky and N. Vasilevski, Toeplitz operators on the Fock space: Radial component effects, Integral Equations and Operator Theory 44(2002), 10–37. [10] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, GTM 199, Springer-Verlag, New York, 2000. [11] B. Korenblum and K. Zhu, An application of Tauberian theorems to Toeplitz operators, J. Operator Theory 44(2002), 10–37. [12] G. Pederson, Analysis NOW, GTM 118, Springer-Verlag, New York, 1989. [13] R. Quiroga-Barranco and N. Vasilevski, Commutative algebras of Toeplitz operators on the Reinhardt domains, Integral Equations and Operator Theory 59(2007), 67–98. [14] R. Quiroga-Barranco and N. Vasilevski, Commutative C*-algebras of Toeplitz operators on the unit ball, I. Bargmann-type transforms and spectral representations of Toeplitz operators, Integral Equations and Operator Theory 59(2007), 379–419. [15] R. Quiroga-Barranco and N. Vasilevski, Commutative C*-algebras of Toeplitz operators on the unit ball, II. Geometry of the level sets of symbols, Integral Equations Operator Theory 60(2008), 89–132. [16] W. Rudin, Real and Complex Analysis, China Machine Press, 3rd Edition. [17] W. Rudin, Function Theory in the Unit Ball of Cn , Springer-Verlag, New York, Heidelberg, Berlin, 1980. [18] L. Trieu, Diagonal Toeplitz operators on weighted Bergman spaces, preprint, available at http://www.math.uwaterloo.ca/∼t29le/Papers.htm. [19] L. Trieu, The commutants of certain Toeplitz operators on weighted Bergman spaces, preprint, available at http://www.math.uwaterloo.ca/∼t29le/Papers.htm. [20] N. Vasilevski, Quasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators, Reporte Interno # 386, Departamento de Matematicas, CINVESTAV del I.P.N., Mexico, 2008, 13 p. [21] N. Vasilevski, Parabolic quasi-radial quasi-homogeneous symbols and commutative algebras of Toeplitz operators, Reporte Interno # 387, Departamento de Matematicas, CINVESTAV del I.P.N., Mexico, 2008, 19 p. Hansong Huang Fudan University Mathematical Science Department East Building 1903 220 Handan Road Shanghai 200433 China e-mail: [email protected] Submitted: August 16, 2007. Revised: May 19, 2009.

Integr. equ. oper. theory 64 (2009), 399–408 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030399-10, published online July 3, 2009 DOI 10.1007/s00020-009-1697-7

Integral Equations and Operator Theory

On Some Product of Two Unbounded Self-Adjoint Operators Mohammed Hichem Mortad Abstract. We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). Mathematics Subject Classification (2000). Primary 47B15; Secondary 47B25, 47A05. Keywords. Normal operator, self-adjoint operator, closed operator, product of unbounded operators.

1. Introduction If we consider a self-adjoint operator, then it is normal regardless of its boundedness. For the converse to be true, one has to add some conditions. One simple property is that an unbounded normal operator which is symmetric is self-adjoint. Another one is that a normal operator with real spectrum is self-adjoint. Less obvious is the following: If N = HK is a normal product of two self-adjoint operators, then when is it self-adjoint? This was answered by Albrecht and Spain [1] who showed that if a product of two bounded self-adjoint operators H, K is normal, then it is self-adjoint provided that the spectrum of K satisfies σ(K)∩σ(−K) ⊆ {0} (this condition will be referred to as the condition “C”). The author, in a previous work [7], extended their result (using the same condition) to the case when one or both of the operators are unbounded (the proofs were mainly based on the Fuglede-Putnam theorem [10, 4]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).

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It is worth mentioning that W. Rehder [12] proved in 1982 the same result for bounded operators, where one of them was positive, which is included in the condition “C”. Neither the author nor Albrecht-Spain were aware of this reference. So it is just fair now to call this result the Rehder-Albrecht-Spain theorem. A generalization of the Fuglede-Putnam theorem is also proved in [7] subject to some extra hypotheses. In the end, a counterexample that showed that the result is no longer true in the most naive form of generalization is also given in the same reference, i.e., if H and K are self-adjoint operators such that K is positive and if HK has a normal closure then HK need not have a self-adjoint closure (and so it need not be essentially self-adjoint). This paper constitutes mainly the follow-up of [7]. Here we examine the counterexample in detail and identify what goes wrong and why the proof of Theorem 5 in [7], which in this paper is called Theorem A (see below), cannot be applied to the counterexample. In order to do this, we find the spectral measure for the normal operator in question which will be the key point. In the end of this paper we treat a natural question left open in [7], although not explicitly mentioned. Before this we recall briefly the main results in [7]. Let K be a bounded self-adjoint operator satisfying the condition “C” and let H be an unbounded selfadjoint operator. Then if HK (respectively KH) is normal then HK (respectively KH) is self-adjoint. The proof for the case KH was deduced from that of HK. If we no longer assume that K is bounded, then only HK normal implies that HK is self-adjoint. Nothing was said about the case KH. So in the penultimate section we give a counterexample which shows the failure of this property in general and we also give additional assumptions that make it true. We digress to say that W. Rehder [12] gave some references to some applications of products of self-adjoint operators in both Quantum Mechanics and Mathematics. We now recall the following theorem (which, for bounded K, is an immediate consequence of the Fuglede-Putnam-Rosenblum theorem) Theorem A ([7]). Let N and K be two unbounded operators. Assume that K is self-adjoint and that N is normal such that D(N ) ⊂ D(K). Then KN ⊂ N ∗ K implies KN ∗ ⊂ N K. Theorem A was mainly used to prove Theorem B ([7]). Let K, H be two unbounded self-adjoint operators such that σ(K) ∩ σ(−K) ⊆ {0}. If HK is normal then it is self-adjoint. We recall briefly the counterexample. Let H and K be defined as follows: d : H 1 (R) → L2 (R), K = |x| : D(K) → L2 (R). dx These two operators are known to be self-adjoint (see e.g. [11]) on their respective domains D(H) = H 1 (R) = {f ∈ L2 (R) : f 0 ∈ L2 (R)} and D(K) = {f ∈ L2 (R) : H = −i

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|x|f ∈ L2 (R)}. Then setting N = HK we proved that N is normal but not selfadjoint, i.e., N is not essentially self-adjoint. Since N : f 7→ N f = −i(|x|f )0 (we actually mean N ) is normal on D(N ) = {f ∈ L2 (R) : xf 0 ∈ L2 (R)}, there is a unitary transformation, say U , that diagonalizes N (see e.g. [3]). In other words, via U , N will be unitarily equivalent to a multiplication operator by a complex-valued function. Notations and Definitions. An operator A is said to be an extension of B, and we write B ⊂ A, if D(B) ⊂ D(A) and A and B coincide on each element of D(B). A stands for the closure of the operator which is by definition the smallest closed extension of A. An operator A is called normal if it is densely defined, closed and satisfies AA∗ = A∗ A. It is known that if A is normal, then so is its adjoint. An operator is said to be self-adjoint if A = A∗ . For the sake of simplicity, the operator N , which was introduced above, is denoted by N in this paper. The spectral measure (on the ball BR ) of the normal operator N is denoted by PBR . For any R > 0 put HR = ran PBR . All operators considered in this paper are unbounded linear operators on a complex Hilbert space. Any other notion or result which will be used will be assumed to be known by the reader. There is a vast literature concerning this subject. We cite [3] and [13] among others.

2. Spectral Analysis of N = HK First, we find an explicit unitary operator U which diagonalizes N in the way we will need it. Theorem 2.1. Let N be the normal operator defined on D(N ) = {f ∈ L2 (R) : xf 0 ∈ L2 (R)} by N f = −i(|x|f )0 . Then N is unitarily equivalent to M = M+ ⊕M− where M+ is defined on L2 (R) by M+ f (s) = (s − 12 i)f (s) and M− is defined on L2 (R) by M− f (s) = (s + 12 i)f (s). The required unitary transformation is given by U f = U+ f+ ⊕ U− f− where f+ is the restriction of f to R+ , f− is the restriction of f to R− . The operator U+ is defined by U+ = F −1 V where F −1 is the inverse L2 -Fourier transform and V : L2 (R+ ) → L2 (R) is the unitary operator defined by t

(V f )(t) = e 2 f (et ) and U− is defined by U− = F −1 W where W : L2 (R− ) → L2 (R) is defined by t

(W f )(t) = e− 2 f (e−t ). Proof. Since we have the decomposition L2 (R) = L2 (R+ ) ⊕ L2 (R− ), N may be written as N+ ⊕ N− where ∗ N+ h = N+ h − ih

∗ and N− h = N− h + ih.

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Let λ ∈ σ(N+ ). Then λ = λ − i which gives =λ = − 12 , that is σ(N+ ) ⊆ {α − 21 i|α ∈ R} (it is actually equal to this set as we will see below). The idea now is to try to find the eigenvalues of the operator N+ . We have −ixh0 (x) − ih(x) = λh(x)

or

h0 (x) i(λ + i) = , h(x) x

1

hence h(x) = cx− 2 +iα where c is arbitrary and where α = λ + 21 i. This h is clearly not in L2 (R+ ). Thus there are no eigenvalues but this is sufficient to find the unitary equivalence of N . Define Z ∞ 1 1 x− 2 +iu f (x)dx where f ∈ L2 (R+ ). (U+ f )(u) = √ (1) 2π 0 The previous equation is a well-defined Fourier transform in L2 (R) (it may be considered as a form of the Mellin transform). One can verify this by using the change of variable x = et in (1). We then get Z Z 1 1 1 1 [e 2 t f (et )]eiut dt = √ f (et )eiut+ 2 t dt. (U+ f )(u) = √ 2π 2π R

R

The inversion formula is given by 1 F (t) = f (e ) = √ 2π t

Z

1

(U+ f )(u)e− 2 t−iut du.

(2)

R

Let us check using the formula in (2) that N+ is unitarily equivalent to M+ . We have F 0 (t) = et f 0 (et ) = xf 0 (x) and at the same time Z 1 1 1 F 0 (t) = √ − − iu (U+ f )(u)e− 2 t−iut du. 2 2π R

Hence

1 −iF 0 (t) − iF (t) = √ 2π

Z

1 1 −u − i (U+ f )(u)e− 2 t−iut du. 2

R

Then 1 N+ f (x) = −ixf (x) − if (x) = √ 2π 0

Z

1 1 −u − i (U+ f )(u)e− 2 t−iut du. 2

R

Thus

1 U+ N+ f (s) = s − i (U+ f )(s) = (M+ U+ f )(s). 2 So N+ is unitarily equivalent to M+ and the unitary operator is given by (1) and hence σ(N+ ) = σ(M+ ) = {s − 21 i|s ∈ R}. The proof for the case L2 (R− ) is just a matter of “cut and paste”. So we just give the unitary operator in this case, that is Z 1 1 f (e−t ) = √ (U− f )(u)e+ 2 t−iut du 2π R

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and hence σ(N− ) = σ(M− ) = {s + 12 |s ∈ R}. In the end N = N+ ⊕ N− is unitarily equivalent to M = M+ ⊕ M− where M+ f (α) = (−α − 21 i)f (α) and M− f (α) = (−α + 12 i)f (α). Thus n o n o 1 1 σ(N ) = σ(N+ ) ∪ σ(N− ) = s − i s ∈ R ∪ s + i s ∈ R . 2 2 Now we have the necessary tools to investigate what goes wrong in the proof of Theorem A when applied to our counterexample. The operator N is not closed (see [7]) but there is something else that makes the result untrue. The crucial point in the proof Theorem A is that we restricted K to HR and unfortunately this is not possible when using this counterexample, i.e., HR is not a subset of D(K). This is shown in the following proposition. Proposition 2.2. Let PBR be the spectral measure of the normal operator N . Then HR is not a subset of D(K). Proof. We need to find an f that is in HR and not in D(K) i.e. xf ∈ / L2 (R). 2 + It suffices to do this in L (R ). We also denote the spectral projection for N + by PBR . The operator M+ has R × {− 12 } as its spectrum. So it lies in some line parallel to the x-axis. Also since the multiplication operator M+ has the multiplication by a characteristic function, say 1Im , as its spectral measure and since N+ is unitarily equivalent to M+ , it follows that PBR is unitarily equivalent to 1Im (m and −m represent the intersection of the disc of radius R and the line y = − 21 ) via the transform defined in (2). Then we have F PBR F −1 = 1Im

where

Im = [−m, m].

Hence PBR F −1 = F −1 1Im . So for g ∈ L2 (R+ ) one has f = PBR F −1 g = F −1 1Im g. We observe that to say that f ∈ HR or F f = 1Im g, g ∈ L2 (R+ ) is the same thing. Hence we seek an f such that F f (s) = 1 on [0, m] and zero otherwise (we have taken g = 1[0,m] ) such that xf ∈ / L2 (R+ ) or et f (et ) ∈ / L2 (R). By (2) we have Z m 1 1 1 1 f (et ) = √ e− 2 t−ist ds = √ e− 2 t (1 − e−imt ). 2π 0 it 2π Obviously, f ∈ L2 (R+ ),

but

et f (et ) ∈ / L2 (R)

since Z Z t Z t 2 e e 1 12 t −imt ) dt = (2 − 2 cos(mt))dt ≥ (2 − 2 cos(mt))dt = ∞. e (1 − e it t t + R R R

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There is nothing special with this counterexample. We may take other pairs d of self-adjoint operators such as K = x2 , H = −i dx and we will encounter the same problem, i.e., HK not being closed. To sort out this point, we use the following elementary theorem about the product of closed operators (there are more results in the literature). Theorem 2.3. Let H and K be two closed operators in a Hilbert space. Then HK (in this order) is closed if one of the following occurs: 1. K bounded; 2. H invertible with a bounded inverse. Now we can state with ease the following corollary Corollary 2.4. Let K, H be two unbounded self-adjoint operators with K satisfying the condition “C”. Suppose further that one of the two properties listed in the previous theorem is satisfied. If HK has a normal closure, then it is essentially self-adjoint.

3. Normality of KH Now, we give a counterexample that shows that the order of the operators H and K cannot be interchanged in Theorem B above. Proposition 3.1. There exist two unbounded self-adjoint operators H and K such that K satisfies condition “C”, KH is normal but not self-adjoint. d with D(H) = H 1 (R), the Sobolev space. Then H is selfProof. Let H = −i dx adjoint. Take Kf (x) = (1 + |x|)f (x) with domain

D(K) = {f ∈ L2 (R) : (1 + |x|)f ∈ L2 (R)}. Then it is well known that K has a bounded inverse. Besides, K is self-adjoint since it is a multiplication operator by a real-valued function and obviously K satisfies σ(K) ∩ σ(−K) ⊆ {0} since it is a positive operator. So, it only remains to verify that M = KH is normal, i.e. M is closed and M M ∗ = M ∗ M and that M is not self-adjoint. For f in D(M ) = {f ∈ L2 (R) : (1 + |x|)f 0 ∈ L2 (R)}, where the derivative is considered in distributional sense, one has M f (x) = −i(1 + |x|)f 0 (x). The closedness of M then follows from Theorem 2.3 (property 2). To find its adjoint we proceed exactly as in [7] by first doing it for f ∈ C0∞ (R \ {0}), the space of smooth functions with compact support away from the origin, and then approximating using the graph norm of M since C0∞ (R \ {0}) is dense in D(M ) (the interested reader finds all the details about a similar question in [7]). We find M ∗ f (x) = sgn(x)if (x) − i(1 + |x|)f 0 (x) where “sgn” is the usual sign function.

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Clearly M is not self-adjoint but M M ∗ f (x) = M ∗ M f (x) = −(1 + |x|)(2 sgn(x)f 0 (x) − (1 + |x|)f 00 (x)). The previous equation combined with (as one can verify) D(M M ∗ ) = D(M ∗ M ) = {f ∈ L2 (R) : (1 + |x|)f 0 ∈ L2 (R), (1 + |x|)f 00 ∈ L2 (R)} allow us to conclude that M = KH is a normal operator.

Remark. If we interchange the roles of H and K in the previous example and set Kf (x) = −if 0 (x)

and

Hf (x) = (1 + |x|)f (x),

then HK is normal and not self-adjoint. We also observe that σ(K) = R, and hence σ(K) ∩ σ(−K) = R 6⊆ {0}. This shows that Condition “C” cannot merely be dropped in Theorem B. Remark. In the spirit of the previous section we can verify that D(KM ) = {f ∈ L2 (R) : (1 + |x|)2 f 0 ∈ L2 (R)} and D(M ∗ K) = {f ∈ L2 (R) : (1 + |x|)2 f, (1 + |x|)2 f 0 ∈ L2 (R)}. Hence KM ∗ 6⊂ M K (this will be shown by giving an explicit function in D(KM ) which does not belong to D(M ∗ K), and a way of doing this is to use a method used in [8] in a different setting by a simple linear interpolation). We also observe that D(M ) 6⊂ D(K) and this means that the assumptions of Theorem A are not all fulfilled. This shows the power of Theorem A. So it appears that for the result to be true one has to add some hypotheses. We give two possible assumptions which will guarantee the self-adjointness of KH (given KH is normal). They are both a consequence of Theorem B but with two different approaches and a slightly different method of proof. Here is the first one: Corollary 3.2. Let K, H be two unbounded self-adjoint operators such that σ(K) ∩ σ(−K) ⊆ {0}. If KH and HK are both normal, then KH is self-adjoint. Proof. Since KH is normal, then so is (KH)∗ . Since K and H are self-adjoint and using a known property about the adjoint of a product of two unbounded operators, one has HK = H ∗ K ∗ ⊂ (KH)∗ . Since HK is normal and it has a normal extension which is (KH)∗ , we have HK = (KH)∗ (as normal operators are maximally normal, see e.g., [13]). Now by Theorem B we deduce that HK or (KH)∗ is self-adjoint. Since KH is closed, we obtain (KH)∗ = (KH)∗∗ = KH = KH, and hence the result. Now we exploit the following result due to K. Gustafson to give the second approach to the problem.

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Lemma 3.3 ([5]). Let A and B be two (unbounded) self-adjoint operators in a Hilbert space H and suppose that D(AB) is dense, that R(B) ⊃ D(A), and D(B) ⊃ R(A). Then (AB)∗ = B ∗ A∗ . Proposition 3.4. Let H and K be two unbounded self-adjoint operators such that σ(K) ∩ σ(−K) ⊆ {0}. Let R(H) ⊃ D(K), and D(H) ⊃ R(K). If KH is normal, then it is self-adjoint. Proof. Since KH is normal, then its adjoint (KH)∗ is also normal. By the previous lemma and the self-adjointness of H and K we get (KH)∗ = HK so that HK is also normal and hence Theorem B allows us to deduce the self-adjointness of HK. This means that (HK)∗ = HK which, in turn, gives: (KH)∗ is self-adjoint. This, combined with the fact that KH is closed (it is normal!), yields (KH)∗∗ = KH = KH = (KH)∗ , and hence the self-adjointness of KH.

Remark. Now, since there are similar results to that of K. Gustafson, one can use them to give different versions of the foregoing proposition. See for instance [2, 6].

4. A Question Let K, N and M be bounded operators on a Hilbert space H (N and M being normal) such that KN = M K. Then the following assertions are equivalent: 1. KN ∗ = M ∗ K. 2. N commutes with K ∗ K. 3. M commutes with KK ∗ .

∗

4. The subspace of those x ∈ H for which supλ∈C KeλN −λN x < ∞ is dense in H.

∗

5. The subspace of those x ∈ H for which supλ∈C K ∗ eλM −λM x < ∞ is dense in H. This is partly based on the Fuglede-Putnam theorem in its classical form. Now, can we prove Theorem A for two different unbounded normal operators M and N (as for the Fuglede-Putnam theorem [4, 10])? I.e., if one has KN ⊂ M K then does it follow that KN ∗ ⊂ M ∗ K (probably under some extra conditions)? The general form does not hold, i.e., we cannot have a fourth operator involved in this question, i.e., if (we just consider the bounded case) AN = M B then it is not necessary that AN ∗ = M ∗ B. For if A = M = I (the identity operator) and N = B, then it holds only if B is self-adjoint! Some answers to this question should be found in [9].

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Acknowledgments The spectral analysis section appeared in the author’s Ph.D. thesis at Edinburgh University under the supervision of Professor Alexander M. Davie. The author takes this opportunity to thank him again for the great supervision work he did. The author also owes thanks to Professor K. Gustafson for providing him with his reference [5]. Finally, I also thank the referee for his comments and for the first paragraph of the section “A Question”.

References [1] E. Albrecht, P. G. Spain, When Products of Selfadjoints Are Normal, Proc. Amer. Math. Soc., 128/8 (2000) 2509–2511. [2] J. A. W. van Casteren, S. Goldberg, The Conjugate of the Product of Operators, Studia Math., 38 (1970) 125–130. [3] J. B. Conway, A Course in Functional Analysis, Springer, 1990 (2nd edition). [4] B. Fuglede, A Commutativity Theorem for Normal Operators, Proc. Nati. Acad. Sci., 36 (1950) 35–40. [5] K. Gustafson, A Composition Adjoint Lemma, Stochastic processes, physics and geometry: new interplays, II (Leipzig), (1999), 253–258, CMS Conf. Proc., 29, Amer. Math. Soc., Providence, RI, (2000). [6] S. S. Holland Jr., On the Adjoint of the Product of Operators, J. Functional Analysis, 3 (1969) 337–344. [7] M. H. Mortad, An Application of the Putnam-Fuglede Theorem to Normal Products of Self-adjoint Operators, Proc. Amer. Math. Soc., 131/10 (2003) 3135–3141. [8] M. H. Mortad, Self-adjointness of the Perturbed Wave Operator on L2 (Rn ), n ≥ 2, Proc. Amer. Math. Soc., 133/2, (2005) 455–464. [9] M. H. Mortad, Yet More Versions of The Fuglede-Putnam Theorem, Glasg. Math. J. (to appear). doi:10.1017/S0017089509005114 [10] C. R. Putnam, On Normal Operators in Hilbert Space, Amer. J. Math., 73 (1951) 357–362. [11] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, 1972. [12] W. Rehder, On the Product of Self-adjoint Operators, Internat. J. Math. and Math. Sci., 5/4 (1982) 813–816 . [13] W. Rudin, Functional Analysis, McGraw-Hill, 1991 (2nd edition). Mohammed Hichem Mortad D´epartement de Math´ematiques Universit´e d’Oran (Es-senia) B.P. 1524, El Menouar Oran 31000 Algeria

408 Postal address: Dr. Mohammed Hichem Mortad B.P. 7085 Seddikia Oran 31025 Algeria e-mail: mortad [email protected] [email protected] Submitted: March 26, 2008. Revised: June 15, 2009.

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Integr. equ. oper. theory 64 (2009), 409–428 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030409-20, published online June 2, 2009 DOI 10.1007/s00020-009-1688-8

Integral Equations and Operator Theory

Positive Schatten(-Herz) Class Toeplitz Operators on Pluriharmonic Bergman Spaces Kyunguk Na Abstract. We study characterizations of arbitrary positive Toeplitz operators of Schatten (or Schatten-Herz) type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman space. Mathematics Subject Classification (2000). Primary 47B35; Secondary 31B05. Keywords. Toeplitz operator, Schatten-Herz type, pluriharmonic Bergman space.

1. Introduction For a fixed integer n ≥ 2, let B = Bn denote the open unit ball in Cn . Given α > −1, the weighted pluriharmonic Bergman space b2α = b2α (B) is the set of all complex-valued pluriharmonic functions f on B such that Z 1/2 2 kf k2 = |f | dVα <∞ B 2 α

where dVα (z) = (1 − |z| ) dV (z) and V denotes the Lebesgue volume measure on B. For 1 ≤ p ≤ ∞, let Lpα = Lp (Vα ) be the weighted Lebesgue spaces on B and Rzα (w) = Rα (z, w) be the reproducing kernel for b2α . It is known that a pluriharmonic function f belongs to B if and only if we can decompose f = g + h, where g, h are holomorphic. Furthermore, if f ∈ L2α , then the boundedness of the Bergman projection implies that both g and h are in A2α = A2α (B), the holomorphic Bergman space with respect to the weight (1−|z|2 )α to simplify the notation. Thus, we see that b2α = A2α + A2α This work was supported by a Hanshin University Research Grant.

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and the explicit formula of Rzα (w) is as follows: Rzα (w) =

1 1 + −1 n+α+1 (1 − w · z) (1 − z · w)n+α+1

(1.1)

for z, w ∈ B. Here and subsequently, z · w = z1 w1 + · · · + zn wn denotes the Hermitian inner product on Cn . We let Qα be the Hilbert space orthogonal projection from L2α onto b2α . It is known that Qα is an integral operator given by Z Qα ψ(z) = ψ(w)Rzα (w) dVα (w), z ∈ B, (1.2) B 2

for functions ψ ∈ L . From (1.1), one can see that |Rzα (z)| ≈

1 (1 − |z|2 )n+α+1

and

|Rzα (w)| .

1 |1 − z · w|n+α+1

(1.3)

for all z, w ∈ B so that Rzα is bounded for fixed z ∈ B. Thus, the projection Qα can be extended to an integral operator via (1.2) from L1α into the space of all pluriharmonic functions on B. Moreover, for 1 < p < ∞, one can see from (1.3) and Lemma 2.3 (see Section 2) that Qα is bounded projection from Lpα onto bpα . The integral transform Qα can be extended to M, the space of all complex Borel measures on B. That is to say, for each µ ∈ M, the integral Z Qα µ(z) = Rzα (w) dµ(w), z ∈ B, B

defines a function pluriharmonic on B. For µ ∈ M, the Toeplitz operator Tµ with symbol µ is defined by Tµ f = Qα (f dµ) for f ∈ b2α ∩ L∞ . In case dµ = ϕ dVα , we write Tµ = Tϕ . Note that Tµ is defined on a dense subset of b2α , because bounded pluriharmonic functions form a dense subset of b2α . A Toeplitz operator Tµ is called positive if µ ∈ M is a positive (finite) Borel measure (we will simply write µ ≥ 0). In the case 1 ≤ p ≤ ∞, Choi ([4]) studied boundedness, compactness and the membership of the Schatten classes Sp (see Section 3) for positive Toeplitz operators on pluriharmonic Bergman spaces. Also, Choi and Na ([5]) studied another aspect of positive Toeplitz operators, namely, the notion of the so-called Schatten-Herz classes Sp,q (see Section 4). This concept was introduced and studied in [7] in the holomorphic case on the unit disk. Harmonic and pluriharmonic analogues were subsequently studied in [2] and [5]. However, these works only considered the restricted range of parameter of p with 1 ≤ p, q ≤ ∞. Recently, Choe et al. ([1]) extended the characterizations in [3] and [2] for Schatten(-Herz) classes of positive Toeplitz operators to the full range of parameters p and q. We also extend such characterizations in [4] and [5] to the full range of parameters p and q.

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To state our main results, we let λ denote the measure on B defined by dλ(z) = Rz0 (z)dV (z). Note that λ = V−n−1 . Also, we let Kqp (λ) denote the so-called Herz spaces (see Section 4). We also refer to Section 2 for definitions of the averaging functions µ bδ and the Berezin transforms µ e of µ ∈ M. The following theorems are the main results of this paper. In the case 1 ≤ p < ∞ and α = 0, Theorems 1.1 and 1.2 below have been proved in [4] and [5], respectively. Theorem 1.1. Let 0 −1 and r ∈ (0, 1). If µ ≥ 0 then the following conditions are equivalent: (a) Tµ ∈ Sp . (b) µ br ∈ Lp (λ). Moreover, if n/(n + α + 1) < p, then the above statements are also equivalent to (c) µ e ∈ Lp (λ). Theorem 1.2. Let 0 −1 and r ∈ (0, 1). If µ ≥ 0 then the following conditions are equivalent: (a) Tµ ∈ Sp,q . (b) µ br ∈ Kqp (λ). Moreover, if n/(n+α+1) < p ≤ ∞, then the above statements are also equivalent to (c) µ e ∈ Kqp (λ). In Section 2 we recall some basic properties of automorphisms and some basic estimates for pseudohyperbolic distance and the kernel functions. We also investigate weighted Lp -behavior of averaging functions and Berezin transforms. In Section 3 we introduce Schatten class Toeplitz operators and prove Theorem 1.1. In the last section we define Herz and Schatten-Herz spaces and prove Theorem 1.2. We often abbreviate inessential constants involved in inequalities by writing X . Y for positive quantities X and Y if the ratio X/Y has a positive upper bound. Also, we write X ≈ Y if X . Y and Y . X.

2. Preliminaries In this section we introduce some notations and collect several basic lemmas which will be used in later sections. We first recall automorphisms on B. For z, w ∈ B, define ϕz (w) =

z − Pz (w) − (1 − |z|2 )1/2 Qz (w) , 1 − w · z¯

z ∈ B,

(2.1)

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where Pz is the orthogonal projection of Cn onto the subspace [z] generated by z, namely, P0 = 0, w · z¯ Pz (w) = z if z 6= 0 |z|2 and Qz (w) = w − Pz (w). Then each ϕz is a automorphism on B and ϕz ◦ ϕz is the identity on B. Note that 1 − |ϕz (w)|2 =

(1 − |z|2 )(1 − |w|2 ) , |1 − z · w|2

z, w ∈ B,

(2.2)

and for a ∈ B, 1 − ϕa (z) · ϕa (w) =

(1 − |a|2 )(1 − z · w) , (1 − z · a)(1 − a · w)

z, w ∈ B,

(2.3)

;see Section 2 of [10] for details. We also recall the hyperbolic metric β(z, w) on B defined by β(z, w) =

1 1 + ρ(z, w) log 2 1 − ρ(z, w)

where ρ(z, w) = |ϕz (w)| is the pseudohyperbolic metric on B. Note that ρ is automorphism invariant, namely ρ(ϕa (z), ϕa (w)) = ρ(z, w). For z ∈ B and r ∈ (0, 1), let Er (z) be the pseudohyperbolic ball with center z ∈ B and radius r ∈ (0, 1) is defined by Er (z) = ϕz (rB). Since ϕz is an involution, w ∈ Er (z) if and only if |ϕz (w)| < r. Using this with (2.1), a little manipulation shows that Er (z) consists of all w ∈ B that satisfy |Pz (w) − cz |2 |Qz (w)|2 + <1 2 2 r ρz r 2 ρz where center cz and radius ρz defined by cz =

(1 − r2 )z 1 − |z|2 r2

and ρz =

1 − |z|2 . 1 − |z|2 r2

(2.4)

(2.5)

Thus we can see that Er (z) is an ellipsoid with center at cz . It is also known that Vα [Er (z)] ≈ (1 − |z|2 )n+α+1 .

(2.6)

The constants suppressed above depend only on r (see [10] for details). Using the invariance of ρ under automorphism and (2.2), we have the following result. Lemma 2.1. The inequality 1 − |z|2 1 − ρ(z, w) 1 + ρ(z, w) ≤ ≤ 2 1 + ρ(z, w) 1 − |w| 1 − ρ(z, w) holds for z, w ∈ B.

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Proof. Let z, w ∈ B with ρ(z, w) < r for some r ∈ (0, 1). Write w = ϕz (a) for some a ∈ B with ρ(0, a) < r. Since 1 − |a| ≤ |1 − z · a| ≤ 1 + |a|, we have 1 − |a| 1 + |a| |1 − z · a|2 ≤ ≤ 1 + |a| 1 − |a|2 1 − |a| for ρ(0, a) < r. Since ρ is invariant under automorphisms, we obtain 1+r 1−r |1 − z · a|2 ≤ ≤ 1+r 1 − |a|2 1−r for ρ(z, w) < r. Thus (2.2) implies the desired result.

Note that since ρ(z, w) < r is symmetric, we have 1−|z| ≈ |1−z ·w| ≈ 1−|w| for ρ(z, w) < r. The constants suppressed above depend only on r. Lemma 2.2. The inequality |1 − z · a| 1 + ρ(z, w) 1 − ρ(z, w) ≤ ≤ 1 + ρ(z, w) |1 − w · a| 1 − ρ(z, w) holds for z, w, a ∈ B. Proof. Note that 1 − |a| ≤ |1 − z · a| ≤ 1 + |a| and 1 − |a| ≤ |1 − w · a| ≤ 1 + |a|. Using these facts, the result follows as in the proof of Lemma 2.1. The following lemma is taken from Proposition 1.4.10 of [10]. Here and elsewhere, dS denotes the surface area measure on ∂B. Lemma 2.3. For −1 < a < ∞, c real and z ∈ B, let Z dS(ζ) Jc (z) = n+c ∂B |1 − z · ζ| and

(1 − |w|)a dV (w) n+1+a+c B |1 − z · w| for z ∈ B. Then the following estimates hold:  if c < 0,  1 1 if c = 0, Jc (z) ≈ Ia,c (z) ≈ 1 + log 1−|z|2   1 if c > 0, (1−|z|2 )c Z

Ia,c (z) =

as |z| → 1. The constants suppressed above depend only on n and c. Lemma 2.4. Given α > −1, there exist some rα = rα (n) ∈ (0, 1) and a constant C = C(n, α) such that C −1 ≤ Rzα (w)(1 − |z|2 )n+α+1 ≤ C whenever w ∈ Er (z) and 0 < r ≤ rα .

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Proof. We just prove the lower estimate by (2.6). Let z ∈ B and w ∈ Er (z) where r is to be chosen later. We have by (1.1) and (2.3) Rzα (w)(1 − |z|2 )n+α+1 = (1 − ϕz (w) · z)n+α+1 + (1 − z · ϕz (w))n+α+1 − (1 − |z|2 )n+α+1 = F (ϕz (w), z) where F (a, z) = (1 − a · z)n+α+1 + (1 − z · a)n+α+1 − (1 − |z|2 )n+α+1 . Since F is uniformly continuous on B × B, there exists rα ≥ r in (0, 1) such that if |ϕz (w)| ≤ rα , then |F (ϕz (w), z) − F (0, z)| < 1/2 for z ∈ B. It follows that 1 3 1 Rzα (w)(1 − |z|2 )n+α+1 = F (ϕz (w), z) ≥ F (0, z) − = − (1 − |z|2 )n+α+1 ≥ . 2 2 2 Thus we have the desired result.

3. Schatten class of positive Toeplitz operators In this section we collect the Lp (λ)-behavior of the averaging functions (as well as its discretized version) and the Berezin transforms. Let α > −1 and z ∈ B. Given µ ≥ 0 and r ∈ (0, 1), the averaging function µ br and Berezin transform µ e are defined by Z µ[Er (z)] and µ e(z) = |rzα (w)|2 dµ(w) µ br (z) = Vα [Er (z)] B where rzα (·) = Rzα (·)/kRzα k2 is the normalized reproducing kernel. Note that these notations are slightly different from those of [4] and [5]. Actually, the authors of [4] and [5] used the above two notations with α = 0. Also, we let µ br = ϕ br for dµ = ϕ dVα . For measurable functions f , we define fbr and fe similarly, whenever they are well defined. Also, given a sequence a = {am } in B, we let `p,α (a) denote the p-summable sequence space weighted by {(1 − |am |2 )α }. For α = 0, we let `p = `p,0 (a). Lemma 3.1. Let α > −1, µ ≥ 0 and 0 < r ≤ rα where rα is the number provided by Lemma 2.4. Then there exists a constant C = C(n, α, r) such that Z µ br (z) ≤ C |Rzα (w)|2 dµ(w), z ∈ B. B

In particular, µ br ≤ C µ e for 0 < r ≤ r0 . Proof. Given z ∈ B, we have by Lemma 2.4 and (2.6) Z |Rzα (w)|2 dµ(w) & (1 − |z|2 )−2(n+α+1) µ(Er (z)) & (1 − |z|2 )−(n+α+1) µ br (z) Er (z)

where the constants suppressed depend only on n, α and r. This implies the first part of the lemma. Using (1.3), the second part follows from this. The proof is complete.

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Let {am } be a sequence in B and r ∈ (0, 1). We say that {am } is a r-separated if the balls Er (am ) are pairwise disjoint or simply say that {am } is separated if it is r-separatedSfor some r. Also, we say that {am } is an r-lattice if it is 2r -separated and B = m Er (am ). For a ∈ B, we note that if {am } maximal 2r -separated sequence, thenSthere exists am such that a ∈ Er (am ) by the maximality, which implies B ⊂ m Er (am ). From this with Er (am ) ⊂ B for all m, we have the following: Any ’maximal’ 2r -separated sequence is an r-lattice. The next two lemmas are basic properties involving separated sequences we use later. Before proceeding, we note the intersection of Er (z) with [z], the subspace generated by z, is a disk of radius rρz and the intersection of Er (z) with the real (2n − 2)-dimensional space perpendicular to [z] at cz is a ball of much larger √ radius r ρz . Here cz and ρz are provided by (2.5). Thus (2.4) implies that B1 ⊂ Er (z) ⊂ B2 where B1 and B2 are the balls as follows: B1 = {w ∈ B : |Pz (w) − cz |2 + |Qz (w)|2 < r2 ρ2z } and B2 = {w ∈ B : |Pz (w) − cz |2 + |Qz (w)|2 < r2 ρz }. p √ Note that rρz ≈ r(1 − |z|2 )2 and r ρz ≈ r 1 − |z|2 when r is small; see [8] for details. Using the facts above, the following results can be proved by the same argument as in Lemma 3.2 and Lemma 3.3 of [1], respectively. Lemma 3.2. For α > 0 and r ∈ (0, 1) with αr < 1, there exists a constant N = N (α, r) with the following property: If {am } is an r-separated sequence, then more than N of the balls Eαr (am ) contain no point in common. Lemma 3.3. Given 0 < r < δ < 1, there exists a positive integer N = N (δ, r) with the following property: Any r-separated sequence can be decomposed into N δ-separated subsequences. We now turn to Schatten class operators. For a positive compact operator T on a separable Hilbert space H, there exist an orthonormal set {em } in H and a decreasing sequence {sm } tending to 0 such that X Tx = sm hx, em iem m

for all x ∈ H where h, i denotes the inner product on H. For 0 < p < ∞, we say that a positive operator T is in the Schatten p-class Sp (H) if ( ) p1 X p kT kp := sm < ∞. m

More generally, given a compact operator T on H, we say that T ∈ Sp (H) if the positive operator |T | = (T T ∗ )1/2 belongs to Sp (H) and we define kT kp = k |T | kp .

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Also, we denote by S∞ the class of all bounded linear operators on H and let kT k∞ denote the operator norm kT k of T ∈ S∞ . In what follows, we take H = b2α and, in that case, we put Sp = Sp (b2α ). If a = {am } is an r-lattice in the Bergman metric, one can obtain the following result extending Theorem 7.16 of [9] for the holomorphic Bergman spaces. Theorem 3.4. Let 0 < p ≤ 1 and µ ≥ 0. For r ∈ (0, 1), suppose a = {am } is an r-lattice. Then the following conditions are equivalent: (a) Tµ ∈ Sp . (b) µ br ∈ Lp (λ). (c) {b µr (am )} ∈ `p (a). Remark. In the case 1 ≤ p ≤ ∞ and α = 0, the above result was proved in Theorems 3.9 and 3.12 of [4]. The same conclusion can be drawn for general α > −1 in much the same way as Theorems 3.9 and 3.12 of [4]. Proof. This can be proved by the same method as in Theorem 7.16 of [9] with the hyperbolic metric. Note that a simple calculation gives the following relation between the hyperbolic metric β and pseudohyperbolic metric ρ: ρ(z, w) = tanh β(z, w) for z, w ∈ B. From (3.1), it follows with pseudohyperbolic metric.

(3.1)

In Theorem 3.4, the Lp (λ)-behavior of averaging functions of positive measure is independent of the radii. Thus, we have the following result. Corollary 3.5. Let 0 −1 and r, δ ∈ (0, 1). Assume µ ≥ 0. Then µ br ∈ Lp (λ) if and only if µ bδ ∈ Lp (λ). We establish the boundedness of the Berezin transform on the weighted Lebesgue spaces Lp (Vγ ). Given a and b real, we let Z f (w) Ψa,b f (z) = (1 − |z|2 )a dVb (w) n+1+a+b B (1 − z · w) and Z

f (w) dVb (w) w|n+1+a+b |1 − z · B for z ∈ B. The following result is taken from Proposition 3.5 of [5]. 2 a

Φa,b f (z) = (1 − |z| )

Proposition 3.6. Let 1 ≤ p < ∞ and a, b, c be real. Then the following conditions are equivalent: (a) Ψa,b is bounded on Lp (Vc ). (b) Φa,b is bounded on Lp (Vc ). (c) −pa < c + 1 < p(b + 1). For the discretized version of Corollary 3.5, we first need the following which is a special case of Proposition 3.6.

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Lemma 3.7. Let 1 ≤ p ≤ ∞, α > −1 and γ be real. Then the Berezin transform is bounded on Lpγ if and only if −(n + α + 1) < (γ + 1)/p < α + 1. As in Lemma 2.24 of [10], using subharmonicity and Fubini’s theorem, we get the following: given δ ∈ (0, 1), there is a constant C = C(δ) such that Z Z f dµ ≤ C fµ bδ dVα (3.2) B

B

for all f ≥ 0 subharmonic on B and µ ≥ 0. The following is an immediate consequence of (3.2). Corollary 3.8. Given δ ∈ (0, 1), there exists a positive constant C = C(δ) such g that µ e ≤ C (b µδ ) for µ ≥ 0. The following is our first main result. Theorem 3.9. Let 0 −1 and µ ≥ 0. Assume that {am } is an r-lattice with r ∈ (0, 1). Then the following statements are equivalent: (a) Tµ ∈ Sp . (b) {b µr (am )} ∈ `p . (c) µ br ∈ Lp (λ). Moreover, if n/(n + α + 1) < p, then the above statements are also equivalent to (d) µ e ∈ Lp (λ). Proof. From Theorem 3.4, we only need to show that (d) =⇒ (c) and either (b) or (c) =⇒ (d). (d) =⇒ (c): This follows from Lemma 3.1 and Corollary 3.5. We now assume n/(n + α + 1) < p and prove that either (b) or (c) implies (d). If p ≥ 1, one may use Lemma 3.7 and Corollary 3.8 to see that (c) implies (d). If p < 1 we have n < p < 1. n+α+1 Now we assume this and prove below that (b) implies (d). Assume {b µr (am )} ∈ `p (a). We have by (2.6) and Lemma 2.2 XZ dµ(w) µ e(z) ≤ (1 − |z|2 )n+α+1 |1 − z · w|2(n+α+1) E (a ) r m m X (1 − |z|2 )n+α+1 . µ[Er (am )] |1 − z · am |2(n+α+1) m X (1 − |z|2 )n+α+1 ≈ (1 − |am |2 )n+α+1 µ br (am ) |1 − z · am |2(n+α+1) m

(3.3)

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for all z ∈ B. Now, since p(n + α + 1) > n, we obtain by Lemma 2.3 and (3.3) Z X µ e(z)p dλ(z) . (1 − |am |2 )p(n+α+1) µ br (am )p B

m

(1 − |z|2 )p(n+α+1)−n−1 dV (z) × 2p(n+α+1) B |1 − z · am | X ≈ µ br (am )p . Z

m

Thus, we have the desired result.

We also mention that corresponding characterizations for boundedness and compactness. In case α = 0, (a) ⇐⇒ (b) ⇐⇒ (d) was proved in Theorems 3.1 and 3.2 (Theorems 3.3 and 3.4 resp.) of [4]. Here, L0 denote the space of all bounded functions f on B such that f (z) → 0 as |z| → 1. Theorem 3.10. Let r ∈ (0, 1), α > −1 and µ ≥ 0. Then the following conditions are equivalent: (a) Tµ is bounded (compact) on b2α . (b) µ br ∈ L∞ (L0 ). (c) {b µr (am )} ∈ `∞ (`0 ). (d) µ e ∈ L∞ (L0 ). Proof. (a) ⇐⇒ (b) ⇐⇒ (d): These statements can be proved as Theorems 3.1 and 3.2 (Theorems 3.3 and 3.4 resp.) of [4]. (b) =⇒ (c): It follows from |am | → 1 as m → ∞. (c) =⇒ (d): Note that 1 − |a| ≤ |1 − z · a| for all a, z ∈ B. Given a positive integer j, put Mj = supm≥j µ br (am ) and let N be the positive integer provided by Lemma 3.2. By the proof of (b) =⇒ (d) of Theorem 3.9, Lemma 2.2 and Lemma 2.4 we have for each j X (1 − |z|2 )n+α+1 (1 − |am |2 )n+α+1 µ e(z) . µ br (am ) |1 − z · am |2(n+α+1) m n+α+1 X 1 − |z|2 . µ br (am ) 1 − |am |2 m<j XZ (1 − |z|2 )n+α+1 + Mj dV (w) |1 − z · w|2(n+α+1) m≥j Er (am ) n+α+1 X 1 − |z|2 . µ br (am ) + Mj N. 1 − |am |2 m<j Now, letting |z| → 1 for fixed j, we have lim µ e(z) . sup µ br (am )

|z|→1

m≥j

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for each j. Thus, taking the limit j → ∞, we have the desired result.

Since the averaging functions of positive measure in the above theorem is independent of the radii, we obtain the following result. Corollary 3.11. Let 0 −1 and r, δ ∈ (0, 1). Assume µ ≥ 0. Then µ br ∈ L0 if and only if µ bδ ∈ L0 .

4. Schatten-Herz class of positive Toeplitz operators In this section we prove Theorem 1.2. First, we introduce the Herz spaces. Let Am = {z ∈ B : rm ≤ |z| < rm+1 } −m

where rm = 1 − 2 for each integer m ≥ 0. Recall that we write χm for the characteristic function of Am for each m. Also, given µ ∈ M, we let µχm stand for the restriction of µ to Am for each m. Let α > −1. Given γ real and 0 < p, q ≤ ∞, the Herz space Kqp,γ is the space consisting of all locally Lpα -integrable functions f such that

kf kKqp,γ = 2−mγ kf χm kLpα q < ∞. `

Equipped with the norm above, the space Kqp,γ is a Banach space. Also, we let p,γ p,γ K0p,γ be the subspace of K∞ consisting of all functions f ∈ K∞ such that lim 2−mγ kf χm kLpα = 0.

m→∞

Note that Kqp,γ ⊂ K0p,γ for all q < ∞. For more information on the Herz spaces, see [6] and references therein. For α > −1, let 0 < p < ∞ and γ real. Then, given m ≥ 0, we have 1 − |z|2 ≈ 2−m for z ∈ Am and thus we obtain Z p p −mγp 2 kf χm kLpα = 2−mγ |f (z)| dVα (z) A Z m ≈ (1 − |z|2 )γp |f (z)|p dVα (z) Am

≈ kf χm kpLp

α+γp

and this estimate is uniform in m. It follows that

n o

kf kKqp,γ ≈ kf χm kLpα+γp

(4.1)

`q

for 0 < q ≤ ∞. In particular, since λ = V−n−1 , we have

kf kKp,−(n+α+1)/p ≈ kf χm kLp (λ)

`q

q

and this estimate is easily seen to be valid even for p = ∞ since −(n + α + 1)/p = 0 if p = ∞. For this reason, we put p,− n+α+1 p

Kqp (λ) = Kq

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for the full range 0 < p ≤ ∞ and 0 ≤ q ≤ ∞. Note that Kpp (λ) ≈ Lp (λ) for 0 < p ≤ ∞. That is, these two spaces are the same as sets and have equivalent norms as Banach spaces. Note that H¨ older’s inequality holds in the Herz space as follows : Z f g dVα ≤ kf kKp,γ kgk p0 ,−γ (4.2) q Kq0 B

0

for functions f ∈ and g ∈ Kqp0 ,−γ for the full range 1 ≤ p, q ≤ ∞ and arbitrary γ real (see [2] for the details). Here and in what follows, p0 is the conjugate exponent of p. Given 0 −(α+1)/p; or γ = −(α + 1)/p and q = ∞. Hence we obtain by (4.2), Kqp,γ

p,γ 1 ≤ p ≤ ∞ and γ < (α + 1)/p0 =⇒ K∞ ⊂ L1α .

The similar conclusion can be obtained for the case γ = (α + 1)/p0 , namely, 1 ≤ p ≤ ∞ and γ = (α + 1)/p0 =⇒ K1p,γ ⊂ L1α . Now, we can see that if 1 ≤ p ≤ ∞ and if either γ < (α + 1)(1 − 1/p); Kqp,α

or γ = (α + 1)(1 − 1/p) and 0 < q ≤ 1,

(4.3)

L1α .

then the Herz space ⊂ Next, we introduce a discrete version of Herz spaces. Let a = {am } be an arbitrary lattice. Given a complex sequence ξ = {ξm } and a positive integer k, let ξχk denote the sequence defined by (ξχk )m = ξm χk (am ). Now, given 0 < p, q ≤ ∞, α > −1 and γ real, we let `p,γ q (a) be mixed-norm space of all complex sequences ξ such that

n o

kξk`p,γ (a) = 2−kγ kξχk k`p,α (a) q < ∞. q `

So, we have ) pq

( kξkq`p,γ (a) = q

X k

2−kγp

X

|ξm |p (1 − |am |2 )α

,

0 < p, q < ∞.

am ∈Ak

−kγ p ξχk k`p,α (a) } ∈ `0 . Finally, we let `p,0 Also, we say ξ ∈ `p,γ q (a) = `q (a). 0 (a) if k{2 We now introduce the so-called Schatten-Herz class of Toeplitz operators. Let S∞ denote the class of all bounded linear operators on b2 and k k∞ denote the operator norm. Given 0 < p, q ≤ ∞, the Schatten-Herz class Sp,q is the class of all Toeplitz operators Tµ such that Tµχk ∈ Sp for each k and the sequence {kTµχk kLpα } belongs to `q . The norm of Tµ ∈ Sp,q is defined by

kTµ kp,q = {kTµχk kLpα } q . `

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Also, we say Tµ ∈ Sp,0 if Tµ ∈ Sp,∞ and {kTµχk kLpα } ∈ `0 . Note that Sp,q ⊂ Sp,0 for all q < ∞. Theorem 4.1. Let 0 < p, q ≤ ∞, r ∈ (0, 1), α > −1 and γ real. Then the following statements hold for µ, τ ≥ 0. (a) {2−kγ kb µr χk kLp (τα ) } ∈ `q if and only if {2−kγ k(µχ dk )r kLp (τα ) } ∈ `q . −kγ −kγ (b) {2 kb µr χk kLp (τα ) } ∈ `0 if and only if {2 k(µχ dk )r kLp (τα ) } ∈ `0 . Here, dτα (z) = (1 − |z|2 )α dτ (z). Before proceeding to the proof, we note the following covering property (see Lemma 4.1 of [5]): If r ∈ (0, 1) and N = N (r) is a positive integer such that 1+r N − 1 ≤ log2 ≤ N, (4.4) 1−r then k+N [ Er (z) ⊂ Aj , z ∈ Ak . (4.5) j=k−N

Here and in the proof below, we let Aj = ∅ if j < 0. Proof. Let µ, τ ≥ 0 be given. We prove the theorem only for q < ∞; the case q = ∞ is implicit in the proof below. Choose N = N (r) as in (4.4) and put γp = γ(p, N ) = max{1, (2N + 1)p−1 }. We note by (4.5) k+N X

µ br χk ≤

(µχ dj )r

j=k−N

for all k. Thus, we have 1

kb µr χk kLp (τα ) ≤ γpp

k+N X

k(µχ dj )r kLp (τα )

j=k−N

for all k. Since 2−kγ ≤ 2N |γ| 2−jγ for k − N ≤ j ≤ k + N , it follows that k+N X

2−kγ kb µr χk kLp (τα ) .

2−jγ k(µχ dj )r kLp (τα )

j=k−N

for all k. This implies one direction of (b). Also, it follows that −kγq

2

kb µr χk kqLp (τα )

. γq

k+N X

2−jγq k(µχ dj )r kqLp (τα )

j=k−N

for all k. Thus, summing up the both sides of the above over all k, we have ∞ ∞ X X 2−kγq kb µr χk kqLp (τα ) . (2N + 1) 2−jγq k(µχ dj )r kqLp (τα ) , k=0

which gives one direction of (a).

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We now prove the other directions. Note that Er (z) can intersect Ak with k ≥ 0 only when z ∈ ∪k+N j=k−N Aj by (4.5). Thus we have (µχ dk )r =

k+N X

(µχ dk )r χj ≤

j=k−N

k+N X

µ br χj .

j=k−N

Thus, a similar argument yields the other directions of (a) and (b).

As an immediate consequence of Corollary 3.5 and Theorem 4.1 with τ = V , we have the following Herz space version of Corollary 3.5. Corollary 4.2. Let 0 < p ≤ ∞, 0 ≤ q ≤ ∞ and δ, r ∈ (0, 1). Let µ ≥ 0. Then µ br ∈ Kqp (λ) if and only if µ bδ ∈ Kqp (λ). Also, applying Theorem 4.1 with discrete measures τ = denotes the point mass at z ∈ B, we have the following.

P

m δam

where δz

Corollary 4.3. Let 0 < p, q ≤ ∞, r ∈ (0, 1) and γ be real. Let µ ≥ 0. Assume \ that a = {am } is an r-lattice and put ξk = 2−kγ k{(µχ k )r (am )}m k`p,α (a) for k ≥ 0. Then the following statements hold. q (a) {b µr (am )} ∈ `p,γ q (a) if and only if {ξk } ∈ ` . p,γ (b) {b µr (am )} ∈ `0 (a) if and only if {ξk } ∈ `0 .

The following shows that Kqp (λ)-boundedness of the Berezin transform. Proposition 4.4. Let 1 ≤ p < ∞, 1 ≤ q ≤ ∞, α > −1 and γ be real. If −(n + α + 1) − (α + 1)/p < γ < (α + 1)(1 − 1/p), then the Berezin transform is bounded on Kqp,γ . In particular, the Berezin transform is bounded on Kqp (λ). Proof. Assume −(n + α + 1) − (α + 1)/p < γ < (α + 1)(1 − 1/p), or equivalently, −(n + α + 1)p < γp + α + 1 < (α + 1)p. Note that Kqp,γ ⊂ L1α whenever γ < (α+1)(1−1/p). Thus the Berezin transform is well defined on Kqp,γ by the condition of p, α and γ. Choosing an > 0 such that −(n + α + 1)p < γp ± + α + 1 < (α + 1)p, we see by Proposition 3.6 (with a = n+α+1 and b = α) that the Berezin transform is actually bounded on Lp (Vα+γp± ). In other words, Z Z |fg χm |p dVα+γp± ≤ C1 |f |p dVα+γp± B

Am

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for each m ≥ 0. Note that 1 − |z| ≈ 2−m for z ∈ Am and m ≥ 0. Thus the above estimate yields Z Z p ∓k g |f χm | dVα+γp ≈ 2 |fg χm |p dVα+γp± Ak Ak Z ∓k .2 |f |p dVα+γp± Am Z ∓(k−m) ≈2 |f χm |p dVα+γp B

so that k(fg χm )χk kLpα+γp . 2−|k−m|/p kf χm kLpα+γp

(4.6)

for all integers m, k ≥ 0. So, for each k = 0, 1, · · · , we have by (4.6) X X kfeχk kLpα+γp ≤ k(fg χm )χk kLpα+γp . 2−|k−m|/p kf χm kLpα+γp m

m

and therefore conclude by (4.1) ( )

X

kfekKp,γ . 2−k/p {kf χm kLp

}

α+γp

q

k

`q

≈ kf kKqp,γ ,

where the first inequality holds by Young’s inequality. This completes the proof of the first part of the proposition. Taking γ = −(n + α + 1)/p, we have the second part of the proposition. The proof is complete. We now extend the parameter q to the full range 0 ≤ q ≤ ∞ in the above proposition. Lemma 4.5. Let 1 ≤ p ≤ ∞, α > −1 and γ be real. For 0 ≤ q ≤ ∞, if −(n + α + 1) < γ + (α + 1)/p < α + 1, then the Berezin transform is bounded on Kqp,γ . Proof. Assume −(n + α + 1) − γ < (α + 1)/p < α + 1 − γ and let f ≥ 0 be a given measurable function. For 1 ≤ p ≤ ∞, the proof of Proposition 3.6 and Lemma 3.7 of [5] yield that there exist positive constants = (p, α, γ) and C = C(p, α, γ) such that −kγ

2 for all k ≥ 0.

kfeχk kLpα ≤ C

∞ X 2−mγ kf χm kLpα 2|m−k| m=0

(4.7)

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Now, for 0 < q ≤ 1, we have by (4.7) kfekqKqp,γ ≤ C q ≤ Cq ≤ Cq

∞ X

∞ X 2−mγ kf χm kLpα 2|m−k| m=0

k=0 ∞ X ∞ X

k=0 m=0 ∞ X

!q

2−mqγ kf χm kqLpα 2q|m−k| ! 2−q|k|

kf kqKqp,γ .

k=−∞

Combining this with Proposition 4.4, we have kfekKqp,γ .

∞ X

!1/q −q|k|

2

kf kKqp,γ

k=−∞

for 0 < q ≤ ∞. Finally, we consider the case q = 0. Assume f ∈ K0p,γ and let an integer k ≥ 0 be given. Then we have by (4.7) −kγ

2

∞ X X X 2−mγ kf χm kLpα = + 2|m−k| m=0 m>d m≤d X −mγ p,γ . sup 2 kf χm kLpα + kf kK∞

kfeχk kLpα .

1 |m−k| 2 m≤d

m>d

for each integer d ≥ 1. Now, taking the limit k → ∞ with d fixed, we obtain lim sup 2−kγ kfeχk kLpα . sup 2−mγ kf χm kLpα k→∞

m>d

for all d. So, taking another limit d → ∞, we conclude fe ∈ K0p,γ .

Theorem 4.6. Let 0 −1. Assume that µ ≥ 0 and a = {am } is an r-lattice with r ∈ (0, 1). Then the following statements are equivalent. (a) Tµ ∈ Sp,q . (b) µ br ∈ Kqp (λ). (c) {b µr (am )} ∈ `pq . Moreover, if n/(n + α + 1) < p ≤ ∞, then the above statements are equivalent to (d) µ e ∈ Kqp (λ). Proof. The proofs of Theorems 3.9 and 4.1 show that all the associated norms are equivalent. We have (a) ⇐⇒ (b) by Theorem 3.9 and Corollary 4.3. (d) =⇒ (a): This follows from Lemma 3.1 and Corollary 3.5. We now assume n/(n + α + 1) < p ≤ ∞ and prove that (a) or (b) implies (d). For 1 ≤ p ≤ ∞, one may use Lemma 4.5 and Corollary 3.8 to see that (a) implies (d). So, we may further assume that p < 1 in the proof below.

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(b) =⇒ (d): Assume (b). First, consider the case 0 < q < ∞. By the proof of Theorem 3.9, we have X (1 − |z|2 )p(n+α+1) µ e(z)p . (4.8) (1 − |am |2 )p(n+α+1) µ br (am )p |1 − z · am |2p(n+α+1) m for z ∈ B. Let j and k be given. Note that by an integration in polar coordinates and Lemma 2.3 Z 2−j dV (z) . 2p(n+α+1) [1 − (1 − 2−j−1 )|a|]2p(n+α+1)−n Aj |1 − z · a| ≈

2−j (2−j

+

2−k )2p(n+α+1)−n

,

a ∈ Ak ;

this estimate is uniform in j and k. Hence, if am ∈ Ak , then we have Z (1 − |z|2 )p(n+α+1) 2 p(n+α+1) (1 − |am | ) (1 − |z|2 )−n−1 dV (z) 2p(n+α+1) Aj |1 − z · am | Z dV (z) ≈ 2−kp(n+α+1) 2−j{p(n+α+1)−n−1} 2p(n+α+1) Aj |1 − z · am | .

2−kp(n+α+1) 2−j{p(n+α+1)−n} (2−j + 2−k )2p(n+α+1)−n

=

2−kp(n+α+1) 2jp(n+α+1) (1 + 2j−k )2p(n+α+1)−n

=

2p(j−k)(n+α+1) . (1 + 2j−k )2p(n+α+1)−n

P Thus, setting ξkp = am ∈Ak µ br (am )p and integrating both sides of (4.8) over Aj against the measure dλ(z), we obtain Z X p 2p(j−k)(n+α+1) µ e(z)p (1 − |z|2 )−n−1 dV (z) . ξk . (1 + 2j−k )2p(n+α+1)−n Aj k Note that for k ≥ j 1 2p(j−k)(n+α+1) ≤ 2p(j−k)(n+α+1) = p(n+α+1)|k−j| (1 + 2j−k )2p(n+α+1)−n 2 and for k < j 2p(j−k)(n+α+1) 1 1 ≈ (j−k){p(n+α+1)−n} = {p(n+α+1)−n}|k−j| . j−k 2p(n+α+1)−n (1 + 2 ) 2 2 Therefore, combining these estimates, we have Z X µ e(z)p (1 − |z|2 )−n−1 dV (z) . Aj

k

ξkp 2γ|k−j|

for all j where γ = p(n + α + 1). Note that γ > 0 since n/(n + α + 1) < p.

(4.9)

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We first consider the case p < q ≤ ∞. From (4.9) and Young’s inequality, we have ke µkpKqp (λ) . k{ξkp }k

q

`p

= k{ξk }kp`q

Now, for 0 < q ≤ p, we have again by (4.9) ( ) pq p X X X qX ξ 1 k ke µkpKqp (λ) . ≈ k{ξk }kq`q . ξk γ|k−j| γq|k−j|/p 2 2 j j k k as desired. The case q = 0 also easily follows from (4.9).

5. Remarks In this section, we prove the cut-off point n/(n + α + 1) in Theorems 3.9 and 4.6 is sharp. In order to prove this, we set Γa,b,θ = {z = (z1 , · · · zn ) ∈ B : a < |z| < b, | arg(zi )| < θ}. Then we have the following result. Lemma 5.1. Let α > −1 and (n + α + 1)δ = cos−1 34 . Then there exists a constant C > 0 such that C Rzα (w) ≥ |1 − z · w|n+α+1 for z ∈ Γ1/2,1,δ/2 , w ∈ Γ1/4,1/2,δ/2 . Proof. Let z ∈ Γ1/2,1,δ/2 . Then one can see that arg(z · w) < δ

and

1 1 < |z||w| < 8 2

for w ∈ Γ1/4,1/2,δ/2 so that arg(1 − z · w) < δ

and |1 − z · w| < 1.

This implies that 1 − z · w ∈ Γ1/2,1,δ . Now we let u = (1 − z · w)n+α+1 . Then |u| < 1

and

cos(arg u) <

3 . 4

Using these above facts, we have Rzα (w) =

2Re u − |u|2 1 1 = (2 cos(arg u) − |u|) ≥ 2 |u| |u| |u|

for z ∈ Γ1/2,1,δ/2 and w ∈ Γ1/4,1/2,δ/2 .

3 1 − |u| ≥ 2 2|u|

We let β be real with β > n/p and consider the function fβ (z) = (1 − |z|)β . d Note that fβ ≈ (f β )r for each r ∈ (0, 1). We also have Z Z 1 p 2 −n−1 |fβ (z)| (1 − |z| ) dV (z) ≤ (1 − r)pβ−n−1 dr < ∞ (5.1) B

0

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for pβ − n > 0. Note Lemma 5.1 implies that 2 n+α+1 ff β (z) & (1 − |z| )

for z ∈ Γ1/2,1,δ/2 . From this, we have Z Z p 2 −n−1 |ff dV (z) & C β (z)| (1 − |z| ) B

(5.2)

(1 − |z|2 )p(n+α+1)−n−1 dV (z) = ∞

Γ1/2,1,δ/2

(5.3) for p(n + α + 1) ≤ n. Thus we have by (5.1) and (5.3) p d (f β )r ∈ L (λ),

but ff / Lp (λ) β ∈

for (n + α + 1) ≤ n/p < β, which implies that the cut-off point n/(n + α + 1) in Theorem 3.9 is sharp. Finally, we prove n/(n + α + 1) is sharp in Theorem 4.6. First we have q/p X n oq/p X Z p 2 −n−1 2−m(pβ−n) <∞ |fβ (z)| (1 − |z| ) dV (z) ≈ m

Am

m

for pβ − n > 0. In the case q < ∞, we also have by (5.2) q/p X X Z p 2 −n−1 f |fβ (z)| (1 − |z| ) dV (z) & 2−mq/p(p(n+α+1)−n) = ∞ m

Am

m

for p(n + α + 1) ≤ n. Thus, we have p d (f β )r ∈ Kq (λ),

but feβ ∈ / Kqp (λ)

for (n + α + 1) ≤ n/p < β and q < ∞. Similarly, one can obtain the same result in the case q = ∞. Acknowledgments The author thanks Professor Boo Rim Choe for giving some ideas concerning Theorem 3.4 and also thanks the referee for helpful comments.

References [1] B. R. Choe, H. Koo and Y. J. Lee, Positive Schatten class Toeplitz operators on the ball, Studia Math. 189 (2008), 65–90. [2] B. R. Choe, H. Koo and K. Na, Positive Toeplitz operators of Shatten-Herz type, Nagoya Math. J. 185 (2007), 31–62. [3] B. R. Choe, Y. J. Lee and K. Na, Toeplitz operators on harmonic Bergman spaces, Nagoya Math. J. 174 (2004), 165–186. [4] E. S. Choi, Positive Toeplitz operators on pluriharmonic Bergman space, J. Math. Kyoto. Univ. 47 (2007), 93–111. [5] E. S. Choi and K. Na, Schatten-Herz type positive Toeplitz operators on pluriharmonic Bergman spaces, J. Math. Anal. Appl. 327 (2007), 679–694.

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Na

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[6] E. Hern´ andez and D. Yang, Interpolation of Herz spaces and applications, Math. Nachr. 205 (1999), 69–87. [7] M. Loaiza, M. L´ opez-Garc´ıa and S. P´erez-Esteva, Herz classes and Toeplitz operators in the disk, Integr. equ. oper. theory 53 (2005), 287–296. [8] W. Rudin, Function theory in the unit ball of Cn , Springer Verlag, 1980. [9] K. Zhu, Operator theory in function spaces, Marcel Dekker, New York and Basel, 1989. [10] K. Zhu, Spaces of holomorphic functions in the unit ball of C n , Springer-Verlag (GTM 226), 2004. Kyunguk Na General Education, Mathematics Hanshin University Gyeonggi 447-791 Korea e-mail: [email protected] Submitted: September 10, 2008. Revised: April 22, 2009.

Integr. equ. oper. theory 64 (2009), 429–453 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/030429-25, published online May 12, 2009 DOI 10.1007/s00020-009-1682-1

Integral Equations and Operator Theory

Riesz Systems and Controllability of Heat Equations with Memory L. Pandolfi Abstract. The main result we derive is the proof that a particular set of functions related to the controllability of the heat equation with memory and finite signal speed, with suitable kernel, is a Riesz system. Riesz systems are important tools in applied mathematics, for example for the solution of inverse problems. In this paper we shows that the Riesz system we identify can be used to give a constructive method for the computation of the control steering a given initial condition to a prescribed target. Mathematics Subject Classification (2000). Primary 44A60; Secondary 93B05. Keywords. Heat equation with memory, controllability, moment problem.

1. Introduction Let us consider the following heat equation with memory in one space dimension, Z t θt = N (t − s)∆θ(s) ds . (1.1) 0

Here, θ = θ(t, x) with x ∈ (0, π) and t > 0. We associate the following initial and boundary conditions to Eq. (1.1): θ(0) = θ(0, x) = ξ(x),

x ∈ (0, π) ,

θ(t, 1) = 0 ,

t > 0.

θ(t, 0) = u(t) ,

(1.2)

The function u(·) is locally square integrable for t > 0 and the initial condition ξ belongs to L2 (0, π). The operator ∆ in Eq. (1.1) is the laplacian in one space dimension, ∆θ(x) = θxx (x) . Assumptions on the kernel N (t) are described below. This paper fits into the research program of the GNAMPA-INDAM .

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Equation (1.1) has been independently proposed by several authors as a version of the “heat equation” with finite signal speed, noticeably in [8] and [14]. The goal of this paper is the identification of a special sequence {zn (t)} of functions, related to Eq. (1.1), which forms a Riesz system in L2 (0, π) (the definition is in Section 1.1). The functions zn (t) are the solution of the integrodifferential equation Z t N (t − s)zn (s) ds , zn (0) = 1 . zn0 (t) = −n2 0

The introduction of these functions is suggested by a control problem (described below) and in this paper we use these functions (and known results on the controllability of Eq. (1.1)) in order to represent a function u, a “control”, which drives the initial condition ξ to a final target (both in L2 (0, π)) in time T (it will be T ≥ π); i.e. a control which solves the problem θ(T ) = η .

(1.3)

So, we can interpret the results in this paper as a “constructive” solution of the control problem (1.1)–(1.3). In fact, this is only a part of the story since moment problems appear often in applied mathematics, for example in the solution of inverse problems, see [1, 4], so that the identification of a suitable Riesz system which is naturally associated to Eq. (1.1) is interesting by itself, in particular because the solution of moment problems posed with respect to Riesz systems are well posed. Algorithms for the solution of moment problems are described in [1]. Applications of the Riesz system introduced here to the solution of inverse problems will appear elsewhere, see [22]. Let us now comment on the solvability of the problem (1.1)-(1.2). Existence and uniqueness of the solution for every locally square integrable control u is proved in [20], provided that the kernel N (t) is twice differentiable and N (0) > 0 (so we can assume N (0) = 1.) In that paper it is proved that for every square integrable control u, the solution is a continuous L2 (0, π)-valued function, so that evaluation of the solution at a certain time T is permissible. The controllability problem for Equation (1.1) has been studied and solved firstly in [5]. It was proved in that paper that the controllability problem is solvable under an additional regularity condition on η, provided that T > π. The solution rests on Carleman estimates and it can’t be considered constructive. The case that the space variable belongs to IRn has been considered in [27], still using Carleman estimates, and in [20]. The proof in this last paper (see also [21]) is based on Baire’s theorem and compactness arguments, and it is not constructive. In this paper we first show that the controllability problem can be reduced to a suitable moment problem with respect to the Riesz system {zn }. The known results on the controllability problem shortly recalled above (and described with more details in Section 3.2) are derived under the assumption that N (t) is of class C 3 . So, as in the previous papers on controllability, the standing

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assumptions in this paper are that the kernel N (t) is of class C 3 and N (0) = 1. Controllability for a different class of kernels is studied in [19]. Finally, let us note that even if η = 0 then we don’t have controllability to rest: we construct a control which forces the solution to hit the 0 target at a certain time T but we cannot force the solution to remain at rest in the future: in fact, in the case of heat equations with memory, controllability to rest can be achieved only in exceptional cases, see [15]. 1.1. Preliminaries on the abstract moment problem Key references for this section are [1, 2, 12, 28]. We consider a separable Hilbert space H. A sequence {zn }n≥1 in H is a Schauder basis when every element h ∈ H can be represented in a unique way as h=

+∞ X

αn zn .

(1.4)

n=1

The convergence of the series is in the norm of H. A Schauder basis has the additional property of being a minimal basis, i.e. for every j we have zj ∈ / cl span{zr , r 6= j} , see [12, p. 312]. A sequence which is a minimal basis in its span is called a minimal sequence. An abstract moment problem in H is as follows. Let zn be elements of H and let {cn } ∈ l2 . We want to know whether it is possible to find v ∈ H such that hv, zn i = cn

(1.5)

(the inner product is that of H.) A moment problem like (1.5) can be defined in a Banach space too and in this case the crochet represents duality. Moment problems are mostly studied in the case that the functions zn have a special form (polynomials or exponentials) and this study has been at the core of functional analysis. However, even if the functions zn do not have any special form, conditions for solvability are in [2, Theorem I.2.1]. In particular, if the moment problem is solvable for every element {cn } of a dense subset of l2 then the sequence {zn } is ω-(linearly) independent, i.e. the following property holds: X {αn } ∈ l2 and αn zn = 0 =⇒ {αn } = 0 . (1.6) If {zn } is a minimal sequence, then it is possible to give a formula for the coefficients associated to h in (1.4). It is possible to construct a biorthogonal sequence of {zn }. That is, it is possible to find a sequence {ζn } such that hzn , ζk i = δn,k (δn,k is the Kronecker delta) and the (unique) element αn in the representation (1.4) of h is αn = hh, ζn i .

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So, when {zn } is a minimal sequence and problem (1.5) is solvable, its solution is v=

+∞ X

cn ζn .

n=1

In spite of this, practical computations with minimal sequences do not lead to well posed problems, since the biorthogonal sequence can be unbounded. A more restrictive condition, under which practical computations are feasible, is that the sequence {zn } be a Riesz sequence (or “Riesz system”. A Riesz sequence is called an L-sequence in [2].) The meaning of this is as follows: let L ⊆ H be the closed linear space spanned by the vectors zn . The sequence {zn } is a Riesz sequence when there exists an orthogonal basis {n } (of a Hilbert space H 0 ) and a linear bounded operator T 0 from H 0 onto L which is boundedly invertible and such that T 0 n = zn . If L = H then the sequence {zn } is called a Riesz basis of H. Riesz systems share the following properties with Fourier series: 1) every Riesz system is bounded; 2) there exist positive numbers m and M such that

2

X X X

αn zn ≤ M |αn |2 . m |αn |2 ≤ H

Note that the biorthogonal sequence of a Riesz basis is a Riesz basis too, hence it is bounded. If a system is a Riesz system in the Hilbert space H, which is not a basis, then it may have an unbounded “biorthogonal system” due to the fact that the set of the projections of ζn on [span {zn }]⊥ can be unbounded but it is possible to find such biorthogonal sets which are bounded. In particular, the one which belongs to cl span {zn } is bounded and it is a Riesz system too. Moreover, when {zn } is a Riesz system, then the series X αn zn converges if and only if {αn } ∈ l2 and every f ∈ L = cl span{zn } can be represented as X f= αn zn , αn = hf, ζn i where {ζn } is biorthogonal to {zn }. So, we wish not only that our moment problem is solvable, but also that our sequence {zn } is a Riesz sequence. Several tests have been given for this. We shall use the following one: Theorem 1.1 (Bari Theorem). If {n } is a Riesz basis in H and if the sequence {zn } satisfies conditions (1.6) and +∞ X

kzn − n k2 < +∞

n=1

then the sequence {zn } is a Riesz sequence too.

(1.7)

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See [28, p. 45] and [12, p. 322] for the proof (the proof in [28, p. 45] assumes that {n } is an orthonormal basis.) As we said, if condition (1.6) holds, then the sequence {zn } is called ωindependent; if condition (1.7) holds, the sequence {zn } is called quadratically close to {n } (when H = L2 (0, T ) we also say “L2 –close” to {n }) and the Riesz basis {zn } is called in particular a “Bari sequence”. In fact, we need a minor variant of Bari’s theorem, which is as follows: Theorem 1.2. Let {n }n≥0 be a Riesz sequence of H. If {zn }n≥1 is ω-independent and quadratically close to {n }n≥1 , then {zn }n≥1 is a Riesz sequence in H. The proof is outlined in [12, Remarque 2.1, p. 323] and it is reported in the Appendix for the sake of completeness. A different version of Bari’s theorem used in control theory is in [13]. 1.2. Preliminaries on the wave equation In order to put this paper in the proper setting, we recall here the very well known problem of the controllability of the wave equation in one space dimension, wtt = wxx ,

0 < x < π,

t>0

(1.8)

with conditions w(t, 0) = u(t) ,

( w(0, x) = ξ0 , wt (0, x) = ξ1 .

w(t, π) = 0 ,

The problem we consider is as follows: a target η ∈ L2 (0, π) is given and we want to find a suitable time T > 0 and a square integrable control u(·) such that w(T, ·) = η(·). Note that we are controlling the final shape but not the final velocity. We shall see that it is possible to choose T = π, the same time for every initial conditions and target (if instead we want to control both the configuration and the velocity then it must be T = 2π.) The proof of this result goes as follows: we consider the functions φn (x) = sin nx, n ∈ IN, which solve the eigenvalue problem φ00n (x) = −n2 φn (x) , φn (0) = φn (π) = 0 . (1.9) We compute the scalar product (in L2 (0, π)) of both the sides of (1.8) with φn . Integration by parts in x gives the equality d2 hw, φn i = −n2 hw, φn i + φ0n (0)u(t) dt2 so that Z hw(t), φn i = An cos nt + Bn sin nt +

t

[sin ns]u(t − s) ds . 0

The coefficients An and Bn are given by An = hξ0 , φn i ,

Bn =

1 hξ1 , φn i n

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(so that w(t, ·) ∈ H 1 (0, π) if ξ0 (·) ∈ H 1 (0, π) and ξ1 (·) ∈ L2 (0, π).) The condition w(π, ·) = η(·) is then equivalent to the moment problem Z π [sin ns]v(s) ds = hη, φn i − (−1)n hξ0 , φn i = cn (1.10) 0

where v(s) = u(π − s): the control u(·) exists if and only if there exists a function v which solves the equalities (1.10) for every n. The known theory of the Fourier series shows that this problem is very easily solved: v(t) =

+∞ 2X ck sin kt . π

(1.11)

k=1

We can go the opposite way: if controllability of the wave equation has been independently proved, then the previous arguments are a proof of the fact that the moment problem (1.10) is solvable. This is the turn of ideas we follow in this paper: we first prove that the sequence {zn } is L2 -close to a Riesz system; we then use the fact that control problem (1.1)–(1.3) is known to be solvable in order to prove that the sequence {zn } is ω-independent so that Bari Theorem can be applied. So, it will turn out that the sequence {zn } is a Riesz sequence. We then use this property in order to derive a formula for the control which drives the initial condition ξ to the target η, which extends (1.11). 1.3. References It seems that one of the first papers which uses moment problems in control theory is [7], followed by several papers in particular by Russel and Fattorini (see [10, 11, 23].) Among the numerous recent papers we confine ourselves to cite the papers [3, 16]. Apart from the papers too numerous to be cited, the applications of the moment problem to control theory has been examined in three books: [2, 17, 18]. As we noted, Riesz systems are crucial in the solution of a large class of inverse problems. As an example of this, we cite the paper [4]. We note that the “classical” moment problems, when {zn } is a sequence of polynomials, do not correspond to Riesz basis and in fact the truncated problems obtained by considering only finitely many equation is severely ill conditioned, see [25]. In contrast with this, moment problems which correspond to Riesz basis are well posed problems. An application of the results proved here to the solution of an identification problem can be found in [22].

2. Reduction to a moment problem In this section we are going to prove that the controllability problem we presented for the equation (1.1) is equivalent to a certain moment problem. As in the case of the wave equation, we consider the functions φn (x) = sin nx, n ∈ IN, which solve the eigenvalue problem (1.9). We note thatp{φn } is a complete orthogonal system in L2 (0, π) (not a normal system: kφn k = π/2.)

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Let λ be one of the numbers −n2 and let φ be the corresponding eigenfunction sin nx. Let h(t) solve Z T h0 (t) = −λ N (r − t)h(r) dr , h(T ) = 1 (2.1) t

(here T is any fixed number. Its value for the controllability problem will be specified later on.) Let the control function u(t) and the initial datum ξ be fixed. We consider the integral Z π Z T d [h(t)θ(t, x)] dt dx . I= φ(x) dt 0 0 Integration by parts shows that this integral is Z T Z π Z T Z π h(t)θt (t, x) dt dx φ(x) h0 (t)θ(t, x) dt dx + φ(x) I= 0 0 0 0 # Z Z " Z π

=

T

T

−λ

φ(x) 0

0 π

Z +

φ(x)

π

=

Z

T

0

0

Z

=

T

"0

0

Z

−λ

φ(x) 0

0 T

Z + So we have Z π

T

π

φ(x)θxx (r, x) dx dr dt #

N (r − t)h(r) dr θ(t, x) dt dx t

Z h(t)

0

Z N (t − r)

h(t) Z

N (r − t)h(r) dr θ(t, x) dt dx

t

Z

0 π

dt dx

t

T

+

T

−λ

N (t − r)θxx (r, x) dr #

h(t)

0

Z

t

Z

"0

φ(x)

Z

T

Z

0

Z

N (r − t)h(r) dr θ(t, x) dt dx t

t

Z 0 N (t − r) φ (0)u(r) + λ

0

π

φ(x)θ(r, x) dx dr dt .

0

Z

π 0

φ(x)θ(T, x) dx − h(0) φ(x)ξ(x) dx = φ (0) 0 0 Z t v˜(t) = N (t − r)u(r) dr .

Z

T

h(t)˜ v (t) dt , 0

(2.2)

0

Smoothness assumptions needed to justify integration by parts hold provided that u is smooth and certain compatibility conditions between u and ξ hold, see [20]. These conditions are satisfied by the pairs (ξ, u) in a dense subset of L2 (0, π) × L2 (0, T ). Both the sides of (2.2) are continuous functions of u and ξ so that equality (2.2) holds for every initial condition ξ ∈ L2 (0, π) and every control u ∈ L2 (0, T ). This proves the necessity part of the following result:

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Lemma 2.1. Let hn (t) be the solution of problem (2.1) with λ = −n2 . Let ξ be an initial condition and let η be the prescribed target, both in L2 (0, π). A square integrable control u which transfer ξ to η in time T exists if and only if for every n we have Z π Z T 1 φn (x) [η(x) − hn (0)ξ(x)] dx (2.3) hn (r)˜ v (r) dr = 0 φn (0) 0 0 where Z

r

N (r − s)u(s) ds .

v˜(r) =

(2.4)

0

Proof. We need to prove the sufficiency part. We assume that there exists a control u such that the function v˜ defined in (2.4) solves (2.3) for every n. Repeating the computations above we see that Z T Z π 1 hn (r)˜ v (r) dr = 0 φn (x) [θ(T, x) − hn (0)ξ(x)] dx . φn (0) 0 0 Our assumption is that (2.3) holds for the function v˜(r) given by (2.4) so that we have also Z π Z T φn (x)θ(T, x) dx = φn (x)η(x) dx . 0

0 2

The set {φn (x)} being complete in L (0, π), we get θ(t, x) = η(x), as wanted. In the next section, we shall concentrate on the solution of the problem (2.3) in terms of v˜ and we disregard the fact that v˜ should have a special expression in terms of u. I.e., we study a different problem: the problem of finding a square integrable function v˜ which solves (2.3). Once this problem is solved and the regularity properties of the solution v˜ have been studied, we shall see that also the original control problem (in terms of u) can be solved. Let us consider now the right hand side of (2.3). The integral on the right hand side is cn = hφn , ηiL2 (0,π) − hn (0)hφn , ξiL2 (0,π) . We shall see in Section 3 that the sequence {hn (0)} is bounded so that the sequence {cn } belongs to l2 : the problem of determining a function v˜ which solves (2.3) is a moment problem in L2 (0, T ), Z

T

hn (r)˜ v (r) dr = 0

cn , n

{cn } ∈ l2 .

(2.5)

The sequence {hn (0)} being bounded, the sequence {cn /n} fills a dense subspace of l2 , which is a proper subspace of l2 .

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3. The Riesz system In this section we prove that the sequence {hn } which appears in (2.5) is a Riesz system in a suitable bounded interval identified below (and, in the course of the proof, we shall also see boundedness of the sequence of functions {hn (t)} on every bounded interval.) Computations have a more natural appearance if we use the following transformation: zn (t) = hn (T − t) (note that T will be π later on. For the moment, T is a any fixed number.) The function zn (t) solves Z t zn0 (t) = −n2 N (t − s)zn (s) ds , zn (0) = 1, (3.1) 0

and the moment problem takes the form Z π Z T 1 φn (x)[η(x) − zn (T )ξ(x)] dx, zn (t)v(t) dt = cn , cn = n 0 0

(3.2)

where now v(t) = v˜(T − t) and u(t) has to be determined from the equation Z t N (t − s)u(s) ds = v(T − t) . (3.3) 0

The proof that {zn (t)}n≥1 is a Riesz sequence in L2 (0, π) is based on Theorem 1.2 and it is divided in two parts: we first prove (in Subsection 3.1) that {zn (t)}n≥1 is quadratically close to the sequence√{eαt cos nt}n≥1 , where α = N 0 (0)/2, (the sequence {cos nt} plus the element 1/ 2 is a complete orthogonal system of elements of constant norm in L2 (0, π) so that the sequence {eαt cos nt}n≥1 is a Riesz system.) The property of ω-independence is in Subsection 3.2. In order to illustrate the ideas in this paper as clearly as possible, the computations in Section 3.1 uses the restrictive assumption N 0 (0) = 0. At the expenses of more involved computations, the same ideas can be used in the general case, even if N 0 (0) 6= 0, as shown in the Appendix. 3.1. L2 -closedness to a Riesz sequence In this section we prove that {zn (t)} is quadratically close to a Riesz sequence. The sole conditions on the kernel N (t) needed in this proof are the standing assumptions of this paper: N (t) is of class C 3 with N (0) = 1. The computations in this case are involved and relegated to the appendix. For the sake of clarity, we present here the computations under the restrictive condition N 0 (0) = 0 and we prove that {zn (t)}n≥1 is L2 -close to the the Riesz sequence {cos nt}n≥1 . We compute the derivatives of both sides of (3.1) and we see that Z t zn00 (t) = −n2 zn (t) − n2 N 0 (t − s)zn (s) ds , zn (0) = 1 , zn0 (0) = 0 0

so that Z zn (t) = cos nt − n

t

Z sin n(t − s)

0

0

s

N 0 (s − r)zn (r) dr ds .

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Integration by parts gives Z s Z t d zn (t) = cos nt − N 0 (s − r)zn (r) dr ds cos n(t − s) ds 0 0 Z t Z t 0 = cos nt + N (0) cos n(t − s)zn (s) ds − N 0 (t − r)zn (r) dr 0 0 Z t Z s + cos n(t − s) N 00 (s − r)zn (r) dr ds 0 0 Z t = cos nt + [N 0 (0) cos n(t − r) − N 0 (t − r)]zn (r) dr 0 Z t Z t cos n(t − s)N 00 (s − r) ds zn (r) dr . (3.4) + 0

r

The Gronwall inequality shows the existence of a constant M such that |zn (t)| < M ,

t ∈ [0, T ] .

The number M does depend on T but not on n. This shows: Lemma 3.1. The sequence {cn } in (3.2) belongs to l2 . Note that this holds also if N 0 (0) 6= 0. From now on instead we use the restrictive condition N 0 (0) = 0 (to be removed later on.) Equality (3.4) suggests comparison of zn (t) with cos nt. The sequence {cos nt} being orthogonal on L2 (0, π), with constant norm, from now on we impose T = π. We introduce en (t) = zn (t) − cos nt and, using (3.4), we see that, when N 0 (0) = 0, Z t Z t Z r en (t) = − N 0 (t − r)en (r) dr + cos n(t − r) N 00 (r − s)en (s) ds dr 0

0

0

=A+B where Z t    A = − N 0 (t − s) cos ns ds ,  0 Z t Z r   B = cos n(t − r) N 00 (r − s) cos ns ds dr . 0

(3.5)

0

Both these integrals can be integrated by parts: Z Z 1 t d 1 t 00 1 sin ns N 0 (t − s) ds = N (t − s) sin ns ds , (3.6) −A = n 0 ds n 0 n Z Z r 1 t 1 00 000 B= cos n(t − r) N (0) sin nr + N (r − s) sin ns ds dr (3.7) n 0 n 0

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(the integral in (3.6) can be integrated by parts once more, and (3.6) is of the order 1/n2 ). Using again Gronwall inequality we see the existence of a constant M such that M |en (t)| ≤ . n This shows that the sequence {zn (t)} is quadratically close to the sequence {cos nt}. The inequality above is sufficient for the application of Bari’s theorem, but Section 4 will use a refined version of this inequality. Remark 3.2. Similar computations show that the sequence {zn0 (t)/n} is L2 -close to Rt the sequence {− sin nt} and the fact that the sequence {n 0 zn (s) ds} is L2 -close to the sequence {sin nt}. The inequalities in this section have been derived in the simple case when N 0 (0) = 0. We state now the general result, proved in the appendix: Theorem 3.3. Let N (t) be of class C 3 and let N 0 (0) = 1. We define α = N 0 (0)/2. The sequence {zn (t)} is then L2 (0, T )-close to the sequence {e−αt cos nt} for every T > 0. 3.2. The property of ω-independence In the previous section we stated that the sequence {zn (t)} is L2 (0, T )-close to the Riesz sequence {e−αt cos nt} where α = N 0 (0)/2. Here T > 0 is arbitrary. In this section we use known controllability results in order to prove that if T ≥ π, then {zn (t)} is ω–independent so that, from Theorem 1.2, it is a Riesz system in L2 (0, π). This is the point where finite signal speed has to be taken into account. This property, derived by many authors, see for example [6, 9], is recalled in the form we need in this paper. For completeness, a sketch of the proof is given in the Appendix. We consider system (1.1), but now • on the interval (−, π) with > 0; • with homogeneous Dirichlet boundary condition θ(t, −) = 0, θ(t, π) = 0 and null initial condition, θ(0, x) = 0; • the system is acted upon by a control distributed on (−, 0). So we consider the system  Z t θ (t, x) = N (t − s)∆θ(s, x) ds + χ(x)ν(t, x), x ∈ (−, π), t (3.8) 0  θ(0, x) = 0, x ∈ (−, π) , θ(t, −) = θ(t, π) = 0 . The function χ(x) is the characteristic function of (−, 0). The function ν(t, x) is a “distributed control”, and belongs to L2loc ((0, +∞) × (−, 0)). The property of “finite propagation” we need is that θ(t, x) is supported on x ∈ [−, π − ] for every t < π − . Now we observe that a smooth control ν(t, x) produces a smooth solution θ(t, x) so that evaluation of θ(t, x) at x = 0 is possible and, when we restrict the

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space variable x to [0, π], this corresponds to consider the above control problem with “boundary control” at 0 given by u(t) = θ(t, 0). The conditions under which θ(t, x) is smooth (so that the computation of u(t) = θ(t, 0) makes sense) are in [20, Appendix]. The required (smoothness) conditions on ν(t, x) are satisfied in a dense subset of L2 ((0, T ) × (−, 0)). Now we recall the known controllability results, from [5] (under more restrictive assumptions) and from [26, 27], under the assumption in this paper (and in fact more general, see below.) These papers study controllability under distributed control. The results proved in these paper cover the case of a space interval [−, π] with distributed control supported on [−, 0] and every initial and final condition in L2 (−, π) (actually, [5] imposes further regularity to the target.) Exact controllability is proved in time T ≥ π + . As described in [5], this result on controllability with control acting on [−, 0] implies boundary controllability on [0, π]. Let for simplicity the initial condition ξ, defined on [0, π], be ξ = 0. Extend ξ and the target η(x) (given on [0, π]) with 0 to [−, 0] and consider the distributed system (3.8). Construct the stearing distributed control, in time T + for this new function, defined on (−, π) and then use the trace u(t) = θ(t; 0) as the boundary control. So, we have exact controllability in every time T > π; and, more in general, boundary controllability is possible in an arbitrary time, longer then the width of the interval. However, the steering control ν(t, x) so constructed needs not be smooth so that in principle u(t) might not be well defined. In spite of this, we are going to show that the previous argument implies approximate controllability, and this is sufficient for our needs. Approximate controllability is seen as follows: let RT ⊆ L2 (−, π) be the reachable set at time T for the control system (3.8) and let RT ;0 ⊆ L2 (0, π) be the set of the restrictions to L2 (0, π) of the elements of RT . The previous observations on controllability proves that, for every σ > 0, every function of L2 (0, π) whose support is in [0, π−σ] belongs to Rπ;0 . Hence the subspace Rπ;0 is dense in L2 (0, π). Density is retained if we confine ourselves to consider solely those elements which are reachable by using smooth controls ν(t, x). So, we conclude approximate controllability: every η(x) in a dense subset of L2 (0, π) is a reachable target for the control system (1.1)-(1.2). Let us go back to consider the moment problem (3.2) with ξ = 0. The moment problem is solvable for every reachable target. So, the set of the sequences {ηn } for which the following moment problem is solvable is a dense subspace of l2 . The moment problem is Z π Z T 1 ηn = φn (x)η(x) dx , η ∈ Rπ;0 . (3.9) zn (t)v(t) dt = ηn , n 0 0 Also the set of the sequences {ηn /n} which correspond to solvable moment problems is dense in l2 . This shows that the sequence {zn (t)}n≥1 is ω-independent, see [2, Theorem I.2.1 (d)], as we wanted: Theorem 1.2 can be applied and we conclude that the sequence {zn } is a Riesz sequence in L2 (0, π). In conclusion, we have the following result:

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Theorem 3.4. The sequence {zn (t)}n≥1 is a Riesz system in L2 (0, π), provided that N (t) ∈ C 3 (0, π + ), N (0) = 1. Remark 3.5. The results on controllability in [5] concern a one dimensional space variable and require that N (t) is continuous for t ≥ 0 and completely monotonic, i.e. of class C ∞ with derivatives of alternating sign; in particular, N (t) ≥ 0. The result in [27] considers the case of elliptic operators with variable coefficients in a region Ω of IRn . The kernel N can depend on x, N = N (t, x). The coefficients and N (t, x) have to be of class C 3 and N (0, x) > 0. Both these papers identify the controllability time as a consequence of Carleman estimates. Also the papers [20, 21] studies the problem in a region of IRn but the controllability time is not identified there. Finally, we note that the controllability result which is really needed in this section is approximate controllability.

4. Back to the control problem In the previous section we proved that the moment problem (3.2), i.e. (2.3), is solvable for a suitable square integrable function v. The proof is based on the fact that controllability of our system has been already proved with different methods but, as we noted, these methods do not provide a real construction of the control u(t) which steers the initial datum ξ to the prescribed target η. This problem is examined now. The moment method gives a formula for the solution v(t) of problem (3.2), i.e. +∞ X cn ζn (t) (4.1) v(t) = n n=1 where {ζn (t)} is biorthogonal to {zn (t)}. The steering control solves the Volterra integral equation of the first kind (3.3). We are now going to prove that the function v(t) in (4.1) is of class W 1,2 (0, π) so that the control u(t) can be computed from the Volterra integral equation of the second kind Z t u(t) + N 0 (t − s)u(s) ds = v 0 (t) . (4.2) 0

In order to prove this additional regularity of v(t), we need the following estimate: Lemma 4.1. Let α = N 0 (0)/2. There exists a sequence of square integrable functions {an (t)} such that for t ∈ [0, π] we have: a (t) eαt n [2α + N 00 (0)t] sin nt ≤ , (4.3) zn (t) − eαt cos nt − 2n n " # Z +∞ +∞ π X X |an (t)|2 < M < +∞ , |an (t)|2 dt < +∞ . (4.4) n=1

0

n=1

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Proof. We present the proof in the case N 0 (0) = 0 (see the appendix for the general case). We observed already that the row (3.6) is of the order 1/n2 . Instead, row (3.7) gives a term of the order 1/n2 plus the contribution a ˜n (t) N 00 (0) t sin nt + . 2n n For fixed t, a ˜n (t) is Z t a ˜n (t) = cos n(t − r)bn (r) dr ,

Z bn (r) =

0

r

N (3) (r − s) sin ns ds .

0

Now t

Z

Z cos n(t − r)bn (r) dr =

a ˜n (t) = 0

=

1 2

t

N (3) (s)

0

Z

t

Z

cos n(t − r) sin n(r − s) dr ds s

t

N (3) (s)(t − s) sin n(t − s) ds .

0

Expanding sin n(t − s), we get the sum of several terms. One of them is Z t sin nt t (3) N (s) cos ns . 2 0 So, using Parseval’s equality, 2 2 Z +∞ Z t +∞ X X t sin nt t (3) (3) ≤M N (s) cos ns N (s) cos ns 2 0 0 n=1 n=1 Z T ≤M |N (3) (s)|2 ds . 0

The remaining terms are estimated in a similar way so that +∞ X

|˜ an (t)|2 ≤ M

∀t ∈ [0, π] .

n=1

We introduce eˆn (t) = [en (t) − (N 00 (0)/2n)t sin nt] in (3.5) so that, with certain kernels Mn (t), we get Z t Z t N 00 (0) eˆn (t) = Mn (t − s)ˆ en (s) + − N 0 (t − r)r sin nr 2n 0 0 Z t Z r 1 1 + ˜n (t) + 2 Ψn (t) . cos n(t − r) N 00 (r − s)s sin ns ds dr + a n n 0 0 The functions Mn (t) and Ψn (t) are bounded on [0, π], uniformly with respect to n. Moreover, the integrals in brackets are of the order of 1/n so that (with a suitable constant M ) the following inequality holds for every n and every t ∈ [0, π]: Z t M 1 |ˆ en (t)| ≤ M |ˆ en (s)| ds + 2 + |˜ an (t)| . (4.5) n n 0

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Now we proceed as in the proof of the generalized Gronwall inequality in [24, p. 11]. Multiplying both the sides of (4.5) with M e−M t we see that Z 1 d −M t t M + e M |ˆ en (s)| ds ≤ e−M t |˜ a (t)| . n dt n2 n 0 So, with suitable constants H and K we get on [0, π] Z t Z H K t M |ˆ en (s)| ds ≤ 2 + |˜ an (s)| ds . n n 0 0 We replace this estimate in (4.5) and we see that Z t an (t) M +H |ˆ en (t)| ≤ |˜ an (s)| ds . , an (t) = + a ˜n (t) + K n n 0 The sequence {an (t)} inherits properties (4.4) from the corresponding properties of {˜ an (t)}. Remark 4.2. Note that when N (t) ∈ W 4,2 (0, π) then |bn (t)| < M/n and in this case we easily find |ˆ en (t)| < M/n2 . Now we present an additional piece of information from the proof of Bari’s theorem: let n = eαt cos nt (this is a Riesz sequence, its biorthogonal sequence being {e−2αt n }). We know that {zn (t)} is ω-independent and L2 -close to {n }. It is then seen from the proof of Bari’s theorem that ζk = T ∗ ζk + k where T is the operator Tf =−

+∞ X

hf, k iek ,

T ∗g = −

k=1

+∞ X

k hg, ek i .

k=1

Hence, each function ζn (t) has the following representation ζn (t) = e−αt cos nt −

+∞ X k=1

αs

Z

π

ζn (s)ˆ ek (s) ds −

k 0

+∞ X k=1

Z k

π

Φn (s) 0

sin ks ds , (4.6) k

00

Φn (s) = e [α + (N (0)s)/2]ζn (s) . We recall that the biorthogonal sequence of a Riesz basis is a Riesz basis, hence it is bounded. So, also the sequence {Φn (t)} is bounded in L2 (0, π). Finally, we introduce the sequence of the functions 1 0 cos kt αt rk (t) = k (t) = e α − sin kt . (4.7) k k

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We shall see in the appendix the existence of a number M such that the following inequalities hold, for every f ∈ L2 (0, π) and {γn } ∈ l2 :  2 Z π X Z π   ≤M  |f (s)|2 ds , f (s)r (s) ds k   0 0 k 2 (4.8) Z π X X   2   |γk | . γk rk (t) dt < M  0 k

k

We compute: Z π Z Φn (s) sin ks ds = − 0

π

−αs e Φn (s) rk (s) ds + α

Z

0

π

Φn (s) 0

cos ks ds k

so that Z

0

π

2 M 2 +2 2 Φn (s) sin ks ds ≤ 2γn,k k

(4.9)

where M does not depend on n and Z π −αs γn,k = e Φn (s) rk (s) ds. 0

Hence, from (4.8), +∞ X k=1

2

π

Z

e2αs |Φn (s)|2 ds < M

|γn,k |

(4.10)

0

(M does not depend on n since {ζn (t)} is bounded in L2 (0, π).) We are now ready to study the regularity of the function v(t). We first prove a result which is of independent interest: Lemma 4.3. We have ζn (t) ∈ W 1,2 (0, π) for every n. Proof. We need to see that the series of the derivatives of the two series in (4.6) converge in L2 (0, π). Using (4.7), the derivative of the first series has the following form Z π +∞ X ζn (s)kˆ ek (s) ds ≤ kζn kL2 (0,π) kak kL2 (0,π) . − (kγn,k )rk (t) , |kγn,k | = k=1

0

Using (4.4), we see that {γn,k } ∈ l2 so that the series converges in L2 (0, π). P+∞ The derivative of the second series has the form − k=1 γn,k rk (t) and now Z π γn,k = Φn (s) sin ks ds , 0

the Fourier coefficients of Φn (s) so that also in this case we have {γn,k } ∈ l2 and the series converges.

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We now consider the function v(t), which is a linear combination of the three series +∞ X cn n , (4.11) n n=1 "+∞ Z # +∞ π X sin ks cn X k Φn (s) ds , (4.12) n k 0 n=1 k=1 "+∞ Z # +∞ π X cn X k ζn (s)ˆ ek (s) ds . (4.13) n 0 n=1 k=1

We have to prove that the series of the derivatives converge in L2 (0, π). This is clear for the series (4.11). We consider the series (4.12) and we prove that the series which is obtained with a formal termwise differentiation converges in L2 (0, π). So we consider "+∞ # 2 Z π Z π X R sin ks cn X 0 k (t) ds dt Φn (s) n k 0 0 n=M k=1 "+∞ # 2 Z π X Z π R cn X = rk (t) Φn (s) sin ks ds dt n 0 0 n=M k=1  2  ! Z π X Z π R R +∞ X X 1 |cn |2  ≤ r (t) Φ (s) sin ks ds dt k n n2 0 0 n=M n=M k=1 ! R ! +∞ R R R X X X 1 X X 1 M 2 2 γ + ≤ (const) · |c | ≤ |cn |2 n n,k n2 k2 n2 n=M

n=M

n=M

k=1

n=M

(here we used inequalities (4.8)-(4.10).) This shows L2 (0, π)-convergence of the sequence of the partial sums, as wanted. We now consider the series of the derivatives of (4.13), i.e. the series 2 Z π Z π X +∞ R cn X 0 k (t) ζn (s)ˆ ek (s) ds dt n 0 0 n=M k=1 ! 2 Z π X Z π R +∞ cn X = rk (t) ζn (s)[kˆ ek (s)] ds dt 0 n=M n 0 k=1   ! 2 Z π X R R +∞ X X 1 ≤ |cn |2  γn,k rk (t) dt n2 0 n=M n=M k=1 ! ! +∞ R R X X 1 X |γn,k |2 . (4.14) ≤M |cn |2 n2 n=M

n=M

k=1

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Here we used (4.8) with Z

π

γn,k =

ζn (s)[kˆ ek (s)] ds . 0

Hence, Z π 2 |ζn (s)|2 ds |kˆ ek (s)| ds 0 Z π 0 Z 2 |ak (s)|2 ds ≤ M ≤ kζn kL2 (0,π)

|γn,k |2 ≤

Z

π

0

π

|ak (s)|2 ds .

0

Property (4.4) shows that +∞ X

|γk,n |2 < M

k=1

and the number M does not depend on n. Going back to (4.14), we see that the partial sums of the series of the derivatives of (4.13) converges in L2 (0, π), as wanted. This concludes the proof that the solution to the control problem is the solution u(t) of (4.2), where v(t) is given by (4.1). In this sense, this paper provides a constructive approach to the computation of the solution u of the control problem (1.1)–(1.3). We have also a minor improvement on the existing controllability results: the sequence {zn }n≥1 being a Riesz sequence in L2 (0, π), the moment problem can be solved for every sequence {cn } ∈ l2 so that, thanks to Lemma 2.1, the controllability time with boundary control is π. This result has been proved in [19] under different assumptions on the kernel N (t). Remark 4.4. Note the role of the factor 1/n in (3.2) and see [22] for further applications of the results in this paper.

5. Appendix In this appendix we remove the assumption N 0 (0) = 0, which was used solely to show in a simple case the ideas in this paper and, for completeness, we outline the proof of the finite signal speed and of the preliminary result Theorem 1.2. Proof of Theorem 1.2 The proof of Theorem 1.2 is as follows: if we can find a vector z0 ∈ H which is orthogonal to the sequence {zn }n≥1 , then the new sequence {zn }n≥0 is ωindependent and quadratically close to the basis {}n≥0 . Theorem 1.1 can be applied and we see that {zn }n≥0 is a Riesz basis; hence, {zn }n≥1 is a Riesz system. So, we have to prove the existence of such orthogonal element z0 . For this, we add any element z˜ as the first element of the sequence {zn }n≥1 and we consider this new sequence.

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As in the proof of Theorem 1.1, we consider the operator T defined by " +∞ # +∞ X X αn n = α0 [˜ T z − 0 ] + αn [zk − n ] . n=0

n=1

It is possible to prove that this operator is compact. We then consider the operator T 0 = (I + T ) which has the following property: T 0 k = zk . If T 0 is boundedly invertible, we are done. Otherwise, thanks to the compactness of T , the rank of T 0 is closed and different from H. Hence, there exists an element z0 ∈ [im T 0 ]⊥ . In particular, this element is orthogonal to every zk and we are done. The proof of inequalities (4.8) The sequence {rk (t)} is L2 (0, π)-close to the Riesz sequence {eαt sin kt}. Using [28, Theorem 13], it is possible to prove the existence of a number N such that {rk (t)}k≥N is a Riesz system. This proves the existence of a number M such that 2 X Z π f (s)rk (s) ds ≤ M kf k2L2 (0,π) . k≥N

0

So, the series is convergent and it is easy to estimate also the first term from above, and to get the required inequality. The second inequality is obtained as follows: 2 2 Z π NX Z π NX −1 X −1 γ r (t) + γ r (t) dt ≤ 2 γ r (t) dt k k k k k k 0 k=1 0 k≥N k=1 2 Z π X +2 γk rk (t) dt 0 k≥N   −1  X NX X ≤M |γk2 | + |γk |2 ,   k=1

k≥N

the finite sum being directly estimated. The estimate on the series is obtained from the properties of Riesz systems. The case N 0 (0) 6= 0 The proofs in Section 3.1 (and that of Lemma 4.1) used the artificial assumption N 0 (0) = 0 in order to have short computations and to show clearly the key ideas of this paper. Now, we present the proofs in the general case N 0 (0) 6= 0.

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It is convenient to introduce the following transformation. We define α = ˜ = e2αt θ(t) (compare with Theorem 3.3 and Lemma 4.1 and −N 0 (0)/2 and θ(t) ˜ solves note the minus sign we use here.) Clearly, θ(t) Z t ˜ + ˜ ds , ˜ (t − s)∆θ(s) ˜ (t) = e2αt N (t) . θ˜0 (t) = 2αθ(t) N N (5.1) 0

˜ (t) = e2αt N (t) satisfies N ˜ (0) = 1, N ˜ 0 (0) = 0. Controllability The new kernel N of the original system and the system so modified being equivalent, we study the controllability of system (5.1). The corresponding moment problem has the same form as in Section 2, i.e. Z T Z T Z T ˜ (T − r − s)u(s) ds dr z˜n (r)v(r) dr = z˜n (r) N 0 0 T −r Z π 1 φn (x) [θ(T, x) − zn (T )θ(0, x)] dx = 0 φn (0) 0 where φn (x) still solves (1.9) while zn (t) now solves Z t z˜n0 (t) = 2α˜ zn (t) − n2 N (t − s)˜ zn (s) ds,

z˜n (0) = 1 .

(5.2)

0

Of course, z˜n (t) = e2αt zn (t) . We must prove estimates similar to those in Section 3.1 when z˜n (t) solves Eq. (5.2) ˜ 0 (0) = 0. In fact we shall prove the following and now, after this transformation, N estimate for z˜n (t): M 0 |˜ zn (t) − eαt cos nt| = ˜ zn (t) − e−N (0)t/2 cos nt ≤ , n which is equivalent to M 0 |zn (t) − e−αt cos nt| = zn (t) − eN (0)t/2 cos nt < n (with a different constant M , which depends on the interval [0, T ] we are considering). This being understood, we work with Eq. (5.2) and for simplicity of notations we drop the ˜. We note that Z t zn00 (t) = 2αzn0 (t) − n2 zn (t) − n2 zn (0) = 1 , zn0 (0) = 2α . N 0 (t − s)zn (s) ds , 0

The zeros of the characteristic polynomial λ2 − 2αλ + n2 are p σ1 = α + iβn , σ2 = α − iβn , βn = n 1 − α2 /n2 . Note that βn is real for large n and that |n − βn | ≤

α2 . n

(5.3)

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The solution z(t) of the problem z 00 − 2αz 0 + n2 z = f ,

z(0) = 1 ,

z 0 (0) = 2α

is z(t) = eαt cos βn t +

α αt 1 e sin βn t + βn βn

Z

t

eα(t−s) sin βn (t − s)f (s) ds .

0

We apply this formula and find α αt zn (t) = eαt cos βn t + e sin βn t βn Z Z t−r n2 t αr − e sin βn r N 0 (t − r − s)zn (s) ds dr βn 0 0 α αt = eαt cos βn t + e sin βn t βn Z t−r 2 Z t d n cos βn r eαr N 0 (t − r − s)zn (s) ds dr + 2 βn 0 dr 0 Z t α αt n2 αt = e cos βn t + e sin βn t − 2 N 0 (t − s)zn (s) ds βn βn 0 Z t Z t−r +α eαr cos βn r N 0 (t − r − s)zn (s) ds dr 0 0 Z t Z t−r αr 00 − e cos βn r N (t − r − s)zn (s) ds dr . 0

(5.4)

0

We introduce en (t) = zn (t) − eαt cos βn t and see that Z t n2 α αt e sin βn t − 2 N 0 (t − s)en (s) ds en (t) = βn βn 0 Z t Z t−r +α eαr cos βn r N 0 (t − r − s)en (s) ds dr 0 0 Z t Z t−r − eαr cos βn r N 00 (t − r − s)en (s) ds dr 0 0 Z t 2 n − 2 N 0 (t − s)eαs cos βn s ds βn 0 Z t Z t−r +α eαr cos βn r N 0 (t − r − s)eαs cos βn s ds dr 0 0 Z t Z t−r αr 00 αs − e cos βn r N (t − r − s)e cos βn s ds dr . 0

0

(5.5)

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We recall that βn n (see (5.3)) and we integrate by parts the integrals in the last brace, as in Section 3.1. We see that |en (t)| ≤

M n

∀t ∈ [0, T ] .

(5.6)

The constant M does depend on T but not on n. Using (5.3), we see that there exists a constant M such that |cos nt − cos βn t| <

M n

t ∈ [0, T ]

(5.7)

so that the sequence {zn (t)} is L2 -close to the sequence {eαt cos nt} as we wanted to prove. We now sketch the proof of Lemma 4.1 in the general case N 0 (0) 6= 0. The first and a second integral in the second brace of (5.5) can be integrated twice by parts so that their contribution is of the order 1/n2 . Instead, partial integration of the third integral gives Z t 1 00 αt N (0)e cos βn r sin βn (t − r) dr βn 0 Z t Z t−r 1 + αeαr cos βn r N 00 (t − r − s)eαs sin βn s ds dr βn 0 0 Z t Z t−r 1 − eαr cos βn r N 000 (t − r − s)eαs sin βn s ds dr . βn 0 0 When inserted in (5.5), the first integral above gives terms of the order 1/n2 and the term N 00 (0) n2 αt e t sin βn t , 2 βn3 which has a difference of the order 1/n2 with N 00 (0) αt e t sin nt . 2n The remaining terms are dominated by an (t) M + 2. n n Using (sin βn t − sin nt) ∼ M/n, we see that the sequence {an (t)} has the properties (4.4). This (and the first addendum in (5.5)) suggests the definition eˆn (t) = e(t) −

eαt 1 [α + N 00 (0)t] sin nt . n 2 αt

Inequality (4.3) is obtained by adding and subtracting − en [α + 12 N 00 (0)t] sin nt to en (s) in (5.5), integrating by parts and using the method in Lemma 4.1.

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Finite signal speed The following fact, that we enunciate with reference to system (3.8), has been noted in several papers. We adapt the proof in [6], which considers the case that x is not confined to a bounded set. Using the formula for the solutions given in [20], we see that when ξ = 0, the solution of problem (3.8) is given by the following Volterra equation of the second kind: Z t Z t R+ (t − s)χν(s) ds . (5.8) L(t − s)θ(s) ds , Ψ(t) = θ(t) = Ψ(t) + 0

0

The operator L(t) is defined by L(t)φ = N 0 (0)R+ (t)φ − N 0 (t)φ +

t

Z

R+ (t − s)N 00 (s)φ ds

0

for every φ ∈ L2 (−, π). The operator R+ (t) is given by [R+ (t)φ](·) = w(t, ·) where w(t, x) solves wtt = wxx ,

t > 0,

x ∈ (−, π) ,

with conditions w(t, −) = w(t, π) = 0 ,

w(0, x) = φ(x) ,

wt (0, x) = 0 .

It is known that when the support of φ is in (−, 0) then [R+ (t)φ](x) = w(t, x) = 0 for x ∈ (π − , π) and t < π − . This is the “finite signal speed” property of the wave equation, and we see that it is shared by the solution θ(t) of (3.8), i.e. of (5.8). We proceed as follows: we note that if a function y(t, ·) has support in x ≤ π − Rt for every t ≤ π − , then the same property is retained by its integrals 0 y(s, ·) ds, t ≤ π − . So, Ψ(t, x) = 0 for x ∈ (π − , π) and t < π − because it is obtained as the integral of functions which have this property for every s ≤ t ≤ π − . The Volterra integral equation (5.8) can be solved by successive approximations. If L is the integral operator in (5.8), then θ(t) =

+∞ X

Ln Ψ .

n=0

The result follows since each term in the expression of L(t − s)θ(s) in (5.8) is supported in x ≤ π − for t ≤ π − , a property which is retained by every integral Ln Ψ.

References [1] D.D. Ang, R. Gorenflo, L.V. Khoi and D.D.Trong, Moment theory and some inverse problems in potential theory and heat conduction. Lecture Notes in Mathematics, 1792. Springer-Verlag, Berlin 2002.

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[2] S.A. Avdonin and S.A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, New York 1995. [3] S.A. Avdonin, S.A. Ivanov and D.L. Russell, Exponential bases in Sobolev spaces in control and observation problems for the wave equation. Proc. Roy. Soc. Edinburgh A-130 (2000) 947–970. [4] S.A. Avdonin, L. Suzanne and V. Protopopescu, Solving the dynamical inverse problem for the Schr¨ odinger equation by the boundary control method. Inverse Problems 18 (2002) 349–361. [5] V. Barbu and M. Iannelli, Controllability of the heat equation with memory. Diff. Integral Eq. 13 (2000) 1393–1412. [6] A. Belleni-Morante, An integro-differential equation arising from the theory of heat conduction in rigid materials with memory. Boll. Unione Mat. Ital. (5) 15-B (1978) 470–482. [7] A.G. Butkovski, The method of moments in optimal control theory with distributed parameter systems. Avtomat. i Telemeh. 24 (1963) 1217–1225. [8] C. Cattaneo, Sulla conduzione del calore. Atti del Seminario Matematico e Fisico dell’Universit` a di Modena 3 (1948) 3–21. [9] G.W. Desch, R. Grimmer, and R. Zeman, Wave propagation for abstract integrodifferential equations. In Infinite-dimensional systems (Retzhof, 1983), 62–70, Lecture Notes in Math., 1076, Springer, Berlin (1984). [10] H.O. Fattorini and D.L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quart. Appl. Math. 32 (1974/75) 45–69. [11] H.O. Fattorini and D.L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272–292. [12] I.C. Gohberg and M.G. Krejn, Op`erateurs lin`eairs non auto-adjoints dans un espace hilbertien. Dunod, Paris 1971. [13] B.Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 39 (2001) 1736–1747. [14] M.E. Gurtin and A.G. Pipkin, A general theory of heat conduction with finite wave speed. Arch. Rat. Mech. Anal. 31 (1968) 113–126. [15] S.A. Ivanov and L. Pandolfi, Heat equation with memory: lack of controllability to the rest. J. Math. Anal. Appl. 355 (2009) 1–11, doi:10.1016/j.jmaa.2009.01.008. [16] S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. Differential Equations 145 (1998) 184–215. [17] W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes. Lecture Notes in Control and Information Sciences, 173. Springer-Verlag, Berlin 1992. [18] V. Komornik and P. Loreti, Fourier series in control theory. Springer Monographs in Mathematics. Springer-Verlag, New York 2005. [19] G. Leugering, time optimal boundary controllability of a simple linear viscoelastic liquid. Math. Methods in the Appl. Sci. 9 (1987) 413–430.

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[20] L. Pandolfi, The controllability of the Gurtin-Pipkin equation: a cosine operator approach. Applied Mathematics and Optimization 52 (2005) 143–165. [21] L. Pandolfi, Controllability of the Gurtin-Pipkin equation. SISSA, Proceedings of Science, PoS(CSTNA2005)015. [22] L. Pandolfi, Riesz systems and an identification problem for heat equations with memory. in preparation. [23] D.L. Russell, Nonharmonic Fourier series in the control theory of distributed parameter systems. J. Math. Anal. Appl. 18 (1967) 542–560. [24] G. Sansone, R. Conti, Non-linear differential equations, Pergamon Press, Oxford, 1964. [25] G. Talenti, Recovering a function from a finite number of moments. Inverse Problems 3 (1987) 501–517. [26] X. Fu, J. Yong and X. Zhang, Controllability and observability of the heat equation with hyperbolic memory kernel. Submitted [27] J. Yong and X. Zhang, Exact controllability of the heat equation with hyperbolic memory kernel. In Control of Partial Differential Equations, Control theory of partial differential equations, 387–401, Lect. Notes Pure Appl. Math. 242, Chapman & Hall/CRC, Boca Raton, FL 2005. [28] R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, New York 1980. L. Pandolfi Politecnico di Torino Dipartimento di Matematica Corso Duca degli Abruzzi 24 10129 Torino Italy e-mail: [email protected] Submitted: October 20, 2008. Revised: April 1, 2009.

Integr. equ. oper. theory 64 (2009), 455–486 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040455-32, published online August 3, 2009 DOI 10.1007/s00020-009-1702-1

Integral Equations and Operator Theory

Bounds on Variation of Spectral Subspaces under J -Self-adjoint Perturbations Sergio Albeverio, Alexander K. Motovilov and Andrei A. Shkalikov Abstract. Let A be a self-adjoint operator on a Hilbert space H. Assume that the spectrum of A consists of two disjoint components σ0 and σ1 . Let V be a bounded operator on H, off-diagonal and J-self-adjoint with respect to the orthogonal decomposition H = H0 ⊕ H1 where H0 and H1 are the spectral subspaces of A associated with the spectral sets σ0 and σ1 , respectively. We find (optimal) conditions on V guaranteeing that the perturbed operator L = A + V is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on the variation of the spectral subspaces of A under the perturbation V . Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a PT -symmetric perturbation is discussed. Mathematics Subject Classification (2000). Primary 47A15; Secondary 47A25, 47A62, 47B50. Keywords. Subspace perturbation problem, Krein space, J-symmetric operator, J-self-adjoint operator, P T symmetry, P T -symmetric operator, Riccati equation, Sylvester equation, Davis-Kahan theorems.

1. Introduction Let A be a (possibly unbounded) self-adjoint operator on a Hilbert space H. Assume that V is a bounded operator on H. It is well known that in such a case the spectrum of the perturbed operator L = A+V lies in the closed kV k-neighborhood of the spectrum of A even if V is non-self-adjoint. Thus, if the spectrum of A consists of two disjoint components σ0 and σ1 , that is, if spec(A) = σ0 ∪ σ1 and dist(σ0 , σ1 ) = d > 0,

(1.1)

This work was supported by the Deutsche Forschungsgemeinschaft (DFG), the Heisenberg-Landau Program, and the Russian Foundation for Basic Research.

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then the perturbation V with a sufficiently small norm does not close the gaps between σ0 and σ1 in C. This allows one to think of the corresponding disjoint spectral components σ00 and σ10 of the perturbed operator L = A + V as a result of the perturbation of the spectral sets σ0 and σ1 , respectively. Assuming (1.1), by EA (σ0 ) and EA (σ1 ) we denote the spectral projections of A associated with the disjoint Borel sets σ0 and σ1 , and by H0 and H1 the respective spectral subspaces, H0 = Ran EA (σ0 ) and H1 = Ran EA (σ1 ). If there is a possibility to associate with the disjoint spectral sets σ00 and σ10 the corresponding spectral subspaces of the perturbed (non-self-adjoint) operator L = A + V , we denote them by H00 and H01 . In particular, if one of the sets σ00 and σ10 is bounded, this can easily be done by using the Riesz projections (see, e.g. [24, Sec. III.4]). In the present note we are mainly concerned with bounded perturbations V that possess the property V ∗ = JV J, (1.2) where J is a self-adjoint involution on H given by J = EA (σ0 ) − EA (σ1 ).

(1.3)

Operators V with the property (1.2) are called J-self-adjoint. A bounded perturbation V is called diagonal with respect to the orthogonal decomposition H = H0 ⊕ H1 if it commutes with the involution J, VJ = JV . If V anticommutes with J, i.e. VJ = −JV , then V is said to be off-diagonal. Clearly, any bounded V can be represented as the sum V = Vdiag + Voff of the diagonal, Vdiag , and off-diagonal, Voff , terms. The spectral subspaces H0 and H1 remain invariant under A + Vdiag while adding a non-zero Voff does break the invariance of H0 and H1 . Thus, the core of the perturbation theory for spectral subspaces is in the study of their variation under off-diagonal perturbations (cf. [25]). This is the reason why we add to the hypothesis (1.2) another basic assumption, namely that all the perturbations V involved are off-diagonal with respect to the decomposition H = H0 ⊕ H1 . We recall that if an off-diagonal perturbation V is self-adjoint in the usual sense, that is, V ∗ = V , then the condition d (1.4) 2 ensuring the existence of gaps between the perturbed spectral sets σ00 and σ10 may be essentially relaxed. Generically, if no assumptions on the mutual position of the initial spectral sets σ0 and σ1 are made except (1.1), the sets σ00 and σ10√ remain disjoint for any off-diagonal self-adjoint V satisfying the bound kV k < 23 d (see [27, Theorem 1 (i)] and [49, Theorem 5.7 (ii)]). If, in addition to (1.1), it is known that one of the sets σ0 and σ1 lies in a finite gap of the other set then this bound may be relaxed further: for the√perturbed sets σ00 and σ10 to be disjoint it only suffices to require that kV k < 2d (see [26, Theorem 3.2 and Remark 3.3]; cf. [27, Theorem 2 (i)] and [49, Theorem 5.7 (iii)]). Finally, if the sets σ0 and σ1 are subordinated, say sup σ0 < inf σ1 , then no requirements on kV k are needed kV k <

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at all: the interval (sup σ0 , inf σ1 ) belongs to the resolvent set of the perturbed operator L = A + V for any bounded off-diagonal self-adjoint V (see [2, 17, 33]; cf. [28]) and even for some off-diagonal unbounded symmetric V (see [41, Theorem 1] and [49, Corollary 5.3]). It is easily seen from Example 5.5 below that in the case of J-self-adjoint off-diagonal perturbations the condition (1.4) ensuring the disjointness of the perturbed spectral sets σ00 and σ10 can be relaxed for none of the above dispositions of the initial spectral sets σ0 and σ1 . Assuming that V is a bounded J-self-adjoint off-diagonal perturbation of the (possibly unbounded) self-adjoint operator A we address the following questions: (i) Does the spectrum of the perturbed operator L = A + V remain real under conditions (1.1) and (1.4)? (ii) If yes, is it then true that L is similar to a self-adjoint operator? (iii) What are the (sharp) bounds on variation of the spectral subspaces associated with the spectral sets σ0 and σ1 as well as on the variation of these sets themselves? In our answers to the above questions we distinguish two cases: (G) the generic case where no assumptions on the mutual positions of the spectral sets σ0 and σ1 are made except for the disjointness assumption (1.1); (S) the particular case where the sets σ0 and σ1 are either subordinated, e.g. sup σ0 < inf σ1 , or one of these sets lies in a finite gap of the other set, say σ0 lies in a finite gap of σ1 . We have to underline that this distinction is quite different from the one that arises when the perturbations V are self-adjoint in the usual sense: the case (S) now combines the two spectral dispositions that should be treated separately if V were self-adjoint (see [7, 17, 27, 41]). Our answers to the questions (i) and (ii) are complete and positive in the case (S). In this case the spectrum of the perturbed operator L = A+V does remain real for any off-diagonal J-self-adjoint V satisfying the bound kV k ≤ d/2. Moreover, the operator L turns out to be similar to a self-adjoint operator whenever the strict inequality (1.4) holds. These results combined in Theorem 5.8 (ii) below (see also Remark 5.13) represent an extension of similar results previously known due to [1] and [37] for the spectral dispositions with subordinated σ0 and σ1 . By using the results of [32, 50], we give a positive answer to the question (i) also in the generic case (G) provided that the unperturbed operator A is bounded (see Theorem 5.12). For A unbounded, we prove that in case (G) the spectrum of L = A + V for sure is purely real if V satisfies a stronger bound kV k ≤ d/π. The strict bound kV k < d/π guarantees that, in addition, L is similar to a self-adjoint operator see Theorem 5.8 (i) . The question whether this is true for d/π ≤ kV k < d/2 remains an open problem. We answer the question (iii) by using the concept of the operator angle between two subspaces (for discussion of this notion and references see, e.g., [25]). Recall that if M and N are subspaces of a Hilbert space, the operator angle Θ(M, N) between M and N measured relative to the subspace M is introduced by

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the following formula [26]: Θ(M, N) = arcsin

q

IM − PM PN M ,

(1.5)

where IM denotes the identity operator on M and PM and PN stand for the orthogonal projections onto M and N, respectively. Set 2d/π, case (G), δ= (1.6) d, case (S), and assume that kV k < δ/2. Since in both the cases (G) and (S) under this assumption we have got the positive answer to the question (ii), one can easily identify the spectral subspaces H00 and H01 of L associated with the corresponding perturbed spectral sets σ00 and σ10 (cf. Lemma 5.6). Let Θj = Θ(Hj , H0j ), j = 0, 1, be the operator angle between the unperturbed spectral subspace Hj and the perturbed one, H0j . Our main result (presented in Theorem 5.8) regarding the operator angles Θ0 and Θ1 is that under condition kV k < δ/2 the following bound holds: 2kV k 1 arctanh , j = 0, 1, (1.7) tan Θj ≤ tanh 2 δ which means, in particular, that Θj < π4 , j = 0, 1. Theorem 5.8 also gives the bounds on location of the perturbed spectral sets σ00 and σ10 see formulas (5.20) . In the case (S) the bounds on σ00 and σ10 as well as the bounds (1.7) are optimal (see Remark 5.10). Inequalities (1.7) resemble the sharp norm estimate for the operator angle between perturbed and unperturbed spectral subspaces from the celebrated Davis-Kahan tan 2Θ Theorem (see [17], p. 11; cf. [28, Theorem 2.4] and [41, Theorem 1]). Recall that the latter theorem serves for the case where the unperturbed spectral subsets σ0 and σ1 are subordinated and the off-diagonal perturbation V is self-adjoint. The difference is that the usual tangent of the Davis-Kahan tan 2Θ Theorem is replaced on the right-hand side of (1.7) by the hyperbolic one. Another distinction is that the bound (1.7) holds not only for the subordinated spectral sets σ0 and σ1 but also for the disposition where one of these sets lies in a finite gap of the other set and thus σ0 and σ1 are not subordinated. The results obtained are of particular interest for the theory of operators on Krein spaces [9]. The reason for this is that introducing an indefinite inner product [x, y] = (Jx, y), x, y ∈ H, instead of the initial inner product (·, ·), turns H into a Krein space. The operators V and L = A + V being J-self-adjoint on H appear to be self-adjoint operators on the newly introduced Krein space K. Under the condition kV k < δ/2 in both cases (G) and (S) we establish that the perturbed spectral subspaces H00 and H01 are mutually orthogonal with respect to the inner product [·, ·]. Moreover, these subspaces are maximal uniformly positive and maximal uniformly negative, respectively (see Remark 5.11). The restrictions of L onto H00 and H01 are K-unitary equivalent to self-adjoint operators on H0 and H1 , respectively. This extends similar results previously known from [1] and [37] for the case where the spectral sets σ0 and σ1 are subordinated.

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Another motivation for the present paper is in the spectral analysis of nonself-adjoint Schr¨ odinger operators that involve the so-called PT -symmetric potentials. Starting from the pioneering works [11, 12], these potentials attracted considerable attention because of their property to produce, in some cases, purely real spectra (see, e.g., [3, 4, 10, 29, 39, 53]). Some PT -symmetric potentials appear to be J-self-adjoint with respect to the space parity operator P (see, e.g., [32, 39]), allowing for an embedding the problem into the context of the spectral theory for J-self-adjoint perturbations (this also means that the PT -symmetric perturbations may be studied within the framework of the Krein space theory [4, 32, 48]). The main tool we use in our analysis is a reduction of the problems (i)–(iii) to the study of the operator Riccati equation KA0 + A1 K + KBK = −B ∗ associated with the representation of the perturbed operator L = A + V in the 2 × 2 block matrix form A0 B L= , −B ∗ A1 where A0 = A H0 , A1 = A H1 , and B = V H1 . Assuming (1.6), we prove that the Riccati equation has a bounded (in fact, uniformly contractive) solution K for any B such that kBk < δ/2. The key statement is that the perturbed spectral subspaces H00 and H01 are the graphs of the operators K and K ∗ , respectively, which then allows us to derive the bounds (1.7). The plan of the paper is as follows. In Section 2 we give necessary definitions and present some basic results on the operator Riccati equations associated with a class of unbounded non-self-adjoint 2 × 2 block operator matrices. Section 3 is devoted to the related Sylvester equations. In Section 4 we prove a number of existence and uniqueness results for the operator Riccati equations. In Section 5 we consider J-self-adjoint perturbations and find conditions on their norm guaranteeing the reality of the resulting spectrum. In this section we also prove the bound (1.7) on the variation of the spectral subspaces and discuss the embedding of the problem into the context of the Krein space theory. Finally, in Section 6 we apply some of the results obtained to a quantum-mechanical Hamiltonian describing the harmonic oscillator under a PT -symmetric perturbation. We conclude the introduction with the description of some more notations that are used throughout the paper. By a subspace we always understand a closed linear subset of a Hilbert space. The identity operator on a subspace (or on the whole Hilbert space) M is denoted by IM . If no confusion arises, the index M may be omitted in this notation. The Banach space of bounded linear operators from a Hilbert space M to a Hilbert space N is denoted by B(M, N). For B(M, M) we use a shortened notation B(M). By M ⊕ N we will understand the orthogonal sum of two Hilbert spaces (or orthogonal subspaces) M and N. By Or (M, N), 0 ≤ r < ∞, we denote the closed ball in B(M, N), having radius r and being centered at zero,

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that is, Or (M, N) = {K ∈ B(M, N) kKk ≤ r}. If it so happens that r = +∞, by O∞ (M, N) we will understand the whole space B(M, N). The notation conv(σ) is used for the convex hull of a Borel set σ ⊂ R. By Or (Ω), r ≥ 0, we denote the closed r-neighborhood of a Borel set Ω in the complex plane C, i.e. Or (Ω) = {z ∈ C dist(z, Ω) ≤ r}.

2. Operator Riccati equation We start by recalling the concepts of weak, strong, and operator solutions to the operator Riccati equation (see [5, 6]). Definition 2.1. Assume that A0 and A1 are possibly unbounded densely defined closed operators on the Hilbert spaces H0 and H1 , respectively. Let B and C be bounded operators from H1 to H0 and from H0 to H1 , respectively. A bounded operator K ∈ B(H0 , H1 ) is said to be a weak solution of the Riccati equation KA0 − A1 K + KBK = C (2.1) if (KA0 x, y) − (Kx, A∗1 y) + (KBKx, y) = (Cx, y) for all x ∈ Dom(A0 ) and y ∈ Dom(A∗1 ). A bounded operator K ∈ B(H0 , H1 ) is called a strong solution of the Riccati equation (2.1) if (2.2) Ran K|Dom(A0 ) ⊂ Dom(A1 ) and KA0 x − A1 Kx + KBKx = Cx for all x ∈ Dom(A0 ).

(2.3)

Finally, K ∈ B(H0 , H1 ) is said to be an operator solution of the Riccati equation (2.1) if Ran(K) ⊂ Dom(A1 ), (2.4) the operator KA0 is bounded on Dom(KA0 ) = Dom(A0 ), and the equality KA0 − A1 K + KBK = C

(2.5)

holds as an operator equality, where KA0 denotes the closure of KA0 . Remark 2.2. We will call the equation XA∗1 − A∗0 X − XB ∗ X = −C ∗

(2.6)

the adjoint of the operator Riccati equation (2.1). It immediately follows from the definition that an operator K ∈ B(H0 , H1 ) is a weak solution to the Riccati equation (2.1) if and only if the adjoint of K, X = K ∗ , is a weak solution to the adjoint equation (2.6).

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Clearly, any operator solution K ∈ B(H0 , H1 ) to the Riccati equation (2.1) is automatically a strong solution. Similarly, any strong solution is also a weak solution. But, in fact, by a result of [6] one does not need to distinguish between weak and strong solutions to the Riccati equation (2.1). This is seen from the following statement. Lemma 2.3 ([6], Lemma 5.2). Let A0 and A1 be densely defined possibly unbounded closed operators on the Hilbert spaces H0 and H1 , respectively, and B ∈ B(H1 , H0 ), C ∈ B(H0 , H1 ). If K ∈ B(H0 , H1 ) is a weak solution of the Riccati equation (2.1) then K is also a strong solution of (2.1). If the operators A0 , A1 , B, and C are as in Definition 2.1, then a 2 × 2 operator block matrix A0 B L= , Dom(L) = Dom(A0 ) ⊕ Dom(A1 ), (2.7) C A1 is a densely defined and possibly unbounded closed operator on the Hilbert space H = H0 ⊕ H1 .

(2.8)

The operator L will often be viewed as the result of the perturbation of the block diagonal matrix A = diag(A0 , A1 ),

Dom(A) = Dom(A0 ) ⊕ Dom(A1 ),

by the off-diagonal bounded perturbation 0 B V = . C 0

(2.9)

(2.10)

The operator Riccati equation (2.1) and the block operator matrix L are usually said to be associated to each other. Surely, one can also associate with the matrix L another operator Riccati equation, K 0 A1 − A0 K 0 + K 0 CK 0 = B,

(2.11)

0

assuming that a solution K (if it exists) should be a bounded operator from H1 to H0 . It is well known that the solutions to the Riccati equations (2.1) and (2.11) determine invariant subspaces for the operator matrix L (see, e.g., [5] for the case where the matrix L is self-adjoint or [31] for the case of a non-self-adjoint L). These subspaces have the form of the graphs G(K) = {x ∈ H0 ⊕ H1 | x = x0 ⊕ Kx0 for some x0 ∈ H0 }

(2.12)

and G(K 0 ) = {x ∈ H0 ⊕ H1 | x = K 0 x1 ⊕ x1 for some x1 ∈ H1 } (2.13) 0 of the corresponding (bounded) solutions K and K . Notice that the subspaces of the form (2.12) and (2.13) are usually called the graph subspaces associated with the operators K and K 0 , respectively, while K and K 0 themselves are called the angular operators. Usage of the latter term is explained, in particular, by the fact

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that if a subspace G ⊂ H is a graph G = G(K) of a bounded linear operator K from a subspace M to its orthogonal complement M⊥ , M⊥ = H M, then the following equality holds (see [25]; cf. [17] and [21]): |K| = tan Θ(M, G), (2.14) √ where |K| is the absolute value of K, |K| = K ∗ K, and Θ(M, G) the operator angle (1.5) between the subspaces M and G. The precise statement relating solutions of the Riccati equations (2.1) and (2.11) to invariant subspaces of the operator matrix (2.7) is as follows. Lemma 2.4. Let the entries A0 , A1 , B, and C be as in Definition 2.1 and let a 2 × 2 block operator matrix L be given by (2.7). Then the graph G(K) of a bounded operator K from H0 to H1 satisfying (2.2) is an invariant subspace for the operator matrix L if and only if K is a strong solution to the operator Riccati equation (2.1). Similarly, the graph G(K 0 ) of an operator K 0 ∈ B(H1 , H0 ) such 0 that Ran K |Dom(A1 ) ⊂ Dom(A0 ) is an invariant subspace for L if and only if this operator is a strong solution to the Riccati equation (2.11). The proof of this lemma is straightforward and follows the same line as the proof of the corresponding part in [5, Lemma 5.3]. Thus, we omit it. The next assertion contains two useful identities involving the strong solutions to the Riccati equations (2.1) and (2.11). Lemma 2.5. Let the entries A0 , A1 , B, and C be as in Definition 2.1. Assume that operators K ∈ B(H0 , H1 ) and K 0 ∈ B(H1 , H0 ) are strong solutions to equations (2.1) and (2.11), respectively. Then (2.15) Ran K 0 K|Dom(A0 ) ⊂ Dom(A0 ), Ran KK 0 |Dom(A1 ) ⊂ Dom(A1 ), and (I − K 0 K)(A0 + BK)x = (A0 − K 0 C)(I − K 0 K)x for all x ∈ Dom(A0 ), (I − KK 0 )(A1 + CK 0 )y = (A1 − KB)(I − KK 0 )y for all y ∈ Dom(A1 ).

(2.16) (2.17)

Proof. The inclusions (2.15) follow immediately from the definition of a strong solution to the operator Riccati equation (see condition (2.2)). Let x ∈ Dom(A0 ). Taking into account the first of the inclusions(2.15) as well as the inclusions Ran K|Dom(A0 ) ⊂ Dom(A1 ) and Ran K 0 |Dom(A1 ) ⊂ Dom(A0 ) one can write (A0 − K 0 C)(I − K 0 K)x = (A0 − K 0 C)x − (A0 K 0 − K 0 CK 0 )Kx = (A0 − K 0 C)x − (K 0 A1 − B)Kx,

(2.18)

by making use of the Riccati equation (2.11) itself at the second step. Similarly, (I − K 0 K)(A0 + BK)x = (A0 + BK)x − K 0 (KA0 + KBK)x = (A0 + BK)x − K 0 (C + A1 K)x,

(2.19)

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due to the Riccati equation (2.1). Comparing (2.18) and (2.19) we arrive at the identity (2.16). Identity (2.17) is proven analogously. We will also need the following auxiliary lemma. Lemma 2.6. Suppose that the operators K ∈ B(H0 , H1 ) and K 0 ∈ B(H1 , H0 ) are such that the 2 × 2 operator block matrix I K0 W = (2.20) K I considered on H = H0 ⊕ H1 is boundedly invertible, i.e. the inverse operator W −1 exists and is bounded. Then the graphs G(K) and G(K 0 ) of the operators K and K 0 are linearly independent subspaces of H and H = G(K) u G(K 0 ),

(2.21)

where the sign “u” denotes the direct sum of two subspaces. Proof. The existence and boundedness of W −1 imply that equation W x = y is uniquely solvable for any y ∈ H. This means that there are unique x0 ∈ H0 and unique x1 ∈ H1 such that y = x0 ⊕ Kx0 + K 0 x1 ⊕ x1 and hence H ⊂ G(K) + G(K 0 ). Since both G(K) and G(K 0 ) are subspaces of H, the inclusion turns into equality, H = G(K) + G(K 0 ). The linear independence of G(K) and G(K 0 ) follows from the fact that equation W x = 0 has only the trivial solution x = 0. Remark 2.7. It is well known that the following three statements are equivalent. (i) The operator matrix (2.20) is boundedly invertible. (ii) The inverse (I − KK 0 )−1 exists and is bounded. (iii) The inverse (I − K 0 K)−1 exists and is bounded For a proof of this assertion see, e.g. [22, Theorem 1.1. and Lemma 2.1] where even a Banach-space case of 2 × 2 block operator matrices of the form (2.20) with unbounded entries K and K 0 has been studied. Remark 2.8. The inverse of the operator W is explicitly written as (I − K 0 K)−1 −K 0 (I − KK 0 )−1 −1 W = −K(I − K 0 K)−1 (I − KK 0 )−1 (I − K 0 K)−1 −(I − K 0 K)−1 K 0 = −(I − KK 0 )−1 K (I − KK 0 )−1 .

(2.22)

The (oblique) projections QG(K) and QG(K 0 ) onto the graph subspaces G(K) and G(K 0 ) along the corresponding complementary graph subspaces G(K 0 ) and G(K) are given by I QG(K) = (I − K 0 K)−1 I −K 0 (2.23) K and 0 K QG(K 0 ) = (I − KK 0 )−1 −K I , (2.24) I respectively.

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Corollary 2.9. Assume the hypothesis of Lemma 2.4. Suppose that K ∈ B(H0 , H1 ) and K 0 ∈ B(H1 , H0 ) are strong solutions to the Riccati equations (2.1) and (2.11), respectively. Assume, in addition, that the 2 × 2 operator block matrix W formed with these solutions according to (2.20) is a boundedly invertible operator on H = H0 ⊕ H1 . Then: (i) The operator L is similar to a block diagonal operator matrix Z = diag(Z0 , Z1 ), L = W ZW −1 ,

(2.25)

where Z0 and Z1 are operators on H0 and H1 , respectively, given by Z0 = A0 + BK,

Dom(Z0 ) = Dom(A0 ),

0

Z1 = A1 + CK ,

Dom(Z1 ) = Dom(A1 ).

(2.26) (2.27)

(ii) The Hilbert space H splits into the direct sum H = H00 u H01 of the graph subspaces H00 = G(K) and H01 = G(K 0 ) that are invariant under L. The restrictions L|H00 and L|H01 of L onto H00 and H01 are similar to the operators Z0 and Z1 , W0−1 L|H00 W0 = Z0

and H00

W1−1 L|H01 W1 = Z1 ,

(2.28)

H01

where the entries W0 : H0 → and W1 : H1 → correspond to the respective columns of the block operator matrix W , 0 I K W0 x0 = x0 , x0 ∈ H0 , and W1 x1 = x1 , x1 ∈ H1 . (2.29) K I Proof. First, one verifies by inspection that LW = W Z taking into account that K and K 0 are the strong solutions to the Riccati equations (2.1) and (2.11), respectively. The remaining statements immediately follow from Lemma 2.4 combined with Lemma 2.6. Remark 2.10. The similarity (2.25) of the operators L and Z implies that the spectrum of L coincides with the union of the spectra of Z0 and Z1 , that is, spec(L) = spec(Z0 ) ∪ spec(Z1 ).

3. Operator Sylvester equation Along with the Riccati equation (2.1) we need to consider the operator Sylvester equation XA0 − A1 X = Y (3.1) assuming that the entries A0 and A1 are as in Definition 2.1 and Y ∈ B(H0 , H1 ). The Sylvester equation is a particular (linear) case of the Riccati equation and its weak, strong, and operator solutions X ∈ B(H0 , H1 ) are understood in the same way as in the above definition. Furthermore, by Lemma 2.3 (cf. [8, Lemma 1.3]) one does not need to distinguish between the weak and strong solutions to (3.1). Because of its importance for various areas of mathematics there is an enormous literature on the Sylvester equation (for a review and many references see

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paper [14]). With equation (3.1) one often associates the Sylvester operator S defined on the Banach space B(H0 , H1 ) by the left-hand side of (3.1): S(X) = XA0 − A1 X

(3.2)

with domain n o Dom(S) = X ∈ B(H0 , H1 ) Ran(X Dom(A ) ) ⊂ Dom(A1 ) . 0

(3.3)

Clearly, for any Y ∈ B(H0 , H1 ) the Sylvester equation (3.1) has a unique solution X ∈ Dom(S) if and only if 0 6∈ spec(S). It is known that in general the spectrum of S is larger than the (numerical) difference between the spectra of A0 and A1 . More precisely, provided that spec(A0 ) 6= C or spec(A1 ) 6= C, the following inclusion always holds [8]: spec(A0 ) − spec(A1 ) ⊂ spec(S), (3.4) where we use the notation Σ − ∆ = {z − ζ | z ∈ Σ, ζ ∈ ∆} for the numerical difference between two Borel subsets Σ and ∆ of the complex plane C. The opposite inclusion in (3.4) may fail to hold if both operators A0 and A1 are unbounded. The corresponding example was first given by V. Q. Ph´ong [43] for the Sylvester equation (3.1) where one of the entries A0 and A1 is an operator on a Banach (but not Hilbert) space. An example where both A0 and A1 are operators on Hilbert spaces and spec(S) 6⊂ spec(A0 ) − spec(A1 ) may be found in [8, Example 6.2]. Equality spec(S) = spec(A0 ) − spec(A1 )

(3.5)

holds if both A0 and A1 are bounded operators. This result is due to G. Lumer and M. Rosenblum [34]. Equality (3.5) also holds if only one of the entries A0 and A1 is a bounded operator [8]. In this case (3.5) implies that if the spectra of A0 and A1 are disjoint then 0 6∈ spec(S) and hence the operator S is boundedly invertible. Moreover, a unique solution of the Sylvester equation (3.1) admits an “explicit” representation in the form a contour integral. Lemma 3.1. Let A0 be a possibly unbounded densely defined closed operator on the Hilbert space H0 and A1 a bounded operator on the Hilbert space H1 such that spec(A0 ) ∩ spec(A1 ) = ∅ and let Y ∈ B(H0 , H1 ). Then the Sylvester equation (3.1) has a unique operator solution Z 1 X= dz (A1 − z)−1 Y (A0 − z)−1 , (3.6) 2πi γ where γ is a union of closed contours in C with total winding numbers 0 around spec(A0 ) and 1 around spec(A1 ) and the integral converges in the norm operator topology. Corollary 3.2. Under the hypothesis of Lemma 3.1 the norm of the inverse of the Sylvester operator S may be estimated as kS−1 k ≤ (2π)−1 |γ| sup k(A0 − z)−1 k k(A1 − z)−1 k , z∈γ

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where |γ| denotes the length of the contour γ in (3.6). The result of Lemma 3.1 may be attributed to M. G. Krein who lectured on the operator Sylvester equation in late 1940s (see [14]). Later, it was independently obtained by M. Rosenblum [45]. As for the Sylvester operator (3.2) with both unbounded entries A0 and A1 , we have an important result which is due to W. Arendt, F. R¨abiger, and A. Sourour (see [8, Theorem 4.1 and Corollary 5.4]). Theorem 3.3 ([8]). Let A0 and A1 be closed densely defined operators on the Hilbert spaces H0 and H1 , respectively. Assume that one (or both) of the following holds (hold) true: (i) A0 and (−A1 ) are generators of eventually norm continuous C0 -semigroups; (ii) A0 and (−A1 ) are generators of C0 -semigroups one of which is holomorphic. Then the spectrum of the Sylvester operator (3.2) is given by (3.5). Recall that an operator H on the Hilbert space M is said to be m-dissipative if it is closed and both the spectrum and numerical range of H are contained in the left half-plane {z ∈ C | Im z ≤ 0}. The Lumer-Phillips theorem asserts (see, e.g., [19, Section II.3.b]; cf. [20, Theorem B.21]) that a C0 -semigroup on M is a contraction semigroup if and only if its generator is an m-dissipative operator. The next statement represents a generalization of a well known result by E. Heinz ([23, Satz 5]) to the case of unbounded operators. Notice that the exponential eHt , t ≥ 0, is understood below as the corresponding element of the strongly continuous contraction semigroup generated by an (unbounded) m-dissipative operator H. Theorem 3.4. Let A0 + 2δ I and −A1 + 2δ I, δ > 0, be m-dissipative operators on the Hilbert spaces H0 and H1 , respectively, and Y ∈ B(H0 , H1 ). Then the Sylvester equation (3.1) has a unique weak (and hence unique strong) solution given by Z +∞ X=− dt e−A1 t Y eA0 t , (3.7) 0

where the integral is understood in the weak operator topology. Moreover, the norm of the solution (3.7) satisfies the estimate 1 kXk ≤ kY k. (3.8) δ Proof. Under the hypothesis the operators A0 and (−A1 ) are themselves m-dissipative. Let U0 (t) and U1 (t), t ≥ 0, be contraction C0 -semigroups generated respectively by A0 and (−A1 ), that is, U0 (t) = eA0 t and U1 (t) = e−A1 t . Clearly, δ

δ

kU0 (t)k ≤ e− 2 t and kU1 (t)k ≤ e− 2 t ,

t ≥ 0. ∗

(3.9)

The same bound also holds for the adjoint semigroup U1 (t) , t ≥ 0, whose generator is the m-dissipative operator (−A∗1 ). Pick an arbitrary x ∈ H0 and y ∈ H1 and introduce the orbit maps ξx : t 7→ ξx (t) = U0 (t)x and ζy : t 7→ ζy (t) = U1 (t)∗ y. By the definition of a strongly continuous semigroup, these maps are continuous

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functions of t ∈ [0, ∞). Taking into account the bounds (3.9) one then concludes that the improper integral Z ∞ Z ∞ dt (U1 (t)Y U0 (t)x, y) = dt (Y ξx (t), ζy (t)) (3.10) 0

0

converges and its absolute value is bounded by kY kkxkkyk/δ. Thus, the weak integral on the right-hand side of (3.7) exists and the bound (3.8) holds. Now assume that x ∈ Dom(A0 ) and y ∈ Dom(A∗1 ). In this case the orbit d maps ξx (t) and ζy (t) are continuously differentiable in t and dt ξx (t) = U0 (t)A0 x, d ∗ ∗ ζ (t) = −U (t) A y. For X given by (3.7), an elementary computation shows 1 1 dt y that Z ∞ d d ∗ dt Y ξx (t), ζy (t) + Y ξx (t), ζy (t) (XA0 x, y) − (Xx, A1 y) = − dt dt Z0 ∞ d =− dt (Y ξx (t), ζy (t)) = (Y ξx (0), ζy (0)) , dt 0 taking into account (3.9) in the last step. Since ξx (0) = x and ζy (0) = y, by Definition 2.1 this implies that the integral (3.7) is a weak (and hence strong) solution to the Sylvester equation (3.1). To prove the uniqueness of the weak solution (3.7) it is sufficient to show that the homogeneous Sylvester equation XA0 − A1 X = 0 has the only weak solution X = 0. For a weak solution X to this equation we have (XA0 u, v) − (Xu, A∗1 v) = 0

for all u ∈ Dom(A0 ) and v ∈ Dom(A∗1 ).

(3.11) ∗

Take the vectors u and v of the form u = ξx (t) = eA0 t x, v = ζy (t) = e−A1 t y, t ≥ 0, where the orbit maps ξx (t) and ζy (t) correspond to some x ∈ Dom(A0 ) and y ∈ Dom(A∗1 ) and hence are both continuously differentiable in t ∈ [0, ∞). Notice that the assumption x ∈ Dom(A0 ), y ∈ Dom(A∗1 ) also implies (see, e.g. [19, Lemd d ma 1.3]) that u ∈ Dom(A0 ), v ∈ Dom(A1 ), and A0 u = dt ξx (t), A∗1 v = − dt ζ(t). With such a choice of u and v it follows from (3.11) that d dt (Xξx (t), ζy (t))

=0

whenever x ∈ Dom(A0 ) and y ∈ Dom(A∗1 ).

Hence the function (Xξx (t), ζy (t)), t ≥ 0, is a constant. Moreover, it equals zero since it vanishes as t → ∞. This yields in particular that (Xx, y) = (Xξx (0), ζy (0)) = 0

for all x ∈ Dom(A0 ) and y ∈ Dom(A∗1 ).

The latter implies X = 0, which completes the proof.

Remark 3.5. A statement similar to Theorem 3.4 was previously announced without a proof in [5] (see [5, Lemma 2.6]). The second important example where a bound of the type (3.8) exists is given in [13, Theorem 3.2]. This example is as follows.

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Theorem 3.6. Assume that the operators A0 and A1 are densely defined and closed. Assume, in addition, that there is λ ∈ %(A1 ) such that kA0 − λk ≤ r and k(A1 − λ)−1 k ≤ (r + δ)−1 for some r ≥ 0 and δ > 0. Then for any Y ∈ B(H0 , H1 ) the unique strong solution X to the Sylvester equation (3.1) admits the estimate δkXk ≤ kY k. If the operators A0 and A1 are normal then no reference point λ is needed and the result is stated in a more universal form (see [13, Theorem 3.1]). Corollary 3.7. Let both A0 and A1 be normal operators such that spec(A0 ) is contained in a closed disk of radius r, r ≥ 0, while spec(A1 ) is disjoint from the open disk (with the same center) of radius r+δ, δ > 0. Then for any Y ∈ B(H0 , H1 ) the Sylvester equation (3.1) has a unique strong solution X and δkXk ≤ kY k. The above two theorems and corollary give examples where the bounded inverse of the Sylvester operator S exists and for the norm of S −1 the estimate δkS −1 k ≤ 1 holds with some δ > 0. Moreover, this estimate is universal in the sense that it remains valid for any A0 and A1 satisfying the corresponding hypotheses.

4. Existence results for the Riccati equation In this section we return to the operator Riccati equation (2.1) to prove some sufficient conditions for its solvability. In their proof we will rely just on the assumption that an estimate like (3.8) holds for the solution of the corresponding Sylvester equation. Theorem 4.1. Let A0 and A1 be possibly unbounded closed densely defined operators on the Hilbert spaces H0 and H1 , respectively. Assume that the Sylvester operator S defined on B(H0 , H1 ) by (3.2) and (3.3) is boundedly invertible (that is, 0 6∈ spec(S)) and 1 (4.1) kS−1 k ≤ δ for some δ > 0. Assume, in addition, the operators B ∈ B(H1 , H0 ) and C ∈ B(H0 , H1 ) are such that the following bound holds: p

kBkkCk <

δ . 2

(4.2)

Then the operator Riccati equation (2.1) has a unique strong solution in the ball Oδ/(2kBk) (H1 , H0 ). The strong solution K satisfies the estimate kCk

kKk ≤ δ 2

+

q

δ2 4

− kBk kCk

.

(4.3)

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Proof. If B = 0 then the assertion, including the estimate (4.3), follows immediately from the hypothesis on the invertibility of S on B(H0 , H1 ) taking into account the bound (4.1). Suppose that B 6= 0. In this case the proof is performed by applying Banach’s Fixed Point Theorem. First, we notice that the bounded invertibility of S on B(H0 , H1 ) allows us to rewrite the Riccati equation (2.1) in the form K = F (K) where the mapping F : B(H0 , H1 ) → Dom(S) is given by F (K) = S −1 (C − KBK). By (4.1) we have kF (K)k ≤

1 (kCk + kBk kKk2 ), δ

K ∈ B(H0 , H1 )

(4.4)

and 1 kBk (kK1 k+kK2 k) kK1 −K2 k, K1 , K2 ∈ B(H0 , H1 ). (4.5) δ The bound (4.4) implies that F maps the ball Or (H0 , H1 ) into itself whenever kF (K1 )−F (K2 )k ≤

kBk r2 + kCk ≤ rδ.

(4.6)

At the same time, from (4.5) it follows that F is a strict contraction of the ball Or (H1 , H0 ) whenever 2kBkr < δ. (4.7) Solving inequalities (4.6) and (4.7) one concludes that if the radius r of the ball Or (H1 , H0 ) is within the bounds kCk δ 2

+

q

δ2 4

− kBk kCk

≤r<

δ , 2kBk

(4.8)

then F is a strictly contractive mapping of the ball Or (H0 , H1 ) into itself. Applying Banach’s Fixed Point Theorem one then infers that equation (2.1) has a unique solution within any ball Or (H0 , H1 ) whenever the radius r satisfies (4.8). This means that the fixed point is the same for all the radii satisfying (4.8) and hence it belongs to the smallest of the balls. This conclusion proves the bound (4.3) and completes the whole proof. Remark 4.2. In (4.2)–(4.3) one may set δ = kS −1 k−1 . Remark 4.3. By using the hyperbolic tangent function and its inverse the bound (4.3) (for B 6= 0) can be equivalently written in the hypertrigonometric form s ! p 2 kBkkCk kCk 1 kKk ≤ tanh arctanh . (4.9) kBk 2 δ

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Notice that under condition (4.2) we always have ! p 2 kBkkCk 1 tanh < 1. arctanh 2 δ Remark 4.4. Fixed-point based approaches to prove the solvability of the operator Riccati equation with bounded entries A0 and A1 have been used in many papers (see, e.g., [2], [18], [42], [46], [47]). In the case where at least one of the entries A0 and A1 is an unbounded self-adjoint or normal operator, a fixed-point approach has been employed in [5], [6], [36], and [40]. Theorem 4.1 represents an extension of the fixed-point existence results obtained in [47, Theorem 3.5] and [42, Theorem 3.1] for the Riccati equation (2.1) with both bounded A0 and A1 to the case where the entries A0 and A1 are not necessarily bounded. Theorem 4.5. Assume the hypothesis of Theorem 4.1. Then the block operator matrix L defined by (2.7) is block diagonalizable with respect to the direct sum decomposition H = G(K)uG(K 0 ) where K is the unique strong solution to the Riccati equation (2.1) within the operator ball Oδ/(2kBk) (H0 , H1 ) and K 0 the unique strong solution to the Riccati equation (2.11) within the operator ball Oδ/(2kCk) (H1 , H0 ). Proof. By Theorem 4.1 for K the estimate (4.3) holds. By the same theorem for K 0 we have !−1 r 2 δ δ + − kBk kCk . (4.10) kK 0 k ≤ kBk 2 4 Then the hypothesis kBkkCk < δ/2 also implies that kKkkK 0 k < 1. Hence by Remark 2.7 the operator W in (2.20) is boundedly invertible. Applying Corollary 2.9 completes the proof. Remark 4.6. If, in addition, both the operators A0 and A1 are normal then, for r defined by ! p p 2 kBkkCk 1 kBk kCk q = kBk kCk tanh arctanh , (4.11) r= 2 2 δ δ + δ − kBk kCk 2

4

the spectrum of the block matrix L lies in the closed r-neighborhood of the spectrum of its main-diagonal part A = diag(A0 , A1 ). That is, dist z, spec(A0 ) ∪ spec(A1 ) ≤ r whenever z ∈ spec(L). This immediately follows from the representation (2.25)–(2.27) and the bounds (4.3) p and (4.10) (see also Remark 2.10). Notice that if B 6= 0 and C 6= 0 then r < kBkkCk and hence r < kV k taking into account that kV k = max(kBk, kCk). From now on we assume that the entries A0 and A1 are self-adjoint operators with disjoint spectra and thus adopt the following

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Hypothesis 4.7. Let A0 and A1 be (possibly unbounded) self-adjoint operators on the Hilbert spaces H0 and H1 with domains Dom(A1 ) and Dom(A1 ), respectively. Assume that the spectra of the operators A0 and A1 are disjoint and let d = dist spec(A0 ), spec(A1 ) > 0 . (4.12) Hypothesis 4.7 imposes no restrictions on the mutual position of the spectral sets spec(A0 ) and spec(A1 ) except that they are disjoint and separated from each other by a distance d. Sometimes, however, we will consider particular spectral dispositions described in Hypothesis 4.8. Assume Hypothesis 4.7. Assume, in addition, that either the spectra of A0 and A1 are subordinated, that is, sup spec(A0 ) < inf spec(A1 ) or inf spec(A0 ) > sup spec(A1 ),

(4.13)

or one of the sets spec(A0 ) and spec(A1 ) lies in a finite gap of the other set, that is, conv spec(A0 ) ∩ spec(A1 ) = ∅ or spec(A0 ) ∩ conv spec(A1 ) = ∅. (4.14) Under Hypotheses 4.7 or 4.8 the bound on the norm of the inverse of the Sylvester operator (3.2) may be given in terms of the distance d between spec(A0 ) and spec(A1 ). The following result is well known. Theorem 4.9. Assume Hypothesis 4.7. Let the Sylvester operator S be defined by (3.2) and (3.3). (i) Then the inverse of S exists and is bounded. Moreover, the following estimate holds: π . (4.15) kS−1 k ≤ 2d (ii) Assume Hypothesis 4.8. Then the following stronger inequality holds: kS−1 k ≤

1 . d

(4.16)

Remark 4.10. In the generic case (i), where no assumptions on the mutual position of the sets spec(A0 ) and spec(A1 ) are imposed, the existence of a universal c constant c such that kS −1 k ≤ has been proven in [13]. The proof of the fact d that c = π/2 is best possible is due to R. McEachin [35]. For more details see [5, Remark 2.8]. As for the particular spectral disposition (4.13), the bound (4.16) is an immediate corollary to Theorem 3.4. Since any self-adjoint operator is simultaneously a normal operator, in the case of the spectral disposition (4.14) the bound (4.16) follows from Corollary 3.7. Sharpness of the bound (4.16) in case (ii) is proven by an elementary example where the spaces H0 and H1 are onedimensional, H0 = H1 = C, and the entries A0 = a0 and A1 = a1 are real numbers such that |a1 − a0 | = d > 0.

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Under the assumption that both the entries A0 and A1 are self-adjoint operators, below we present an existence result for the operator Riccati equation (2.1), which is written directly in terms of the distance between the spectra of the entries A0 and A1 (and norms of the operators B and C). The result is an immediate corollary to Theorems 4.1 and 4.9. We only notice that the role of the quantity δ in the bounds like (4.1) (see inequalities (4.18) and (4.20) below) will be played by either π2 d from (4.15) or d from (4.16). Theorem 4.11. Assume Hypothesis 4.7. (i) Then for any B ∈ B(H1 , H0 ) and C ∈ B(H0 , H1 ) such that p d kBkkCk < π the Riccati equation (2.1) has a unique strong solution K in the ball Od/(πkBk) (H0 , H1 ). This solution satisfies the estimate kCk

kKk ≤ d π

+

q

d2 π2

.

(4.17)

(4.18)

− kBk kCk

(ii) If the conditions of Hypothesis 4.8 also hold then the Riccati equation (2.1) has a unique strong solution K in the ball Od/(2kBk) (H0 , H1 ) whenever B ∈ B(H1 , H0 ) and C ∈ B(H0 , H1 ) satisfy the bound p d (4.19) kBkkCk < . 2 The solution K satisfies the estimate kCk

kKk ≤ d 2

+

q

d2 4

.

(4.20)

− kBk kCk

Remark 4.12. The part (i) is a refinement of Theorem 3.6 in [5] that only claimed the existence of a weak (but not strong) solution to the Riccati equation (2.1) within the ball Od/(πkBk) (H0 , H1 ). The result of the part (ii) is new. Remark 4.13. Let r be given by formula (4.11) where δ = π2 d in case (i) and δ = d in case (ii). By Remarks 2.10 and 4.6 one concludes that the spectrum of the block operator matrix L consists of the two disjoint components σ00 = spec(Z0 ) and σ10 = spec(Z1 ) lying in the closed r-neighborhoods Or spec(A0 ) and Or spec(A1 ) of the corresponding spectral sets spec(A0 ) and spec(A1 ). Remark 4.14. Examples 4.15 and 4.16 below show that the bound (4.20) is sharp in the following sense. Given a number d > 0 and values of the norms kBk and kCk satisfying (4.19) one can always present self-adjoint (and even rank one or two) entries A0 , A1 and bounded B and C such that in case (ii) the bound (4.20) turns into equality. Notice that Examples 4.15 and 4.16 serve for the spectral dispositions (4.13) and (4.14), respectively.

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Example 4.15. Let H0 = H1 = C. In this case the entries A0 , A1 , B and C of (2.1) are simply the operators of multiplication by numbers. Set A0 = − d2 , A1 = d2 , √ B = b, and C = −c where b, c, and d are positive numbers such that bc < d/2. The Riccati equation (2.1) turns into a numeric quadratic equation whose solutions K (1) and K (2) are given by c c q q K (1) = , K (2) = . (4.21) d d2 d d2 + − bc − − bc 2 4 2 4 The right-hand sides of the equalities in (4.21) also represent the norms of the corresponding solutions K (1) and K (2) . Obviously, only the solution K (1) satisfies d . Also notice that the eigenvalues of the associated 2 × 2 the bound kKk < 2kBk p matrix L (which is given by (2.7)) read λ− = − d2 /4 − bc and λ+ = −λ− . One observes, in particular, that λ− = A0 + BK (1) . Example 4.16. Let H0 = C and H1 = C2 . Assume that −d 0 0 A0 = 0, A1 = , B = (0 b), and C = , 0 d −c √ where b, c, and d are positive numbers such that bc < d/2. In this case the Riccati ! (1) k− (1) and equation (2.1) is easily solved explicitly. It has two solutions K = (1) k+ ! (2) k− (1) (2) (2) with k− = k− = 0 and K = (2) k+ c c (1) (2) q q k+ = , k+ = . d d2 d d2 + − bc − − bc 2 4 2 4 Clearly, kBk = b, kCk = c, and only the solution K (1) belongs to the ball Od/(2kBk) (H0 , H1 ). Its norm is given by the equality kCk

kK (1) k = d 2

+

q

d2 4

.

− kBk kCk

5. J -symmetric perturbations In this section we deal with perturbations of spectral subspaces of a self-adjoint operator under off-diagonal J-self-adjoint perturbations. For notational setup we adopt the following hypothesis. Hypothesis 5.1. Assume that A0 and A1 are self-adjoint operators on the Hilbert spaces H0 and H1 with domains Dom(A0 ) and Dom(A1 ), respectively. Let B be a bounded operator from H1 to H0 and C = −B ∗ . Also assume that A and V are operators on H = H0 ⊕ H1 given by (2.9) and (2.10), respectively, and L = A + V with Dom(L) = Dom(A).

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By J, J=

I 0

0 −I

,

(5.1)

cf. (1.3) we denote a natural involution on the Hilbert space H associated with its orthogonal decomposition H = H0 ⊕H1 . Subsequently introducing the indefinite inner product [x, y] = (Jx, y), x, y ∈ H, (5.2) turns H into a Krein space that we denote by K. A (closed) subspace L ⊂ K is called uniformly positive if there is a γ > 0 such that [x, x] ≥ γ kxk2 for any nonzero x ∈ K. (5.3) The subspace L is called maximal uniformly positive if it is not a proper subset of any other uniformly positive subspace of K. Uniformly negative and maximal uniformly negative subspaces of K are defined in a similar way. The only difference is in the replacement of (5.3) by the inequality [x, x] ≤ −γ kxk2 that should also hold for all x ∈ K, x 6= 0. For more definitions related to the Krein spaces we refer to [9] and [30]. Clearly, under Hypothesis 5.1 both V and L are J-self-adjoint operators on H, that is, the products JV and JL are self-adjoint with respect to the initial inner product (·, ·). This means that V and L are self-adjoint on the Krein space K. The statement below provides us with a sufficient condition for a J-selfadjoint block operator matrix L to have purely real spectrum and to be similar to a self-adjoint operator on H. Notice that for the particular case where the spectra of the entries A0 and A1 are subordinated, say sup spec(A0 ) < inf spec(A1 ), closely related results may be found in [1, Theorem 4.1] and [37, Theorem 3.2]. Theorem 5.2. Assume Hypothesis 5.1. Suppose that the Riccati equation KA0 − A1 K + KBK = −B ∗

(5.4)

has a weak (and hence strong) strictly contractive solution K : H0 → H1 , kKk < 1. Then: (i) The operator matrix L has a purely real spectrum and it is similar to a selfadjoint operator on H. In particular, the following equality holds: L = T ΛT −1 ,

(5.5)

where T is a bounded and boundedly invertible operator on H given by −1/2 I K∗ I − K ∗K 0 T = (5.6) K I 0 I − KK ∗ and Λ is a block diagonal self-adjoint operator on H, Λ = diag(Λ0 , Λ1 ),

Dom(Λ) = Dom(Λ0 ) ⊕ Dom(Λ1 ),

(5.7)

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whose entries −1/2 Λ0 = (I − K ∗ K)1/2 (A0 + BK)(I − K ∗ K) , ∗ 1/2 Dom(Λ0 ) = Ran(I − K K) , Dom(A )

(5.8)

0

and

∗ −1/2 Λ1 = (I − KK ∗ )1/2 (A1 − B ∗ K ∗ )(I − KK , ) ∗ 1/2 Dom(Λ1 ) = Ran(I − KK ) , Dom(A1 )

(5.9)

are self-adjoint operators on the corresponding component Hilbert spaces H0 and H1 . (ii) The graph subspaces H00 = G(K) and H01 = G(K ∗ ) are invariant under L and mutually orthogonal with respect to the indefinite inner product (5.2). Moreover, K = H00 [+]H01 where the sign “[+]” stands for the orthogonal sum in the sense of the Krein space K. The subspace H00 is maximal uniformly positive while H01 maximal uniformly negative. The restrictions of L onto the subspaces H00 and H01 are K-unitary equivalent to the self-adjoint operators Λ0 and Λ1 , respectively. Proof. In the case under consideration the second Riccati equation (2.11) associated with the operator matrix L reads K 0 A1 − A0 K 0 − K 0 B ∗ K 0 = B.

(5.10)

Thus, it simply coincides with the corresponding adjoint (2.6) of the Riccati equation (5.4). By Remark 2.2 this means that the adjoint of K, K 0 = K ∗ , is a weak (and hence strong) solution to (5.10). Since kK ∗ k = kKk < 1, the operators I − K ∗ K and I − KK ∗ are strictly positive, I − K ∗ K ≥ I − kKk2 > 0

and I − KK ∗ ≥ I − kKk2 > 0,

(5.11)

and, hence, boundedly invertible. This also means that the operator T in (5.6) is well defined and bounded. In addition, by Remark 2.7 this implies that the operator W in (2.20) is boundedly invertible and, consequently, the same holds for T . Now notice that by Lemma 2.5 we have Ran K ∗ K|Dom(A0 ) ⊂ Dom(A0 ), Ran KK ∗ |Dom(A1 ) ⊂ Dom(A1 ), and (I − K ∗ K)(A0 + BK)x = (A0 + K ∗ B ∗ )(I − K ∗ K)x for all x ∈ Dom(A0 ), (I − KK ∗ )(A1 − B ∗ K ∗ )y = (A1 − KB)(I − KK ∗ )y for all y ∈ Dom(A1 ),

(5.12) (5.13)

from which one easily infers that both Λ0 and Λ1 are self-adjoint operators. By using (5.8) and (5.9) one expresses the operators Z0 = A0 + BK and Z1 = A1 − B ∗ K ∗ with Dom(Zi ) = Dom(Ai ), i = 0, 1, in terms of the operators Λ0 and Λ1 , respectively. Then combining the expressions obtained with equality

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(2.25) from Corollary 2.9 we obtain formula (5.5). The similarity (5.5) means, in particular, that spec(L) is a Borel subset of R. This completes the proof of part (i). The J-orthogonality of the subspaces H00 and H01 is obvious since for any x, y ∈ H of the form x = x0 ⊕ Kx0 , x0 ∈ H0 , and y = K ∗ y1 ⊕ y1 , y1 ∈ H1 ,

(5.14)

we have [x, y] = (Jx, y) = (x0 , K ∗ y1 ) − (Kx0 , y1 ) = 0. Thus, the fist two assertions of part (ii) follow from Corollary 2.9 (ii). On the other hand, (5.14) yields kxk2 ≤ (1 + kKk2 )kx0 k2 and kyk2 ≤ (1 + kKk2 )ky1 k2 , and, hence, combined with (5.11), it implies [x, x] ≥ γkxk2 and [y, y] ≤ −γkyk2 where γ = (1 − kKk2 )(1 + kKk2 )−1 > 0. This means that H00 and H01 are maximal uniformly positive and maximal uniformly negative subspaces, respectively. Now introduce the operators T0 = W0 (I − K ∗ K)−1/2 and T1 = W1 (I − KK ∗ )−1/2 , where W0 and W1 are given in (2.29) assuming that K 0 = K ∗ . Taking into account (5.8) and (5.9), the identities (2.28) of Corollary 2.9 (ii) then imply T0−1 L|H00 T0 = Λ0 H00 ,

and T1−1 L|H01 T1 = Λ1 .

(5.15)

H01 ,

Clearly, Ran T0 = Ran T1 = [T0 x0 , T0 y0 ] = (x0 , y0 ) for any x0 , y0 ∈ H0 , and [T1 x1 , T1 y1 ] = −(x1 , y1 ) for any x1 , y1 ∈ H1 . This means that both T0 : H0 → H00 and T1 : H1 → H01 are K-unitary operators. Therefore, equalities (5.15) prove the remaining statement of part (ii). Remark 5.3. By equalities (5.8) and (5.9) the self-adjoint operators Λ0 and Λ1 are similar to the operators Z0 = A0 + BK, Dom(Z0 ) = Dom(A0 ),

(5.16)

Z1 = A1 − B ∗ K ∗ , Dom(Z1 ) = Dom(A1 ),

(5.17)

and respectively and, thus spec(Λ0 ) = spec(Z0 ) and spec(Λ1 ) = spec(Z1 ).

(5.18)

Notice that identities (5.12) and (5.13) imply that the operators Z0 and Z1 are selfadjoint on the corresponding Hilbert spaces H0 and H1 equipped with the new inner products hf0 , g0 iH0 = (I −K ∗ K)f0 , g0 H0 and hf1 , g1 iH1 = (I −KK ∗ )f1 , g1 H1 , respectively. Remark 5.4. The requirement kKk < 1 is sharp in the following sense: If there is no strictly contractive solution to the Riccati equation (5.4) then the operator matrix L may not be similar to a self-adjoint operator at all. This is clearly seen from the simple example below.

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Example 5.5. Let H0 = H1 = C. Set A0 = − d2 , A1 = d2 , and B = b where b and d are positive numbers such that equation (5.4) has q b ≥ d/2. If b > d/2, the Riccati q two solutions K (1) =

d 2b

+i

d2 4b2

− 1 and K (2) =

d 2b − (2)

i

d2 4b2

− 1. Both K (1) and

K (2) are not strictly contractive since kK (1) k = kK

k = 1. At the sameqtime the 2 spectrum of the matrix L consists of the two complex eigenvalues λ1 = i b2 − d4 q 2 and λ2 = −i b2 − d4 . If b = d2 , the equation (5.4) has the only solution K = 1. In this case the spectrum of the matrix L is real (it consists of the only point zero) but one easily verifies by inspection that the only eigenvalue of L has a nontrivial Jordan chain and, thus, L is not diagonalizable. Therefore, in both cases b > d/2 and b = d/2 the matrix L cannot be made similar to a self-adjoint operator. The next assertion represents a quite elementary corollary to Theorem 5.2. Lemma 5.6. Let the assumptions of Theorem 5.2 hold. Assume, in addition, that the spectra σ00 = spec(Z0 ) and σ10 = spec(Z1 ) of the operators Z0 and Z1 given by (5.16) and (5.17) are disjoint, that is, σ00 ∩ σ10 = ∅. Then σ00 and σ10 are complementary spectral subsets of the block operator matrix L, spec(L) = σ00 ∪ σ10 , and the graphs H00 = G(K) and H01 = G(K ∗ ) are the spectral subspaces associated with the subsets σ00 and σ10 , respectively. Proof. By the assumption the spectra spec(Λ0 ) = spec(Z0 ) = σ00 and spec(Λ1 ) = spec(Z1 ) = σ10 (see Remark 5.3) of the self-adjoint operators Λ0 and Λ1 given by (5.8), (5.9) are disjoint. Hence, the spectral projections EΛ (σ00 ) and EΛ (σ10 ) of the self-adjoint diagonal block operator matrix Λ = diag(Λ0 , Λ1 ) associated with its spectral subsets σ00 and σ10 read simply as I 0 0 0 0 0 EΛ (σ0 ) = and EΛ (σ1 ) = 0 0 0 I By Theorem 5.2 (i) the operator L is similar to the operator Λ. This means that the similarity transforms EL (σ00 ) = T EΛ (σ00 )T −1 and EL (σ10 ) = T EΛ (σ10 )T −1 of the spectral projections EΛ (σ00 ) and EΛ (σ10 ) with T given by (5.6) represent the corresponding spectral projections of L. One verifies by inspections that EL (σ00 ) = QG(K) and EL (σ10 ) = QG(K ∗ ) where QG(K) and QG(K ∗ ) are given by (2.23) and (2.24) assuming that K 0 = K ∗ . That is, EL (σ00 ) and EL (σ10 ) are the (oblique) projections onto the graph subspaces G(K) and G(K ∗ ), respectively, which completes the proof. Remark 5.7. The spectral projections EL (σ00 ) = QG(K) and EL (σ10 ) = QG(K ∗ ) are orthogonal projections with respect to the Krein space inner product (5.2). From now on we will assume that the spectra of the entries A0 and A1 are disjoint and, thus, the sets σ0 = spec(A0 ) and σ1 = spec(A1 ) appear to be complementary disjoint spectral subsets of the total self-adjoint operator A. In such a case for any bounded perturbation V satisfying the bound kV k < d/2, d = dist(σ0 , σ1 ), the spectrum of the perturbed operator L = A + V consists of

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two disjoint subsets σ00 and σ10 , lying in the closed kV k-neighborhoods OkV k (σ0 ) and OkV k (σ1 ) of the spectral sets σ0 = spec(A0 ) and σ1 = spec(A1 ), respectively. One can think of the sets σ00 and σ10 as the result of the perturbation of the corresponding spectral sets σ0 and σ1 . Provided that the perturbation V is J-symmetric and kV k < d/2, Theorem 5.8 below gives sufficient a priori conditions for the perturbed operator L = A + V to remain similar to a self-adjoint operator. Hence, this theorem also gives sufficient conditions for the perturbed spectral sets σ00 and σ10 to remain on the real axis. Furthermore, the theorem presents the main result of the section giving for such V an a priori norm bound on variation of the spectral subspaces of A associated with the disjoint spectral subsets σ0 and σ1 . Theorem 5.8. Assume Hypothesis 5.1 and choose one of the following: (i) Assume (4.12) and set δ = π2 d; (ii) Assume (4.13) or (4.14) and set δ = d. Also suppose that δ (5.19) kV k < . 2 Then the spectrum of the operator L is purely real and consists of two disjoint components σ00 and σ10 such that σ00 ⊂ Or spec(A0 ) and σ10 ⊂ Or spec(A1 ) , (5.20) where

1 2kV k arctanh < kV k. 2 δ Moreover, the operator L is similar to a self-adjoint operator and the same is true for the parts of L associated with the spectral subsets σ00 and σ10 . Furthermore, the following bound holds: 2kV k 1 arctanh , (5.21) tan Θ0 ≤ tanh 2 δ

r = kV k tanh

where Θ0 = Θ(H0 , H00 ) denotes the operator angle between the subspace H0 and the spectral subspace H00 of L associated with the spectral subset σ00 . Exactly the same bound holds for the operator angle Θ1 = Θ(H1 , H01 ) between the subspace H1 and the spectral subspace H01 of L associated with the spectral subset σ10 . Proof. Under either assumption (i) or (ii) from Theorem 4.11 it follows that the Riccati equation (5.4) associated with the block operator matrix L has a solution K ∈ B(H0 , H1 ) that is unique in the ball Oδ/2kBk (H0 , H1 ) and satisfies the bound see formulas (4.18) and (4.20) kV k 1 2kV k q kKk ≤ = tanh arctanh . (5.22) 2 δ δ δ2 2 + − kV k 2 4 Here we have taken into account that kBk = kV k. We refer to Remark 4.3 regarding the use of the hyperbolic tangent in (5.22).

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Clearly, the bound (5.22) yields that the solution K is a strict contraction, kKk < 1. Then by Theorem 5.2 the block operator matrix L is similar to the self-adjoint operator Λ given by (5.7)–(5.9). Hence spec(L) ⊂ R and spec(L) = σ00 ∪ σ10 where σ00 = spec(Λ0 ) and σ10 = spec(Λ1 ). By Remark 5.3 we also have σ00 = spec(Z0 ) and σ10 = spec(Z1 ) where Z0 and Z1 are given by (5.16) and (5.17). Since kBKk ≤ kV kkKk ≤ r and kB ∗ K ∗ k ≤ kV kkKk ≤ r, for the spectral sets σ00 = spec(Z0 ) and σ10 = spec(Z1 ) the inclusions (5.20) hold and these sets are disjoint, dist(σ00 , σ10 ) ≥ δ − 2r > δ − 2kV k > 0. To prove the remaining statements of the theorem one only needs to apply Lemma 5.6 and then to notice that due to (2.14) we have k tan Θ0 k = kKk and hence tan Θ0 ≤ kKk. Similarly, tan Θ1 ≤ kK ∗ k = kKk. Remark 5.9. By the upper continuity of the spectrum, the inclusion spec(L) ⊂ R also holds for kV k = d/π in case (i) and for kV k = d/2 in case (ii). Remark 5.10. In case (ii) the bounds (5.20) on the location of spec(L) and the bound (5.21) on the angle Θ0 are optimal. The optimality of both (5.20) and (5.21) is seen from Examples 4.15 and 4.16 where one sets c = b. Remark 5.11. Under condition (5.19) in both cases (i) and (ii) the perturbed spectral subspaces H00 and H01 are mutually orthogonal with respect to the Krein space inner product (5.2) and, thus, K = H00 [+]H01 . These subspaces are maximal uniformly positive and maximal uniformly negative, respectively. The restrictions of L onto H00 and H01 are K-unitary equivalent to the self-adjoint operators Λ0 and Λ1 given by (5.8) and (5.9), respectively. By Theorem 5.2 (ii) all this follows from the fact that kKk < 1 which we established in the proof of Theorem 5.8. Theorem 5.8 claims that the spectrum of the block operator matrix L is purely real whenever the off-diagonal J-self-adjoint perturbation V satisfies the bounds kV k < d/2 in case (i) or kV k < d/π in case (ii). Recall that case (ii) corresponds to the generic spectral situation where no constraints are imposed on the mutual positions of the spectra spec(A0 ) and spec(A1 ) except for the condition (4.12). Now we want to prove that, in fact, the spectrum of the operator L remains purely real under condition (4.12), even if d/π ≤ kV k < d/2, at least in the case where the entries A0 and A1 are bounded. Our proof will be based on results from [32] and [50]. Theorem 5.12. Assume Hypothesis 5.1. Assume, in addition, that both the entries A0 and A1 are bounded and such that dist spec(A0 ), spec(A1 ) = d > 0. Also suppose that kV k < d/2. Then the spectrum of the block operator matrix L is real, that is, spec(L) ⊂ R. Proof. Under Hypothesis 4.8 and condition kV k < d/2 the inclusion spec(L) ⊂ R has been already proven in Theorem 5.8 (ii). Thus, let us only consider the case that is not covered by Hypothesis 4.8. In this case, because of the separation condition dist spec(A0 ), spec(A1 ) = d, the spectrum of A0 consists of several (at least two) nonempty subsets isolated from each other at least by the distance 2d. Denote

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(i)

these isolated spectral subsets of A0 by σ0 , i = 1, 2, . . . , n0 , n0 ≥ 2, assuming (i) (i+1) that they are numbered from left to right (i.e. sup σ0 < inf σ0 ) and that the (i) (i+1) gap between sup σ0 and inf σ0 contains a nonempty subset of the spectrum Sn0 (i) σ0 = spec(A0 ). In exactly the same way, divide the spectrum of of A1 and i=1 Sn1 (j) (j) σ1 = spec(A1 ), A1 into the subsets σ1 , j = 1, 2, . . . , n1 , n1 ≥ 2, so that j=1 (j)

(j+1)

(j)

(j+1)

(i)

sup σ1 < inf σ1 , and (sup σ1 , inf σ1 ) ∩ spec(A0 ) 6= ∅. Denote by H0 , (j) i = 1, 2, . . . , n0 , and H1 , j = 1, 2, . . . , n1 , the spectral subspaces of the operators (i) (j) A0 and A1 associated with the corresponding spectral subsets σ0 and σ1 . Surely, (i) (i) 1 0 H1 = H1 . H0 = H0 and ⊕ni=1 ⊕ni=1 Now take arbitrary unit vectors (i)

(i)

(i)

e0 ∈ H0 , ke0 k = 1, i = 1, 2, . . . , n0 , (j) (j) (j) e1 ∈ H1 , ke1 k = 1, j = 1, 2, . . . , n1 ,

(5.23)

and construct numerical matrices A0 , A1 , and B with the entries (k)

(i)

A0,ik = (A0 e0 , e0 ),

(l)

(j)

A1,jl = (A1 e1 , e1 ),

(j)

(i)

and Bij = (Be1 , e0 ),

b 0 = Cn0 and respectively. Consider the matrices A0 and A1 as operators resp. on H n1 b b b H1 = C , and B as an operator from H1 to H0 . Out of the matrices A0 and A1 ∗ construct the block diagonal matrix A = diag(A0 , A1 ) and out of B and B the 0 B off-diagonal matrix V = . Both matrices A and V have dimension −B∗ 0 n × n where n = n0 + n1 , and we consider them as operators on the n-dimensional b=H b0 ⊕ H b 1. space H Our nearest goal is to prove that the spectrum of the operator L = A + V is real. To this end, first, introduce the indefinite inner product [x, y] = (x0 , y0 )H b 0 − (x1 , y1 )H b1 , b b 1, x = x0 ⊕ x1 , y = y0 ⊕ y1 , x0 , y0 ∈ H0 , x1 , y1 ∈ H

(5.24)

b into a Krein (Pontrjagin) space. We denote the which turns the Hilbert space H b The operator A is self-adjoint both on H b and K b while B only on K. b latter by K. (0) (0) Then notice that for different i and k the vectors ei and ek belong to the different (and mutually orthogonal) spectral subspaces of A0 and, hence, A0,ik = (0) (0) (i) (i) λi δik where λi = (A0 e0 , e0 ) and δik is the Kronecker’s delta. Similarly, A1,jl = (1) (1) (1) (1) (0) (1) λj δjl where λj = (A1 ej , ej ). Clearly, both λi , i = 1, 2, . . . , n0 , and λj , j = 1, 2, . . . , n1 , are simple eigenvalues of A and, by construction of A, one has (0) (i) (1) (j) λi ∈ conv(σ0 ) and λj ∈ conv(σ1 ). This yields (0)

min |λi

i,k, i6=k

(0)

(1)

(1)

(0)

− λk | ≥ 2d, min |λj − λl | ≥ 2d, min |λi j,l, j6=l

i,j

(1)

− λj | ≥ d.

(5.25)

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It is also obvious that, with respect to the inner product (5.24), the eigenvalues λi , (1) i = 1, 2, . . . , n0 , are of positive type, while the eigenvalues λj , j = 1, 2, . . . , n1 , are of negative type. Now to prove the inclusion spec(L) ⊂ R it only remains to observe that kVk ≤ kV k < d/2 and then to apply [32, Corollary 3.4] (cf. [16, Theorem 1.2]). Since the inclusion spec(L) ⊂ R holds for any choice of the vectors (5.23), one then concludes that also W n (L) ⊂ R where W n (L) denotes the block numerical range (see [50, Definition 2.1]) of the operator L with respect to the decomposition (1)

(n0 )

H = H0 ⊕ . . . ⊕ H 0

(1)

(n1 )

⊕ H1 ⊕ . . . ⊕ H1

.

(5.26)

By [50, Theorem 2.5] we have spec(L) ⊂ W n (L). Hence, spec(L) ⊂ R, which completes the proof. Remark 5.13. By the upper continuity of the spectrum, under the hypothesis of Theorem 5.12 the spectrum of L = A + V is real also for kV k = d/2 (cf. Remark 5.9). Remark 5.14. Under the assumptions of Theorems 5.8 (ii) or 5.12 the requirement kV k ≤ d/2 guaranteeing the inclusion spec(L) ⊂ R is sharp. This is seen from Example 5.5 with b > d/2.

6. Quantum harmonic oscillator under a PT -symmetric perturbation Let A be the Schr¨ odinger operator for a one-dimensional quantum harmonic oscillator (see, e.g., [38, Chapter 12]). The corresponding Hilbert space is H = L2 (R). Assuming that the units are chosen in such a way that ~ = m = ω = 1, the operator A reads 1 d2 f (x) + 12 x2 f (x), (Af )(x) = − 2 2 dx Z Dom(A) = f ∈ W22 (R) dx x4 |f (x)|2 < ∞ ,

(6.1)

R

W22 (R)

where denotes the Sobolev space of those L2 (R)-functions that have their second derivatives in L2 (R). The subspaces H0 = L2,even (R) and H1 = L2,odd (R)

(6.2)

of even and odd functions appear to be the spectral subspaces of the (self-adjoint) operator A associated with the spectral subsets σ0 = spec(A H0 ) = {n + 1/2 n = 0, 2, 4, . . . } and σ1 = spec(A H ) = {n + 1/2 n = 1, 3, 5 . . .}, 1

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respectively (see, e.g., [44, p. 142]). Clearly, H = H0 ⊕ H1 , the spectral sets σ0 and σ1 are disjoint, d = dist(σ0 , σ1 ) = 1, and σ0 ∪ σ1 = spec(A).

(6.3)

Let P be the parity operator on L2 (R), (Pf )(−x) = f (−x), and T the (antilinear) operator of complex conjugation, (T f )(x) = f (x), f ∈ L2 (R). An operator V on L2 (R) is called PT -symmetric if it commutes with the product PT , that is, PT V = V PT (see, e.g. [15, 16] and references therein). In a particular case where the PT -symmetric potential V is an operator of multiplication by a function V (·) of L∞ (R), the following equality holds (see, e.g., [3]; cf. [32]): V (x) = V (−x) for a.e. x ∈ R (6.4) and hence V ∗ = PV P.

(6.5)

Observe that the parity operator P represents nothing but the involution (1.3) associated with the complementary spectral subspaces (6.2) of the oscillator Hamiltonian (6.1). Therefore, the equality (6.5) implies that the PT -symmetric multiplication operator V is J-self-adjoint with respect the involution J = P. Any bounded complex-valued function V on R possessing the property (6.4) admits the representation V (x) = a(x) + ib(x) (6.6) where both a and b are real-valued functions such that a(−x) = a(x) and b(−x) = −b(x) for any x ∈ R. The terms Vdiag (x) = a(x) and Voff (x) = ib(x) represent the corresponding parts of the multiplication operator V that are diagonal and off-diagonal with respect to the orthogonal decomposition H = H0 ⊕ H1 , that is, with respect to the decomposition L2 (R) = L2,even (R) ⊕ L2,odd (R). Now assume that V is an arbitrary bounded off-diagonal operator on H = L2 (R) being J-self-adjoint with respect to the involution J = P. One can choose in particular a PT -symmetric potential (6.6) with a = 0. By taking into account (6.3), from [16, Theorem 1.2] it follows that the spectrum of the perturbed oscillator Hamiltonian L = A + V , Dom(L) = Dom(A), remains real (and discrete) whenever kV k ≤ 1/2. If, in addition, the bound kV k < 1/π is satisfied then one can tell much more: Under such a bound Theorem 5.8 (i) implies that L is similar to a self-adjoint operator. This theorem also gives bounds on the variation of the spectral subspaces (6.2): 1 arctanh(πkV k) < 1, j = 0, 1, tan Θj ≤ tanh 2 where Θj = Θ(Hj , H0j ) stands for the operator angle between the subspace Hj and the spectral subspace H0j of the perturbed oscillator Hamiltonian L = A + V associated with the spectral subset σj0 = spec(L) ∩ OkV k (σj ), j = 0, 1.

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Note added in proof. After having submitted this paper the authors became aware of the papers [51, 52] by K. Veseli´c which contain a number of important results on the spectral properties of J-self-adjoint operators. In particular, from [52, Corollary 4] it follows that under Hypothesis 5.1 the perturbed operator L = A + V is similar to a self-adjoint operator provided that kV k < d/2 and either the number of gaps in R separating the sets spec(A0 ) and spec(A1 ) is finite or, otherwise, the (infinite) sum of the inverse lengths of these gaps is finite. Thus, the result of Theorem 5.12 is implied by [52, Corollary 4]. Nevertheless, we decided to leave our proof of Theorem 5.12 in the paper since it is very different from the one found in [52]. One also needs to notice an essential difference between the results of [52, Corollary 4] and Theorem 5.8 (i). Except for condition (5.19), the hypothesis of Theorem 5.8 (i) imposes no requirements on the lengths of the gaps between spec(A0 ) and spec(A1 ). Unlike [52, Theorem 3 and Corollary 4], it allows all these lengths to be uniformly bounded from above even if the number of the gaps is infinite. Acknowledgments. The authors thank Shao-Ming Fei for his useful remarks on PT -symmetric operators and Heinz Langer for pointing out the papers [51, 52]. The authors are also thankful to the anonymous referee for suggestions concerning possible further developments. A. K. Motovilov and A. A. Shkalikov gratefully acknowledge the kind hospitality of the Institut f¨ ur Angewandte Mathematik, Universit¨ at Bonn, where the main part of this research has been performed.

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[9] T. Y. Azizov and I. S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, Chichester, 1989. [10] C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70 (2007), 947–1018; arXiv: hep-th/0703096. [11] C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT Symmetry, Phys. Rev. Lett. 80 (1998), 5243–5246; arXiv: physics/9712001. [12] C. M. Bender, S. Boettcher, and P. N. Meisinger, PT -symmetric quantum mechanics, J. Math. Phys. 40 (1999), 2201–2229; arXiv: quant-ph/9809072. [13] R. Bhatia, C. Davis, and A. McIntosh, Perturbation of spectral subspaces and solution of linear operator equations, Linear Algebra Appl. 52/53 (1983), 45 - 67. [14] R. Bhatia and P. Rosenthal, How and why to solve the operator equation AX −XB = Y , Bull. London Math. Soc. 29 (1997), 1–21. [15] E. Caliceti, F. Cannata, and S. Graffi, Perturbation theory of P T symmetric Hamiltonians, J. Phys. A 39 (2006), 10019 – 10027; arXiv: math-ph/0607039. [16] E. Caliceti, S. Graffi, and J. Sj¨ ostrand, Spectra of P T -symmetric operators and perturbation theory, J. Phys. A 38 (2005), 185–193; arXiv: math-ph/0407052. [17] C. Davis and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal. 7 (1970), 1–46. [18] J. W. Demmel, Three methods for refining estimates of invariant subspaces, Computing 38 (1987), 43–57. [19] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution equations, Graduate Texts in Mathematics 194, Springer–Verlag, New York, 2000. [20] M. Haase, The Functional Calculus for Sectorial Operators and Similarity Methods, Dr. rer. nat. thesis, Universit¨ at Ulm, 2003. [21] P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389. [22] V. Hardt, A. Konstantinov, and R. Mennicken, On the spectrum of product of closed operators, Math. Nachr. 215 (2000), 91–102. [23] E. Heinz, Beitr¨ age zur St¨ orungstheorie der Spektralzerlegung, Math. Annalen 123 (1951), 415–438. [24] T. Kato, Perturbation Theory for Linear Operators, Springer–Verlag, Berlin, 1966. [25] V. Kostrykin, K. A. Makarov, and A. K. Motovilov, Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach, Contemporary Mathematics (AMS) 327 (2003), 181–198; arXiv: math.SP/0207125. [26] V. Kostrykin, K. A. Makarov, and A. K. Motovilov, On the existence of solutions to the operator Riccati equation and the tan Θ theorem, Integr. equ. oper. theory 51 (2005), 121–140; arXiv: math.SP/0210032 v2. [27] V. Kostrykin, K. A. Makarov, and A. K. Motovilov, Perturbation of spectra and spectral subspaces, Trans. Amer. Math. Soc. 359 (2007), 77–89; arXiv: math.SP/0306025. [28] V. Kostrykin, K. A. Makarov, and A. K. Motovilov, A generalization of the tan 2Θ Theorem, Operator Theory: Adv. Appl. 149 (2004), 349–372; arXiv: math.SP/ 0302020. [29] D. Krejˇciˇr´ık, Calculation of the metric in the Hilbert space of a PT -symmetric model via the spectral theorem, J. Phys. A 41 (2008), 244012 (6 pp.); arXiv:0707.1781.

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[30] H. Langer, Krein space, in: Encyclopaedia of Mathematics (Ed. M. Hazewinkel); http://eom.springer.de/k/k055840.htm. [31] H. Langer, A. Markus, V. Matsaev, and C. Tretter, A new concept for block operator matrices: The quadratic numerical range, Linear Algebra Appl. 330 (2001), 89–112. [32] H. Langer and C. Tretter, A Krein space approach to PT-symmetry, Czech. J. Phys. 54 (2004), 1113–1120; Corrigendum, Ibid. 56 (2006), 1063–1064. [33] H. Langer and C. Tretter, Diagonalization of certain block operator matrices and applications to Dirac operators, Operator Theory: Adv. Appl. 122 (2001), pp. 331– 358. [34] G. Lumer and M. Rosenblum, Linear operator equations, Proc. Amer. Math. Soc. 10 (1959), 32–41. [35] R. McEachin, Closing the gap in a subspace perturbation bound, Linear Algebra Appl. 180 (1993), 7–15. [36] R. Mennicken and A. K. Motovilov, Operator interpretation of resonances arising in spectral problems for 2 × 2 operator matrices, Math. Nachr. 201 (1999), 117–181; arXiv: funct-an/9708001. [37] R. Mennicken and A. A. Shkalikov, Spectral decomposition of symmetric operator matrices, Math. Nachr. 179 (1996), 259–273. [38] A. Messiah, Quantum Mechanics, Vol. I, Wiley & Sons, 1963. [39] A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. Math. Phys. 43 (2002), 205 – 214; arXiv: math-ph/0107001. [40] A. K. Motovilov, Removal of the resolvent-like energy dependence from interactions and invariant subspaces of a total Hamiltonian, J. Math. Phys. 36 (1995), 6647–6664; arXiv: funct-an/9606002. [41] A. K. Motovilov and A. V. Selin, Some sharp norm estimates in the subspace perturbation problem, Integr. equ. oper. theory 56 (2006), 511–542; arXiv: math.SP/0409558. [42] M. T. Nair, An iterative procedure for solving the Riccati equation A2 R − RA1 = A3 + RA4 R, Studia Math. 147 (2001), 15–26. [43] V. Q. Ph´ ong, The operator equation AX − XB = C with unbounded operators A and B and related abstract Cauchy problems, Math. Z. 208 (1991), 567–588. [44] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, 1980. [45] M. Rosenblum, On the operator equation BX − XA = Q, Duke Math. J. 23 (1956), 263–269. [46] G. W. Stewart, Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Review 15 (1973), 727–764. [47] G. W. Stewart, Error bounds for approximate invariant subspaces of closed linear operators, SIAM J. Numer. Anal. 8 (1971), 796–808. [48] T. Tanaka, General aspects of P T -symmetric and P-self-adjoint quantum theory in a Krein space, J. Phys. A 39 (2006), 14175–14203; arXiv: hep-th/0605035. [49] C. Tretter, Spectral inclusion for unbounded block operator matrices, J. Funct. Anal. 256 (2009), 3806 – 3829.

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[50] C. Tretter and M. Wagenhofer, The block numerical range of an n × n block operator matrix, SIAM J. Matrix Anal. Appl. 24 (2003), 1003–1017. [51] K. Veseli´c, On spectral properties of a class of J-selfadjoint operators. I, Glasnik Mat. 7:2 (1972), 229–248. [52] K. Veseli´c, On spectral properties of a class of J-selfadjoint operators. II, Glasnik Mat. 7:2 (1972), 249–254. [53] M. Znojil, Solvable PT-symmetric Hamiltonians, Phys. Atom. Nucl. 65 (2002), 1149– 1151; arXiv: quant-ph/0008125. Sergio Albeverio Institut f¨ ur Angewandte Mathematik Universit¨ at Bonn Endenicher Allee 60 D-53115 Bonn Germany; SFB 611 and HCM, Bonn; BiBoS, Bielefeld-Bonn; CERFIM, Locarno; Accademia di Architettura, USI, Mendrisio e-mail: [email protected] Alexander K. Motovilov Bogoliubov Laboratory of Theoretical Physics, JINR Joliot-Curie 6 141980 Dubna Moscow Region, Russia e-mail: [email protected] Andrei A. Shkalikov Faculty of Mathematics and Mechanics Moscow Lomonosov State University Leninskie Gory 119992 Moscow Russia e-mail: [email protected] Submitted: October 28, 2008. Revised: June 2, 2009.

Integr. equ. oper. theory 64 (2009), 487–494 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040487-8, published online August 3, 2009 DOI 10.1007/s00020-009-1700-3

Integral Equations and Operator Theory

On the Orbit of an m-Isometry Teresa Berm´ udez, Isabel Marrero and Antonio Martin´on To Professor Jos´ e Rodr´ıguez Exp´ osito on his 60th birthday

Abstract. A bounded linear operatorPT on a Hilbert space ∗k Hk is called an mm−k m isometry for a positive integer m if m T T = 0. We prove k=0 (−1) k some properties concerning the behaviour of the orbit of an m-isometry. For example, every orbit of an m-isometry is eventually norm increasing and some m-isometries can not be N -supercyclic, that is, there does not exist an N dimensional subspace EN such that the orbit of T at EN is dense in H. Mathematics Subject Classification (2000). 47A16. Keywords. Isometry, m-isometry, supercyclic, N -supercyclic.

1. Introduction Throughout this paper H will denote a separable infinite-dimensional Hilbert space and L(H) the space of all bounded linear operators on H. Definition 1.1. An operator T ∈ L(H) is called an m-isometry for a positive integer m if it satisfies the following identity: m X m ∗k k (−1)m−k T T = 0, (1.1) k k=0

∗

where T denotes the adjoint operator of T . The theory of these operators was studied by Agler and Stankus ([1, 2, 3, 4]). A simple manipulation proves that (1.1) is equivalent to m X m−k m (−1) kT k xk2 = 0, (1.2) k k=0

Supported by Ministerio de Educaci´ on y Ciencia (Spain), MTM2007-65604 and by Universidad de La Laguna, MGC/08/17.

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for all x ∈ H. It is clear that an 1-isometry is an isometry and that any m-isometry is also an (m + 1)-isometry. The converse of the last property is not true (see [7, Proposition 8]). Given T ∈ L(H), the orbit of a subset E ⊂ H under T is defined by Orb(T, E) := {T n x : x ∈ E, n ∈ N}. An operator T is said to be supercyclic if there exists an one-dimensional subspace E of H such that Orb(T, E) is dense in H. This class of operators was introduced by Hilden and Wallen in [12]. There are different generalizations of supercyclicity, for example: • T is finitely supercyclic if there exists a finite subset E = {x1 , x2 , . . . , xn } such Sn that i=1 C Orb(T, xi ) is dense. Recently, it has been proven that finitely supercyclic operators are supercyclic [14]. • T is weakly supercyclic if there exists an one-dimensional subspace E such that the closure with respect to the weak topology of Orb(T, E) is dense. See for example [16], [15]. • T is N -supercyclic if there exists an N -dimensional subspace E such that Orb(T, E) is dense. There are N -supercyclic operators that are not supercyclic [11, Example 3.3] (see also [9]). In 1974, Hilden and Wallen [12] proved that no normal operator on a complex Hilbert space can be supercyclic. Later, Ansari and Bourdon [5] extended this to the class of all isometries on a Banach space; notice that supercyclic operators have dense range, hence a supercyclic isometry on a Hilbert space must be unitary, and therefore normal. Recently, Faghih and Hedayatian [10] proved that no m-isometry can be supercyclic; they base their proof in the fact that the orbit of each vector is norm increasing or norm decreasing, except possibly for a finite number of terms. However, these results are not true for weak supercyclic operators, since Sanders gave a weakly supercyclic isometry in a Banach space [15, Theorem 2] and Bayart and Matheron gave a weakly supercyclic unitary operator [8, Example 3.6]. The purpose of this paper is to study the behaviour of the orbit of an misometry. In fact, we prove that the orbit of each vector under an m-isometry is eventually norm increasing, that is, it is norm increasing except possibly for a finite number of terms. Also, if T is an m-isometry, then kT n k2 and nm−1 have the same behaviour as n → ∞. Finally, those results allow us to prove that no m-isometry with covariance operator injective is N -supercyclic.

2. Properties of m-isometries For n = 0, 1, 2, . . . and k = 0, 1, 2, . . . , we denote 1 if n = 0 or k = 0, (k) n := n(n − 1) . . . (n − k + 1) otherwise.

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Given T ∈ L(H) and k = 0, 1, 2, . . . , we consider the operator k 1 X k ∗j j βk (T ) := (−1)k−j T T . k! j=0 j Observe that if T is an m-isometry, then βk (T ) = 0, for every k ≥ m. From [2, page 388] we have that ∞ X T ∗n T n = n(k) βk (T ). k=0

Hence, if T is an m-isometry then T ∗n T n =

m−1 X

n(k) βk (T ),

k=0

and consequently kT n xk2 =

m−1 X

n(k) hβk (T )x, xi

(2.1)

k=0

for all x ∈ H, where h·, ·i denotes the inner product on H. If T is an m-isometry, then the covariance operator ∆T is defined by ∆T := βm−1 (T ). Remark 2.1 ([2, Propositions 1.5 & 1.6]). If T is an m-isometry then the following properties are satisfied: 1. ∆T is a positive operator; 2. the null space Ker(∆T ) of ∆T is an invariant subspace for T and the restriction operator T |Ker(∆T ) is an (m − 1)-isometry; 3. for every x ∈ H, h∆T x, xi =

m−1 X

(−1)m−k−1

k=0

1 kT k xk2 . k!(m − k − 1)!

(2.2)

Faghih and Hedayatian proved that the orbit Orb(T, x) of an m-isometry T , for every vector x, is norm increasing or norm decreasing, except possibly for a finite number of terms [10, Theorem 2]. In the following result we obtain that the orbit is always eventually norm increasing. Proposition 2.2. Let T be an m-isometry and x ∈ H. Then Orb(T, x) is eventually norm increasing. Proof. We prove the statement by induction on m. For m = 1, it is clear. Suppose it is true for m − 1 and let us prove it for m. If T is an m-isometry, from (2.1) we obtain m−1 X kT n+1 xk2 − kT n xk2 = (n + 1)(k) − n(k) hβk (T )x, xi. (2.3) k=0

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Note that (n + 1)(k) − n(k) is a polynomial in n of degree k − 1. We consider two cases, namely h∆T x, xi = 0 and h∆T x, xi = 6 0. Assume that h∆T x, xi = 0. Since the operator ∆T is positive we have that x ∈ Ker(∆T ), hence Orb(T, x) = Orb(T |Ker(∆T ) , x). As T |Ker(∆T ) is an (m − 1)isometry, we obtain kT n+1 xk2 − kT n xk2 ≥ 0, except possibly for a finite number of terms. If h∆T x, xi 6= 0, then kT n+1 xk2 − kT n xk2 is a polynomial in n of degree m − 1 with leading coefficient h∆T x, xi > 0, hence kT n+1 xk2 − kT n xk2 > 0 for large enough n. The following property will allows us to study the N -supercyclicity of misometries in the next section. Denote by BH the closed unit ball of H, that is, BH := {x ∈ H : kxk ≤ 1}. Proposition 2.3. If T is an m-isometry, then the following properties are satisfied: kT n xk2 converges uniformly on BH to h∆T x, xi. 1. nm−1 n 2

√ 2 √ kT k 2. m−1 converges to ∆T , where ∆T denotes the square root of the n operator ∆T . Proof. By (2.1) we have that (m−1) m−2 X n(k) kT n xk2 n −h∆ x, xi = − 1 h∆ x, xi+ hβk x, xi → 0 (n → ∞). T T nm−1 nm−1 nm−1 k=0

Moreover, the convergence is uniform on BH . Indeed, given ε > 0 and x ∈ BH , n 2 (m−1) m−2 X n(k) kT xk n ≤ − h∆ x, xi − 1 |h∆ x, xi| + |hβk x, xi| T T nm−1 m−1 n nm−1 k=0 (m−1) m−2 X n(k) n ≤ − 1 C + C<ε nm−1 nm−1 k=0

for large enough n, where C = max0≤k≤m−1 kβk (T )k. So part 1 is proved. n

2

xk By part 1 we get that kT nm−1 converges uniformly on BH , hence we can exchange the limit with the supremum to obtain

kT n k2 kT n xk2 kT n xk2 = lim sup = sup lim m−1 n→∞ nm−1 n→∞ x∈BH nm−1 x∈BH n→∞ n

p 2

= sup h∆T x, xi = ∆T . lim

x∈BH

Corollary 2.4 ([10, Theorem 1]). If T is a power bounded operator (that is, there exists M > 0 such that kT n k ≤ M for all n ∈ N) and T is also an m-isometry, then T is an isometry.

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3. N -supercyclicity of an m-isometry The proof of the following result is similar to the hypercyclic case [13, Lemma 2.1], but we include it for the sake of completeness. Lemma 3.1. Let Ti : Xi → Xi be a (linear and continuous) operator on the Banach space Xi (i = 1, 2) and let S : X1 → X2 be an operator with dense range, such that T2 S = ST1 , that is, such that the following diagram commutes: X1

T1 -

X1

S

S ? X2

T2 -

? X2

If T1 is N -supercyclic, then T2 is N -supercyclic. Proof. Assume that T1 is N -supercyclic; then there exists an N -dimensional subspace E of X1 generated by the vectors x1 , x2 , . . . , xN , and such that Orb(T1 , E) is dense in X1 . Then Orb(T2 , S(E)) is dense in X2 and S(E) is generated by the vectors S(x1 ), S(x2 ), . . . , S(xN ), since the diagram commutes and S has dense range. Hence T2 is at least an N -supercyclic operator. Lemma 3.2. Suppose that T is an operator on a Hilbert space H with the following properties: 1. There is a sequence (an )n∈N of positive real numbers such that an+1 /an converges to some positive real number and there exist M, ε > 0 such that, for every n, kT n k ε< <M . an 2. For each non zero x ∈ H, there exists δx > 0 such that, for every n, δx <

kT n xk . kT n k

e an isometry Te on H e and an operator S : H → H e Then there is a Hilbert space H, with dense range such that ST = TeS, that is, such that the following diagram commutes: T H H S

S ? e H

Te

-

? e H

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IEOT

Proof. In [5, Theorem 2.1 and Remarks] this lemma is proved when H is a Banach e space. Moreover, in [6, Remark 5] it is noted that if H is a Hilbert space, then H is also a Hilbert space. Lemma 3.3. No isometry operator on a Hilbert space H can be N -supercyclic. Proof. Assume that T is an N -supercyclic isometry. Then T is an invertible isometry, so that T is an unitary operator. In particular, T is normal. This is a contradiction, since Bayart and Matheron proved that normal operators are not N -supercyclic (in fact, hyponormal operators are not N -supercyclic) [8, Theorem 3.2]. Below we prove that certain class of m-isometries are not N -supercyclic. Theorem 3.4. If T is an m-isometry such that its covariance operator ∆T is injective, then T is not N -supercyclic. Proof. First, we prove that T satisfies the hypotheses of Lemma 3.2, with an := √ nm−1 . By Proposition 2.3, since T is an m-isometry with Ker(∆T ) = {0} (in particular, k∆T k = 6 0 ), there exist ε > 0 and M > 0 such that, for all n, kT n k ε< √ < M. nm−1 Then, for each non zero vector x ∈ H, x 6= 0 and for every n,

(3.1)

kT n xk2 kT n xk2 < . M 2 nm−1 kT n k2 Moreover, it was proved in Proposition 2.3 that kT n xk2 → h∆T x, xi (n → ∞). nm−1 kT n xk Since h∆T x, xi > 0, taking δx := inf √ > 0 we obtain n M nm−1 kT n xk δx < kT n k for each n. Therefore, the hypotheses of Lemma 3.2 are satisfied. Hence there exist e an isometry Te on H e and an operator S : H → H e with dense a Hilbert space H, range such that ST = TeS. Taking into account Lemma 3.3 we have that Te is not N -supercyclic, and considering Lemma 3.1 we obtain that T is not N -supercyclic. Proposition 3.5. If T is an m-isometry with m even and Ker(∆T ) 6= H, then T is not N -supercyclic. Proof. By hypothesis T is not an (m − 1)-isometry, since Ker(∆T ) 6= H. Assume that T is an m-isometry with m even and that T is also N -supercyclic. Then T is an invertible m-isometry with m even, hence T is an (m − 1)-isometry [2, Proposition 1.23], which yields a contradiction.

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On the Orbit of an m-Isometry

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The fact that no 2-isometries are N -supercyclic can be obtained without supplementary hypotheses. Corollary 3.6. If T is a 2-isometry, then T is not N -supercyclic. Proof. If T is an isometry, then Lemma 3.3 gives the result. If T is a 2-isometry and it is not an isometry, then it suffices to apply the above proposition. The following example gives a 3-isometry with Ker(∆T ) 6= {0} which is not N -supercyclic. Example. Let T2 en := αn en+1 and T3 enq:= βn en+1 be unilateral weighted shift q n+2 operators defined on `2 (N), where αn := n+1 and β := . It is not difficult n n n to prove that T2 is a 2-isometry with Ker(4T2 ) = {0} and T3 is a 3-isometry with Ker(4T3 ) = {0} (see [7, Proposition 8]). By Theorem 3.4, the operators T2 and T3 are not N -supercyclic. Moreover, T := T2 ⊕ T3 defined on H := `2 (N) ⊕ `2 (N) is a 3-isometry with Ker(4T ) 6= {0}. Assume that T is N -supercyclic. Let E be the N -dimensional subspace on H generated by x1 , x2 , . . . , xN where xj = x2j ⊕ x3j ∈ `2 (N)⊕`2 (N) for j = 1, . . . , N , such that Orb(T, E) is dense in H. Then Orb(Ti , Ei ) is dense on `2 (N), where Ei denotes the linear space generated by xi1 , . . . , xiN for i = 2, 3, which is a contradiction. So, T is not N -supercyclic. This leads naturally to the following conjecture. Conjecture. No m-isometry can be N -supercyclic.

References [1] J. Agler, A disconjugacy theorem for Toeplitz operators. Amer. J. Math. 112 (1990), no. 1, 1–14. [2] J. Agler, M. Stankus, m-isometric transformations of Hilbert space. I. Integral Equations and Operator Theory 21 (1995), no. 4, 383–429. [3] J. Agler, M. Stankus, m-isometric transformations of Hilbert space. II. Integral Equations and Operator Theory 23 (1995), no. 1, 1–48. [4] J. Agler, M. Stankus, m-isometric transformations of Hilbert space. III. Integral Equations and Operator Theory 24 (1996), no. 4, 379–421. [5] S. Ansari, P. Bourdon, Some properties of cyclic operators. Acta Sci. Math. (Szeged) 63 (1997), no. 1–2, 195–207. [6] S. Ansari, P. Enflo, Extremal vectors and invariant subspaces. Trans. Amer. Math. Soc. 350 (1998), no. 2, 539–558. [7] A. Athavale, Some operator theoretic calculus for positive definite kernels. Proc. Amer. Math. Soc., 112 (1991), no. 3, 701–708. [8] F. Bayart, E. Matheron, Hyponornal operators, weighted shifts and weak forms of supercyclicity. Proc. Edinburgh Math. Soc. 49 (2006) 1–15. [9] P. S. Bourdon, N. Feldman, N. J. H. Shapiro, Some properties of N -supercyclic operators. Studia Math. 165 (2004), no. 2, 135–157.

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[10] M. Faghih Ahmadi, K. Hedayatian, Hypercyclicity and supercyclicity of m-isometric operators, Preprint. [11] N. Feldman, n-supercyclic operators. Studia Math. 151 (2002), no. 2, 141–159. [12] H. M. Hilden, L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators. Indiana Univ. Math. J. 23 (1973/74), 557–565. [13] F. Mart´ınez-Gim´enez, A. Peris, Chaos for backward shift operators. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 12 (2002), no. 8, 1703–1715. [14] A. Peris, Multi-hypercyclic operators are hypercyclic. Math. Z. 236 (2001), no. 4, 779–786. [15] R. Sanders, Weakly supercyclic operators. J. Math. Anal. Appl. 292 (2004), no. 1, 148–159. [16] R. Sanders, An isometric bilateral shift that is weakly supercyclic. Integral Equations and Operator Theory 53 (2005), no. 4, 547–552. Teresa Berm´ udez, Isabel Marrero and Antonio Martin´ on Departamento de An´ alisis Matem´ atico Universidad de La Laguna 38271 La Laguna (Tenerife) Spain e-mail: [email protected] [email protected] [email protected] Submitted: July 28, 2008. Revised: April 27, 2009.

Integr. equ. oper. theory 64 (2009), 495–537 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040495-43, published online June 26, 2009 DOI 10.1007/s00020-009-1694-x

Integral Equations and Operator Theory

Mixed Boundary Value Problems of Thermopiezoelectricity for Solids with Interior Cracks T. Buchukuri, O. Chkadua and D. Natroshvili Abstract. We investigate asymptotic properties of solutions to mixed boundary value problems of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks. Using the potential methods and theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the mechanical, thermal and electric fields are analysed near the crack edges and near the curves, where the types of boundary conditions change. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are treated in detail as well. Mathematics Subject Classification (2000). Primary 35J55; Secondary 74F05, 74F15, 74B05. Keywords. Thermoelasticity, piezoelasticity, thermopiezoelectricity, mixed boundary value problems, cracks, potential method, pseudodifferential equations, asymptotics of solutions.

1. Introduction The paper deals with three-dimensional mixed type boundary value problems (BVP) arising in the theory of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks. Piezoelectric effects were discovered in the 1880’s and the first mathematical model describing this phenomena was created in 1909 [30]. However, the practical use of piezoelectricity became This research was supported by the Georgian National Science Foundation grant GNSF/ST07/3170 and by the German Research Foundation grant DFG 436 GEO113/8/0-1.

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possible only when piezoceramics and other materials (metamaterials) with pronounced piezoelectric properties were constructed. In recent times sensors and actuators made of such materials are widely used in medicine, in aerospace, in various industrial and domestic appliances. Due to great theoretical and practical importance, problems of piezoelectricity became very popular among mathematicians and engineers. According to the references [18]–[21], during the last years more than 1000 scientific papers have been published annually! Most of them are engineering-technical papers dealing with the two-dimensional case. In a piezoceramic material, due to its brittleness, cracks arise often, especially when a piezoelectric device works under an intensive thermomechanical loading. The influence of the electric field on the crack growth has a very complex character. Experiments revealed that the electric field can either promote or retard the crack growth, depending on the direction of polarization and can even close an open crack [25]. As it is well known from the classical mathematical physics and the classical elasticity theory, in general, solutions to crack type and mixed boundary value problems have singularities near the crack edges and near the lines where the types of boundary conditions change, regardless of the smoothness of given boundary data. Throughout the paper we shall refer to such lines as exceptional curves. The same effect can be observed in thermoelectroelasticity. In this paper, our main goal is a detailed theoretical investigation of regularity and asymptotic properties of mechanical, thermal and electric fields near the exceptional curves. By explicit calculations we show that the stress singularity exponents essentially depend on the material parameters, in general. We draw a special attention to the problem of oscillating singularities which is very important in engineering applications. Such singularities usually lead to some mechanical contradictions, e.g., overlapping of materials (see, e.g., [9] and the references therein). It turned out that there are classes of anisotropic media for which the oscillating singularities near the exceptional curves do not occur. In particular, calcium phosphate based bioceramics, such as hydroxyapatite, possess the above property. These materials are extensively used in medicine and dentistry [16]. Our main tools are the potential methods and the theory of pseudodifferential equations, which proved to be very efficient in deriving the asymptotic formulas. They allow us to calculate effectively the field singularity exponents by means of the characteristics related to the symbol matrices of the corresponding pseudodifferential operators. In our analysis we essentially apply the results obtained it the references [4], [7], [8], [13], [23], [27]. To demonstrate the dependence of the singularity exponents on the material parameters let us compare the behaviour of solutions to the crack type mixed boundary value problems near the exceptional curves for the Laplace equation (Zaremba type problem), for equations of the classical elasticity (e.g., the Lam´e equations for an isotropic solid) and for the equations of transversally-isotropic

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thermopiezoelectric media. Near the crack edge the asymptotic formulae for solutions of all the above three problems have the same form, namely, c1 r1/2 + c2 r3/2 + · · · , where r is the distance from the reference point x to the crack edge [2], [3]. We have quite a different situation near the exceptional curve, where the types of boundary conditions (for example, the Dirichlet and Neumann type conditions) change. The asymptotic expansions of solutions for the Laplace and Lam´e equations are of type b1 r1/2 + b2 r1/2+i δ + b3 r1/2−i δ + O(r3/2 ), while the asymptotic expansion of a solution to the thermoelectroelasticity equations for the transversally-isotropic case reads as d1 rγ1 + d2 r1/2+i δ + d3 r1/2−i δ + d4 r1/2 + O(rγ2 ), e

e

where γ1 ∈ (0, 0.5), γ2 > 0.5, and δ and δe are real numbers. Note that γ1 − 1 represents the dominant stress singularity exponent. The parameter γ1 depends on the material constants and may take on an arbitrary value from the interval (0, 0.5) (for details see Section 5). Thus, the stress singularity exponent essentially depends on the material constants and is less than −0.5, in general. Consequently, in the classical elasticity, we have oscillating stress singularities, while in the piezoelectricity theory we have no oscillating stress singularities for the transversally isotropic case due to the inequality γ1 < 1/2 (for details see Section 5 and Subsection 6.2). Beside a pure theoretical interest, our investigation is strongly motivated by its relevance to a wide variety of applications. The problems considered in the paper appear in many engineering and industrial applications as well as in biological and medical sciences, since due to the contemporary technological and industrial developments and recent important progress in material sciences they require the use of more generalized and refined models of continuum mechanics. For example, the analysis in the recent paper [28] shows that the anionic collagen and collagen-hydroxyapatite composites have, due to their physical, chemical, dielectric and piezoelectric properties, potential applications in support for cellular growth and in systems for bone regeneration - a collagenous structure of reconstituted collagen fibers could act as nucleators for the formation of apatite crystal similar to those of bone. Therefore, the questions related to mathematical modelling and investigation of the boundary value problems for solids with complex microstructure are very challenging. The paper is organized as follows. In Section 2, we formulate the mixed boundary value problem in appropriate function spaces, derive Green’s formulae and prove the uniqueness theorem. In Section 3, we describe the properties of potentials and the corresponding boundary integral operators. In Section 4, we prove existence theorems of solutions to the mixed crack type boundary value problems

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and study the regularity properties of the corresponding physical fields. In Section 5, we derive asymptotic expansion formulae of solutions near the exceptional curves and, finally, in Section 6, we study the qualitative and quantitative properties of dominant singularity exponents of physical fields. At the end of the paper, in the Appendix, we describe how to calculate explicitly the principal homogeneous symbol matrices of the pseudodifferential operators involved in our analysis. Finally, we would like to express our thanks to the anonymous referee for his/her valuable remarks and comments.

2. Formulation of the problem 2.1. Thermopiezoelastic field equations In this subsection, we collect the field equations of the linear theory of thermopiezoelasticity for a general anisotropic case and introduce the corresponding matrix partial differential operators (cf. [24], [26]). In the thermopiezoelasticity, we have the following governing equations: Constitutive relations: σij = σji = cijkl skl − elij El − γij ϑ = cijkl ∂l uk + elij ∂l ϕ − γij ϑ, i, j = 1, 2, 3, −1

S = γij sij + gl El + α [ T0 ]

ϑ,

(2.1) (2.2)

Dj = ejkl skl + εjl El + gj ϑ = ejkl ∂l uk − εjl ∂l ϕ + gj ϑ, j = 1, 2, 3.

(2.3)

Fourier Law: qi = −κil ∂l T, i = 1, 2, 3, Equations of motion: ∂i σij + Xj = % ∂t2 uj ,

T = T0 + ϑ. j = 1, 2, 3.

(2.4) (2.5)

Equation of static electric field: ∂i Di − X4 = 0.

(2.6)

T ∂t S = −∂j qj + X5 .

(2.7)

Equation of entropy balance: >

Here u = (u1 , u2 , u3 ) is the displacement vector, σkj is the mechanical stress tensor in the theory of thermoelectroelasticity, skj = 12 (∂k uj + ∂j uk ) is the strain tensor, T is the temperature, T0 is the initial reference temperature, that is the temperature in the natural state in the absence of deformation and electromagnetic fields, ϑ = T − T0 is the temperature increment, ϕ is the electric potential, D is the electric displacement vector, E = (E1 , E2 , E3 ) := −grad ϕ is the electric field vector, q = (q1 , q2 , q3 ) is the heat flux vector, S is the entropy density, % is the mass density, cijkl are the elastic constants, ekij are the piezoelectric constants, εkj are the dielectric (permittivity) constants, γkj are thermal strain constants, κkj are thermal conductivity constants, α := % e c with e c as the specific heat per unit mass,

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gi are pyroelectric constants characterizing the relation between thermodynamic processes and piezoelectric effect, X = (X1 , X2 , X3 )> is a mass force density, X4 is a charge density, X5 is a heat source density; we also employ the notation ∂ = ∂x = (∂1 , ∂2 , ∂3 ), ∂j = ∂/∂xj , ∂t = ∂/∂t; the superscript (·)> denotes transposition. Throughout the paper, the summation over the repeated indices is meant from 1 to 3, unless stated otherwise. From (2.1)–(2.7) one can derive the system of dynamics corresponding to the linear model of thermopiezoelasticity (see, e.g., [24]): cijlk ∂i ∂l uk + elij ∂l ∂i ϕ − γij ∂i ϑ + Xj = % ∂t2 uj , j = 1, 2, 3, − eikl ∂i ∂l uk + εil ∂i ∂l ϕ − gi ∂i ϑ + X4 = 0, − T0 γil ∂t ∂l ui + T0 gi ∂t ∂i ϕ + κil ∂i ∂l ϑ − α ∂t ϑ + X5 = 0. If all the functions involved in these equations are harmonic time dependent, that is they can be represented as product of a function of the spatial variables (x1 , x2 , x3 ) and the multiplier exp{τ t}, where τ = σ + iω is a complex parameter, we have the pseudo-oscillation equations of the theory of thermopiezoelasticity. Note that the pseudo-oscillation equations can be obtained from the corresponding dynamical equations by the Laplace transform. If τ = iω is a pure imaginary number, with the so-called frequency parameter ω ∈ R, we obtain the steady state oscillation equations. Finally, if τ = 0 we get the equations of statics. In particular, the corresponding pseudo-oscillation equations read as cijlk ∂i ∂l uk − % τ 2 uj + elij ∂l ∂i ϕ − γij ∂i ϑ + Xj = 0,

j = 1, 2, 3,

− eikl ∂i ∂l uk + εil ∂i ∂l ϕ − gi ∂i ϑ + X4 = 0,

(2.8)

− τ T0 γil ∂l ui + τ T0 gi ∂i ϕ + κil ∂i ∂l ϑ − τ α ϑ + X5 = 0, or in matrix form e A(∂, τ ) U (x) + X(x) = 0 in Ω, (2.9) > > e = (X1 , X2 , X3 , X4 , X5 ) , X = (X1 , X2 , X3 ) is a given where U := (u, ϕ, ϑ) , X mass force density, X4 is a given charge density, X5 is a given heat source density, A(∂, τ ) is the matrix differential operator generated by equations (2.8) >

A(∂, τ ) = [Apq (∂, τ )]5×5 , Ajk (∂, τ ) = cijlk ∂i ∂l − % τ 2 δjk ,

Aj4 (∂, τ ) = elij ∂l ∂i ,

Aj5 (∂, τ ) = −γij ∂i ,

A4k (∂, τ ) = −eikl ∂i ∂l ,

A44 (∂, τ ) = εil ∂i ∂l ,

A45 (∂, τ ) = −gi ∂i ,

A5k (∂, τ ) = −τ T0 γkl ∂l ,

A54 (∂, τ ) = τ T0 gi ∂i ,

A55 (∂, τ ) = κil ∂i ∂l − α τ,

j, k = 1, 2, 3.

(2.10)

Clearly, we obtain the equations and operators of statics if τ = 0. Constants involved in these equations satisfy the symmetry conditions: cijkl = cjikl = cklij , eijk = eikj , εij = εji , γij = γji , κij = κji , i, j, k, l = 1, 2, 3.

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Moreover, from physical considerations it follows that cijkl ξij ξkl ≥ c0 ξij ξij for all ξij = ξji ∈ R, 2

2

εij ηi ηj ≥ c1 |η| , κij ηi ηj ≥ c2 |η|

(2.11) 3

for all η = (η1 , η2 , η3 ) ∈ R ,

where c0 , c1 , and c2 are positive constants. In addition, we require that α | ζ|2 − 2 < ( ζ gl η l ) ≥ c3 ( | ζ|2 + | η|2 ) εij ηi η j + T0 for all ζ ∈ C and η ∈ C3

(2.12)

(2.13)

with a positive constant c3 (for the physical sense of these inequalities see, e.g., [24]). A sufficient condition for (2.13) to be satisfied reads as α c1 − g 2 > 0, 3 T0 where g = max {|g1 |, |g2 |, |g3 |} and c1 is the constant involved in (2.12). With the help of the inequalities (2.11) and (2.12) it can easily be shown that A(∂, τ ) is a strongly elliptic nonselfadjoint operator, that is there holds the inequality < −A(0) (ξ) ζ · ζ ≥ c |ζ|2 |ξ|2 for all ξ ∈ R3 and ζ ∈ C5 , (2.14) where A(0) (ξ) is the principal homogeneous symbol matrix of the operator A(∂, τ ); here and in what follows a · b denotes the scalar product of two vectors a, b ∈ Ck . By A∗ (∂, τ ) := [A(−∂, τ )]> we denote the operator formally adjoint to A(∂, τ ). In the theory of thermopiezoelasticity the components of the three-dimensional mechanical stress vector acting on a surface element with a normal n = (n1 , n2 , n3 ) have the form σij ni = cijlk ni ∂l uk + elij ni ∂l ϕ − γij ni ϑ for j = 1, 2, 3, while the normal components of the electric displacement vector and the heat flux vector (with opposite sign) read as −Di ni = −eikl ni ∂l uk + εil ni ∂l ϕ − gi ni ϑ,

−qi ni = κil ni ∂l ϑ.

Let us introduce the following matrix differential operator T (∂, n) = Tjk (∂, n) 5×5 ,

(2.15)

where, for j, k = 1, 2, 3, Tjk (∂, n) = cijlk ni ∂l ,

Tj4 (∂, n) = elij ni ∂l ,

Tj5 (∂, n) = −γij ni ,

T4k (∂, n) = −eikl ni ∂l ,

T44 (∂, n) = εil ni ∂l ,

T45 (∂, n) = −gi ni ,

T5k (∂, n) = 0 ,

T54 (∂, n) = 0 ,

T55 (∂, n) = κil ni ∂l .

For a vector U = (u, ϕ, ϑ)> we have T (∂, n) U = ( σi1 ni , σi2 ni , σi3 ni , −Di ni , −qi ni )> .

(2.16)

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The components of the vector T U given by (2.16) have the physical sense: the first three components correspond to the mechanical stress vector in the theory of thermoelectroelasticity, the forth and fifth ones are the normal components of the electric displacement vector and the heat flux vector with opposite sign, respectively. In Green’s formulas, there appears also the following boundary operator associated with the adjoint differential operator A∗ (∂, τ ), (2.17) Te (∂, n, τ ) = Tejk (∂, n, τ ) 5×5 , where, for j, k = 1, 2, 3, Tejk = Tjk ,

Tej4 = −Tj4 ,

Tej5 = −τ T0 Tj5 ,

Te4k = −T4k ,

Te44 = T44 ,

Te45 = −τ T0 gi ni ,

Te5k = 0 ,

Te54 = 0 ,

Te55 = T55 .

2.2. Green’s formulas Let Ω be a bounded 3-dimensional domain in R3 with a compact, C ∞ -smooth boundary ∂Ω. Assume that the domain Ω is filled with an anisotropic homogeneous thermopiezoelectric material. s By Lp , Wpr , Hps , and Bp,q (with r ≥ 0, s ∈ R, 1 < p < ∞, 1 ≤ q ≤ ∞) we denote the well-known Lebesgue, Sobolev-Slobodetski, Bessel potential, and r Besov function spaces, respectively (see, e.g., [29]). Recall that H2r = W2r = B2,2 , k k t s s t H2 = B2,2 , Wp = Bp,p , and Hp = Wp , for any r ≥ 0, for any s ∈ R, for any positive and non-integer t, and for any non-negative integer k. In our analysis, we essentially employ also the spaces: e ps (M) := {f : f ∈ Hps (M0 ), supp f ⊂ M}, H e s (M) := {f : f ∈ B s (M0 ), supp f ⊂ M}, B p,q p,q Hps (M) := {rM f : f ∈ Hps (M0 )},

s s Bp,q (M) := {rM f : f ∈ Bp,q (M0 ) },

where M0 is a compact surface without boundary, M ⊂ M0 and rM is the restriction operator to a set M. 5 For arbitrary vector-functions U = (u1 , u2 , u3 , u4 , u5 )> ∈ C 2 (Ω) and V = 5 (v1 , v2 , v3 , v4 , v5 )> ∈ C 2 (Ω) , we have the following Green’s formulae [22]: Z Z h i A(∂, τ ) U · V + E(U, V ) dx = { T U }+ · { V }+ dS , (2.18) Ω ∂Ω Z h i A(∂, τ ) U · V − U · A∗ (∂, τ ) V dx Ω Z h i = { T U }+ · { V }+ − { U }+ · { Te V }+ dS, (2.19) ∂Ω

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Z hX 3 Ω

[ A(∂, τ ) U ]j uj +[ A(∂, τ ) U ]4 u4 +

j=1

=−

Z h

IEOT

i τ [ A(∂, τ ) U ]5 u5 dx 2 |τ | T0

cijlk ∂i uj ∂l uk + % τ 2 |u|2 + εjl ∂l u4 ∂j u4 − 2< { gl u5 ∂l u4 }

Ω

Z hX 3 i + + α τ + |u5 |2 + 2 κjl ∂l u5 ∂j u5 dx − T U j uj T0 |τ | T0 ∂Ω j=1 + + + i τ + + T U 4 u4 T U 5 u5 + 2 dS, (2.20) |τ | T0 where E(U, V ) = cijlk ∂i uj ∂l vk + % τ 2 u · v + elij (∂l u4 ∂i vj − ∂i uj ∂l v4 ) + εjl ∂j u4 ∂l v4 + κjl ∂j u5 ∂l v5 + τ α u5 v5 + γjl ( τ T0 ∂j ul v5 − u5 ∂j vl ) − gl ( τ T0 ∂l u4 v5 + u5 ∂l v4 ) >

>

(2.21)

+

with u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ) . The symbol { · } denotes the onesided limit (the trace operator) on ∂Ω from Ω. Note that the above Green’s formulae can be generalized, by a standard limiting procedure, to Lipschitz domains and to vector-functions U ∈ [Wp1 (Ω)]5 and V ∈ [Wp10 (Ω)]5 with A(∂, τ )U ∈ [Lp (Ω)]5 , A∗ (∂, τ )V ∈ [Lp 0 (Ω)]5 , 1 < p < ∞, 0 1/p + 1/p = 1. With the help of these Green’s formulae we can determine a 5 −1/p generalized trace vector {T (∂, n)U }+ ∈ [Bp,p (∂Ω)]5 for a function U ∈ Wp1 (Ω) 5 with A(∂, τ )U ∈ [Lp (Ω)] , Z Z h

+ + {T (∂, n)U } , {V } ∂Ω := A(∂, τ ) U · V dx + E(u, v ) + % τ 2 u · v Ω

Ω

+ γjl ( τ T0 ∂j ul v5 −u5 ∂j vl )+κjl ∂j u5 ∂l v5 +elij (∂l u4 ∂i vj −∂i uj ∂l v4 ) i + τ α u5 v5 − gl ( τ T0 ∂l u4 v5 + u5 ∂l v4 ) + εjl ∂j u4 ∂l v4 dx, where V = (v, v4 , v5 )> ∈ [Wp10 (Ω)]5 is an arbitrary vector-function. Here the symbol h · , · i∂Ω denotes the duality between the function spaces 1/p −1/p [Bp,p (∂Ω)]5 and [Bp 0 ,p 0 (∂Ω)]5 which extends the usual L2 scalar product Z X N h f , g i∂Ω = fj gj dS for f, g ∈ [L2 (∂Ω)]N . ∂Ω j=1

2.3. Formulation of mixed BVPs for solids with cracks Let us assume that a piezoelectric elastic solid occupying the simply connected domain Ω contains an interior crack. We identify the crack surface as a twodimensional, two-sided manifold S with the crack edge `c := ∂S. We assume that S is a submanifold of a closed surface S0 ⊂ Ω surrounding a domain Ω0 ⊂ Ω. Denote ΩS := Ω \ S.

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Further, we assume that ∂Ω is divided into two disjoint parts SD and SN : ∂Ω = S D ∪ S N . Put ∂SD = ∂SN =: `m . In what follows, for simplicity we assume that ∂Ω, SD , SN , S, S0 , `c , `m are C ∞ -smooth. Throughout the paper n = (n1 , n2 , n3 ) stands for the exterior unit normal vector to ∂Ω and S0 = ∂Ω0 . This agreement defines the positive direction of the normal vector on the crack surface S. We will consider the following problem: (i) The thermopiezoelectric elastic solid under consideration is mechanically fixed along the subsurface SD , and at the same time there are given the temperature and the electric potential functions (i.e., on SD there are given the components of the vector {U }+ -Dirichlet conditions). (ii) On the subsurface SN there are prescribed the mechanical stress vector and the normal components of the heat flux and the electric displacement vectors (i.e., on SN there are given the components of the vector {T U }+ -Neumann conditions). (iii) The crack surface S is mechanically traction free and we assume that the temperature, electric potential, and the normal components of the heat flux and the electric displacement vectors are continuous. Reducing the nonhomogeneous differential equations (2.9) to the corresponding homogeneous ones, we can formulate the above problem mathematically as follows: 5 Find a vector U = (u, ϕ, θ)> = (u1 , u2 , u3 , u4 , u5 )> ∈ Wp1 (ΩS ) satisfying the differential equation A(∂, τ ) U = 0 in ΩS ,

τ = σ + i ω, σ > 0,

(2.22)

the crack conditions on S, {[ T U ]j }+ = Fj+

on S,

j = 1, 3,

(2.23)

Fj−

on S,

j = 1, 3,

(2.24)

−

{[ T U ]j } = + {u4 − u4 }− = f4

on S,

(2.25)

−

on S,

(2.26)

−

on S,

(2.27)

−

on S,

(2.28)

{U }+ = g (D) on SD ,

(2.29)

+

{[ T U ]4 } − { [ T U ]4 } = F4 +

{u5 } − {u5 } = f5 +

{[ T U ]5 } − { [ T U ]5 } = F5 and the mixed boundary conditions on ∂Ω,

+

{T U } = g

(N )

on SN .

(2.30)

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We require that the boundary data possess the natural smoothness properties associated with the trace theorems, −1/p e 1/p 0 (S), Fj+ , Fj− ∈ Bp,p (S), j = 1, 2, 3; f4 , f5 ∈ B p,p 1/p 0 5 −1/p 5 −1/p (D) ep,p (S), g (2.31) F4 , F5 ∈ B ∈ Bp,p (SD ) , g (N ) ∈ Bp,p (SN ) , 1 < p < ∞, 1/p + 1/p 0 = 1. Moreover, the following compatibility conditions e −1/p (S), j = 1, 2, 3, Fj+ − Fj− ∈ rS B p,p

(2.32)

are to be satisfied. Here and in what follows rM denotes the restriction operator to a set M. The differential equation (2.22) is understood in the distributional sense, in general. We remark that if U ∈ [Wp1 (ΩS )]5 solves the homogeneous differential equation then actually we have the inclusion U ∈ [C ∞ (ΩS )]5 due to the ellipticity of the corresponding differential operators. In fact, U is a complex valued analytic vector function of spatial real variables (x1 , x2 , x3 ) in ΩS . The Dirichlet-type conditions (2.25), (2.27) and (2.29) are understood in the usual trace sense, while the Neumann-type conditions (2.23), (2.24), (2.26), (2.28) and (2.30) involving boundary limiting values of the components of the vector T U are understood in the functional sense related to Green’s formulas and described above. Now, we prove a uniqueness theorem for p = 2. The similar uniqueness theorem for p 6= 2 will be proved later. Theorem 2.1. Let τ = σ + iω with σ > 0 and ω ∈ R. Then the problem (2.22)– (2.30) with homogeneous boundary conditions has only the trivial solution in the 5 space W21 (ΩS ) . 5 Proof. Let U ∈ W21 (ΩS ) be a solution to the problem (2.22)–(2.30) with homogeneous boundary conditions, Fj± = 0, Fk = fk = 0, j = 1, 2, 3, k = 4, 5, g (D) = 0 and g (N ) = 0. Green’s formula (2.20) along with the homogeneous boundary conditions then imply Z h cijlk ∂i uj ∂l uk + % τ 2 |u|2 + εjl ∂l u4 ∂j u4 − 2 < { gl u5 ∂l u4 } ΩS (2.33) i τ α κjl ∂l u5 ∂j u5 + |u5 |2 dx = 0. + 2 |τ | T0 T0 Due to the relations (2.11) and (2.12) we have cijlk ∂i uj ∂l uk ≥ 0,

εjl ∂l u4 ∂j u4 ≥ 0, κjl ∂l u5 ∂j u5 ≥ 0

(2.34)

with the equality only for a complex rigid displacement vectors, a constant temperature distribution and a constant potential field: u = a × x + b, u4 = a4 , u5 = a5 ,

(2.35)

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where a, b ∈ C3 , a4 , a5 ∈ C; the symbol × denotes the usual cross product of two vectors. Take into account the above inequalities and separate the real and imaginary parts of (2.33) to obtain Z h cijlk ∂i uj ∂l uk + % (σ 2 − ω 2 ) |u|2 + εjl ∂l u4 ∂j u4 ΩS i σ α − 2 < {gl u5 ∂l u4 } + |u5 |2 + 2 κjl ∂l u5 ∂j u5 dx = 0 , (2.36) T0 |τ | T0 Z h i ω 2 % σ ω |u|2 + 2 (2.37) κjl ∂l u5 ∂j u5 dx = 0. |τ | T0 ΩS First, let us assume that σ > 0 and ω 6= 0. With the help of the homogeneous boundary conditions on SD we easily derive from (2.37) that u = 0 and u5 = 0 in ΩS . From (2.36) we then conclude Z εjl ∂l u4 ∂j u4 dx = 0, ΩS

whence u4 = 0 in ΩS follows immediately due to (2.34), (2.35) and the homogeneity of the boundary conditions on SD . Suppose now, that σ > 0 and ω = 0. Then it follows from (2.36), (2.13), (2.34) and (2.35) that u = 0, u4 = a4 , u5 = a5 , and the homogeneous boundary condition on SD implies U = 0 in ΩS .

3. Properties of potentials 3.1. Mapping and Fredholm properties The full symbol of the pseudo-oscillation differential operator A(∂, τ ) is elliptic provided <τ 6= 0, i.e., det A(−i ξ, τ ) 6= 0,

∀ ξ ∈ R3 \{0}.

Moreover, the entries of the inverse matrix A−1 (−i ξ, τ ) are locally integrable functions decaying at infinity as O(|ξ|−2 ). Therefore, we can construct the fundamental matrix H(x, τ ) = [Hkj (x, τ )]5×5 of the operator A(∂, τ ) by means of the Fourier transform technique, −1 −1 H(x, τ ) = Fξ→x A (−iξ, τ ) . (3.1) The far field and near field properties of this fundamental matrix and the corresponding layer potentials are studied in the references [5], [22]. Here we collect some results which are necessary for our further analysis. Detailed proofs of the theorems below can be found in [22] (see also [12]). Let us put Ω+ := Ω and Ω− := R3 \ Ω. Clearly, ∂Ω = ∂Ω+ = ∂Ω− .

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Furthermore, we introduce the single and double layer potentials: Z H(x − y, τ ) h(y) dy S, Z∂Ω h > i> W (h)(x) = h(y) dy S, Te (∂y , n(y), τ ) H(x − y, τ ) V (h)(x) =

∂Ω

where h = (h1 , h2 , · · · , h5 )> is a density vector function. Theorem 3.1. Let 1 < p < ∞, 1 ≤ q ≤ ∞, s ∈ R. Then the single and double layer potentials can be extended to the following continuous operators s 5 s+1+1/p + 5 V : Bp,q (∂Ω) → Bp,q (Ω ) , s 5 s+1+1/p − 5 : Bp,q (∂Ω) → Bp,q, loc (Ω ) , s 5 5 : Bp,p (∂Ω) → Hps+1+1/p (Ω+ ) , s 5 s+1+1/p − 5 : Bp,p (∂Ω) → Hp, loc (Ω ) ,

s 5 s+1/p + 5 W : Bp,q (∂Ω) → Bp,q (Ω ) , s 5 s+1/p − 5 : Bp,q (∂Ω) → Bp,q, loc (Ω ) , s 5 5 : Bp,p (∂Ω) → Hps+1/p (Ω+ ) , s 5 s+1/p 5 : Bp,p (∂Ω) → Hp, loc (Ω− ) .

Theorem 3.2. Let −1+s 5 s 5 h(1) ∈ Bp,q (∂Ω) , h(2) ∈ Bp,q (∂Ω) , 1 0. Then for any z ∈ ∂Ω V (h

(1)

± )(z) =

Z

H(z − y, τ )h(1) (y) dy S, Z h > i> (2) ± 1 Te (∂y , n(y), τ ) H(z − y, τ ) h (y) dy S. W (h(2) )(z) = ± h(2) (z) + 2 ∂Ω

∂Ω

5 s (∂Ω) . The equalities are understood in the sense of the space [Bp,q −1/p 5 1−1/p 5 Theorem 3.3. Let h(1) ∈ Bp,q (∂Ω) , h(2) ∈ Bp,q (∂Ω) , 1 < p < ∞, 1 ≤ q ≤ ∞. Then for any z ∈ ∂Ω Z ± 1 (1) T V (h )(z) = ∓ h (z) + T (∂z , n(z))H(z − y, τ ) h(1) (y) dy S, 2 ∂Ω + − T W (h(2) )(z) = T W (h(2) )(z) .

(1)

5 −1/p The equalities are understood in the sense of the space [Bp,q (∂Ω) .

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We introduce the following notation for the boundary operators generated by the single and double layer potentials: Z H(h)(z) = H(z − y, τ ) h(y) dy S, z ∈ ∂Ω, (3.2) ∂Ω Z K(h)(z) = T (∂z , n(z))H(z − y, τ ) h(y) dy S, z ∈ ∂Ω, (3.3) Z∂Ω h i> e K(h)(z) = Te (∂y , n(y), τ ) H > (z − y, τ ) h(y) dy S, z ∈ ∂Ω, (3.4) ∂Ω

+ − L(h)(z) = T W (h)(z) = T W (h)(z) ,

z ∈ ∂Ω.

(3.5)

Actually, H is a weakly singular integral operator (pseudodifferential operator of e are singular integral operators (pseudodifferential operator order −1), K and K of order 0), and L is a singular integro-differential operator (pseudodifferential operator of order 1). These operators possess the following mapping and Fredholm properties [22], [17]. Theorem 3.4. Let 1 < p < ∞, 1 ≤ q ≤ ∞, s ∈ R. Then the operators s 5 s+1 5 H : Bp,q (∂Ω) → Bp,q (∂Ω) , 5 5 : Hps (∂Ω) → Hps+1 ∂(Ω) , s 5 s 5 e : Bp,q K, K (∂Ω) → Bp,q (∂Ω) , 5 5 : Hps (∂Ω) → Hps (∂Ω) , s 5 s−1 5 L : Bp,q (∂Ω) → Bp,q (∂Ω) , 5 5 : Hps (∂Ω) → Hps−1 (∂Ω) , are continuous. The operators H and L are strongly elliptic pseudodifferential operators, while e are elliptic, where I5 stands for the 5 × 5 the operators ± 12 I5 + K and ± 12 I5 + K unit matrix. e and 1 I5 + K are invertible, whereas the Moreover, the operators H, 21 I5 + K 2 e and L are Fredholm operators with zero index. operators − 12 I5 + K, − 12 I5 + K There hold the following operator equalities 1 L H = − I5 + K2 , 4

1 e2 . H L = − I5 + K 4

(3.6)

3.2. Some results for pseudodifferential equations on manifolds with boundary In our investigation we need some results describing the Fredholm properties of strongly elliptic pseudodifferential operators on a compact manifold with boundary. They can be found in [1], [13], [15], [27].

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Let M ∈ C ∞ be a compact, n-dimensional, nonselfintersecting manifold with boundary ∂M ∈ C ∞ and let A be a strongly elliptic N × N matrix pseudodifferential operator of order ν ∈ R on M. Denote by σA (x, ξ) the principal homogeneous symbol matrix of the operator A in some local coordinate system (x ∈ M, ξ ∈ Rn \ {0}). Let λ1 (x), · · · , λN (x) be the eigenvalues of the matrix [ σA (x, 0, · · · , 0, +1) ]−1 σA (x, 0, · · · , 0, −1) ,

x ∈ ∂M,

(3.7)

and let δj (x) = < (2π i)−1 ln λj (x) , j = 1, · · · , N. Here ln ζ denotes the branch of the logarithm analytic in the complex plane cut along (−∞, 0]. Due to the strong ellipticity of A we have the strict inequality −1/2 < δj (x) < 1/2 for x ∈ M. The numbers δj (x) do not depend on the choice of the local coordinate system. In particular, if the eigenvalue λj is real, then it is positive. Note that when σA (x, ξ) is a positive definite matrix for every x ∈ M and ξ ∈ Rn \ {0} or when it is an even matrix in ξ we have δj (x) = 0 for j = 1, . . . , N, since all the eigenvalues λj (x) (j = 1, N ) are positive numbers for any x ∈ M. The Fredholm properties of strongly elliptic pseudo-differential operators are characterized by the following theorem. Theorem 3.5. Let s ∈ R, 1 < p < ∞, 1 ≤ q ≤ ∞, and let A be a strongly elliptic pseudodifferential operator of order ν ∈ R, that is, there is a positive constant c0 such that < σA (x, ξ)ζ · ζ ≥ c0 |ζ|2 for x ∈ M, ξ ∈ Rn with |ξ| = 1, and ζ ∈ CN . Then e s (M) → H s−ν (M), A : H p p

e s (M) → B s−ν (M), A : B p,q p,q

(3.8)

are Fredholm operators with index zero if 1 ν 1 −1+ sup δj (x) < s − < + inf δj (x). p 2 p x∈∂M, 1≤j≤N x∈∂M, 1≤j≤N

(3.9)

Moreover, the null-spaces and indices of the operators (3.8) are the same (for all values of the parameter q ∈ [1, +∞]) provided p and s satisfy the inequality (3.9). We will essentially use this theorem in the next subsection to prove the existence and regularity results for our boundary value problem.

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4. Existence and regularity of solutions We start with the remark that the boundary conditions on the crack surface S, (2.23) and (2.24), can be transformed equivalently as + − −1/p ep,p TU j − TU j = Fj+ − Fj− ∈ rS B (S), j = 1, 3 − + −1/p = Fj+ + Fj− ∈ Bp,p (S), j = 1, 3. + TU j TU j Thus, the boundary conditions can be rewritten as + T U + U + − TU j + TU j + − u4 − u4 + − u5 − u5 + − TU j − TU j + − [ T U ]4 − [ T U ]4 + − [ T U ]5 − [ T U ]5

(2.23)–(2.30) of the problem under consideration = g (N )

on SN ,

(4.1)

(D)

on SD ,

(4.2)

=g

= Fj+ + Fj−

on S,

= f4

on S,

(4.4)

= f5

on S,

(4.5)

= Fj+ − Fj−

on S,

= F4

on S,

(4.7)

= F5

on S.

(4.8)

j = 1, 3,

j = 1, 3,

(4.3)

(4.6)

We look for the solutions of the boundary value problem (2.22)–(2.30) in the following form: U = V (H−1 h) + Wc (h(2) ) + Vc (h(1) )

in ΩS ,

(4.9)

where H−1 is the operator inverse to the integral operator H defined by (3.2), Z Vc (h(1) )(x) := H(x − y, τ ) h(1) (y) dy S, (4.10) S Z h > i> (2) Te (∂y , n(y), τ ) H(x − y, τ ) h (y) dy S, (4.11) Wc (h(2) )(x) := ZS V (H−1 h)(x) := H(x − y, τ ) (H−1 h)(y) dy S, (4.12) ∂Ω (i)

(i)

h(i) = (h1 , . . . , h5 )> , i = 1, 2, and h = (h1 , . . . , h5 )> are unknown densities, −1/p 5 e h(1) ∈ B p,p (S) ,

1/p0 5 e h(2) ∈ B p,p (S) ,

1/p0 5 h ∈ Bp,p (∂Ω) .

(4.13)

Due to the above inclusions, clearly, in Vc and Wc we can take the closed surface S0 as an integration manifold instead of the crack surface S. Recall that S is assumed to be a proper part of S0 = ∂Ω0 ⊂ Ω (see Subsection 2.3).

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The boundary and transmission conditions (4.1)–(4.8) lead to the equations rSN A h + rSN T Wc (h(2) ) + rSN T Vc (h(1) ) = g (N ) rSD h + rSD Wc (h(2) ) + rSD Vc (h(1) ) = g (D) rS T V H−1 h j + rS Lc h(2) j + rS Kc (h(1) ) j = 2−1 Fj+ + Fj− , j = 1, 2, 3,

on SN ,

(4.14)

on SD ,

(4.15)

on

(4.16)

S,

(2)

h4 = f4 , (2) h5

= f5 ,

(1) hj

Fj−

=

(4.17) (4.18) −

Fj+ ,

j = 1, 2, 3,

(4.19)

(1)

h4 = −F4 ,

(4.20)

(1) h5

(4.21)

= −F5 ,

where A := − 2−1 I5 + K H−1 is the Steklov-Poincar´e type operator on ∂Ω, and + − Lc (h(2) )(z) := T Wc (h(2) )(z) = T Wc (h(2) )(z) , z ∈ S, Z (1) Kc (h )(z) := T (∂z , n(z))H(z − y, τ ) h(1) (y) dy S, z ∈ S. S

As we see the sought for density h(1) and the last two components of the vector h(2) are determined explicitly by the data of the problem. Hence, it remains to find the density h and the first three components of the vector h(2) . The operator generated by the left hand side expressions of the above simultaneous equations read as 

rS N A

  rS I5  D  −1 ]3×5 P :=   rS [T V H   [0] 2×5  [ 0 ]5×5

rS N T W c

rS N T V c



rS D W c

rS D V c

rS [Lc ]3×5

rS [Kc ]3×5

∗ rS I2×5

[ 0 ]2×5

        

[ 0 ]5×5

rS I5

,

(4.22)

20×15

where [ M ]3×5 denotes the first three rows of a 5 × 5 matrix M , [ 0 ]m×n stands for the corresponding zero matrix, ∗ I2×5 :=

0 0

0 0

0 0

1 0

0 . 1 2×5

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This operator possesses the following mapping properties e ps (S)]5 ×[H e ps−1 (S)]5 P : [Hps (∂Ω)]5 ×[H e ps (S)]2 ×[H e ps−1 (S)]5 , → [Hps−1 (SN )]5 ×[Hps (SD )]5 ×[Hps−1 (S)]3 ×[H s s s−1 ep,q ep,q P : [Bp,q (∂Ω)]5 ×[B (S)]5 ×[B (S)]5

(4.23)

s−1 s s−1 s s−1 ep,q ep,q → [Bp,q (SN )]5 ×[Bp,q (SD )]5 ×[Bp,q (S)]3 ×[B (S)]2 ×[B (S)]5 , 1 < p < ∞, 1 ≤ q ≤ ∞, s ∈ R.

Our main goal is to establish invertibility of the operators (4.23). To this end, by introducing a new additional unknown vector we extend equation (4.15) from SD onto the whole of ∂Ω. We will do this in the following (D) way. Denote by g0 some fixed extension of g (D) from SD onto the whole of ∂Ω preserving the space and introduce a new unknown vector ψ on ∂Ω (D) ψ = h + r∂Ω Wc (h(2) ) + r∂Ω Vc (h(1) ) − g0 .

(4.24)

1/p0 ep,p (SN ) 5 in accordance with (4.15), (4.13), (2.31) and It is evident that ψ ∈ B 1/p0 5 (D) the imbedding g0 ∈ Bp,p (∂Ω) . Moreover, the restriction of equation (4.24) on SD coincide with equation (4.15). Therefore, we replace equation (4.15) in the system (4.14)–(4.21) by equation (4.24). Finally, we arrive at the following simultaneous equations with respect to unknowns h, ψ, h(2) and h(1) : rSN A h + rSN T Wc (h(2) ) + rSN T Vc (h(1) ) = g (N ) (D) h − ψ + r∂Ω Wc (h(2) ) + r∂Ω Vc (h(1) ) = g0 rS T V H−1 h j + rS Lc h(2) j + rS Kc (h(1) ) j = 2−1 Fj+ + Fj− , j = 1, 2, 3, (2) h4

on SN ,

(4.25)

on ∂Ω,

(4.26)

on

(4.27)

S,

= f4 ,

(4.28)

h5 = f5 ,

(4.29)

(2)

(1) hj

= Fj− − Fj+ ,

(1) h4

= −F4 ,

(4.31)

h5 = −F5 .

(4.32)

(1)

j = 1, 2, 3,

(4.30)

In what follows, for the zero vector g (D) = 0 on SD we always choose the fixed (D) extension vector g0 = 0 on ∂Ω.

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Rewrite the system (4.25)–(4.32) in the equivalent form rSN A ψ + rSN T Wc (h(2) ) − rSN A [r∂Ω Wc (h(2) )] + rSN T Vc (h(1) ) (D)

− rSN A [r∂Ω Vc (h(1) )] = g (N ) − rSN A g0 (D) − ψ + h + r∂Ω Wc (h(2) ) + r∂Ω Vc (h(1) ) = g0 rS T V H−1 h j + rS Lc h(2) j + rS Kc (h(1) ) j = 2−1 Fj+ + Fj− , j = 1, 2, 3,

on SN ,

(4.33)

on ∂Ω,

(4.34)

on S,

(4.35)

(2)

h4 = f4 , (2) h5

= f5 ,

(1) hj

Fj−

=

(4.36) (4.37) −

Fj+ ,

j = 1, 2, 3,

(4.38)

(1)

h4 = −F4 ,

(4.39)

(1) h5

(4.40)

= −F5 .

Remark 4.1. The systems (4.14)–(4.21) and (4.33)–(4.40) are equivalent in the following sense: (i) If (h, h(2) , h(1) ) solves the system (4.14)–(4.21), then (ψ, h, h(2) , h(1) ) with ψ (D) given by (4.24) where g0 is some fixed extension of the vector g (D) from SD onto the whole of ∂Ω involved in the right hand side of the equation (4.34), solves the system (4.33)–(4.40). (ii) If (ψ, h, h(2) , h(1) ) solves the system (4.33)–(4.40), then (h, h(2) , h(1) ) solves the system (4.14)–(4.21). The operator generated by the system (4.33)–(4.40) reads as 

rS N A

  −r∂Ω I5   P1 :=   [ 0 ]3×5   [0] 2×5  [ 0 ]5×5

[ 0 ]5×5

rS N R 2

rS N R 1



r∂Ω I5

r∂Ω Wc

r∂Ω Vc

rS [T V H−1 ]3×5

rS [Lc ]3×5

rS [Kc ]3×5

[ 0 ]2×5

∗ rS [ I2×5 ]2×5

[ 0 ]2×5

        

[ 0 ]5×5

[ 0 ]5×5

rS I5

,

(4.41)

20×20

where R1 = T Vc − A [r∂Ω Vc ],

R2 = T Wc − A [r∂Ω Wc ].

(4.42)

Here and in what follows [ M ]5×k with k < 5 denotes the first k columns of a 5 × 5 matrix M , while [ M ]k×5 denotes the first k rows of the same matrix, and [ M ]k×k stands for the upper left k × k block of M .

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This operator possesses the following mapping properties e ps (SN )]5 × [Hps (∂Ω)]5 × [H e ps (S)]5 × [H e ps−1 (S)]5 P1 : [H e ps (S)]2 ×[H e ps−1 (S)]5 , → [Hps−1 (SN )]5 ×[Hps (∂Ω)]5 ×[Hps−1 (S)]3 ×[H e s (SN )]5 × [B s (∂Ω)]5 × [B e s (S)]5 × [B e s−1 (S)]5 P1 : [B p,q p,q p,q p,q

(4.43)

s−1 s s−1 s s−1 ep,q ep,q → [Bp,q (SN )]5 ×[Bp,q (∂Ω)]5 ×[Bp,q (S)]3 ×[B (S)]2 ×[B (S)]5 , 1 < p < ∞, 1 ≤ q ≤ ∞, s ∈ R.

Due to the above agreement about the extension of the zero vector we see that if the right hand side functions of the system (4.14)–(4.21) vanish then the same holds for the system (4.33)–(4.40) and vice versa. The uniqueness Theorem 2.1 and properties of the single and double layer potentials imply the following technical lemma. Lemma 4.2. The null spaces of the operators P and P1 are trivial for s = 1/2 and p = 2. Now we start to analyse Fredholm properties of the operator P1 . From the structure of the operator P1 it is evident that we only need to study Fredholm properties of the operator generated by the upper left 13 × 13 block of the matrix operator (4.41),   rS N A [ 0 ]5×5 rSN [ R2 ]5×3   r∂Ω [ Wc ]5×3  N :=  . (4.44)  −r∂Ω I5 r∂Ω I5  −1 [ 0 ]3×5 rS [T V H ]3×5 rS [ Lc ]3×3 13×13 This operator has the following mapping properties: e s (SN )]5 × [H s (∂Ω)]5 × [H e s (S)]3 N : [H p

p

p

→ [Hps−1 (SN )]5 × [Hps (∂Ω)]5 × [Hps−1 (S)]3 , e s (SN )]5 × [B s (∂Ω)]5 × [B e s (S)]3 N : [B p,q p,q p,q

(4.45)

s−1 s s−1 → [Bp,q (SN )]5 × [Bp,q (∂Ω)]5 × [Bp,q (S)]3 , 1 < p < ∞, 1 ≤ q ≤ ∞, s ∈ R.

For the principal part N0 of the operator N we have  rS N A [ 0 ]5×5 [ 0 ]5×3  N0 :=   −r∂Ω I5 r∂Ω I5 [ 0 ]5×3 [ 0 ]3×5 [ 0 ]3×5 rS L(1)

   

,

(4.46)

13×13

where L(1) := k[Lc ]kj k3×3 , Lc = k[Lc ]kl k5×5 . (4.47) Clearly, the operator N0 has the same mapping properties as N and the difference N − N0 is compact. Actually, N − N0 is an infinitely smoothing operator.

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The operators Lc and A are strongly elliptic pseudodifferential operators of order 1 due to strong ellipticity of the differential operator A(∂, τ ). From (4.47) we get then that L(1) is a strongly elliptic pseudodifferential operator as well. Moreover, we have the following invertibility results. Theorem 4.3. Let 1 < p < ∞, 1 ≤ q ≤ ∞, 1/p − 1/2 < s < 1/p + 1/2. Then the operators s s 3 e (S) 3 → B s−1 (S) 3 (4.48) e (S) → H s−1 (S) 3 , rS L(1) : B rS L(1) : H p,q p,q p p are invertible. Proof. With the help of the first equality in (3.6) we derive that the principal homogeneous symbol matrix of the strongly elliptic pseudodifferential operator Lc reads as 1 2 (x, ξ) [ σHS0 (x, ξ) ]−1 σLc (x, ξ) = σLS0 (x, ξ) := − I5 + σK S0 4 1 2 (x, ξ) [ σHc (x, ξ) ]−1 , x ∈ S, ξ ∈ R2 \ {0}, = − I5 + σK c 4 where HS0 and KS0 are integral operators given by (3.2) and (3.3) with S0 for ∂Ω. One can show exactly as in [6] that the principal homogeneous symbol matrix of the operator Kc is an odd matrix function in ξ, whereas the principal homogeneous symbol matrix of the operator Hc is an even matrix function in ξ. Consequently, the matrix σLc (x, ξ) is even in ξ. From these results it follows that L(1) is a strongly elliptic pseudodifferential operator with even principal homogeneous symbol. Therefore the matrix (see (3.7)) [ σL(1) (x, 0, +1) ]−1 σL(1) (x, 0, −1) is the unit matrix and the corresponding eigenvalues equal to 1. Now, from Theorem 3.5 it follows that the operators (4.48) are Fredholm with zero index for 1 < p < ∞, 1 ≤ q ≤ ∞ and 1/p − 1/2 < s < 1/p + 1/2. It remains to show that the corresponding null spaces are trivial. In turn, due to the same Theorem 3.5, it suffices to establish that the operator 1/2 3 e (S) → H −1/2 (S) 3 rS L(1) : H 2 2 is injective, i.e, we have to prove that the homogeneous equation rS L(1) χ = 0 on S (4.49) 1/2 3 e (S) . possesses only the trivial solution in the space H 2 1/2 3 e (S) solve equation (4.49) and construct the double layer Let χ ∈ H 2 potential U = Wc (e χ), χ e = (χ, 0, 0)> . In view of properties of the double layer potential and the equation (4.49), it can 5 easily be verified that the vector U = (u1 , · · · , u5 )> ∈ W21 (R3 \ S ) is a solution

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to the following crack type boundary transmission problem: A(∂, τ ) U + − [ T U ]j = [ T U ]j + − uk − uk + − [ T U ]k − [ T U ]k

in R3 \ S,

=0 = 0,

j = 1, 2, 3,

on S,

= 0,

k = 4, 5,

on S,

= 0,

k = 4, 5,

on S.

Applying Green’s identity (2.20) by standard arguments we arrive at the equality U = 0 in R3 \ S. Whence χ = (χ1 , χ2 , χ3 )> = 0 on S follows due to the equalities + − uj − uj = χj on S, j = 1, 2, 3. This completes the proof. Let λk , k = 1, 5, be the eigenvalues of the matrix a0 (x) := [ σA (x, 0, +1) ]−1 σA (x, 0, −1),

x ∈ `m ,

where σA (x, ξ) with x ∈ S N and ξ = (ξ1 , ξ2 ) ∈ R2 is the principal homogeneous symbol of the Steklov-Poincar´e operator A. It is shown in [22] that λ5 = 1. Let us introduce the notations 1 1 δ 0 = inf arg λj (x), δ 00 = sup arg λj (x). 1≤j≤5 2π 1≤j≤5 2π x∈`m

x∈`m

Due to strong ellipticity of the operator A we easily derive that 0 00 1 1 <δ ≤0≤δ < . 2 2 Applying again Theorem 3.5, we get

−

Theorem 4.4. Let 1 < p < ∞, 1 ≤ q ≤ ∞, 1/p − 1/2 + δ 00 < s < 1/p + 1/2 + δ 0 . Then the Steklov-Poincar´e operators s e p (SN ) 5 → Hps−1 (SN ) 5 , rS N A : H (4.50) s s−1 5 ep,q (SN ) 5 → Bp,q (SN ) , rS N A : B are invertible. Proof. From Theorem 3.5 it follows that the operators (4.50) are Fredholm with zero index for 1 < p < ∞, 1 ≤ q ≤ ∞, 1/p − 1/2 + δ 00 < s < 1/p + 1/2 + δ 0 . It remains to show that the corresponding null spaces are trivial. Since s = 1/2 and p = 2 satisfy the last inequality, due to the same Theorem 3.5, it suffices to establish that the operator 1/2 e (SN ) 5 → H −1/2 (SN ) 5 rS N A : H 2 2 is injective, i.e, we have to prove that the homogeneous equation rSN A ψ = 0 on SN

(4.51)

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1/2 1/2 e (SN ) 5 . Let ψ ∈ H e (SN ) 3 possesses only the trivial solution in the space H 2 2 be a solution of equation (4.51) and construct the single layer potential U = V (H−1 ψ). With the help of equation (4.51) and properties of the single layer potential it can 5 easily be verified that the vector U ∈ W21 (Ω, ) is a solution to the following mixed boundary value problem: A(∂, τ ) U = 0 in Ω,

{ T U }+ = 0 on SN ,

{ U }+ = 0 on SD .

Applying Green’s identity (2.20) by standard arguments we derive U = 0 in Ω. Thus ψ = 0 on ∂Ω follows immediately due to the equality { U }+ = ψ on ∂Ω. These theorems imply Theorem 4.5. Let 1 < p < ∞, 1 ≤ q ≤ ∞, 1/p − 1/2 + δ 00 < s < 1/p + 1/2 + δ 0 .

(4.52)

Then the operators e ps (SN )]5 × [Hps (∂Ω)]5 × [H e ps (S)]3 N : [H → [Hps−1 (SN )]5 × [Hps (∂Ω)]5 × [Hps−1 (S)]3 , s s s ep,q ep,q N : [B (SN )]5 × [Bp,q (∂Ω)]5 × [B (S)]3

(4.53)

s−1 s s−1 → [Bp,q (SN )]5 × [Bp,q (∂Ω)]5 × [Bp,q (S)]3 ,

are Fredholm with zero index. Proof. From Theorems 4.3 and 4.4 we conclude that for arbitrary p, q and s satisfying the conditions (4.52), the operators e ps (SN )]5 × [Hps (∂Ω)]5 × [H e ps (S)]3 N 0 : [H → [Hps−1 (SN )]5 × [Hps (∂Ω)]5 × [Hps−1 (S)]3 , s s s ep,q ep,q N0 : [B (SN )]5 × [Bp,q (∂Ω)]5 × [B (S)]3 s−1 s s−1 → [Bp,q (SN )]5 × [Bp,q (∂Ω)]5 × [Bp,q (S)]3 ,

are invertible. Therefore the operators (4.53) are Fredholm with zero index.

Now we are in a position to prove the main invertibility property of the operators P. Theorem 4.6. Let the conditions (4.52) be satisfied. Then the operators e ps (S)]5 ×[H e ps−1 (S)]5 P : [Hps (∂Ω)]5 ×[H e ps (S)]2 ×[H e ps−1 (S)]5 , → [Hps−1 (SN )]5 ×[Hps (SD )]5 ×[Hps−1 (S)]3 ×[H s e s (S)]5 ×[B e s−1 (S)]5 P : [Bp,q (∂Ω)]5 ×[B p,q p,q s−1 s s−1 s s−1 ep,q ep,q → [Bp,q (SN )]5 ×[Bp,q (SD )]5 ×[Bp,q (S)]3 ×[B (S)]2 ×[B (S)]5 ,

are invertible.

(4.54)

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Proof. From Theorem 4.5 it follows that the operator P1 is Fredholm with index zero if (4.52) holds. By Lemma 4.2 we conclude then that for s = 1/2 and p = 2 it is invertible. Due to Theorem 3.5 the null-spaces and indices of the operators (4.43) are the same for all values of the parameter q ∈ [1, +∞], provided p and s satisfy the inequality (4.52). Therefore, for these values of the parameters p and s they are invertible. In particular, the nonhomogeneous system (4.33)–(4.40) is uniquely solvable in the corresponding spaces. Moreover, it can be easily shown that the solution vectors h, h(2) , h(1) do not depend on the extension of the vector (D) g (D) , while ψ does. However, the sum ψ + g0 is defined uniquely. Due to Remark 4.1 we conclude that the operators (4.54) are invertible if p, q and s satisfy the conditions (4.52). With the help of this theorem we arrive at the following existence result for the original mixed BVP. Theorem 4.7. Let 4 4
(4.55)

5 Then the BVP (2.22)–(2.30) has a unique solution U in the space Wp1 (ΩS ) , which can be represented as U = V (H−1 h) + Wc (h(2) ) + Vc (h(1) ) in ΩS , where h, h(2) and h(1) are defined by the system (4.14)–(4.21). Proof. The condition (4.55) follows from the inequality (4.52) with s = 1 − 1/p. Now existence of a solution U ∈ [Wp1 (Ω)]5 with p satisfying (4.55) follows from 0 00 Theorem 4.6. Due to the inequalities − 21 < δ ≤ 0 ≤ δ < 21 we have p = 2 ∈ 4 4 3−2δ 00 , 1−2δ 0 . Therefore the unique solvability for p = 2 is a consequence of Theorem 2.1. To show the uniqueness result for all other values of p from the interval (4.55) we proceed as follows. Let a vector U ∈ [Wp1 (Ω)]5 with p satisfying (4.55) be a solution to the homogeneous boundary value problem (2.22)–(2.30). Then, it is evident that + + −1 1− 1 U ∂Ω ∈ [Bp,p p (∂Ω)]5 , T U ∂Ω ∈ [Bp,pp (∂Ω)]5 , ± ± 1− 1 −1 U S ∈ [Bp,p p (S)]5 , T U S ∈ [Bp,pp (S)]5 , (4.56) 1 + − 1− p + − ep,p (S)]5 , U S − U S ∈ [B T U S − T U S = 0 on S. By the general integral representation formula, the so called Green’s third formula, the vector U can be represented in ΩS as − + − + + U = Wc ({U }+ S −{U }S )−Vc ({T U }S −{T U }S )+W ({U }∂Ω )−V ({T U }∂Ω ) , (4.57)

that is, U = U ∗ + Wc (h(2) ) + Vc (h(1) )

in ΩS ,

(4.58)

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where − h(1) = {T U }+ S − {T U }S ,

− h(2) := {U }+ S − {U }S on S,

+ U ∗ := W ({U }+ ∂Ω ) − V ({T U }∂Ω ) in Ω.

(4.59)

Note that U ∗ solves the homogeneous equation A(∂, τ )U ∗ = 0 in Ω.

(4.60)

1−1/p

Denote h := [U ∗ ]+ (∂Ω). Since the Dirichlet problem pos∂Ω . Clearly, h ∈ Bp,p sesses a unique solution in the space Wp1 (Ω) for arbitrary p, we can represent U ∗ uniquely in the form of a single layer potential, U ∗ = V (H−1 h) in Ω. Therefore from (4.58) we get U = V (H−1 h) + Wc (h(2) ) + Vc (h(1) )

in ΩS .

(4.61)

Now, the homogeneous boundary and transmission conditions for U lead to the homogeneous system (cf. (4.14)–(4.21)) PΨ = 0, where Ψ = (h, h(2) , h(1) )> . Whence, Ψ = 0 follows immediately due to invertibility of P (see Theorem 4.7). Consequently, U = 0 in ΩS . Let us now present some regularity results for solutions of the boundary value problem (2.22)–(2.30). Theorem 4.8. Let 1 < t < ∞, 1 ≤ q ≤ ∞, 4 1 1 1 1 4
et,s−1 e s−1 Fj+ − Fj− ∈ B t (S), j = 1, 2, 3, Fk ∈ Bt, t (S), 5 e s (S), k = 4, 5, g (D) ∈ B s (SD ) 5 , g (N ) ∈ Bt,s−1 fk ∈ B t (SN ) , t, t t, t s+1/t 5 then U ∈ Ht (ΩS ) . (ii) If Fj+ , Fj− ∈ Bt,s−1 q (S),

et,s−1 e s−1 Fj+ − Fj− ∈ B q (S), j = 1, 2, 3, Fk ∈ Bt, q (S), 5 et,s q (S), k = 4, 5, g (D) ∈ Bt,s q (SD ) 5 , g (N ) ∈ Bt,s−1 fk ∈ B q (SN ) , s+1/t 5 then U ∈ Bt, q (ΩS ) . (iii) If α > 0 and α−1 Fj+ , Fj− ∈ B∞, ∞ (S),

e α−1 (S), j = 1, 2, 3, Fj+ − Fj− ∈ B ∞, ∞

α−1 α e∞, Fk ∈ B ∞ (S), fk ∈ C (S), r`c fk = 0, k = 4, 5, 5 α−1 5 g (D) ∈ C α (S D ) , g (N ) ∈ B∞, ∞ (SN ) ,

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then U∈

\

0

C α (Ωj ),

519

j = 0, 1,

α 0 <γ 0

where γ = min{α, 1/2 + δ }, −1/2 < δ 0 ≤ 0 and Ω0 is an arbitrary proper subdomain of Ω such that S ⊂ ∂Ω0 and Ω1 = Ω \ Ω0 . Moreover, in one-sided interior and exterior neighbourhoods of the surface S0 the 0 vector U has C γ −ε -smoothness with γ 0 = min {α, 1/2}, while in a one-sided 00 interior neighbourhood of the surface ∂Ω the vector U has C γ − ε -smoothness with γ 00 = min {α, 1/2 + δ 0 }; here ε is an arbitrarily small positive number. Proof. The proof is exactly the same as that of Theorem 4.3 in [22] (see also the proof of [4, Theorem 4.4]). Remark 4.9. Theorem 4.8 describes global smoothness properties of solutions. In Subsection 6.1, with the help of the asymptotic analysis, we will show that actually in a neighbourhood of the line `c the functions u and ϕ have C 1/2 regularity and the temperature function ϑ possesses C 3/2 smoothness.

5. Asymptotic expansion of solutions Here we investigate the asymptotic behaviour of the solution of the problem (2.22)– (2.30) near the curves `c and `m . For simplicity of description of the method applied below, we assume that the boundary data of the problem are infinitely smooth, Fj+ , Fj− ∈ C ∞ (S), Fj+ − Fj− ∈ C0∞ (S), j = 1, 2, 3, fk , Fk ∈ C0∞ (S), k = 4, 5, 5 5 g (D) ∈ C ∞ (S D ) , g (N ) ∈ C ∞ (S N ) , where C0∞ (S) denotes a space of functions vanishing along with all tangential derivatives on `c = ∂S (see Remark 5.1 below concerning finitely smooth data). In Section 4, we have shown that the boundary value problem (2.22)–(2.30) is uniquely solvable and the solution U can be represented by (4.9), where the densities are defined by equations (4.14)–(4.21) or by the equivalent system (4.33)– (4.40). Let Φ := (ψ, h, h(2) , h(1) )> be a solution of the system (4.33)–(4.40): P1 Φ = G, where G is the vector constructed by the right hand side data of the system G ∈ [C ∞ (S N )]5 × [C ∞ (∂Ω)]5 × [C ∞ (S)]3 × [C0∞ (S)]7 . To establish the asymptotic behaviour of the vector U near the curves `c and `m , we rewrite (4.9) as U = V H−1 ψ + Wc (e χ) + R, (5.1) where h i (D) + Wc (f0 ) + Vc (h(1) ), R := −V H−1 r∂Ω Wc (h(2) ) + r∂Ω Vc (h(1) ) − g0

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with f0 = (0, 0, 0, f4 , f5 )> . Note that rΩj R ∈ C ∞ (Ωj ), where Ωj , j = 0, 1, are as in Theorem 4.9, item (iii). The vector χ e involved in (5.1) is defined as follows: (2) (2) (2) χ e = (χ, 0, 0)> , where χ = (χ1 , χ2 , χ3 )> ≡ (h1 , h2 , h3 )> , and χ solves the pseudodifferential equation rS L(1) χ = Ψ(1) on S with Ψ(1) =

(1) (1) (1) (Ψ1 , Ψ2 , Ψ3 )> .

(5.2)

Clearly,

(1) Ψj = 2−1 (Fj+ + Fj− )−rS [T V H−1 h ]j −rS [Kc (h(1) )]j ∈ C ∞ (S), j = 1, 2, 3. Finally, the vector ψ involved in (5.1) solves the pseudodifferential equation rSN A ψ = Ψ(2) on SN ,

(5.3)

where (D)

Ψ(2) = g (N ) − rSN A g0

− rSN T Wc (h(2) ) + rSN A [r∂Ω Wc (h(2) )]

− rSN T Vc (h(1) ) + rSN A [r∂Ω Vc (h(1) )] ∈ [C ∞ (S N )]5 . The principal homogeneous symbol σL(1) (x, ξ), x ∈ S, ξ = (ξ1 , ξ2 ) ∈ R2 \ {0}, of the pseudodifferential operator L(1) is even with respect to the variable ξ and, therefore, the matrix −1 σL(1) (x, 0, +1) σL(1) (x, 0, −1), x ∈ `c , is the unit matrix I3 . Consequently, all eigenvalues of this matrix are equal to one: λj (x) = 1,

j = 1, 5, x ∈ `c .

Applying a partition of unity, natural local co-ordinate systems and local diffeomorphisms, we can straighten `c in the standard way. Then we identify a one-sided neighbourhood (in S) of an arbitrary point x e ∈ `c as a part of the half-plane x2 > 0. Thus, we assume that (x1 , 0) ∈ `c and (x1 , x2,+ ) ∈ S for 0 < x2,+ < ε. Clearly, x2,+ = dist(x, `c ). Applying the results from the references [7] and [10] we can derive the following asymptotic expansion of the solution χ of the strongly elliptic pseudodifferential equation (5.2): 1 2 χ(x1 , x2,+ ) = c0 (x1 ) x2,+ +

M X

1

+k

2 ck (x1 ) x2,+ + χM +1 (x1 , x2,+ ) ,

(5.4)

k=1

3 where ck ∈ C ∞ (`c ) , k = 0, 1, . . . , M, and the remainder term 3 + χM +1 ∈ C M +1 (`+ c,ε ) , `c,ε = `c × [0, ε]. To derive analogous asymptotic expansion for the solution vector ψ of equation (5.3), we apply the same local technique as above to a one-sided neighbourhood (in SN ) of the curve `m and preserve the same notation for the local coordinates. We proceed as follows. Consider a 5 × 5 matrix a0 (x1 ) related to the

Vol. 64 (2009) Mixed Boundary Value Problems of Thermopiezoelectricity

principal homogeneous symbol of the Steklov-Poincar´e operator A −1 a0 (x1 ) := σA (x1 , 0, +1) σA (x1 , 0, −1), (x1 , 0) ∈ `m .

521

(5.5)

Denote by λ1 (x1 ), . . . , λ5 (x1 ) the eigenvalues of the matrix a0 . Denote by µj , j = 1, · · · , l, 1 ≤ l ≤ 5, the distinct eigenvalues and by mj their algebraic multiplicities, m1 + · · · + ml = 5. It is well known that the matrix a0 (x1 ) admits the following decomposition (see, e.g., [14]) a0 (x1 ) = D(x1 ) Ja0 (x1 ) D−1 (x1 ), (x1 , 0) ∈ `m , where D is 5 × 5 nondegenerate matrix with infinitely differentiable entries and Ja0 is block diagonal Ja0 (x1 ) := diag {µ1 (x1 )B (m1 ) (1) , · · · , µl (x1 )B (ml ) (1)}. Here B (ν) (t), ν ∈ {m1 , · · · , ml }, are upper triangular matrices:  k−j t   (k−j)! , j < k, (ν) (ν) (ν) B (t) = kbjk (t)kν×ν , bjk (t) = 1, j = k,   0, j > k, that is,   1    0  B (ν) (t) =   ...    0 0

t 1 ... 0 0

t2 2!

···

t

···

... ... 0 ··· 0 ···

tν−2 (ν − 2)! tν−3 (ν − 3)! ... 1 0

tν−1 (ν − 1)! tν−2 (ν − 2)! ... t 1

          

.

ν×ν

Denote B0 (t) := diag {B (m1 ) (t), · · · , B (ml ) (t)}.

(5.6)

Again, applying the results from the reference [7] we derive the following asymptotic expansion of the solution ψ of the strongly elliptic pseudo-differential equation (5.3): 1 1 2 +∆(x1 ) ψ(x1 , x2,+ ) = D(x1 ) x2,+ B0 − log x2,+ D−1 (x1 ) b0 (x1 ) 2πi M X 1 2 +∆(x1 )+k + D(x1 ) x2,+ Bk x1 , log x2,+ + ψM +1 (x1 , x2,+ ), (5.7) k=1

5 5 + where b0 ∈ C ∞ (`m ) , ψM +1 ∈ C ∞ (`+ m,ε ) , `m,ε = `m × [0, ε], and Bk (x1 , t) = B0 −

t 2πi

k(2m0 −1)

X j=1

tj dkj (x1 ).

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5 Here m0 = max {m1 , · · · , m5 }, the coefficients dkj ∈ C ∞ (`m ) and ∆ := (∆1 , . . . , ∆5 )> , 1 1 1 log λj (x1 ) = arg λj (x1 ) + log |λj (x1 )|, ∆j (x1 ) = 2π i 2π 2π i − π < arg λj (x1 ) < π, (x1 , 0) ∈ `m , j = 1, 5. Furthermore, 1

+∆(x1 )

2 x2,+

1 21 +∆1 (x1 ) 2 +∆5 (x1 ) := diag x2,+ , · · · , x2,+ .

Now, having at hand the formulae (5.4) and (5.7) with the help of the asymptotic expansion of potential-type functions obtained in [8] we can write the following spatial asymptotic expansions of the solution vector U of the boundary value problem (2.22)–(2.30) near the crack edge `c and near the line `m , where the types of boundary conditions change. (a) Asymptotic expansion near the crack edge `c : U (x) =

l0 nX s −1 X hX µ=±1

+

1

−j

2 dsj (x1 , µ) xj3 zs,µ

s=1 j=0

M +2 MX +2−l X

1

+p+k

2 xl2 xj3 zs,µ

i dslkjp (x1 , µ) + UM +1 (x)

(5.8)

k,l=0 j+p=0 k+l+j+p≥1

with the coefficients 5 dsj ( · , µ), dslkjp ( · , µ) ∈ C ∞ (`c ) and UM +1 ∈ C M +1 (Ωj )]5 , j = 0, 1. Here Ωj , j = 0, 1, are as in Theorem 4.9, item (iii), and zs,+1 = −(x2 + x3 τs,+1 ), − π < arg zs,±1 < π,

zs,−1 = x2 − x3 τs,−1 ,

(5.9)

∞

τs,±1 ∈ C (`c ),

0 where {τs,±1 }ls=1 are the different roots of multiplicity ns , s = 1, . . . , l0 , of the −1 polynomial in ζ, det A0 Jκ> (x1 , 0, 0) η± with η± = (0, ±1, ζ)> , satisfying the condition < τs,±1 < 0. The matrix Jκ stands for the Jacobian matrix corresponding to the canonical diffeomorphism κ related to the local co-ordinate system. Under this diffeomorphism `c and S are locally rectified and we assume that (x1 , 0, 0) ∈ `c , x2 = dist(x(S) , `c ), x3 = dist(x, S), where x(S) is the projection of the reference point x ∈ ΩS onto the plane corresponding to the image of S under the diffeomorphism κ. Note that the coefficients dsj ( · , µ) can be expressed by the first coefficient c0 in the asymptotic expansion (5.4).

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(b) Asymptotic expansion near the line `m : U (x) =

l0 nX s −1 X nX µ=±1

+

s=1 j=0

M +2 MX +2−l X

h 1 i 1 2 +∆(x1 )−j B0 − xj3 dsj (x1 , µ) zs,µ log zs,µ e cj (x1 ) 2πi 1

+∆(x1 )+p+k

2 xl2 xj3 dsljp (x1 , µ) zs,µ

Bskjp x1 , log zs,µ

o

k,l=0 j+p=0 k+l+j+p≥1

+ UM +1 (x),

(5.10)

where dsj ( · , µ) and dsljp ( · , µ) are matrices with entries belonging to the space 5 5 C ∞ (`m ), e cj ∈ C ∞ (`m ) , UM +1 ∈ C M +1 (Ω1 ) and κ+∆1 (x1 ) κ+∆(x1 ) κ+∆5 (x1 ) zs,µ := diag zs,µ , · · · , zs,µ , κ ∈ R, µ = ±1, x1 ∈ `m ; Bskjp (x1 , t) are polynomials with respect to the variable t with vector coefficients which depend on the variable x1 and have the order νkjp = k(2m0 − 1) + m0 − 1 + p + j, in general,where m0 = max{m1 , . . . , ml } and m1 + · · · + ml = 5. Note that the coefficients dsj ( · , µ) can be calculated explicitly, whereas the coefficients e cj can be expressed by means of the first coefficient b0 in the asymptotic expansion (5.7) (for details see [8]). Remark 5.1. Note that the above asymptotic expansions hold also true for finitely smooth data. In this case the asymptotic expansions can be obtained as in the references [13], [7] and [8] with the help of the theory of anisotropic weighted Sobolev and Bessel potential spaces.

6. Analysis of singularities of solutions (c)

Let x0 ∈ `c and Π x0 be the plane passing trough the point x0 and orthogonal to the curve `c . We introduce the polar coordinates (r, α), r ≥ 0, −π ≤ α ≤ π, in the (c) plane Π x0 with pole at the point x0 . Denote by S ± the two different faces of the crack surface S. It is clear that (r, ±π) ∈ S ± . (m) Denote the similar orthogonal plane to the curve `m by Π x0 at the point x0 ∈ `m and introduce there the polar coordinates (r, α), with pole at the point x0 . (m) The intersection of the plane Π x0 and ΩS can be identified with the half-plane r ≥ 0 and 0 ≤ α ≤ π. In these coordinate systems, the functions zs,±1 given by (5.9) read as follows: zs,+1 = −r(cos α + τs,+1 (x0 ) sin α), zs,−1 = r(cos α − τs,−1 (x0 ) sin α), where x0 ∈ `c ∪ `m , s = 1, . . . , l0 . We can rewrite asymptotic expansions (5.8) and (5.10) in more convenient forms, in terms of the variables r and α. Moreover, we establish more refined asymptotic properties.

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6.1. Asymptotic analysis of solutions near the crack edge `c The asymptotic expansion (5.8) yields U = (u, ϕ, ϑ)> = a0 (x0 , α) r1/2 + a1 (x0 , α) r3/2 + · · · ,

(6.1)

where aj = (aj1 , . . . , aj5 )> , j = 0, 1, . . . . From this representation it follows that in a neighbourhood of the line `c there holds the imbedding U = (u, ϕ, ϑ)> ∈ [ C 1/2 ]5 .

(6.2)

More detailed analysis shows that a05 = 0 and therefore for the temperature function we have the following asymptotic expansion ϑ = a15 (x0 , α) r3/2 + a25 (x0 , α) r5/2 + · · · .

(6.3)

Indeed, we can see that u5 = ϑ solves the mixed transmission problem κij ∂i ∂j u5 + − u5 − u5 + − [T U ]5 − [T U ]5 + u5 + [T U ]5

= Q∗

in Ω \ S0 ,

(6.4)

= fe5

on S0 ,

(6.5)

= Fe5

on S0 ,

(6.6)

on SD ,

(6.7)

on SN

(6.8)

=

(D) g5 (N )

= g5

with Q∗ = τ T0 γil ∂l ui − τ T0 gi ∂i ϕ + τ α ϑ, fe5 ∈ C ∞ (S0 ), Fe5 ∈ C ∞ (S0 ),

(D)

g5

[T U ]5 = κil ni ∂l ϑ,

∈ C ∞ (S D ),

(N )

g5

∈ C ∞ (S N ),

where fe5 and Fe5 are extensions of the functions f5 and F5 from S onto the whole (D) (N ) of S0 by zero, and g5 and g5 are the fifth components of the vectors g (D) and (N ) g , respectively. The problem (6.4)–(6.8) is a classical transmission problem where transmission conditions are given on the closed interface surface S0 . Regularity of solutions to this problem near the line `c depends on smoothness of the right hand side function Q∗ . Let 1 < t < ∞, 1/t−1/2+δ 00 < s < 1/t+1/2+δ 0 . Then due to Theorem 4.8, item (i), we deduce s+1/t 5 U = (u1 , u2 , u3 , ϕ, ϑ)> ∈ Ht (ΩS . s−1+1/t

s+1+1/t

Whence Q∗ ∈ Ht (ΩS ) follows. Therefore u5 = ϑ ∈ Ht in a one-sided neighbourhood of S0 . From the embedding theorem (see [29]) it follows that for sufficiently large t there holds the inclusion ϑ ∈ C 1+ε in some neighborhood of S0 with some positive ε. Due to this regularity result, from the expansion ϑ = a05 (x0 , α) r1/2 + a15 (x0 , α) r3/2 + · · ·

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it follows that a05 = 0, i.e., for ϑ we actually have ϑ = a15 (x0 , α) r3/2 + a25 (x0 , α) r5/2 + · · · and, consequently, ϑ ∈ C 3/2 in some neighbourhood of the curve `c . 6.2. Asymptotic analysis of solutions near the curve `m The asymptotic expansion (5.10) yields U (x) =

s −1 X nX

cjµ (x0 , α) rγ+i δ B0 −

µ=±1 j=0

1 log r e cjµ (x0 , α) + · · · , 2πi

(6.9)

where rγ+i δ := diag rγ1 +i δ1 , · · · , rγ5 +i δ5 , 1 1 1 + arg λj (x0 ), δj = log |λj (x0 )|, x0 ∈ `m , j = 1, 5, 2 2π 2π and λj , j = 1, 5, are eigenvalues of the matrix −1 a0 (x0 ) = σA (x0 , 0, +1) σA (x0 , 0, −1), x0 ∈ `m . γj =

(6.10)

(6.11)

Here σA (x0 , ξ) is the principal homogeneous symbol of the Steklov-Poincar´e operator A = − 2−1 I5 + K H−1 . Moreover, the eigenvalues λj , j = 1, 5, can be expressed in terms of the eigenvalues + = σK (x0 , 0, +1), where σK (x0 , ξ) is the principal βj , j = 1, 5, of the matrix σK homogeneous symbol matrix of the singular integral operator K. Indeed, we have the following Lemma 6.1. The principal homogeneous symbol σK (x0 , ξ), x0 ∈ S N , ξ = (ξ1 , ξ2 ), is an odd matrix-function with respect to ξ and σK (x0 , ξ) = i σ eK (x0 , ξ), where the entries of the matrix σ eK (x0 , ξ) are real-valued functions. Proof. It is quite similar to the proof of Lemma 6.4 in [6].

Remark 6.2. It is not difficult to check that the principal homogeneous symbol σH (x0 , ξ) of the pseudodifferential operator H is a real even matrix-function with respect to ξ. Theorem 6.3. Let λj , j = 1, 5, be the eigenvalues of the matrix (6.11). Then λj =

1 − 2βj , 1 + 2βj

j = 1, 5,

+ where βj , j = 1, 5, are the eigenvalues of the matrix σK = σK (x0 , 0, +1).

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Proof. The characteristic equation of the matrix a0 given by (6.11) has the form nh i−1 h i o + + −1 − − −1 det (−2−1 I5 + σK ) [σH (−2−1 I5 + σK ] ) [σH − λI5 = 0, (6.12) ] where ± ± σK = σK (x0 , 0, ±1), σH = σH (x0 , 0, ±1). Since the matrix σK (x0 , ξ) is odd and the matrix σH (x0 , ξ) is even in ξ (see Lemma + − + − 6.1 and Remark 6.2), we have σK = −σK and σH = σH . Then the characteristic equation (6.12) can be rewritten as + −1 + −1 −1 + + −1 det σH [2 I5 − σK ] [2 I5 + σK ] [σH ] − λI5 = 0. + + Since the matrices σH and 2−1 I5 ± σK are non-singular, from the previous equality we derive + + det [2−1 I5 + σK ] − λ[2−1 I5 − σK ] = 0. Consequently, 11 − λ + I5 = 0. (6.13) det σK − 2 1+λ + . Then it follows from (6.13) Let βj , j = 1, 5, be the eigenvalues of the matrix σK + that the eigenvalues λj of the matrix a0 and the eigenvalues βj of σK are related by the equation 1 − λj = 2 βj , j = 1, 5, 1 + λj which completes the proof.

It can be shown that λ5 = 1, i.e., β5 = 0 (see [22] for details). Therefore, γ5 = 1/2 and δ5 = 0 in accordance with (6.10). This implies that one could not expect better smoothness for solutions than C 1/2 , in general. More detail analysis leads to the following refined asymptotic behaviour for the temperature function. Proposition 6.4. Near the line `m the function ϑ possesses the following asymptotics: ϑ = b0 r1/2 + R, (6.14) 0

where R ∈ C 1+δ −ε in a neighbourhood of `m and 1 + δ 0 − ε > 1/2 for sufficiently small ε > 0. Indeed, u5 = ϑ is a solution of the problem (6.4)–(6.8). Since the symbol of the differential operator (6.4) is positive-definite, this problem can be reduced to a system of integral equations where the principal part is described by the scalar positive-definite Steklov-Poincar´e type operator on SN possessing an even principal homogeneous symbol. Hence we can establish refined explicit asymptotics (6.14) for solution u5 = ϑ in a neighbourhood of `m (cf. [7], [8]). From (6.14) it follows that: (i) The leading exponent for u5 = ϑ in a neighborhood of line `m equals to 1/2.

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(ii) Logarithmic factors are absent in the first term of the asymptotic expansion of ϑ. (iii) The temperature function ϑ does not oscillate in a neighbourhood of line `m and for the heat flux vector we have no oscillating singularities. In what follows, we will analyze for particular piezoelectric elastic materials the exponents γj +iδj which determine the behaviour of u = (u1 , u2 , u3 ) and ϕ near the line `m . Non-zero parameters δj lead to the so called oscillating singularities for the first order derivatives of u and ϕ, in general. In turn, this yields oscillating stress singularities which sometimes lead to mechanical contradictions, for example, to an overlapping of materials. So, from the practical point of view, it is important to single out classes of solids for which the oscillating effects do not occur. To this end, we will consider a special class of bodies belonging to the 422 (Tetragonal) or 622 (Hexagonal) class of crystals for which the corresponding system of differential equations read as follows (see, e.g., [11]) (c11 ∂12 + c66 ∂22 + c44 ∂32 )u1 + ( c12 + c66 ) ∂1 ∂2 u2 + ( c13 + c44 ) ∂1 ∂3 u3 − e14 ∂2 ∂3 ϕ − γ e1 ∂1 ϑ − % τ 2 u1 = F1 , ( c12 + c66 ) ∂2 ∂1 u1 + ( c66 ∂12 + c11 ∂22 + c44 ∂32 ) u2 + ( c13 + c44 ) ∂2 ∂3 u3 + e14 ∂1 ∂3 ϕ − γ e1 ∂2 ϑ − % τ 2 u2 = F2 , ( c13 + c44 ) ∂3 ∂1 u1 + ( c13 + c44 ) ∂3 ∂2 u2 + ( c44 ∂12 + c44 ∂22 + c33 ∂32 ) u3

(6.15)

2

−γ e3 ∂3 ϑ − % τ u3 = F3 , e14 ∂2 ∂3 u1 − e14 ∂1 ∂3 u2 + ( ε11 ∂12 + ε11 ∂22 + ε33 ∂32 ) ϕ − g3 ∂3 ϑ = F4 , − τ T0 ( γ e1 ∂1 u1 + γ e1 ∂2 u2 + γ e3 ∂3 u3 ) + τ T0 g3 ∂3 ϕ + ( κ11 ∂12 + κ11 ∂22 + κ33 ∂32 ) ϑ − τ α ϑ = F5 , where c11 , c12 , c13 , c33 , c44 , c66 , are the elastic constants, e14 is the piezoelectric constant, ε11 and ε33 are the dielectric constants, γ e1 and γ e3 are the thermal strain constants, κ11 and κ33 are the thermal conductivity constants, g3 is the pyroelectric constant. Note that in the case of the Hexagonal crystals (622 class), we have c66 = (c11 − c12 )/2. It turned out that some important polymers and bio-materials are modelled by the above partial differential equations, for example, the collagen-hydroxyapatite is such a material. As it has been mentioned above, in the introduction, this material is widely used in biology and medicine (see [28]). The other important example is T eO2 [11]. In this model the thermoelectromechanical stress operator is defined as

T (∂, n) = Tjk (∂, n) (6.16) 5×5

with T11 (∂, n) = c11 n1 ∂1 + c66 n2 ∂2 + c44 n3 ∂3 , T12 (∂, n) = c12 n1 ∂2 + c66 n2 ∂1 , T13 (∂, n) = c13 n1 ∂3 + c44 n3 ∂1 , T14 (∂, n) = −e14 n3 ∂2 , T15 (∂, n) = −e γ1 n1 ,

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T21 (∂, n) = c66 n1 ∂2 + c12 n2 ∂1 , T22 (∂, n) = c66 n1 ∂1 + c11 n2 ∂2 + c44 n3 ∂3 , T23 (∂, n) = c13 n2 ∂3 + c44 n3 ∂2 , T24 (∂, n) = e14 n3 ∂1 , T25 (∂, n) = −e γ1 n2 , T31 (∂, n) = c44 n1 ∂3 + c13 n3 ∂1 , T32 (∂, n) = c44 n2 ∂3 + c13 n3 ∂2 , T33 (∂, n) = c44 n1 ∂1 + c44 n2 ∂2 + c33 n3 ∂3 , T34 (∂, n) = 0, T35 (∂, n) = −e γ3 n3 , T41 (∂, n) = e14 n2 ∂3 , T42 (∂, n) = −e14 n1 ∂3 , T43 (∂, n) = e14 (n2 ∂1 − n1 ∂2 ), T44 (∂, n) = ε11 ( n1 ∂1 + n2 ∂2 ) + ε33 n3 ∂3 , T45 (∂, n) = −g3 n3 , T5j (∂, n) = 0 for j = 1, 2, 3, 4, T55 (∂, n) = κ11 ( n1 ∂1 + n2 ∂2 ) + κ33 n3 ∂3 . The material constants satisfy the following inequalities which follow from positive definiteness of the internal energy form (see (2.11)–(2.13)): c11 > |c12 |, c44 > 0, c66 > 0, c33 (c11 + c12 ) > 2c213 , ε11 > 0, ε33 > 0, κ11 > 0, κ33 > 0. From (2.13) it follows also that ε33 α > g32 . T0 Under these conditions the mixed problem in question is uniquely solvable. Furthermore, we assume that the surface ∂Ω is parallel to the plane of isotropy in some neighbourhood of `m (i.e., to the plane x3 = 0). + In this case the symbolic matrix σK = σK (x0 , 0, +1) is calculated explicitly and has the form (see the Appendix):   0 0 0 A14 0  0 0 A23 0 0   +  A32 0 0 0 (6.17) σK =  0 , A41 0 0 0 0 0 0 0 0 0 where e14 c66 (b2 − b1 ) √ A14 = −i , 2 b 1 b2 B s √ A− B b1 = , 2 c44 ε33

e14 ε33 (b2 − b1 ) √ A41 = −i , 2 B s √ A+ B b2 = , 2 c44 ε33

A = e214 + c44 ε11 + c66 ε33 > 0,

B = A2 − 4 c44 c66 ε11 ε33 > 0, √ A > B.

A14 A41 < 0,

(6.18) (6.19) (6.20) (6.21)

To calculate the entries A23 and A32 , we have to consider two cases. We set C := c11 c33 − c213 − 2 c13 c44 ,

D := C 2 − 4 c244 c33 c11 .

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First, let D > 0.√Then it follows from the positive definiteness of the internal energy that C > D and we have c44 (d2 − d1 ) (c11 − c13 d1 d2 ) √ , 2 d1 d2 D c44 (d2 − d1 ) (c33 d1 d2 − c13 ) √ , = −i 2 d1 d2 D

A23 = i A32

s d1 =

√

C− D , 2 c44 c33

s d2 =

√ C+ D . 2 c44 c33

These equalities imply A23 A32 > 0. Now, let D < 0. We get √ √ √ a c44 ( c11 c33 − c13 ) c33 a c44 ( c11 c33 − c13 ) √ √ , A32 = −i A23 = i , √ c11 −D −D

(6.22)

(6.23)

(6.24)

where 1 a= 2

s

√ −C + 2c44 c11 c33 > 0. c44 c33

One can easily check that again A23 A32 =

√ √ c244 a2 ( c11 c33 − c13 )2 c33 > 0. √ −D c11

+ The characteristic polynomial of the matrix σK can be represented as   −β A14 0 0 0 A41 −β 0 0 0    +  0 −β A23 0  det(σK − β I) = det  0 .  0 0 A32 −β 0  0 0 0 0 −β

(6.25)

+ Therefore, we have the following expressions for the eigenvalues of the matrix σK : p p β1,2 = ±i −A14 A41 , β3,4 = ± A23 A32 , β5 = 0. (6.26)

Then by Theorem 6.3 √ √ 1 − 2i −A14 A41 1 1 − 2 A23 A32 1 √ √ λ1 = , λ2 = , λ3 = , λ4 = , λ5 = 1. λ1 λ3 1 + 2i −A14 A41 1 + 2 A23 A32 Note that |λ1 | = |λ2 | = 1. Moreover, since λ3 and λ4 are real, they are positive (see Subsection 3.2).

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Applying the above results we can explicitly write the exponents of the first terms of the asymptotic expansions of solutions (see (6.10)): p 1 1 γ1 = − arctan 2 −A14 A41 , δ1 = 0, (6.27) 2 π p 1 1 δ2 = 0, (6.28) γ2 = + arctan 2 −A14 A41 , 2 π √ 1 1 1 − 2 A23 A32 √ γ3 = γ4 = , , (6.29) δ3 = −δ4 = δe = log 2 2π 1 + 2 A23 A32 1 γ5 = , δ5 = 0. (6.30) 2 Now we can draw the following conclusions: 1. The solutions of the problem possess the following asymptotics near the line `m : 1

1

1

(u, ϕ)> = c0 rγ1 + c1 r 2 +i δ + c2 r 2 −i δ + c3 r 2 + c4 rγ2 + · · · e

e

ϑ = b3 r1/2 + b4 rγ2 + · · · where

(6.31) (6.32)

1 1 1 1 − arctan b, γ2 = + arctan b 2 π 2 π r ε33 c66 |e14 | q b= 4 . √ √ ε11 c44 e2 + ( ε c + ε c )2 γ1 =

and

14

11 44

(6.33)

33 66

As we see, the exponent γ1 characterizing the behaviour of u and ϕ near the line `m depends on the elastic constants, piezoelectric constants, and dielectric constants and does not depend on the thermal constants. Moreover, γ1 takes on values from the interval (0, 1/2), and the following mathematical limit relations hold: (i) if c66 → 0, or c66 → ∞, or c44 → ∞, or ε11 → ∞, or ε33 → 0, or ε33 → ∞, or e14 → 0, then γ1 → 1/2;r ε33 c66 1 1 (ii) if e14 → ∞, then γ1 → − arctan 4 ; ε11 c44 2 π (iii) if c44 → 0, or ε11 → 0, then γ1 → 0. Note also that, if ε11 = ε33 and c44 = c66 , then γ1 =

1 1 |e14 | − arctan p 2 2 π e14 + 4ε11 c44

and 1/4 < γ1 < 1/2. For the general anisotropic case these exponents also depend on the geometry of the line `m , in general. 2. B0 (t) = I, i.e., the first five terms of the asymptotic expansion do not contain logarithmic factors. 3. Since γ1 < 1/2, we have no oscillating singularities for physical fields in some neighbourhood of the curve `m .

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Note that in the classical elasticity theory (for both isotropic and anisotropic solids) for mixed BVPs the dominant exponents are 1/2, 1/2 ± i δe with δe 6= 0 and consequently we have oscillating stress singularities. The following questions arise naturally: a) Does there exist a class of piezoelectric media for which the real part of the principal exponent defining the dominant stress singularity near the line `m does not depend on the material constants? b) Does there exist a class of piezoelectric media for which the real part of the principal exponent equals 1/2? As we will see below, both questions have positive answers. Indeed, let us consider the class of piezoelectric media with cubic anisotropy. Note that such materials as Bi12 GeO20 and GaAs belong to this class (see, e.g., [11]). The latter material is widely used in the electronic industry. The corresponding system of differential equations in this case reads as (c11 ∂12 + c44 ∂22 + c44 ∂32 )u1 + ( c12 + c44 ) ∂1 ∂2 u2 + ( c12 + c44 ) ∂1 ∂3 u3 + 2e14 ∂2 ∂3 ϕ − γ e1 ∂1 ϑ − % τ 2 u1 = F1 , ( c12 + c44 ) ∂2 ∂1 u1 + ( c44 ∂12 + c11 ∂22 + c44 ∂32 ) u2 + ( c12 + c44 ) ∂2 ∂3 u3 + 2e14 ∂1 ∂3 ϕ − γ e1 ∂2 ϑ − % τ 2 u2 = F2 , ( c12 + c44 ) ∂3 ∂1 u1 + ( c12 + c44 ) ∂3 ∂2 u2 + ( c44 ∂12 + c44 ∂22 + c11 ∂32 ) u3 + 2e14 ∂1 ∂2 ϕ − γ e3 ∂3 ϑ − % τ 2 u3 = F3 ,

(6.34)

− 2e14 ∂2 ∂3 u1 − 2e14 ∂1 ∂3 u2 − 2e14 ∂1 ∂2 u3 + ( ε11 ∂12 + ε11 ∂22 + ε11 ∂32 ) ϕ − g3 ∂3 ϑ = F4 , − τ T0 ( γ e1 ∂1 u1 + γ e1 ∂2 u2 + γ e3 ∂3 u3 ) + τ T0 g3 ∂3 ϕ + ( κ11 ∂12 + κ11 ∂22 + κ33 ∂32 ) ϑ − τ α ϑ = F5 . The elastic, piezoelectric and thermal constants involved in the governing equations satisfy the following conditions: −1/2 < c12 /c11 < 1,

c11 > 0,

c44 > 0,

ε11 > 0,

ε11 α/T0 > g32 ,

κ11 > 0,

κ33 > 0.

(6.35)

+ In this case, the matrix σK is self-adjoint and reads as

 0 0 0 0  + 0 A σK = 32  0 0 0 0

0 A23 0 0 0

0 0 0 0 0

 0 0  0 , 0 0

(6.36)

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where c44 (d2 − d1 )(c11 − c12 ) √ A23 = A32 = i for D > 0, 2 D s s √ √ C− D C+ D d1 = , d2 = , 2c44 c11 2c44 c11 c44 a (c11 − c12 ) √ for D < 0. A23 = A32 = i 2 −D Here 2

D=C −

4 c211 c244 ,

C=

c211

−

c212

− 2 c12 c44 ,

1 a= 2

s

√ −C + 2c44 c11 > 0. c44 c11

By simple calculations we derive βj = 0, j = 1, 2, 5,

β3,4 = ±|A23 |,

λj = 1, j = 1, 2, 5,

λ3 =

1 − 2|A23 | > 0, 1 + 2|A23 |

λ4 =

1 , λ3

and γj =

1 1 − 2|A23 | 1 , j = 1, 5, δj = 0, j = 1, 2, 5, δ3 = −δ4 = δe = log . (6.37) 2 2π 1 + 2|A23 |

From Lemma 6.1, Remark 6.2 and the equalities (6.36) and (6.11) we derive h i−1 h i + + −1 − − −1 + + −1 a0 = (−2−1 I5 + σK ) [σH ] (−2−1 I5 + σK ) [σH ] = σH e a0 [σH ] , where + −1 + e a0 = [2−1 I5 − σK ] [2−1 I5 + σK ].

This matrix is self-adjoint due to the equality (6.36) and it is similar to a diagonal matrix, i.e., there is a unitary matrix D such that D e a0 [D]−1 is diagonal. Therefore the matrix a0 can be reduced to a diagonal matrix by the non-degenerate matrix + σH D−1 . In turn, this implies that the leading terms of the asymptotic expansion (6.9) near the curve `m do not contain logarithmic factors. As a result we obtain the asymptotic expansion (u, ϕ)> = c0 r1/2 + c1 r1/2+i δ + c2 r1/2−i δ + · · · e

e

ϑ = b0 r1/2 + b1 r3/2 + · · · . Consequently, the solution is C 1/2 -smooth in a closed neighbourhood of `m .

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7. Appendix: Calculation of the symbolic matrices + In this section we calculate the principal homogeneous symbol matrix σK = σK (x1 , 0, +1) corresponding to the system (6.15) (422 and 622 classes). To this end we write the fundamental matrix (3.1) in the form

−1 −1 H(x, τ ) = Fξ→x A (−iξ, τ ) Z h −1 −iζx3 i 1 dζ , A(−iξ 0 , −iζ, τ ) e = Fξ−1 ± 0 →x0 2π l±

(A.1)

where the sign “−” corresponds to the case x3 > 0 and the sign “+” to the case x3 < 0. Here x0 = (x1 , x2 ), ξ 0 = (ξ1 , ξ2 ), ξ = (ξ 0 , ξ3 ), l+ (l− ) is a closed contour with positive orientation enveloping all the roots of the polynomial det A(−iξ 0 , −iζ, τ ) with respect to the variable ζ in the half-plane = ζ > 0 ( = ζ < 0 ). First, we write the principal homogeneous symbols A(0) and T (0) of the operators A(∂, τ ) and T (∂, n) at a point ζe = (0, 1, ζ). Choosing a local coordinate system appropriately, we can assume that the exterior unit normal vector at this point reads as n = (0, 0, 1). Then we have  A11 0 0 A14 0  0 A22 A23 0 0    (0) e  A23 A33 0 0  A (ζ) = −  0 , −A14 ζ 0 0 A44 0  0 0 0 0 A55   ic44 ζ 0 0 −ie14 0  0 ic44 ζ ic44 0 0    (0) e  ic13 ic33 ζ 0 0  T (ζ, n) = −  0 ,  0 0 0 iε33 ζ 0  0 0 0 0 iκ33 ζ 

(A.2)

(A.3)

where A11 = c44 ζ 2 + c66 ,

A14 = −e14 ζ,

A22 = c44 ζ 2 + c11 ,

A23 = (c13 + c44 )ζ,

A33 = c33 ζ 2 + c44 ,

A44 = ε33 ζ 2 + ε11 ,

A55 = κ33 ζ 2 + κ11 . From (3.3), (A.1)–(A.3) and Theorem 3.3 it follows that Z 1 1 + e n) A(0) (ζ) e −1 e−iζx3 dζ − I + σK = lim T (0) (ζ, x3 →0 2π l+ 2 Z 1 e n) A(0) (ζ) e −1 dζ = kAkj k5×5 , T (0) (ζ, = 2π l+

(A.4)

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where Z c44 ε33 ζ 3 + (c44 ε11 + e214 )ζ i c66 e14 dζ, A14 = − dζ, P1 (ζ) 2π l+ P1 (ζ) l+ Z c33 c44 ζ 3 − c13 c44 ζ i dζ, A1j = 0, j = 2, 3, 5, A22 = 2π l+ P2 (ζ) Z i c13 c44 ζ 2 − c11 c44 A23 = − dζ, A2j = 0, j = 1, 4, 5, 2π l+ P2 (ζ) Z i c33 c44 ζ 2 − c13 c44 A3j = 0, j = 1, 4, 5, A32 = − dζ, 2π l+ P2 (ζ) Z c33 c44 ζ 3 + (c11 c33 − c13 c44 − c213 )ζ i dζ, A33 = 2π l+ P2 (ζ) Z i e14 ε33 ζ 2 A41 = − dζ, A4j = 0, j = 2, 3, 5, 2π l+ P1 (ζ) Z i c44 ε33 ζ 3 + c66 ε33 ζ A44 = dζ, 2π l+ P1 (ζ) Z κ33 ζ i A5j = 0, j = 1, 2, 3, 4, A55 = dζ, 2π l+ κ33 ζ 2 + κ11 P1 (ζ) = c44 ε33 ζ 4 + (c44 ε11 + c66 ε33 + e214 )ζ 2 + c66 ε11 , A11 =

i 2π

Z

P2 (ζ) = c33 c44 ζ 4 + (c11 c33 − 2c13 c44 − c213 )ζ 2 + c11 c44 . (j)

(j)

Denote by ζ1 , ζ2 , j = 1,p2, the roots of the polynomials Pj with positive imaginary part and let ζ (3) = i κ11 /κ33 . We have s s √ √ A − B A+ B (1) (1) ζ1 = ib1 = i , ζ2 = ib2 = i , 2c44 ε33 2c44 ε33 s s √ √ C− D C+ D (2) (2) ζ1 = id1 = i , ζ2 = id2 = i , 2c44 c33 2c44 c33 A = e214 + c44 ε11 + c66 ε33 > 0,

B = A2 − 4c44 c66 ε11 ε33 > 0,

C = c11 c33 − c213 − 2c13 c44 ,

D = C 2 − 4c244 c33 c11 . (2)

(2)

Note that, if D > 0, then the roots ζ1 and ζ2 are purely imaginary. For D < 0 the roots are complex numbers with opposite real parts and equal imaginary parts: (2)

ζ1

= a + i b,

(2)

ζ2

= −a + i b,

a > 0,

b > 0.

To calculate the above curvilinear integrals, we apply theory of residues and Cauchy’s theorem. Taking into account the equalities Z Z (1) (2) (1) (2) i π (ζ2 − ζ1 ) dζ i π (ζ2 − ζ1 ) dζ = , = , √ √ (1) (1) (2) (2) l+ P2 (ζ) l+ P1 (ζ) ζ1 ζ2 B ζ1 ζ2 D

Vol. 64 (2009) Mixed Boundary Value Problems of Thermopiezoelectricity

Z l+

Z l+

Z l+

ζ dζ = 0, P1 (ζ) ζ2 i π (1) (1) dζ = − √ (ζ2 − ζ1 ), P1 (ζ) B iπ ζ3 dζ = , P1 (ζ) c44 ε33

Z l+

Z l+

Z l+

535

ζ dζ = 0, P2 (ζ) i π (2) ζ2 (2) dζ = − √ (ζ2 − ζ1 ), P2 (ζ) D iπ ζ3 dζ = , P2 (ζ) c44 c33

we obtain (1)

(1)

e14 c66 (ζ2 − ζ1 ) , (1) (1) √ 2ζ1 ζ2 B

1 Ajj = − , j = 1, 5, 2

A14 =

A2j = 0, j = 1, 4, 5,

A23 = −

c44 (ζ2 − ζ1 )(c11 + c13 ζ1 ζ2 ) , (2) (2) √ 2ζ1 ζ2 D

A3j = 0, j = 1, 4, 5,

A32 = −

c44 (ζ2 − ζ1 )(c33 ζ1 ζ2 + c13 ) , (2) (2) √ 2ζ1 ζ2 D

(1)

A41 = −

(2)

(2)

(2)

(2)

A1j = 0, j = 2, 3, 5, (2) (2)

(2) (2)

(1)

e14 ε33 (ζ2 − ζ1 ) √ , 2 B

A4j = 0, j = 2, 3, 5,

A5j = 0, j = 1, 4.

Now, taking into account that ζj

(1)

= i bj , bj > 0, j = 1, 2,

(2) ζj

= i dj , dj > 0, j = 1, 2,

(2)

ζ1

= a + i b,

(2)

ζ2

= −a + i b,

if D > 0, a > 0,

b > 0,

(2) (2)

ζ 1 ζ2

r c11 , =− c33

if D < 0,

we immediately obtain (6.18), (6.19), (6.22)–(6.24). + One can calculate the homogeneous symbol matrix σK = σK (x1 , 0, +1) corresponding to the system (6.34) quite similarly.

References [1] M. Agranovich, Elliptic singular integro-differential operators. Uspekhi Mat. Nauk, 20, No. 5 (1965), 3–120. [2] T. Buchukuri, O. Chkadua, Boundary problems of thermopiezoelectricity in domains with cuspidal edges. Georgian Math. J., 7, No. 3 (2000), 441–460. [3] T. Buchukuri, O. Chkadua, R. Duduchava, Crack-type boundary value problems of electro-elasticity. Oper. Theory Adv. Appl., 147 (2004), 189–212. [4] T. Buchukuri, O. Chkadua, D. Natroshvili, A.-M. S¨ andig, Solvability and regularity results to boundary-transmission problems for metallic and piezoelectric elastic materials. Preprint 2005/004, IANS, University of Stuttgart, 2005. (Accepted for publication in “Math. Nachr.”)

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[5] T. Buchukuri, O. Chkadua, D. Natroshvili, A.-M. S¨ andig, Interaction problems of metallic and piezoelectric elastic materials with regard to thermal stresses. Memoirs on Differential Equations and Mathematical Physics, 45 (2008), 7–74. [6] O. Chkadua, The Dirichlet, Neumann and mixed boundary value problems of the theory of elasticity in n-dimensional domains with boundaries containing closed cuspidal edges. Math. Nachr., 189 (1998), 61–105. [7] O. Chkadua, R. Duduchava, Pseudodifferential equations on manifolds with boundary: Fredholm property and asymptotic. Math. Nachr., 222 (2001), 79–139. [8] O. Chkadua, R. Duduchava, Asymptotic of functions represented by potentials. Russian J. Math. Phys., 7, No. 1 (2000), 15–47. [9] M. Comninou,The interface crack. ASME J. Appl. Mech., 44 (1977), 631–636. [10] M. Costabel, M. Dauge, R. Duduchava, Asymtotics without logarithmic terms for crack problems. Comm. Partial Differential Equations, 28, Nos. 5&6 (2003), 869– 926. [11] E. Dieulesaint, D. Royer, Ondes ´elastiques dans les solides - Application au traitement du signal. Masson & Cie , 1974. [12] R. Duduchava, The Green formula and layer potentials. Integral Equations and Operator Theory, 41, No. 2 (2001), 127–178. [13] G. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations. Transl. of Mathem. Monographs, Amer. Math. Soc., 52, Providence, Rhode Island, 1981. [14] F. Gantmakher, Theory of Matrices. Nauka, Moscow, 1967. [15] G. Grubb, Pseudo-differential boundary problems in Lp spaces. Comm. Partial Differential Equations, 15 No. 3 (1990), 289–340. [16] L. L. Hench, Bioceramics. J. Am. Ceram. Soc., 81 (7) (1998), 1705–1728. [17] L. H¨ ormander, The Analysis of Linear Partial Differential Operators, PseudoDifferential Operators, Vol. 3. Springer Verlag, Heidelberg 1985. [18] S. Lang, Guide to the Literature of Piezoelectricity and Pyroelectricity. 20, Ferroelectrics, 297 (2003), 107–253. [19] S. Lang, Guide to the Literature of Piezoelectricity and Pyroelectricity. 22, Ferroelectrics, 308 (2004), 193–304. [20] S. Lang, Guide to the Literature of Piezoelectricity and Pyroelectricity. 23, Ferroelectrics, 321 (2005), 91–204. [21] S. Lang, Guide to the Literature of Piezoelectricity and Pyroelectricity. 24, Ferroelectrics, 322 (2005), 115–210. [22] D. Natroshvili, T. Buchukuri, O. Chkadua, Mathematical modelling and analysis of interaction problems for piezoelectric composites. Rendiconti Academia Nazionale delle Scienze detta dei XL, Memorie di Matematica e Applicazioni, 124◦ , Vol. XXX, fasc. 1 (2006), 159–190. [23] D. Natroshvili, O. Chkadua, E. Shargorodsky, Mixed type boundary value problems of the anisotropic elasticity. Tr. In-ta Prikl. Mat. Tbilisi Univ., 39 (1990), 133–181. [24] W. Nowacki, Efecty electromagnetyczne w stalych cialach odksztalcalnych. Warszawa, Panstwowe Widawnictwo Naukowe, 1983. [25] Y. E. Pak, Linear electro-elastic fracture mechanics of piezoelectric materials. International Journal of Fracture, 54 (1992), 79–100.

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[26] Q. H. Qin, Fracture Mechanics of Piezoelastic Materials. WIT Press, 2001. [27] E. Shargorodsky, An Lp -Analogue of the Vishik-Eskin Theory. Memoirs on Differential Equations and Mathematical Physics, 2 (1994), 41–146. [28] C. C. Silva, D. Thomazini, A. G. Pinheiro, N. Aranha, S. D. Figueir´ o, J. C. G´ oes, A. S. B. Sombra, Collagen-hydroxyapatite films: Piezoelectric properties. Materials Science and engineering, B86 (2001), 210–218. [29] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam, 1978. [30] W. Voight, Lehrbuch der Kristall-Physik. Leipzig, Teubner, 1910. T. Buchukuri and O. Chkadua A. Razmadze Mathematical Institute 1 M. Aleksidze st. Tbilisi 0193 Georgia e-mail: t [email protected] [email protected] D. Natroshvili Georgian Technical University 77 M. Kostava st. Tbilisi 0175 Georgia e-mail: [email protected] Submitted: October 23, 2008. Revised: April 25, 2009.

Integr. equ. oper. theory 64 (2009), 539–552 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040539-14, published online August 3, 2009 DOI 10.1007/s00020-009-1705-y

Integral Equations and Operator Theory

Ideal-Triangularizability of Semigroups of Positive Operators Roman Drnovˇsek and Marko Kandi´c Abstract. Let S be a multiplicative semigroup of positive operators on a Banach lattice E such that every S ∈ S is ideal-triangularizable, i.e., there is a maximal chain of closed subspaces of E that consists of closed ideals invariant under S. We consider the question under which conditions the whole semigroup S is simultaneously ideal-triangularizable. In particular, we extend a recent result of G. MacDonald and H. Radjavi. We also introduce a class of positive operators that contains all positive abstract integral operators when E is Dedekind complete. Mathematics Subject Classification (2000). Primary 47A15; Secondary 47B65. Keywords. Invariant subspaces, Banach lattices, closed ideals, positive operators, abstract integral operators, semigroups of operators.

1. Introduction and preliminaries Let E be a normed Riesz space, and let E + denote the positive cone of E. By an operator on E we mean a bounded linear transformation from E into itself. Let L(E) denote the algebra of all operators on E. An operator T ∈ L(E) is called positive whenever T x ∈ E + for all x ∈ E + . A family F ⊆ L(E) is said to be reducible if there exists a nontrivial closed subspace of E that is invariant under every member of F. Otherwise, we say that F is irreducible. If there exists a maximal subspace chain (i.e., a maximal totally ordered set of closed subspaces) whose elements are invariant under every member of F, then F is said to be triangularizable. The monograph [12] contains an overview of numerous results concerning reducibility and triangularizability of families of operators. For positive operators on E, a natural question is whether there exists a nontrivial invariant closed subspace of a simple geometrical form, i.e., an ideal or even a band. Therefore, it is convenient to introduce the following terminology.

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A family F ⊆ L(E) is said to be ideal-reducible (resp. band-reducible) if there exists a nontrivial closed ideal (resp. band) of E which is invariant under every operator in F. A family F is ideal-triangularizable if there is a chain C that is maximal as a chain of closed ideals of E and that has the property that every ideal in C is invariant under all the operators in F. Any such chain of closed ideals is said to be an ideal-triangularizing chain for the family F. It is proved in [4, Proposition 1.2] that every ideal-triangularizing chain is also a maximal chain of closed subspaces. Therefore, a family of operators is ideal-triangularizable if and only if it is triangularizable with respect to a subspace chain consisting of closed ideals. In particular, for each closed ideal I of an ideal-triangularizing chain C, the quotient space I/I − is at most one-dimensional, where I − = cl(∪(J : J ∈ C, J ⊂ I, J 6= I)) and cl denotes the norm closure. An ideal-triangularizing chain C is said to be continuous if I − = I for each closed ideal I ∈ C. It should be noted that the definition of ideal-triangularizability was already introduced in [6] as a Banach lattice analogue of the concept of triangularizability. In the case of Lp -spaces (1 ≤ p < ∞) this concept is also studied in [12], where it is called completely decomposable, while bands are called standard subspaces. A semigroup of operators is a family of operators closed under multiplication. In 1986, B. de Pagter [11] proved the long standing conjecture that every positive quasinilpotent compact operator on a Banach lattice (of dimension at least 2) is ideal-reducible, from which it follows easily that it is ideal-triangularizable. In [3] this result was extended to semigroups of positive quasinilpotent compact operators by an application of a deep result of Y. V. Turovski [13] asserting that every semigroup of quasinilpotent operators on a Banach space is triangularizable. As a generalization of the Ando-Krieger theorem, one can derive from this that a semigroup of quasinilpotent positive abstract integral operators on a Dedekind complete Banach lattice is ideal-reducible [2]. In the case of Lp -spaces (1 ≤ p < ∞) this was also shown in [12, Theorem 9.4.9]. In [9], G. MacDonald and H. Radjavi have continued this investigation by studying semigroups of more general positive operators on Lp -spaces. Our main result (Theorem 6.3) is not just a Banach lattice generalization of [9, Theorem 3], but it is also its extension in the case of Lp -spaces. In the rest of this section we recall some relevant definitions and results. For the terminology not explained in the text we refer to the books [16], [1], and [10]. Given a positive operator T on a normed Riesz space E, we denote by N (T ) the absolute kernel or null ideal of T , that is, N (T ) = {x ∈ E : T |x| = 0} . It is easy to see that N (T ) is a closed ideal of E. Given a family F of positive operators on E, let N (F) denote the intersection of all absolute kernels of operators from F, that is, \ N (F) = N (T ) . T ∈F

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For the dual space E ∗ of a normed Riesz space E we need the following simple lemma. Lemma 1.1. Let I be a closed ideal in E. If I 6= E then there exists a nonzero positive functional ϕ ∈ E ∗ which vanishes on I. Proof. Choose any vector x ∈ / I. Then at least one of the vectors x+ and x− is not contained in the ideal I. Thus we may assume that x > 0. By [15, Theorem 39.3], there exists a positive linear functional ψ on the quotient normed Riesz space E/I such that ψ(x + I) = kx + Ik > 0. Denote by π the quotient projection of E onto E/I. Then ϕ = ψ ◦ π is a positive linear functional on E such that ϕ(x) = ψ(x + I) = kx + Ik > 0 and ϕ(y) = 0 for all y ∈ I. Let E be a complex Dedekind complete Banach lattice, i.e., E = Re E⊕iRe E, where Re E is a real Dedekind complete Banach lattice. Denote by Lr (E) the space of all regular operators on E, i.e., those operators that can be expressed as linear combinations of positive operators. This subspace of the space L(E) is a complex Banach lattice algebra with respect to the regular norm defined by kT kr := k|T |k. The center Z(E) defined by Z(E) := {T ∈ Lr (E) : |T | ≤ λI for some λ ≥ 0} is a commutative subalgebra of the space L(E). If T ∈ Z(E), then kT k = kT kr = min{λ ≥ 0 : |T | ≤ λI}. Since Z(E) is also a band in Lr (E), we have the band decomposition Lr (E) = Z(E) ⊕ Z(E)d . The associated band projections in Lr (E) onto Z(E) and Z(E)d are denoted by D and N , respectively. By a result of Voigt [14], the diagonal map D is a contraction with respect to the operator norm, i.e., kD(T )k ≤ kT k for all T ∈ Lr (E).

2. Reducibility of semigroups We begin with a generalization of [12, Lemma 8.7.6] that was proved in the case of Lp -spaces. We recall that a subset I of a semigroup S is said to be a semigroup ideal if ST and T S belong to I for all S ∈ S and T ∈ I. Proposition 2.1. Let E be a normed Riesz space, and let S be a nonzero semigroup of positive operators on E. The following statements are equivalent: (1) S is ideal-reducible; (2) there exist a nonzero positive functional ϕ ∈ E ∗ and a nonzero positive vector f ∈ E such that ϕ(Sf ) = {0}; (3) there exist nonzero positive operators A and B on E such that ASB = {0}; (4) some nonzero semigroup ideal of S is ideal-reducible. Proof. The implication (1)⇒(4) follows immediately, while the converse (4)⇒(1) is shown in [4, Lemma 1.3]. To show (1)⇒(2) let L be a nontrivial closed ideal invariant under the semigroup S. If f ∈ L is a nonzero positive vector, then Sf ⊆ L. By Lemma 1.1 there exists a nonzero positive functional ϕ ∈ E ∗ which annihilates the ideal L. Thus ϕ(Sf ) = {0}.

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To show (2)⇒(3) assume that there exist a nonzero positive functional ϕ ∈ E ∗ and a nonzero positive vector f ∈ E such that ϕ(Sf ) = {0}. Define positive operators A = B = f ⊗ ϕ of rank one. It can be easily verified that ASB = {0}. To show (3)⇒(1) assume that there exist nonzero positive operators A, B on E such that ASB = {0} and choose a positive vector g ∈ E such that f = Bg 6= 0. If Sf = {0}, then the closed ideal generated by f is nontrivial and invariant under S. So with no loss of generality one can assume that Sf 6= {0}. Let L denote the ideal generated by the set Sf . We claim that L is invariant under any operator S ∈ S. Let h be an arbitrary positive vector in L. Then there exist n ∈ N, positive scalars λ1 , . . . , λn and S1 , . . . , Sn ∈ S such that n X 0≤h≤ λi Si f. i=1

It follows that 0 ≤ Sh ≤

n X

λi SSi f,

i=1

and so Sh ∈ L as claimed. For any h ∈ L, we also obtain that Ah = 0, since ASf = 0 for all S ∈ S. It follows that the closure cl(L) of L is a proper closed ideal that is invariant under S. We continue with the next analogue of Proposition 2.1 for band-reducibility. Proposition 2.2. Let E be a normed Riesz space with the projection property, and let S be a nonzero semigroup of positive order continuous operators on E. The following statements are equivalent: (1) S is band-reducible; (2) there exist nonzero vectors f, g ∈ E + such that Sf ⊆ {g}d ; (3) there exist nonzero positive order continuous operators A and B on E such that ASB = {0}; (4) some nonzero semigroup ideal in S is band-reducible. These conditions are implied by (5) there exist a nonzero positive order continuous functional ϕ ∈ E ∗ and a nonzero vector f ∈ E + such that ϕ(Sf ) = {0}. Proof. To see that (1) implies (2), let L be a nontrivial band invariant under the semigroup S. Pick any nonzero vector f ∈ L+ . Then the set Sf is contained in L. If g ∈ Ld is any nonzero positive vector, then Sf ⊆ L ⊆ {g}d . Let us prove that (2) implies (3). Since each member of S is order continuous, the band {Sf }dd generated by the set Sf is invariant under the semigroup S. Let P be the band projection on {Sf }dd . Since g is nonzero and {Sf }dd ⊆ {g}d , P is not equal to the identity operator I and we have (I − P )SP = {0}. This proves (3) unless P = 0. In this case the band {f }dd is invariant under the semigroup S, and it is nontrivial, since S 6= {0}. If Q is the corresponding band projection, then 0 6= Q 6= I and (I − Q)SQ = {0}, so that (3) holds.

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To show that (3) implies (1), assume that there exist nonzero positive order continuous operators A, B : E → E such that ASB = {0}. Choose an arbitrary positive vector g ∈ E such that f = Bg 6= 0. If Sf = {0}, then the band {f }dd is nontrivial and invariant under S. Assume that Sf 6= {0}, and let L be the band generated by the set Sf . Then L is invariant under S, since each member of S is order continuous. Since L ⊆ N (A) 6= E, L is also nontrivial, so that (1) holds. The implication (1)⇒(4) follows immediately. To see the converse implication, let J be a nonzero semigroup ideal of S that is band-reducible, and let L be a nontrivial band invariant under J . As in the proof of [4, Lemma 1.3], the order ideal J generated by the set {Sf : S ∈ J , f ∈ L ∩ E + } is invariant under S. The band L1 generated by J is also invariant under S, since each member of S is order continuous. Since J ⊆ L, L1 ⊆ L and therefore L1 6= E. If L1 6= {0}, we are done. If L1 = {0}, then L ⊆ N (S), so that the absolute kernel N (S) is a nontrivial invariant band, and thus (1) holds. To finish the proof, assume that (5) is satisfied. Then the positive operator A = B = f ⊗ ϕ is order continuous. It is easy to verify that ASB = {0}. This proves that (5) implies (3). The condition (5) of the last result is not implied by any of the conditions (1)– (4), because it is possible that there is no nonzero order continuous functional on E. For example, take for E the Dedekind completion of the Banach lattice C([0, 1]). The following proposition asserts that this cannot happen if E is a Banach function space. For a σ-finite measure space (X, µ), we denote by L0 (X, µ) the vector space of all equivalence classes of real µ-measurable functions on X. Proposition 2.3. Let E be a Banach function space in L0 (X, µ). If S is a bandreducible semigroup of positive order continuous operators on E, then the statements (1)–(5) of Proposition 2.2 are equivalent. Proof. We must show that (1) implies (5). Let B be a nontrivial band of E which is invariant under S. Then B consists of all the functions from E that are zero on some measurable subset Y of X such that both Y and Y c have positive measures. We omit the proof of the last statement, since it is very similar to the proof of the proposition describing bands of the space L0 (X, µ) (see [7, Example 4.I]). Let g be an arbitrary nonzero positive function in the associate space E 0 of E R which vanishes on Y c . Then the linear functional ϕg on E defined by ϕg (f ) = X gf dµ is a nonzero positive, order continuous functional on E. If f ∈ B is a nonzero positive function, then ϕg (Sf ) = {0}, so that (5) holds.

3. Ideal-reducibility of abstract integral operators Let E be a Dedekind complete Banach lattice. As in [2], an operator in Lr (E) is said to be an abstract integral operator if it belongs to the band (E ∗ ⊗ E)dd

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in Lr (E) generated by all finite-rank operators. It should be noted that in the usual definition of abstract integral operators the norm dual E ∗ is replaced by the subband En∗ in E ∗ of all order continuous functionals (see e.g. [16, Chap. 13]). The collection of all positive, order continuous, abstract integral operators is a semigroup as the following result implies. Lemma 3.1. Let E be a Dedekind complete Banach lattice and T a positive abstract integral operator on E. Suppose that S and U are positive operators on E, and U is order continuous. Then the operators T S and U T are abstract integral operators. Proof. Since T is a positive abstract operator, there exists an increasing net {Tα } with T as its supremum, and there is a net {Kα } of finite rank operators such that 0 ≤ Tα ≤ Kα for each α. Then the operators Kα S and U Kα are of finite rank, and we have 0 ≤ Tα S ≤ Kα S and 0 ≤ U Tα ≤ U Kα . That 0 ≤ Tα S ↑ T S is immediately, while 0 ≤ U Tα ↑ U T follows because of the order continuity of the operator U . We now introduce the following class of positive operators on a Banach lattice E. By AI(E) we denote the set of all positive operators T on E such that there exist a net {Tα }α∈A of positive operators and a net {Kα }α∈A of positive compact operators with 0 ≤ Tα ≤ T and 0 ≤ Tα ≤ Kα for all α ∈ A, and N ({Tα }α ) = N (T ) (it means that for each positive vector x ∈ E with T x > 0 there exists an α such that Tα x > 0). Note that if E is a Dedekind complete Banach lattice, then AI(E) contains all positive abstract integral operators on E. It is also clear from the definition that if T ∈ AI(E) and S is a positive operator on E, then T S ∈ AI(E). In particular, the set AI(E) is a semigroup of positive operators. The following lemma extends [2, Lemma 2.2]. Lemma 3.2. Let E be a Banach lattice, and let F be a family of operators in AI(E). If N (F) = {0}, then for each nonzero positive vector x0 ∈ E there exist operators S, T and U in F and a positive compact operator K on E such that K ≤ U T S and Kx0 > 0. Proof. Pick any nonzero positive vector x0 ∈ E. Since N (F) = {0}, there exists an operator S ∈ F such that Sx0 > 0. Since S ∈ AI(E), there exist a net {Sα } of positive operators and a net {Kα } of positive compact operators such that 0 ≤ Sα ≤ S and 0 ≤ Sα ≤ Kα for all α, and there exists some index α0 such that Sα0 x0 > 0. Since N (F) = {0}, there exists an operator T ∈ F such that T Sα0 x0 > 0. Since T ∈ AI(E), there exists a net {Tβ } of positive operators, each of them dominated by a positive compact operator, such that 0 ≤ Tβ ≤ T , and there exists β0 such that Tβ0 Sα0 x0 > 0. Repeating this once more, we can find a positive operator operator U ∈ F with U Tβ0 Sα0 x0 > 0, and there is a net {Uγ } of positive operators, each of them dominated by a positive compact operator, such that 0 ≤ Uγ ≤ U , and there exists γ0 such that Uγ0 Tβ0 Sα0 x0 > 0. Now, the positive operator K := Uγ0 Tβ0 Sα0 is compact by [1, Theorem 5.14], and we have K ≤ U T S and Kx0 > 0.

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The following result extends [2, Theorem 2.3] that was proved for abstract integral operators. Theorem 3.3. Let S ⊂ AI(E) be a semigroup of quasinilpotent operators on a Banach lattice E. Then S is ideal-reducible. Proof. We may assume that S = 6 {0}. If the absolute kernel N (S) 6= {0}, then it is a nontrivial closed ideal which is invariant under all the operators in S. Now suppose that N (S) = {0}. By Lemma 3.2, there exists positive operators U, T and S ∈ S such that the operator U T S dominates a nonzero positive compact operator. By [2, Theorem 2.1], S is ideal-reducible, and the proof is complete. Theorem 3.4. Let S be a semigroup of positive order continuous quasinilpotent operators on a Dedekind complete Banach lattice E. Then each of the following conditions implies ideal-reducibility of the semigroup S: (1) there exists a nonzero abstract integral operator S in S; (2) there exists a positive abstract integral operator on E which dominates some nonzero operator from S; (3) there exists a positive abstract integral operator on E which is dominated by some nonzero operator from S. Proof. (1) By Lemma 3.1, the nonzero semigroup ideal generated by S consists of positive quasinilpotent abstract integral operators. This ideal is idealreducible by Theorem 3.3. Now, we apply Proposition 2.1 to conclude that S is ideal-reducible. (2) Since the space of all abstract integral operators is a band in Lr (E), the statement follows from (1). (3) Let S0 be the set of all positive order continuous operators on E which are dominated by some operator in S. Then S0 is a semigroup of positive quasinilpotent order continuous operators on E containing a nonzero abstract integral operator. It follows from (1) that S0 ideal-reducible, and so S as well, since S ⊆ S0 . The last two theorems do not hold without any assumption related to abstract integral operators. Namely, in [5] there was constructed an irreducible semigroup of square-zero positive operators on Lp [0, 1] (1 ≤ p < ∞) in which any finite number of elements generate an ideal-triangularizable semigroup.

4. Atoms and reducibility Let E be a normed Riesz space. A nonzero vector a ∈ E + is said to be an atom whenever for each vector 0 ≤ x ≤ a there exists a λ ≥ 0 such that x = λa, i.e., the one-dimensional subspace Ba generated by a is a band of E. If E does not have any atoms, it is said to be atomless. By [8, Theorem 26.4], Ba is a projection band, and so we have E = Ba ⊕ Bad . Therefore, for every f ∈ E there exist λ ∈ R and g ∈ Bad such that f = λa + g. Note that λ and g are uniquely determined. Let

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ϕa be the positive linear functional on E defined by ϕa (f ) = λ. It follows from |f | = |λ|a + |g| ≥ |λ|a that the functional ϕa is bounded. To each positive operator T on E we associate the zero-set Z(T ) of all atoms a ∈ E such that ϕa (T a) = 0, or equivalently, T a ∧ a = 0. Lemma 4.1. Let E be a normed Riesz space. (1) Let S and T be positive operators on E. Then Z(ST ) ⊆ Z(S) ∪ Z(T ). If, in addition, the pair {S, T } is ideal-triangularizable, then Z(ST ) = Z(S)∪Z(T ). (2) Let T be a positive operator E and n ∈ N. Then Z(T n ) ⊆ Z(T ). If, in addition, T is ideal-triangularizable, then Z(T n ) = Z(T ). Proof. Since the assertion (2) follows from (1) by induction, we need to prove only (1). Let a be an atom of E such that a ∈ / Z(S) ∪ Z(T ). Then Sa ≥ sa for some s > 0 and T a ≥ ta for some t > 0. It follows that ST a ≥ sta, and so a ∈ / Z(ST ). This shows that Z(ST ) ⊆ Z(S) ∪ Z(T ). Suppose that C is an ideal-triangularizing chain for the pair {S, T }, and that a ∈ Z(S) ∪ Z(T ). Then there are closed ideals I and J in C such that I = J ⊕ Ba . (In fact, J = I − .) If a ∈ Z(T ), then T a ∈ J, and so ST a ∈ J, which means that a ∈ Z(ST ). Assume now that a ∈ Z(S), so that Sa ∈ J. Since T a ∈ I, T a = ta + f for some t ≥ 0 and f ∈ J. It follows that ST a = Sf + tSa ∈ J, so that a ∈ Z(ST ), completing the proof of (1). Lemma 4.2. The zero-set Z(T ) of a positive quasinilpotent operator T on a normed Riesz space E is equal to the set of all atoms of E. Proof. Let a be an atom of E. If λ = ϕa (T a), then T a ≥ λa. By an easy induction argument we obtain that T n a ≥ λn a for every n ∈ N. It follows that |λ| ≤ lim inf n→∞ kT n ak1/n = 0, since T is quasinilpotent. This shows that a ∈ Z(T ), completing the proof. The following proposition, proved also in [4, Theorem 2.1], is an easy consequence of Propositions 2.1 and 2.2. Proposition 4.3. Let a be an atom of a normed Riesz space E and let S be a semigroup of positive operators on E such that a ∈ Z(S) for all S ∈ S. Then S is ideal-reducible. If each member of S is order continuous, then S is band-reducible. Proof. Since ϕa is a nonzero positive functional which annihilates the set Sa, the semigroup S is ideal-reducible by Proposition 2.1. The last assertion follows from Proposition 2.2, since Sa ⊆ {a}d . For the following extension of [9, Lemma 2] we find a slightly simpler proof (even in the case of Lp -spaces). Proposition 4.4. Let S be a semigroup of positive operators on a normed Riesz space E such that every operator of S is ideal-triangularizable. If there exists an atom a ∈ E such that a 6∈ Z(S) for any nonzero operator S ∈ S, then the semigroup S is ideal-reducible.

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Proof. If a is an eigenvector of every operator S ∈ S, then the principal band Ba generated by a is a nontrivial closed ideal that is invariant under all S ∈ S. Suppose now that there exists an operator S ∈ S such that a is not an eigenvector for S. Pick any operator T ∈ S. By the assumption, the operator ST ∈ S is ideal-triangularizable, so that there exists an ideal-triangularizing chain C for ST . Necessarily there are two closed ideals I and J in C such that I = J ⊕ Ba . Since a ∈ I, we have ST a ∈ I. From T a ≥ ϕa (T a)a ≥ 0 it follows that ST a ≥ ϕa (T a)Sa ≥ 0, and so Sa ∈ I, since ϕa (T a) > 0. We conclude that f = Sa − ϕa (Sa)a is a nonzero positive vector in J. Since ST f ∈ J and T f ≥ ϕa (T f )a ≥ 0, we have 0 = ϕa (ST f ) ≥ ϕa (T f ) ϕa (Sa) ≥ 0. Since ϕa (Sa) > 0, we obtain that ϕa (T f ) = 0 for all T ∈ S. Therefore, the semigroup S is ideal-reducible by Proposition 2.1. The following proposition will be used more than once. Proposition 4.5. Let E be an atomless Banach lattice with order continuous norm, and let K be an ideal-triangularizable positive compact operator on E. Then K is quasinilpotent. Proof. Since the norm is order continuous, any ideal-triangularizing chain of K consists of projection bands. As E has no atoms, this chain must be continuous. Therefore, K is a quasinilpotent operator by Ringrose’s theorem (see [12, Corollary 7.2.4]). The following example shows that in an atomless Banach lattice an idealtriangularizing chain is not necessarily continuous. Example. Let T be the rank-one operator on E = C([0, 1]) defined by (T f )(x) = f (1) (f ∈ E, x ∈ [0, 1]). Then the maximal ideal J = {f ∈ E : f (1) = 0} is invariant under T . If C is any maximal chain of closed ideals contained in J, then the chain C ∪ E is an idealtriangularizing chain of T . Although E has no atoms, this chain is not continuous, as E − = J.

5. From local ideal-triangularizability to global ideal-reducibility The following two theorems consider conditions under which a semigroup of idealtriangularizable operators is necessarily ideal-reducible. Theorem 5.1. Let E be a normed Riesz space with an atom a, and let S be a semigroup of positive operators on E such that: (1) each element of S is ideal-triangularizable; (2) Z(S) ∪ Z(T ) ⊆ Z(ST ) for all S, T ∈ S. Then the semigroup S is ideal-reducible.

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Proof. If a 6∈ Z(S) for each nonzero operator S ∈ S, then the semigroup S is idealreducible by Proposition 4.4. Otherwise, there exists a nonzero operator S ∈ S such that a ∈ Z(S). Let I be the semigroup ideal of S generated by S. It follows from assumption (2) that a ∈ Z(T ) for all T ∈ I. Then the semigroup ideal I is idealreducible by Proposition 4.3, and so S is ideal-reducible by Proposition 2.1. Theorem 5.2. Let E be an atomless Banach lattice with order continuous norm, and let S be a semigroup of positive abstract integral operators on E. If every member of S is ideal-triangularizable, then S is ideal-reducible. Proof. As in the proof of Theorem 3.3, we may assume that N (S) = {0} and that S contains an operator S that dominates a nonzero positive compact operator K. Since S is ideal-triangularizable, K is ideal-triangularizable as well, and so K is quasinilpotent by Proposition 4.5. The same arguments show that the semigroup ideal I of S generated by K consists of compact quasinilpotent operators. By Theorem 3.3, I is ideal-reducible, and so the semigroup S is also ideal-reducible by Proposition 2.1. We will merge the preceding theorems in the next result. We begin by decomposing a Dedekind complete Banach lattice E into two parts. Let A be the band generated by the set of all atoms of E. Since E is Dedekind complete, A is a projection band. If we denote by C the disjoint complement of A in E, then we have a decomposition E = A ⊕ C. The bands A and C are called the atomic and the continuous (or non-atomic) part of E, respectively. Let PC denote the band projection on C. Theorem 5.3. Let E be a Dedekind complete Banach lattice whose continuous part C has order continuous norm. Let S be a semigroup of positive operators on E such that: (1) each element of S is ideal-triangularizable; (2) Z(S) ∪ Z(T ) ⊆ Z(ST ) for all S, T ∈ S; (3) for each S ∈ S, the compression SC = PC S|C of S to C has the property that N (SC ) is an abstract integral operator. Then the semigroup S is ideal-reducible. Proof. In view of Theorem 5.1 we must consider only the case of an atomless Banach lattice E with order continuous norm. Let SN be the semigroup generated by the set {N (S) : S ∈ S}. By Lemma 3.1, each member of SN is a positive abstract integral oparator. Since 0 ≤ N (S) ≤ S for all S ∈ S and S is a semigroup, we conclude easily that each member of SN is also ideal-triangularizable. Therefore, SN is ideal-reducible by Theorem 5.2. Since any nontrivial closed ideal invariant under SN is also invariant under S, the proof is complete.

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6. Ideal-triangularizability of semigroups In this section we find conditions under which ideal-reducibility in the last result implies ideal-triangularization of given families of operators. From [3] we first recall the triangularization lemma for Banach lattices. Let F be a family of operators on a Banach lattice E, and let I and J be closed ideals of E satisfying J ⊆ I that are invariant under every member of F. Then F induces a family Fˆ of operators on the quotient Banach lattice I/J as follows. For each T ∈ F the operator Tˆ is defined on I/J by Tˆ(x + J) = T x + J . Because of the invariance I and J the operator Tˆ is a well-defined operator on I/J. Any such Fˆ is called a collection of ideal-quotients of the family F. A set P of properties is said to be inherited by ideal-quotients if every family of idealquotients of a family of operators satisfying properties in P also satisfies the same properties. Lemma 6.1 (The Ideal-Triangularization Lemma). Let P be a set of properties inherited by ideal-quotients. If every family of operators on a Banach lattice of dimension greater than one which satisfies P is ideal-reducible, then every such family is ideal-triangularizable. As the first application of this lemma, we prove the following theorem. Theorem 6.2. Let E be an atomic Banach lattice with order continuous norm, and let S be a semigroup of positive quasinilpotent operators on E. Then S is ideal-triangularizable. Proof. By Lemma 4.2, the zero-set Z(S) is equal to the set of all atoms of E for each S ∈ S. It follows that the semigroup S is ideal-reducible by Proposition 4.3. Now, S is ideal-triangularizable by the Ideal-Triangularization Lemma. The following result asserts that semigroups in Theorem 5.3 are even idealtriangularizable provided the Banach lattice norm is order continuous. Theorem 6.3. Let E be a Banach lattice with order continuous norm, and let S be a semigroup of positive operators on E such that: (1) each element of S is ideal-triangularizable; (2) Z(S) ∪ Z(T ) ⊆ Z(ST ) for all S, T ∈ S; (3) for each S ∈ S, the compression SC = PC S|C of S to C has the property that N (SC ) is an abstract integral operator. Then the semigroup S is ideal-triangularizable. Proof. In view of Theorem 5.3 and the Ideal-Triangularization Lemma we must show that conditions (1), (2) and (3) are inherited by ideal-quotients. Let C be an ideal-triangularizing chain for a given operator S ∈ S. Since the norm of E is order continuous, every closed ideal of E is actually a projection band.

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Furthermore, if B1 and B2 are bands of E such that B2 ⊆ B1 , then the quotient Banach lattice B1 /B2 is lattice isometrically isomorphic to the band B := B1 ∩B2d . Therefore, if S leaves the bands B1 and B2 invariant, then the induced operator Sˆ on B1 /B2 can be identified with the compression PB S|B , where PB denotes the band projection on B. Now, using the corresponding band projections it is easy to verify that B ∩ C is an ideal-triangularizing chain for the compression PB S|B , so that the condition (1) is inherited by ideal-quotients. It is obvious that the same holds for the condition (2). To show that the condition (3) is inherited by ideal-quotients, take the band B and the operator S ∈ S from the preceding paragraph. Then the non-atomic part of B is equal to B ∩ C. Since PB∩C S|C ≤ SC and since N (SC ) is an abstract integral operator on C, N (PB∩C S|B∩C ) is then an abstract integral operator on B ∩ C. Lemma 4.1 and Theorem 6.3 imply the following corollary. Corollary 6.4. Let E be a Banach lattice with order continuous norm, and let S be a semigroup of positive operators such that, for each S ∈ S, the compression SC = PC S|C of S to C has the property that N (SC ) is an abstract integral operator. The following statements are equivalent: (1) S is ideal-triangularizable; (2) each pair {S, T } ⊆ S is ideal-triangularizable; (3) each operator of S is ideal-triangularizable and Z(S) ∪ Z(T ) ⊆ Z(ST ) for all S, T ∈ S. Corollary 6.5. Let E be an atomless Banach lattice with order continuous norm, and let S be a semigroup of positive abstract integral operators on E such that every operator from S is ideal-triangularizable. Then S is ideal-triangularizable. Corollary 6.6. Let E be an atomic Banach lattice with order continuous norm, and let S be a semigroup of positive ideal-triangularizable operators such that Z(S) ∪ Z(T ) ⊆ Z(ST ) for all S, T ∈ S. Then S is ideal-triangularizable. For semigroups of compact operators we can prove the following theorem. Theorem 6.7. A semigroup S of positive compact operators on a Banach lattice E with order continuous norm is ideal-triangularizable if and only if every pair {S, T } of operators in S is ideal-triangularizable. Proof. Suppose that every pair {S, T } ⊆ S is ideal-triangularizable. If E is an atomless Banach lattice, then every operator of S is quasinilpotent by Proposition 4.5. Now, S is ideal-triangularizable by [3, Theorem 4.5]. If E contains atoms, then ideal-reducibility of S follows from Lemma 4.1 and Theorem 5.1. Now, we apply the Ideal-Triangularization Lemma.

7. Semigroups of operators on AL- and AM-spaces A Banach lattice E is an AL- (resp. AM-) space if for each pair of disjoint positive vectors f, g ∈ E it holds that kf +gk = kf k+kgk (resp. kf +gk = max{kf k, kgk}).

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Recall that a Banach space X has the Dunford-Pettis property whenever for each sequence {xn }n∈N in X converging weakly to zero and each sequence {ϕn }n∈N in X ∗ converging weakly to 0, the sequence {ϕn (xn )}n∈N converges to zero. By the Grothendieck Theorem [1, Theorem 5.85], AL- and AM-spaces have the DunfordPettis property. The following result extends a result [6, Corollary 2.4] of Jahandideh asserting that every positive weakly compact quasinilpotent operator on an AL-space is ideal-reducible. Proposition 7.1. Let E be a Banach lattice with the Dunford-Pettis property. Then every semigroup of positive quasinilpotent weakly compact operators on E is idealreducible. Proof. By [1, Corollary 5.87], each product of weakly compact operators on E is a compact operator. If there exist S and T ∈ S such that ST 6= 0, then the semigroup ideal generated by the operator ST is a nonzero ideal of positive quasinilpotent compact operators. This ideal is ideal-reducible by [3, Theorem 4.5]. By Proposition 2.1, the semigroup S is ideal-reducible. Assume now that ST = 0 for each S and T ∈ S. It follows that ASB = 0 for each A, B and S ∈ S. By Proposition 2.1 again, the semigroup S is idealreducible. Theorem 7.2. If E is an AL- or an AM-space, then every semigroup S of positive quasinilpotent weakly compact operators on E is ideal-triangularizable. Proof. We consider only the case of an AL-space E, since the proof for AM-spaces is similar. Weakly compactness, positivity and quasinilpotence of operators are properties that are inherited by ideal-quotients. By [1, Exercise 4.1.13] for any ideal I of E, the quotient Banach lattice E/I is again an AL-space. Since the semigroup S is ideal-reducible by Proposition 7.1, ideal-triangularizability of S now follows from the Ideal-Triangularization Lemma. In the case of atomless AL-spaces one can replace the assumption about quasinilpotence of operators with ideal-triangularizability of single operators as the following theorem asserts. Theorem 7.3. A semigroup S of positive weakly compact ideal-triangularizable operators on an atomless AL-space E is ideal-triangularizable. Proof. Let us first show that each S ∈ S is quasinilpotent. Without loss of generality we may assume that S 2 6= 0. By the Dunford-Pettis Theorem [1, Corollary 5.88] the operator S 2 is a nonzero compact operator. Since the operator S 2 is also ideal-triangularizable by assumption, S is a quasinilpotent operator by Proposition 4.5. Ideal-triangularizability of S now follows from Theorem 7.2.

Acknowledgement The authors were supported by the Slovenian Research Agency.

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References [1] C. D. Aliprantis, O. Burkinshaw, Positive operators, reprint of the 1985 original, Springer, Dordrecht, 2006. [2] R. Drnovˇsek, A generalization of the Ando-Krieger theorem, Pub. Math. Debrecen 58 (2001), 515–521. [3] R. Drnovˇsek, Common invariant subspaces for collections of operators, Integral Equations and Operator Theory 39 (2001), 253–266. [4] R. Drnovˇsek, Triangularizing semigroups of positive operators on an atomic normed Riesz spaces, Proc. Edinb. Math. Soc. 43 (2000), 43–55. [5] R. Drnovˇsek, D. Kokol-Bukovˇsek, L. Livshits, G. MacDonald, M. Omladiˇc, H. Radjavi, An irreducible semigroup of non-negative square-zero operators, Integral Equations and Operator Theory 42 (2002), no. 4, 449–460. [6] M. T. Jahandideh, On the ideal-triangularizability of positive operators on Banach lattices, Proc. Amer. Math. Soc. 125 (1997), 2661–2670. [7] E. D. Jonge, A. C. M. V. Rooij, Introduction to Riesz spaces, Mathematical centre tracts 78, Amsterdam, 1977. [8] W. A. J. Luxemburg, A. C. Zaanen, Riesz spaces 1, North-Holland, Amsterdam, 1971. [9] G. MacDonald, H. Radjavi, Standard triangularization of semigroups of non-negative operators, J. Funct. Anal. 219 (2005), 161–176. [10] P. Meyer-Nieberg, Banach lattices, Springer-Verlag, Berlin, 1991. [11] B. de Pagter, Irreducible compact operators, Math. Z. 192 (1986), 149–153. [12] H. Radjavi, P. Rosenthal, Simultaneous triangularization, Universitext, SpringerVerlag, New York, 2000. [13] Yu. V. Turovskii, Volterra semigroups have invariant subspaces, J. Funct. Anal. 162 (1999), 313–322. [14] J. Voigt, The projection onto the center of operators in a Banach lattice, Math. Z. 199 (1988), no. 1, 115–117. [15] A. C. Zaanen, Introduction to operator theory in Riesz spaces, Springer, Berlin–Heidelberg–New York, 1996. [16] A. C. Zaanen, Riesz spaces II, North Holland, Amsterdam, 1983. Roman Drnovˇsek Faculty of Mathematics and Physics, University of Ljubljana Jadranska 19 1000 Ljubljana, Slovenia e-mail: [email protected] Marko Kandi´c Institute of Mathematics, Physics and Mechanics Jadranska 19 1000 Ljubljana, Slovenia e-mail: [email protected] Submitted: July 11, 2008.

Integr. equ. oper. theory 64 (2009), 553–572 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040553-20, published online August 3, 2009 DOI 10.1007/s00020-009-1699-5

Integral Equations and Operator Theory

On the Positive Parts of Second Order Symmetric Pseudodifferential Operators Marco Mughetti and Fabio Nicola Abstract. Using microlocalization, the positive and the negative parts for a class of second order formally self-adjoint pseudodifferential operators are constructed. Mathematics Subject Classification (2000). Primary 35S05; Secondary 35P99. Keywords. Pseudodifferential operators, spectral projections, multiple characteristics, Fefferman-Phong decomposition.

1. Introduction In this paper we deal with the following question: To what extent can we deduce the spectral properties of a formally self-adjoint operator P = p(x, D) ∈ OPS 2 (Rn ) by analyzing its symbol p(x, ξ)? Precisely, we would like to construct explicitly from the symbol p(x, ξ), by microlocalization, an approximate positive part Π+ and an approximate negative part Π− for the operator P ; namely, Π+ and Π− are linear bounded operators in L2 (Rn ) such that Π+ + Π− = Id + “negligible terms” and, for every u ∈ C0∞ (K), K ⊂⊂ Rn , Re(P Π+ u, u) ≥ −CK kuk20 ,

−Re(P Π− u, u) ≥ −CK kuk20 .

This program was proposed by L. Nirenberg and was carried out for a first order pseudodifferential operator Q = q(x, D) by D. Fujiwara [6]. The main tool used in [6] is a kind of cutting and stopping argument introduced by Beals and Fefferman [1] (see also Fefferman [4], Fefferman and Phong [5] and Parmeggiani [15]), that allows a reduction of the operator Q to suitable “normal” forms. The symbol of these normal forms turns out to be either non negative or non positive, or just bounded, or finally, (after a symplectomorphism) of the type q(x, ξ) = ξ1 . For each of these operators the corresponding positive and negative parts Π+ and Π− are easily written; in fact, for a non negative symbol one just takes Π+ = Id and Π− = 0 as a consequence of the Sharp G˚ arding inequality (see [10]), whereas

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for the symbol q(x, ξ) = ξ1 one sets Π+ = Op(H(ξ1 )), Π− = Op((1 − H(ξ1 )), where H(ξ1 ) is the Heaviside function on the real line. Finally these operators are patched together by means of a microlocal partition of unity and one obtains the following result (Theorem 1 of [6]). Theorem 1.1. Using microlocalization, for every > 0 one can construct three bounded linear self-adjoint operators π + , π − and T in L2 (Rn ) such that Π+ + Π− = Id + T,

kT k < ,

and there exists a positive constant C such that Re(Π+ Qv, v) ≥ −Ckvk20 , −

−Re(Π Qv, v) ≥

−Ckvk20 ,

∀v ∈ S(Rn ) n

∀v ∈ S(R )

(1.1) (1.2)

and kT Qk + kQT k + k[Π+ , Q]k + k[Π− , Q]k ≤ C, where [A, B] denotes the commutator of the operators A,B, and kAk represents the norm of A as a bounded linear operator in L2 (Rn ). A similar problem for general second order pseudodifferential operators seems to be quite difficult. As an attempt to attack this problem, we consider the particular case of classical operators, with principal symbol transversally elliptic and satisfying some additional conditions ((H1 ), (H2 ), (H3 ) below). P Precisely, let X ⊂ Rn be an open subset and let p ∼ j≥0 p2−j be the symbol of the classical operator P = P ∗ ∈ OPS 2 (X). We denote by Σ = {(x, ξ) ∈ T ∗ X \ 0 : p2 (x, ξ) = 0} its characteristic set, where T ∗ X \ 0 is the cotangent bundle over X without the zero-section. In the sequel we suppose that (H1 ) Σ is a symplectic sub-manifold of T ∗ X \ 0, namely Tρ Σ ∩ Tρ Σσ = {0} for every ρ ∈ Σ; σ (Tρ Σ is thePorthogonal space to Tρ Σ with respect to the canonical symplectic n 2-form σ = j=1 dξj ∧ dxj ). From now on we assume that the operator P is transversally elliptic, namely its principal symbol p2 vanishes exactly to second order on Σ. Consider now the fundamental matrix F associated with p2 at ρ ∈ Σ, defined by 1 (1.3) σ(v, Fρ w) = hHess p2 (ρ) v, wi, ∀v, w ∈ Tρ T ∗ X. 2 Since the operator P is formally self-adjoint, its principal symbol p2 is real-valued and has locally constant sign because of the transversal ellipticity assumption. Hence, without loss of generality, we may assume that p2 is non negative everywhere; as a consequence, its Hessian map is positive semi-definite. Furthermore the spectrum of the fundamental matrix Fρ consists of the eigenvalue 0 and of eigenvalues ±iµj (ρ), j = 1, . . . , ν, where 0 < µ1 (ρ) ≤ µ2 (ρ) ≤ . . . ≤ µν (ρ),

2ν = codim Σ,

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are real continuous functions of ρ ∈ Σ, positively homogeneous of degree 1 in the fibers. Also, we recall that the positive trace of the fundamental matrix Fρ is given by Tr+ Fρ :=

ν X

µj (ρ),

j=1

and the sub-principal symbol is defined by n

ps1 (ρ) := p1 (ρ) +

i X ∂ 2 p2 (ρ). 2 j=1 ∂xj ∂ξj

Notice that ps1 is invariantly defined at points of Σ and is real valued (for p2 vanishes to second order there and P = P ∗ ). Let us now consider the functions mα (ρ) =

ν X

µj (ρ)(2αj + 1) + ps1 (ρ),

ρ ∈ Σ, α ∈ Zν+ .

(1.4)

j=1

It is well known that the operator P is hypoelliptic with loss of 1 derivative if and only if mα (ρ) 6= 0 for every ρ ∈ Σ and every α ∈ Zν+ (see e.g. [2]). However, these functions can, in general, vanish and in this case we require the following additional conditions. Suppose ρ0 ∈ Σ and mα (ρ0 ) = 0 for some α ∈ Zν+ ; set Aρ0 = {α ∈ Zν+ : mα (ρ0 ) = 0}. Then we assume that (H2 ) µj (ρ0 ) 6= µk (ρ0 ), for j, k = 1, . . . , ν, j 6= k; (H3 ) |α| is either even ∀α ∈ Aρ0 or odd ∀α ∈ Aρ0 . Notice that, under Hypothesis (H2 ), the functions µj , and therefore the mα ’s, α ∈ Aρ0 , are actually smooth in a conic neighborhood of ρ0 in Σ. It is also clear that the conditions (H2 ) and (H3 ) are automatically satisfied if, for example, codim Σ = 2. Furthermore, we observe that, just under these assumptions (H1 )–(H3 ), Helffer ([7], Thm. 4.3.14) characterized the hypoellipticity with loss of 3/2 derivatives of transversally elliptic pseudodifferential operators. We can now state our main result. Theorem 1.2. Let P = P ∗ ∈ OPS 2 (X) be a classical (properly supported) pseudodifferential operator, transversally elliptic. Suppose that Hypothesis (H1 ) holds. Furthermore, if Aρ0 6= ∅ for some ρ0 ∈ Σ, then assume the conditions (H2 ) and (H3 ). Then, for every fixed ρ0 ∈ Σ, there exists a conic neighborhood V of ρ0 for which we can construct, starting directly from the symbol of P , two linear continu−1/2 ous symmetric operators Π+ , Π− : L2comp (X) −→ L2comp (X) and R ∈ OPS1/2,1/2 (X)

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such that Π+ + Π− = Id + R, 1/2

P R, RP ∈ OPS1/2,1/2 (X), and satisfying, for every Ψ = ψ(x, D) ∈ OPS 0 (Rn ) with supp ψ ⊂ V and every compact set K ⊂ X, the estimates Re(P Π+ Ψu, Ψu) ≥ −CK kuk20 , −

−Re(P Π Ψu, Ψu) ≥

−CK kuk20 ,

∀u ∈ C0∞ (K),

(1.5)

C0∞ (K).

(1.6)

∀u ∈

If Aρ0 = ∅ then the construction yields R = 0 and there exist constants cK > 0, CK > 0 such that Re(P Π+ Ψu, Ψu) ≥ cK kΠ+ Ψuk21/2 − CK kuk20 ,

∀u ∈ C0∞ (K),

(1.7)

−Re(P Π− Ψu, Ψu) ≥ cK kΠ− Ψuk21/2 − CK kuk20 ,

∀u ∈ C0∞ (K).

(1.8)

We observe that if ρ0 6∈ Σ and, say, p2 (ρ0 ) > 0, P is elliptic near ρ0 and, upon choosing Π+ = Id and Π− = 0, estimate (1.5) is a trivial consequence of G˚ arding’s inequality (P Ψu, Ψu) ≥ cK kΨuk21 − CK kuk20 ,

∀u ∈ C0∞ (K).

Moreover, from the proof of Theorem 1.2, one will see that if P satisfies the Melin trace+ condition m0 (ρ) = Tr+ Fρ + ps1 (ρ) ≥ 0,

∀ρ ∈ Σ,

(1.9)

then one gets H¨ ormander’s inequality (Thm. 22.3.2 of [10]) (P u, u) ≥ −CK kuk20 ,

∀u ∈ C0∞ (K).

(1.10)

Unfortunately, we are not able in general to patch together the microlocal operators Π+ , Π− so as to obtain a local version of Theorem 1.2, and the problem does not seem to us to be merely a technical one. Remark 1.3. The following weaker local construction can be performed. For any given compact set K ⊂ X, let ψj ∈ S 0 (R2n ), j = 1, . . . , N , be real valued symbols supported in conic neighborhoods Vj where Theorem 1.2 applies and such that PN + − 2 j=1 ψj (x, ξ) = 1 for x near K. Let thus Πj and Πj be the corresponding operPN ators given in Theorem 1.2. Upon defining the operators P+ = j=1 Ψ∗j P Π+ j Ψj , PN − ∗ 0 P− = j=1 Ψj P Πj Ψj , with Ψj = ψj (x, D), we have P+ + P− = P + R , with 1/2

R0 ∈ OPS1/2,1/2 (X), and, for every u ∈ C0∞ (K), Re(P+ u, u) ≥ −CK kuk20 ,

−Re(P− u, u) ≥ −CK kuk20 .

The construction of the operators Π+ and Π− in Theorem 1.2 is based on both the above-mentioned result by Fujiwara [6] and on the theory of Hermiteoperators due to Boutet de Monvel [2] and then developed by Helffer [7] (see also [3]), and recently by Parenti and Parmeggiani [13, 14, 16].

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We now want to illustrate Theorem 1.2 by a very simple example, that however represents a model case. To this purpose, assume that Σ = {(x1 , x2 , ξ1 , ξ2 ) ∈ R4 : x1 = ξ1 = 0, ξ2 6= 0}, and fix ρ0 = (0, x2 , 0, ξ 2 ) ∈ Σ. Let P be any pseudodifferential operator in R2 , taking the following form, microlocally near ρ0 , P = Dx21 + x21 Dx22 + p1 (x2 , Dx2 ), where p1 (x2 , ξ2 ) is real and positively homogeneous of degree 1. We know that P is microlocally hypoelliptic at ρ0 with loss of 1 derivative if and only if its localized operator at ρ = (0, x2 , 0, ξ2 ) (see Definition 2.4 in Section 2) Pρ (y, Dy ) = Dy2 + y 2 |ξ2 |2 + p1 (x2 , ξ2 ) : S(Ry ) → S(Ry ) is invertible when ρ = ρ0 , that is, all its eigenvalues mk (x2 , ξ2 ) = (2k + 1)|ξ2 | + p1 (x2 , ξ2 ),

k ∈ Z+ ,

do not vanish at ρ = ρ0 . Consider the k−th Hermite function k 2 d − t e−t /2 , t ∈ R, hk (t) = π −1/4 (2k k!)−1/2 dt

(1.11)

and notice that hk (|ξ2 |1/2 y) spans the eigenspace of the localized operator Pρ = Pρ (y, Dy ) corresponding to the eigenvalues mk (x2 , ξ2 ). Furthermore, we observe that m0 (x2 , ξ2 ) = min Spec(Pρ ). If the spectrum of Pρ is non negative in a neighborhood U of ρ0 , then (1.9) is fulfilled in U and hence P satisfies the H¨ormander inequality (1.10), microlocally near ρ0 . This fact suggests to construct the operators Π+ and Π− of Theorem 1.2 by suitably “following” the spectral projectors onto the positive and negative eigenspaces of Pρ , as ρ varies near ρ0 . The most interesting case occurs when 0 belongs to the spectrum of Pρ , i.e. there exists k0 ∈ Z+ such that mk0 (x2 , ξ 2 ) = 0. In accordance with our approach, we consider Fujiwara’s approximate projectors π + , π − : L2 (R) → L2 (R) for the first order operator mk0 (x2 , Dx2 ), which is symmetric modulo OPS 0 (R), and set T = π + +π − −Id. For any k ∈ Z+ , we define the operator Hk : L2 (R) → L2 (R2 ) by Z (Hk f )(x1 , x2 ) = (2π)−1 |ξ2 |1/4 eix2 ξ2 hk (|ξ2 |1/2 x1 )fˆ(ξ2 ) dξ2 , in such a way that the localized operator of Hk Hk∗ is the spectral projector onto the eigenspace Ker (Pρ − mk (x2 , ξ2 )Id). Finally we define the operators Π+ and Π− as X Π+ = Hk0 (π + − T )Hk∗0 + Hk Hk∗ , k>k0

Π− = Hk0 π − Hk∗0 +

X 0≤k
with convergence in

0 OPS1/2,1/2 (R2 ).

Hk Hk∗ ,

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The paper is organized as follows: in Section 2 we develop the technical machinery required to prove Theorem 1.2, assuming that the characteristic manifold Σ of P is flat (see (2.1)); in Section 3 we prove Theorem 1.2 by considering the two cases Aρ0 = ∅ and Aρ0 6= ∅.

2. Preliminaries on the theory of Hermite-operators Here we briefly recall some basic definitions and properties of certain classes of pseudodifferential operators considered, for example, in [2, 3, 7, 13, 14]. All the operators are properly supported. γ 2 µ/2−|γ|/2 Define Sµ (R2ν ) = {a ∈ C ∞ (R2ν } (see [8]), z ) : |∂z a(z)| ≤ Cγ (1 + |z| ) and assume that Σ = {(x, ξ) = (x0 , x00 , ξ 0 , ξ 00 ) : x0 = ξ 0 = 0, ξ 00 6= 0},

(2.1)

where we set x = (x0 , x00 ) ∈ Rn = Rνx0 × Rn−ν and accordingly ξ = (ξ 0 , ξ 00 ). x00 Notice that this is actually the microlocal model for any symplectic sub-manifold of codimension 2ν. Moreover we set Σ0 = {(x0 , x00 , ξ 00 ) : x0 = 0}. Definition 2.1. By S m,k (R2n , Σ), (m, k ∈ R), we denote the class of symbols a(x, ξ) which are in S m (R2n ) away from Σ and which satisfy the following estimates 0 00 0 00 |∂xα0 ∂xα00 ∂ξβ0 ∂ξβ00 a(x, ξ)|

00 m−|β|

≤ C|ξ |

1 |ξ 0 | |x | + 00 + 00 1/2 |ξ | |ξ | 0

k−|α0 |−|β 0 | ,

(2.2)

microlocally near Σ (with α = (α0 , α00 ) and β = (β 0 , β 00 )). Moreover we set Hm (R2n , Σ) := ∩j≥0 S m−j,−2j (R2n , Σ). As usual OPS m,k (Rn , Σ) and OPHm (Rn , Σ) denote the classes of the corresponding properly supported ψdo’s. m,k Definition 2.2. For an open conic subset Γ ⊂ Σ, we denote by Shom (Γ, Σ) m−k/2 ∞ k 2ν (resp. Hhom (Γ, Σ) ) the class of all smooth functions in C (Γ, S (R )) (resp. C ∞ (Γ, S(R2ν ))) such that

a(x00 , tξ 00 ; t−1/2 x0 , t1/2 ξ 0 ) = tm−k/2 a(x00 , ξ 00 ; x0 , ξ 0 ), t > 0. t t−m We recall that operators in OPHm (Rn , Σ) map Hcomp (Rn ) → Hcomp (Rn ) continuously for every t ∈ R. m−k/2

m,k Remark 2.3. Microlocally near Σ, symbols in Shom (Γ, Σ) (resp. Hhom m,k 2n m−k/2 2n in S (R , Σ) (resp. H (R , Σ)).

(Γ, Σ)) are

We now define a class of symbols with an asymptotic expansion in quasihomogeneous terms.

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Definition 2.4. We say that a symbol a ∈ S m,k (R2n , Σ) (resp. in Hm (R2n , Σ)) has an asymptotic expansion in quasi-homogeneous terms, if there exist funcm−j/2 m,k+j tions a(k+j) ∈ Shom (Σ, Σ), j ≥ 0, (resp. a(j) ∈ Hhom (Σ, Σ)) such that a − PN (k+j) P N ∈ S m,k+N +1 (R2n , Σ) (resp. a − j=0 a(j) ∈ Hm−(N +1)/2 (R2n , Σ)) j=0 a microlocally near Σ. The symbol a(k) (resp. a(0) ) is called the localized symbol of the operator Op(a) and a(k) (ρ, y, Dy ) ∈ OPSk (Rν ) (resp. a(0) (ρ, y, Dy ) ∈ OPS−∞ (Rν )) is called the localized operator at ρ = (x00 , ξ 00 ) ∈ Σ. Remark 2.5. In the sequel, we use the Sj¨ostrand classes OPN m,k (RnP , Σ) of the classical pseudodifferential operators P with symbol S m (R2n ) 3 p ∼ j≥0 pm−j such that, for every j < k/2, pm−j vanishes on Σ to order k −2j. We point out that OPN m,k (Rn , Σ) is a very important subclass of OPS m,k (Rn , Σ). Hence, according to Definition 2.4, for P we have X 1 α β α β ∂ 0 ∂ 0 pm−l (ρ)x0 ξ 0 . p(k+j) (ρ, x0 , ξ 0 ) := α!β! x ξ |α|+|β|+2l=k+j

Actually, the techniques we are recalling in this section were introduced by Boutet de Monvel [2], Boutet de Monvel, Grigis and Helffer [3] in order to study the hypoellipticity of the operators in OPN m,k (Rn , Σ). The following composition properties (see Section 2 of [7] and Proposition 4.3 of [13]) have several applications in the sequel. Proposition 2.6. (1) If A = Op(a) ∈ OPS m,k (Rn , Σ), and B = OP(b) ∈ 0 0 0 0 OPS m ,k (Rn , Σ), then AB = Op(c) ∈ OPS m+m ,k+k (Rn , Σ). Moreover, if P P 0 a ∼ j≥0 a(k+j) , and b ∼ j≥0 b(k +j) , in the sense of Definition 2.4, with P 0 0 m,k m0 ,k0 a(k+j) ∈ Shom (Σ, Σ), b(k +j) ∈ Shom (Σ, Σ), then c ∼ r≥0 c(k+k +r) , with 0

c(k+k +r) =

X i+j+2|α|=r

0 1 α (k+i) ∂ξ00 a #Dxα00 b(k +j) , α!

(2.3)

0

(∂ξα00 a(k+i) #Dxα00 b(k +j) being the symbol of the composition in S(Rν ) of the 0 operators (∂ξα00 a(k+i) )(ρ, y, Dy ) and (Dxα00 b(k +j) )(ρ, y, Dy )). 0 (2) If A = Op(a) ∈ OPHm (Rn , Σ), and B = Op(b) ∈ OPHm (Rn , Σ), then P 0 (j) AB = Op(c) ∈ OPHm+m (Rn , Σ). Moreover, if a ∼ j≥0 a , and b ∼ P (j) (j) ∈ Hm−j/2 (Σ, Σ), b(j) ∈ j≥0 b , in the sense of Definition 2.4, with a P m0 −j/2 (r) H (Σ, Σ), then c ∼ r≥0 c , with c(r) =

X i+j+2|α|=r

1 α (i) ∂ 00 a #Dxα00 b(j) . α! ξ

(2.4)

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(3) If A = Op(a) ∈ OPHm (Rn , Σ) then A∗ = Op(a∗ ) ∈ OPHm (Rn , Σ). Moreover, P m−j/2 if a ∼ j≥0 a(j) in the sense of Definition 2.4, with a(j) ∈ Hhom (Σ, Σ), then P (r) a∗ ∼ r≥0 a∗ , with X 1 Dα00 ∂ α00 (a(j) )∗ , a∗ (r) = α! ξ x j+2|α|=r

where (a(j) )∗ denotes the symbol of the operator a(j) (ρ, y, Dy )∗ on S(Rνy ). We also need the following classes of Hermite operators. Definition 2.7. We denote by Hm (R2(n−ν) , Σ0 ), m ∈ R, the class of all smooth functions h(x00 , ξ 00 , y), where (x00 , ξ 00 ) ∈ Rn−ν × Rn−ν , y ∈ Rν , which satisfy for any l ≥ 0 the following estimates −2l−|α0 | 00 00 0 00 1 , |∂xα00 ∂yα ∂ξβ00 h(x00 , ξ 00 ; y)| ≤ C|ξ 00 |m+ν/4−l−|β | |y| + 00 1/2 |ξ | uniformly in x00 ∈ K 00 ⊂⊂ Rn−ν , |ξ 00 | ≥ 1, y ∈ Rν . For h ∈ Hm (R2(n−ν) , Σ0 ), the operator Op(h) : C0∞ (Rn−ν ) → C ∞ (Rn ) is defined by Z 00 00 00 −(n−ν) (Op(h)f ) (y, x ) := (2π) eihx ,ξ i f (x00 , ξ 00 ; y)fˆ(ξ 00 ) dξ 00 , and OPHm (Rn−ν , Σ0 ) is the corresponding class of operators, modulo smoothing operators. Moreover, by OPH∗m (Rn−ν , Σ0 ) we denote the class of operators C0∞ (Rn ) → ∞ C (Rn−ν ) that, modulo smoothing operators, are given by ZZ 00 00 ∗ 00 −(n−ν) (Op(h) g)(x ) = (2π) eihx ,ξ i h(x00 , ξ 00 , y)ˆ g (y, ξ 00 )dξ 00 dy, (2.5) R 00 00 with gˆ(y, ξ 00 ) = e−ihx ,ξ i g(y, x00 )dx00 and h ∈ Hm (R2(n−ν) , Σ0 ). m Finally, for an open conic subset Γ ⊂ Σ, we denote by Hhom (Γ, Σ0 ) the class of 00 00 ∞ ν the quasi-homogeneous functions h(x , ξ , y) ∈ C (Γ, S(Ry )) of degree m + ν/4, i.e. h(x00 , tξ 00 , t−1/2 y) = tm+ν/4 h(x00 , ξ 00 , y), t > 0. m Remark 2.8. Functions in Hhom (Γ, Σ0 ) give rise, microlocally near Σ, to symbols m 2(n−ν) 0 that are in H (R , Σ ).

Notice that, if h ∈ Hm (R2(n−ν) , Σ0 ), Op(h)∗ defined in (2.5) coincides with (Op(h))∗ just modulo lower order terms, as shown in the following proposition (see, again, Section 2 of [7] and Proposition 4.3 of [13]). Proposition 2.9. (1) If A = Op(f0 + f−1 ) ∈ OPHm (Rn−ν , Σ0 ), with f−s ∈ Hm−s/2 (R2(n−ν) , Σ0 ), s = 0, 1, then A∗ ∈ OPH∗m (Rn−ν , Σ0 ) with ∗

A∗ = Op (f0 + f−1 ) mod OPH∗m−1 (Rn−ν , Σ0 ).

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(2) If A = Op(f0 + f−1 ) ∈ OPHm (Rn−ν , Σ0 ) and B = Op(g0 + g−1 )∗ ∈ 0 0 OPH∗m (Rn−ν , Σ0 ), with f−s ∈ Hm−s/2 (R2(n−ν) , Σ0 ), g−s ∈ Hm −s/2 (R2(n−ν) , 0 0 Σ0 ), s = 0, 1, then AB ∈ OPHm+m (Rn , Σ) and BA ∈ OPHm+m (Rn−ν , Σ), with 0 AB = Op e−ihy,ηi f0 (ρ, y)gˆ0 (ρ, η) mod OPHm+m −1/2 (Rn , Σ), and Z

BA = Op

(f0 g0 + f−1 g0 + f0 g−1 ) dy

0

mod OPHm+m −1 (Rn−ν , Σ).

Rν 0

(3) If A = Op(h) ∈ OPHm (Rn−ν , Σ0 ) and B ∈ OPS m (Rn−ν ), then AB ∈ 0 OPHm+m (Rn−ν , Σ0 ) with 0

AB = Op (hb) mod OPHm+m −1 (Rn−ν , Σ0 ). As a consequence of Remark 2.5 and Proposition 2.6, we obtain the following results. Proposition 2.10. (1) Let P = Op(p) ∈ OPN m,k (Rn , Σ) and B = Op(b) ∈ P 0 OPHm (Rn , Σ) with b ∼ j≥0 b(j) in the sense of Definition 2.4, with b(j) ∈ m0 −j/2

0

Hhom (Σ, Σ). Then P B and BP are in OPHm+m −k/2 (Rn , Σ) with localized symbols given by p(k) (ρ, ·)#b(0) (ρ, ·) and b(0) (ρ, ·)#p(k) (ρ, ·), respectively. As 0 a result, one gets σ(P B) − p(k) (ρ, ·)#b(0) (ρ, ·) ∈ Hm+m −k/2−1/2 (Rn , Σ) and 0 σ(BP ) − b(0) (ρ, ·)#p(k) (ρ, ·) ∈ Hm+m −k/2−1/2 (Rn , Σ) near Σ. (2) Let P = Op(p) ∈ OPN m,k (Rn , Σ), with classical asymptotic expansion P m0 n−ν p ∼ , Σ0 ) with f−s ∈ j≥0 pm−j , and A = Op(f0 + f−1 ) ∈ OPH (R 0 0 Hm −s/2 (R2(n−ν) , Σ0 ), s = 0, 1, then P A ∈ OPHm+m −k/2 (Rn−ν , Σ0 ) and, microlocally near Σ, we have P A = Op p(k) (ρ; y, Dy )(f0 + f−1 ) + p(k+1) (ρ; y, Dy )f0 0

mod OPHm+m −k/2−1 (Rn−ν , Σ0 ). We end this section by recalling the following lower bound (Corollary 7.2 of [13]), which generalizes the Melin inequality (see [11]) for operators with multiple characteritics due to Mohamed [12]. In the sequel we will apply this result for m = k = 2. We fix the following notation. For any given open conic subset Γ ⊂ Σ and any given positive constant , we define Γ = {(x, ξ) ∈ Rn : (x00 , ξ 00 ) ∈ Γ, |x0 | + |ξ 0 |/|ξ 00 | < , |ξ 00 | ≥ 1/}.

(2.6)

Proposition 2.11. Let P = P ∗ ∈ OPN m,k (Rn , Σ) be transversally elliptic, with non negative principal symbol, and let B = B ∗ ∈ OPHm−k/2 (Rn , Σ) with localized operator b(0) (ρ, x0 , Dx0 ) ∈ OPS−∞ (Rν ) at any ρ ∈ Σ. Fix ρ0 ∈ Σ and suppose that: (1) p(k) (ρ0 , x0 , Dx0 ) + b(0) (ρ0 , x0 , Dx0 ) f, f ≥ 0, ∀f ∈ S(Rν );

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(2) p(k) (ρ0 , x0 , Dx0 ) + b(0) (ρ0 , x0 , Dx0 ) : S(Rν ) → S(Rν ) is invertible. Then there exist a conic neighborhood Γ ⊂ Σ of ρ0 and a constant > 0 such that, for every symbol ψ ∈ S 0 (R2n ) supported in Γ , there exist constants c, C > 0 for which ((P + B)Ψu, Ψu) ≥ ckΨuk2m/2−k/4 − Ckuk2m/2−k/4−1/2 ,

∀u ∈ S(Rn ),

(2.7)

with Ψ = ψ(x, D).

3. Construction of the positive and negative parts In this Section we prove Theorem 1.2 by heavily using the technical machinery developed in the previous section. It is now important to point out that Theorem 1.2 is invariant with respect to canonical changes of variables. To be definite, let χ : T ∗ X \ 0 −→ T ∗ Y \ 0 be a canonical symplectomorphism (homogeneous of degree 1 in the fibres) and F ∈ I 0 (Y × X, Λχ ) be a Fourier integral operator of order 0 associated with χ and such that F ∗ F ≡ IdX , F F ∗ ≡ IdY . Then Theorem 1.2 applies to P if and only if it applies to P˜ := F P F ∗ . Since Σ is symplectic, from Theorem 21.2.4 of [10] there exists a conic neigh˜ ⊂ T ∗ (Rν × Rn−ν ) \ 0, (2ν = borhood Γ ⊂ T ∗ X \ 0 of ρ0 , an open conic set Γ ˜ such that codim Σ), and a smooth homogeneous symplectomorphism χ : Γ −→ Γ ˜ : x0 = ξ 0 = 0}. χ(Γ ∩ Σ) = {(x0 , x00 , ξ 0 , ξ 00 ) ∈ Γ Therefore, in the sequel we may suppose that P = P ∗ ∈ OPS 2 (Rn ) is a classical operator, with principal symbol transversally elliptic with respect to Σ = {(x, ξ) ∈ Rn × Rn : x0 = ξ 0 = 0, ξ 00 6= 0}. In what follows, we simply write ρ = (x00 , ξ 00 ) for any point ρ = (0, x00 , 0, ξ 00 ) ∈ Σ. Since p2 vanishes exactly to 2−nd order on Σ, it is easily seen that Ker Fρ = Tρ Σ = {(x, ξ) : x0 = ξ 0 = 0}, where Fρ denotes the fundamental matrix defined in (1.3). As a consequence, according to Remark 2.5, the localized symbol of P is given by 0 0 x x p(2) (ρ, x0 , ξ 0 ) = σ , F + p1 (ρ), ρ 0 ξ ξ0 where ρ = (x00 , ξ 00 ) ∈ Σ and the vector (x0 , ξ 0 ) is identified with (x0 , 0, ξ 0 , 0); since P ∗ = P , the corresponding localized operator Pρ = p(2) (ρ, y, Dy ) : S(Rν ) −→ S(Rν ) is self-adjoint when regarded as an unbounded operator in L2 (Rν ) with domain B 2 (Rν ) = {f ∈ S 0 (Rν ) : y α Dyβ f ∈ L2 (Rν ) for |α| + |β| ≤ 2}. Furthermore, in view of the transversal ellipticity of P , Pρ is globally elliptic in the sense of Helffer [8]. Hence the spectrum of Pρ consists of a sequence of eigenvalues

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of finite multiplicity, diverging to +∞, and the corresponding eigenfunctions are Schwartz functions. Indeed, in [11] Melin explicitly computed the spectrum of Pρ showing that Spec(Pρ ) = {mα (ρ), α ∈ Zν+ }, where the functions mα are defined in (1.4). It is readily seen that the lowest eigenvalue of Pρ is obtained for α = 0, i.e. m0 (ρ) = Tr+ Fρ + ps1 (ρ). H¨ ormander [9] proved that P satisfies (1.10) if and only if the lowest eigenvalue Tr+ Fρ + ps1 (ρ) is non negative for every ρ ∈ Σ and, hence, if and only if the spectrum of Pρ is non negative for every ρ ∈ Σ. Therefore, the operators Π+ and Π− in Theorem 1.2 will be constructed starting from the exact projectors onto the eigenspaces corresponding to the positive and the negative part of Spec(Pρ ), respectively. More precisely, in order to give an idea of the proof of Theorem 1.2, we point out that two different cases can occur: either mα (ρ0 ) 6= 0 for every α ∈ Zν+ (Case I) or there exists β ∈ Zν+ such that mβ (ρ0 ) = 0 (Case II). Case I is the easiest to study because the condition mα (ρ) 6= 0 remains true for every ρ in a small conic neighborhood Γ ⊂ Σ of ρ0 . Hence the sum of the L negative eigenspaces E − (ρ) = Ker[P − λId] is a vector space with finite ρ λ≤0 constant dimension as ρ = (x00 , ξ 00 ) varies in Γ ⊂ T ∗ Rn−ν . As a consequence, the orthogonal projector π − (ρ) onto E − (ρ) turns out to be a pseudodifferential operator with smooth symbol σ(π − (ρ))(x0 , ξ 0 ) ∈ C ∞ Γ, S(R2ν ) . Then, the approximate negative “projector” Π− in Theorem 1.2 will be obtained by quantizing σ(π − (ρ))(x0 , ξ 0 ) ∈ H0 (R2n , Σ) in all the variables (x, ξ) = (x0 , x00 , ξ 0 , ξ 00 ), whereas the approximate positive “projector” will be defined by setting Π+ = Id − Π− . Case II is more delicate since the eigenvalues mβ (ρ) that vanish at ρ = ρ0 can, in general, change sign near ρ0 ; we thus have to investigate the influence of the positive and the negative parts of such eigenvalues upon the lower bounds satisfied by P . At this point Hypotheses (H2 ), (H3 ) play a crucial role and allow us to apply the construction of Fujiwara [6]. According to the notation of the itroduction, we denote by Aρ0 the set {α ∈ Zν+ : mα (ρ0 ) = 0}. With this definition, Case I reads as Aρ0 = ∅, whereas Case II as Aρ0 6= ∅. 3.1. Case I: Aρ0 = ∅ There exists a small constant δ > 0 and a conic neighborhood Γ ⊂ Σ of ρ0 such that, upon denoting by S∗ Γ = {(x00 , ξ 00 /|ξ 00 |) : (x00 , ξ 00 ) ∈ Γ}, Spec (Pρ ) ∩ ((−∞, −δ −1 ] ∪ [−δ, δ]) = ∅,

∀ρ ∈ S∗ Γ.

We then consider, for ρ ∈ S∗ Γ, the orthogonal projector I 1 − (ζ − Pρ )−1 dζ, π (ρ) = 2πi γ

(3.1)

where γ is a closed path contained in the domain Re ζ < δ/2, around the real interval [−δ −1 , 0]. For every ρ = (x00 , ξ 00 ) ∈ Γ we define π − (x00 , ξ 00 ) = M|ξ00 | π − (x00 , ξ 00 /|ξ 00 |)M|ξ−1 00 | ,

(3.2)

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where Mt , t > 0, is the unitary operator in L2 (Rν ) defined by (Mt f )(y) = − tν/4 f (t1/2 y). It thus follows that, for every L ρ ∈ Γ, π (ρ) is the orthogonal projector on the finite dimensional subspace λ≤0 Ker[Pρ − λId]. In fact, this is an immediate consequence of the following homogeneity property of Pρ P(x00 ,tξ00 ) = tMt P(x00 ,ξ00 ) Mt−1 ,

∀t > 0, ∀(x00 , ξ 00 ) ∈ Σ.

− ν Moreover, we claim that ) → S(Rν ) is a pseudodifferential operator 0π 0 (ρ) : S(R − ∞ with symbol σ π (ρ) (x , ξ ) ∈ C Γ, S(R2ν ) . Indeed, the operator ζ − Pρ has a symbol in C ∞ (γ × S∗ Γ, Sk (R2ν )). Moreover it is globally elliptic (see [8]) and also invertible for ρ ∈ S∗ Γ, ζ ∈ γ. As a consequence of Theorem 4.1 of [3], the inverse operator is pseudodifferential with a symbol in C ∞ (γ × S∗ Γ; S−k (R2ν )). The operator π − (ρ), ρ ∈ S∗ Γ, is therefore pseudodifferential, with a symbol in C ∞ (S∗ Γ; S−k (R2ν )). Since π − (ρ) = π − (ρ)n , for every n = 1, 2, . . ., as an operator, say, on S(Rν ), it follows that its symbol σ(π − ) ∈ C ∞ (S∗ Γ, S−nk (R2ν )) for every n ∈ Z+ , and therefore it is in C ∞ (S∗ Γ, S(R2ν )). This together with (3.2) proves the claim. Choose now as conic neighborhood V in Theorem 1.1 any subset of the type ˜ (see (2.6) for this notation), for some conic neighborhood Γ ˜ ⊂⊂ Γ of ρ0 . Fix two Γ 0 2n ˜ ˜ and ψ˜ ∼ P cut-off functions ψ, ψ˜ ∈ S (R ) such that ψ is supported in Γ j≥0 ψj is a classical symbol with ˜. supp ψ˜ ⊂ Γ2 , ψ˜ = 1 on Γ (3.3) − ˜ ¿From the homogeneity (3.2) and Remark 2.3, it turns out that ψσ(π (ρ)) ∈ 0 2n − ˜ H (R , Σ), with a localized operator given by ψ0 (ρ)π (ρ). In particular, for every ˜ one has ψ˜0 (ρ) = 1, hence such a localized operator coincides with π − (ρ) and ρ∈Γ it turns out that − ˜ ˜ ). ψσ(π ) − σ(π − ) ∈ S −∞ (Γ (3.4) + − Finally the operators Π and Π in Theorem 1.2 are defined by − ˜ Π− = Op(ψσ(π )) ∈ OPH0 (Rn , Σ), Π+ = Id − Π− . (3.5)

Remark 3.1. Actually, with this definition, the operators Π− and Π+ are not self-adjoint. However they are self-adjoint modulo OPH−1 (Rn , Σ), namely S := ∗ Π− − (Π− + Π− )/2 ∈ OPH−1 (Rn , Σ) and similarly for Π+ . By virtue of (1) in 0 0 n Proposition 2.10 with m = k = 2 and m = −1, one gets P S ∈ OPH (R , Σ) ⊂ 2 n L L (R ) . Hence we can limit ourselves to proving (1.5) and (1.6) by setting Π+ and Π− as in (3.5). Before starting with the proof of (1.7) and (1.8), we put in evidence the following useful properties of the operator Π− . Lemma 3.2. One has 2

and

Π− Ψ = Π− Ψ mod OPH−1 (Rn , Σ),

(3.6)

Π− P = P Π− + R, with R ∈ OPH1/2 (Rn , Σ).

(3.7)

∗

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Proof. Formula (3.6) follows from (2) of Proposition 2.6. Indeed, by virtue of (3.4) ˜ ) in the conic region Γ ˜ , whence (2) of one has that σ(Π− ) = σ(π − ) + S −∞ (Γ ˜ Proposition 2.6 yields, in Γ , 2 2 −1 ˜ −1 ˜ σ Π− = σ π − + Hhom (Γ, Σ) = σ π − + Hhom (Γ, Σ). ˜ , (3.6) immediately follows from Remark 2.3. Since supp ψ ⊂ Γ As regards (3.7), it suffices to study what happens in a neighborhood of a point of Σ, because Π− and Π∗ are regularizing outside Σ. Now, in such a neighbor∗ hood both the localized operators of Π− P and P Π− coincide with ψ˜0 (ρ)Pρ π − (ρ) ˜ and Pρ is the localized operator (where, as above, ψ˜0 is the principal symbol of ψ, of P at ρ). Hence, an application of (1) of Proposition 2.10 concludes the proof. Proof of (1.7) and (1.8) (Case I) Let us prove the estimate −Re(P Π− Ψu, Ψu) ≥ ckΠ− Ψuk21/2 − Ckuk20 ,

∀u ∈ C0∞ (K).

(3.8)

As a consequence of Lemma 3.2 and of (1) of Proposition 2.10, one has 2

−(P Π− Ψu, Ψu) = −(P Π− Ψu, Ψu) + O(kuk20 ) = −(P Π− Ψu, Π− Ψu) + (Π− Ψu, R∗ u) + O(kuk20 ), 2

2

= −(P Π− Ψu, Π− Ψu) + (Π− Ψu, R∗ u) + O(kuk20 ),

(3.9)

where R∗ ∈ OPH1/2 (Rn , Σ) by (3) of Proposition 2.6. In order to treat the first term in the r.h.s. of (3.9), we consider the Grushin operator ν n X X G= Dx2j + |x0 |2 Dx2j ∈ OPN 2,2 (Rn , Σ) j=1

j=ν+1

and observe that, again by (3.6) and by (1) of Proposition 2.10, we have 2

2

− (P Π− Ψu, Π− Ψu) ∗

∗

= (((Id − Π− )G(Id − Π− ) − Π− P Π− )Π− Ψu, Π− Ψu) + O(kuk20 ). Now we are going to apply Proposition 2.11 (with m = k = 2) to the operator ∗

∗

(Id − Π− )G(Id − Π− ) − Π− P Π− . ˜ is given by Indeed, one readily sees that its localized operator at ρ = (x00 , ξ 00 ) ∈ Γ (Id − π − (x00 , ξ 00 ))(|Dy |2 + |y|2 |ξ 00 |2 )(Id − π − (x00 , ξ 00 )) − P(x00 ,ξ00 ) (y, Dy )π − (x00 , ξ 00 ), which is non negative and injective at ρ0 = (x000 , ξ000 ). From Proposition 2.11 it thus ˜ , there exist constants c > 0, C > 0 for follows that, possibly after shrinking Γ which 2 2 (3.10) −Re(P Π− Ψu, Π− Ψu) ≥ ckΠ− Ψuk21/2 − Ckuk20 .

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As regards the second term in the r.h.s of (3.9), we observe that, for every > 0 |(Π− Ψu, R∗ u)| ≤ kΠ− Ψuk21/2 + C kR∗ uk2−1/2 = kΠ− Ψuk21/2 + O(kuk20 ). Hence, if is small enough, this term can be reabsorbed by virtue of (3.10). This concludes the proof of (3.8). Let us now prove (1.7), namely Re(P (Id − Π− )Ψu, Ψu) ≥ ck(Id − Π− )Ψuk21/2 − Ckuk20 ,

∀u ∈ C0∞ (K). (3.11)

As a consequence of Lemma 3.2 and (1) of Proposition 2.10 we have (P (Id − Π− )Ψu, Ψu) = (P (Id − Π− )Ψu, (Id − Π− )Ψu) + ((Id − Π− )Ψu, R∗ Ψu) + O(kuk20 ),

(3.12)

with R∗ ∈ OPH1/2 (Rn , Σ). Again by (3.6), for every constant ω, we have (P (Id − Π− )Ψu, (Id − Π− )Ψu) ∗

= ((P + ωΠ− hDx iΠ− )(Id − Π− )Ψu, (Id − Π− )Ψu) + O(kuk20 ), where, as usual, hDx i = Op (1 + |ξ|2 )1/2 . If we choose ω ∈ R such that (Pρ0 v, v) , ω|ξ000 | > − Tr+ Fρ0 + ps1 (ρ0 ) = − min v6=0 kvk20 with ρ0 = (x000 , ξ000 ), we can then apply Proposition 2.11 to the operator P + ∗ ωΠ− hDx iΠ− . In fact, its localized operator at ρ0 is given by Pρ0 (y, Dy ) + ω|ξ000 |π − (ρ0 ), ˜ , there exist which is non negative and injective. Hence, possibly after shrinking Γ constants c, C > 0 such that Re(P (Id − Π− )Ψu, (Id − Π− )Ψu) ≥ ck(Id − Π− )Ψuk21/2 − Ckuk20 .

(3.13)

By arguing as above, one takes advantage of this stronger lower bounds to reabsorb the term ((Id − Π− )Ψu, R∗ Ψu) in the r.h.s. of (3.12). The proof of (3.11) is thus complete. 3.2. Case II: Aρ0 6= ∅ Since the spectrum of Pρ0 consists of eigenvalues with finite multiplicity, Aρ0 is a finite set with cardinality d := #Aρ0 ; hence Aρ0 = {α1 , . . . , αd } ⊂ Zν+ . We can choose a constant δ > 0 and a conic neighborhood Γ ⊂ Σ of ρ0 such that ±δ 6∈ Spec(Pρ ),

∀ρ ∈ S∗ Γ,

dim W (ρ) = d,

∀ρ ∈ S∗ Γ,

and

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where we set W (ρ) :=

M

Ker[Pρ − λId],

∀ρ ∈ S∗ Γ.

(3.14)

|λ| < δ λ ∈ Spec(Pρ )

For any ρ ∈ S∗ Γ, we define π − (ρ) by (3.1), where γ is here a closed path contained in the region Re z ≤ −δ, around the real interval [−ω, −δ], ω 1; then, by using (3.2), we extend this definition to all ρ ∈ Γ. It turns out that π − (ρ) is the orthogonal projector onto the negative eigenspaces of Pρ , except for the ones contained in W (ρ). Finally we set − 0 n ˜ Π− 1 := Op(ψσ(π )) ∈ OPH (R , Σ),

(3.15)

where ψ˜ ∈ S 0 (R2n ) is a classical symbol satisfying (3.3) above. Similarly we define I 1 (ζ − Pρ )−1 dζ M|ξ−1 ρ = (x00 , ξ 00 ) ∈ Γ, π ˜ (ρ) = M|ξ00 | 00 | , 2πi γ with γ taking values in the region Re z ≤ δ and enclosing the real interval [−ω, δ], ω 1, and we set ˜ π )), Π+ (3.16) 1 := Id − Op(ψσ(˜ ˜ π )) ∈ OPH0 (Rn , Σ). where Op(ψσ(˜ The next result is a consequence of Hypotheses (H2 ) and (H3 ). Proposition 3.3. (1) Pρ is regular at ρ0 in the sense of [14, Appendix 1]. Namely, possibly after shrinking Γ, there exists a smooth morphism of hermitian vector bundles F ˜ U S∗ Γ × Cd −−−−→ W := ρ∈S∗ Γ W (ρ)     y y Id

S∗ Γ −−−−→ such that, for every ρ ∈ S∗ Γ,

S∗ Γ

˜ (ρ)ζ, U ˜ (ρ)ζ 0 )L2 (Rν ) = hζ, ζ 0 iCd (U

∀ζ, ζ 0 ∈ Cd ,

and ˜ (ρ)∗ Pρ U ˜ (ρ) = diag[mα (ρ), . . . , mα (ρ)], U 1 d where diag[mα1 (ρ), . . . , mαd (ρ)] is the diagonal matrix with the smooth functions mαj (ρ), j = 1, . . . , d, as diagonal entries. (2) Ker Pρ0 has an orthonormal basis φ1 (y), . . . , φd (y) where the φj ’s are either all even or all odd in y ∈ Rν . Proof. By virtue of (H2 ), from the invariance of the Weyl calculus ([10], Thm. 18.5.9) and from H¨ ormander’s theorem on the classification of semi-definite

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quadratic forms ([10], Thm. 21.5.3), it follows that there exists a conic neighborhood Γ and a smooth family of metaplectic transformations S∗ Γ 3 ρ 7→ U (ρ) in L2 (Rν ), associated with linear symplectic maps, such that ∗

U (ρ) Pρ U (ρ) =

ν X

µj (ρ)(Dy2j + yj2 ) + ps1 (ρ) =: Pρ0 ,

∀ρ ∈ S∗ Γ.

j=1

Let

d −t hk (t) := π (2 k!) dt be the k−th Hermite function, and define −1/4

k

−1/2

ϕβ (y) = Πνj=1 hβj (yj ),

k

2

e−t

/2

,

k = 0, 1, . . . ,

β = (β1 , . . . , βν ) ∈ Zν+ .

We have Spec(Pρ0 ) = {mα (ρ) : α ∈ Zν+ } and Pρ0 ϕα (y) = mα (ρ)ϕα (y). In particular, Ker Pρ0 = Span{ϕα ; α ∈ Aρ }. Therefore it suffices to define ˜ : S∗ Γ × Cd 3 (ρ, ζ) 7→ (ρ, U (ρ)(ζ1 ϕα + . . . + ζd ϕα )) ∈ W (ρ). U 1 d As regards (2), we observe that the statement is trivially true for Pρ0 0 in view of (H3 ). On the other hand, Ker Pρ0 = U (ρ0 )Ker Pρ0 0 and any metaplectic transformation associated with a linear symplectic map is parity preserving. This concludes the proof. The regularity and parity properties established in Proposition 3.3 ((1) and (2) respectively) allow us to apply the following result of [14] (see, precisely, a., b., c., d. of page 71 in [14] and Lemma 4.10 of [14]). Lemma 3.4. Possibly after shrinking Γ, there exist functions fj , gj ∈ C ∞ (Γ, S(Rν )), j = 1, . . . , d, such that a) fj (resp. gj ) are quasi-homogeneous of degree ν/4, (resp. ν/4 − 1/2), i.e. fj (x00 , tξ 00 ; t−1/2 y) = tν/4 fj (x00 , ξ 00 ; y),

∀t > 0, ∀(x00 , ξ 00 ) ∈ Γ, ∀y ∈ Rν ,

gj (x00 , tξ 00 ; t−1/2 y) = tν/4−1/2 gj (x00 , ξ 00 ; y),

∀t > 0, ∀(x00 , ξ 00 ) ∈ Γ, ∀y ∈ Rν ;

b) {fj (ρ, ·)}j=1,...,d is an orthonormal basis of W (ρ); c) the eigenfunctions f0,j satisfies the following parity property either fj (ρ, −y) = fj (ρ, y),

∀ρ ∈ Γ, ∀y ∈ Rν , ∀j = 1, . . . , d, (3.17)

or fj (ρ, −y) = −fj (ρ, y),

∀ρ ∈ Γ, ∀y ∈ Rν , ∀j = 1, . . . , d;

d) the functions gj , j = 1, ..., d, solve the equations Pρ (fj (ρ, ·) + gj (ρ, ·)) + Rρ fj (ρ, ·) = mαj (ρ)(fj (ρ, ·) + gj (ρ, ·)),

∀ρ ∈ Γ, (3.18)

where Rρ := p(3) (ρ, y, Dy ) =

X |α|+|β|+2j=3

1 α β ∂ 0 ∂ 0 p2−j (ρ)y α Dyβ . α!β! x ξ

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We now take a classical symbol ψ 0 ∈ S 0 (R2(n−ν) ) supported in Γ ∩ {|ξ 00 | ≥ 1} ˜ ∩ {|ξ 00 | ≥ 2}. We then define the properly supported and satisfying ψ 0 = 1 in Γ operators Hj = Op(ψ 0 (fj + gj )) ∈ OPH0 (Rn−ν , Σ0 ), and we set Lj = Op(ψ 0 mαj ) ∈ OPS 1 (Rn−ν ),

j = 1, . . . , d.

Remark 3.5. It follows from d) of Lemma 3.4, from (2) of Proposition 2.10 and (3) of Proposition 2.9, that P Hj = Hj Lj

mod OPH0 (Rn−ν , Σ0 ).

(3.19)

Property (3.19) will be crucially used in what follows. Now we observe that the operators Lj ∈ OPS 1 (Rn−ν ) are formally selfadjoint modulo OPS 0 (Rn−ν ). Hence, an application of Theorem 1.1 yields the following result. Proposition 3.6. For every j = 1, . . . , d, using microlocalization, one can construct three bounded linear self-adjoint operators πj+ , πj− and Tj in L2 (Rn−ν ) such that i) for j = 1, . . . , d, πj+ + πj− = Id + Tj ;

(3.20)

ii) there exists a positive constant C such that Re(πj+ Lj v, v) ≥ −Ckvk20 , −Re(πj− Lj v, v)

≥

−Ckvk20 ,

∀v ∈ S(Rn−ν ), ∀v ∈ S(R

n−ν

),

(3.21) (3.22)

and kTj Lj k + kLj Tj k + k[πj+ , Lj ]k + k[πj− , Lj ]k ≤ C,

(3.23)

where [A, B] denotes the commutator of the operators A,B, and kAk represents the norm of A as a bounded linear operator in L2 (Rn−ν ). Upon denoting Π− 0 =

d X

Hj πj− Hj∗ ,

Π+ 0 =

j=1 +

Π =

Hj (πj+ − Tj )Hj∗ ,

j=1

we define the operators Π , Π +

d X

−

Π+ 0

of Theorem 1.2 by + Π+ 1,

− Π− = Π− 0 + Π1 ,

+ where Π− 1 and Π1 are defined in (3.15) and (3.16). Notice that Remark 3.1 also applies to this case. It follows that the localized operator of the pseudodifferential operator Π+ + − Π is just the identity operator in L2 (Rν ) so that R := Id − (Π+ + Π− ) ∈ OPH−1/2 (Rn , Σ) (see Definition 2.4). By virtue of (1) of Proposition 2.10, one has P R, RP ∈ OPH1/2 (Rn , Σ).

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Proof of (1.5) and (1.6) (Case II) + We start by observing that Lemma 3.2 holds true for the operators Π− 1 and Π1 . Inequality (1.6), i.e. − 2 −Re(P (Π− 0 + Π1 )Ψu, Ψu) ≥ −Ckuk0 ,

(3.24)

is a consequence of the following lower bounds 2 −Re(P Π− 1 Ψu, Ψu) ≥ −C1 kuk0 ,

(3.25)

2 −Re(P Π− 0 Ψu, Ψu) ≥ −C2 kuk0 .

(3.26)

and Since the proof of (3.25) goes exactly as the one of (3.8), here we focus only on (3.26). In view of (3.19) we have −Re(P Π− 0 Ψu, Ψu) = −

d X

Re(P Hj πj− Hj∗ Ψu, Ψu)

j=1

=−

d X

Re(Lj πj− Hj∗ Ψu, Hj∗ Ψu) + O(kuk20 )

j=1

≥ −C2 kuk20 , where, for the last inequality, we used the properties (3.22) and (3.23) of Proposition 3.6, and the fact that Hj∗ : L2 (Rn ) → L2 (Rn−ν ) is a linear bounded operator. This concludes the proof of (3.26) and therefore (3.24) is verified. It remains to show (1.5), i.e. + 2 Re(P (Π+ 0 + Π1 )Ψu, Ψu) ≥ −Ckuk0 .

(3.27)

− If we repeat the proof of (3.11) with Π+ 1 in place of Id − Π we obtain 2 Re(P Π+ 1 Ψu, Ψu) ≥ −Ckuk0 .

(3.28)

On the other hand, by Proposition 3.6 and by (3.19) it turns out that Re(P Π+ 0 Ψu, Ψu) =

d X

Re(P Hj (πj+ − Tj )Hj∗ Ψu, Ψu)

j=1

=

d X

Re(Lj πj+ Hj∗ Ψu, Hj∗ Ψu) + O(kuk20 ) ≥ −Ckuk20 .

j=1

This concludes the proof of (3.27) and therefore Theorem 1.2 is proved. Acknowledgment We wish to thank A. Parmeggiani and L. Rodino for helpful discussions.

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References [1] R. Beals, C. Fefferman, On local solvability of linear partial differential operators. Ann. of Math. 97 (1974), 482–298. [2] L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudodifferential operators. Comm. Pure Appl. Math. 27 (1974), 585–639. [3] L. Boutet de Monvel, A. Grigis, B. Helffer, Param´etrixes d’op´erateurs pseudodiff´erentiels ` a caract´eristiques multiples. Ast´erisque 34-35 (1976), 93–121. [4] C. Fefferman, The uncertainty principle. Bull. Amer. Math. Soc. 9 (1983), 129–205. [5] C. Fefferman, D.H. Phong, The uncertainty principle and Sharp G˚ arding inequality. Comm. Pure Appl. Math. 34 (1981), 285–331. [6] D. Fujiwara, A construction of approximate positive parts of essentially self-adjoint pseudodifferential operators. Comm. Pure Appl. Math. 37 (1984), 101–147. [7] B. Helffer, Sur l’hypoellipticit´e des op´erateurs pseudodiff´erentiels ` a caract´eristiques multiples. M´emoires de la SMF 51-52 (1977), 13–61. [8] B. Helffer, Th´eorie spectrale pour des op´erateurs globalement elliptiques. Ast´erisque 112 (1984). [9] L. H¨ ormander, The Cauchy problem for differential operators with double characteristics. J. Anal. Math. 32 (1977), 118–196. [10] L. H¨ ormander, The Analysis of Linear Partial Differential Operators, Vol.I-III, Springer-Verlag, 1983/85. [11] A. Melin, Lower bounds for pseudo-differential operators. Arkiv f¨ or Matematik 9 (1971), 117– 140. [12] A. Mohamed, Etude spectrale d’operateurs hypoelliptiques a caracteristiques multiples II. Comm. Partial Differerential Equations 8 (1983), 247–316. [13] C. Parenti, A. Parmeggiani, A Generalization of H¨ ormander’s Inequality-I. Comm. Partial Differerential Equations 25 (2000), 457–506. [14] C. Parenti, A. Parmeggiani, Lower bounds for systems with double characteristics. J. Anal. Math., 86 (2002), 49–91. [15] A. Parmeggiani, Subunit balls for symbols of pseudodifferential operators. Adv. in Math. 131 (1997), 357–452. [16] A. Parmeggiani, On lower bounds of pseudodifferential systems. In “Hyperbolic problems and related topics”, 269–293, Grad. Ser. Anal., Int. Press, Somerville, MA, 2003. [17] M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, 1987. [18] J. Sj¨ ostrand, Parametrices for pseudodifferential operators with multiple characteristics. Arkiv f¨ or Matematik 12 (1974), 85–130. Marco Mughetti Department of Mathematics University of Bologna Piazza di Porta S. Donato 5 40127 Bologna Italy e-mail: [email protected]

572 Fabio Nicola Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy e-mail: [email protected] Submitted: January 30, 2008.

Mughetti and Nicola

IEOT

Integr. equ. oper. theory 64 (2009), 573–597 c 2009 Birkh¨

auser Verlag Basel/Switzerland 0378-620X/040573-25, published online August 3, 2009 DOI 10.1007/s00020-009-1701-2

Integral Equations and Operator Theory

The Isometric Representation Theory of Numerical Semigroups Sean T. Vittadello Abstract. We study representations of numerical semigroups Σ by isometries on Hilbert space with commuting range projections. Our main theorem says that each such representation is unitarily equivalent to the direct sum of a representation by unitaries and a finite number of multiples of particular concrete representations by isometries. We use our main theorem to identify the faithful representations of the C ∗ -algebra C ∗ (Σ) generated by a universal isometric representation with commuting range projections, and also prove a structure theorem for C ∗ (Σ). Mathematics Subject Classification (2000). Primary 46L05; Secondary 20M30. Keywords. Isometric representation, numerical semigroup, C ∗ -algebra.

1. Introduction Coburn proved that the C ∗ -algebras generated by non-unitary isometries are all canonically isomorphic [1]. Coburn’s result can be viewed as saying that the C ∗ algebras generated by representations of the additive semigroup N by non-unitary isometries are all canonically isomorphic. This result has since been generalised to other semigroups, in particular the positive cones of ordered subgroups of R by Douglas [2], the positive cones of totally ordered abelian groups by Murphy [6], and amenable quasi-lattice ordered groups by Nica [7] and Laca-Raeburn [5]. Murphy [6] and Jang [3, 4] have observed that such a result does not hold for the additive semigroup N \ {1}, however, by finding two representations by non-unitary isometries whose C ∗ -algebras are not canonically isomorphic. This observation motivated an investigation of the isometric representations of N \ {1} and the C ∗ -algebras they generate [8]. The main theorem in [8] says that each isometric representation of N \ {1} with commuting range projections is unitarily equivalent to the direct sum of a unitary representation, a multiple of the Toeplitz This research was supported by the Australian Research Council.

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representation on `2 (N \ {1}), and a multiple of a representation by shifts on `2 (N). This theorem was used to study the C ∗ -algebra C ∗ (N \ {1}) generated by a universal isometric representation with commuting range projections. The faithful representations of C ∗ (N \ {1}) were identified, and the structure of C ∗ (N \ {1}) described in terms of the usual Toeplitz algebra. The purpose of this article is to extend the study in [8] to the whole class of numerical semigroups, of which the semigroup N \ {1} is a particular example. As in [8] we study representations by isometries with commuting range projections. We analyse the isometric representations of numerical semigroups with commuting range projections, and investigate the structure of the C ∗ -algebras they generate. The organisation of this article is as follows. We discuss the necessary background material on numerical semigroups in Section 2. In Section 3 we describe the class of isometric representations that we investigate, and in particular our main class of concrete examples. In Section 4 we associate to every numerical semigroup Σ a certain collection A(Σ) of subsets of N. To each set in A(Σ) we then associate a particular concrete isometric representation of Σ, and the resulting collection of representations is used in our description of a general representation which constitutes our main theorem. We prove our main theorem in Section 5 by extending a special case of this theorem. The proof of the special case is the subject of Section 6, and the basic strategy is analogous to that for [8, Theorem 3.1]. In Section 7 we use our main theorem to investigate the C ∗ -algebra C ∗ (Σ) generated by a universal isometric representation of a numerical semigroup Σ with commuting range projections. We obtain a condition describing precisely when a given representation of C ∗ (Σ) is faithful, and describe the structure of C ∗ (Σ) in terms of the usual Toeplitz algebra.

2. Numerical semigroups Throughout this article, N denotes the additive semigroup of non-negative integers (including 0). A numerical semigroup is a subsemigroup of N containing 0 such that the greatest common divisor of its elements is equal to one. The following proposition is proved in [9, page 105, Proposition 10.2] and provides some equivalent characterisations of numerical semigroups. Proposition 2.1. Let Σ be a subsemigroup of N containing 0. Then the following are equivalent: 1. Σ is a numerical semigroup; 2. Σ generates the additive group Z of integers; 3. Σ has finite complement in N. For a numerical semigroup Σ, the complement N \ Σ is finite by Proposition 2.1, so Z \ Σ contains a largest element (with respect to the usual order on Z), which we call the conductor of Σ and denote by C(Σ).

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Notation 2.2. For a non-empty finite subset M = {n1 , . . . , nr } of N we shall denote by hM i or hn1 , . . . , nr i the subsemigroup of N generated by M and containing 0. Remark 2.3. For a non-empty finite subset M of N, hM i is a numerical semigroup if and only if the greatest common divisor of the elements of M is equal to one. The following straightforward proposition shows that every numerical semigroup is finitely generated and has a unique minimal (with respect to inclusion) system of generators. These properties are also discussed in [9, pages 106 and 107]. Proposition 2.4. Let Σ be a numerical semigroup and define Σ∗ := Σ \ {0}. Then Σ∗ \(Σ∗ +Σ∗ ) is finite, generates Σ, and is contained in every system of generators of Σ.

3. Isometric representations of numerical semigroups An isometric representation of a numerical semigroup Σ on a Hilbert space H is a map V : Σ → B(H) such that each Vn is an isometry, and such that Vm+n = Vm Vn . We say that V has commuting range projections if the range projections Vn Vn∗ pairwise commute. Our main examples are of a particular class. Example 3.1. Let Σ be a numerical semigroup and let A be a non-empty subset of N. Denote by { eA,a | a ∈ A } the standard orthonormal basis for the Hilbert space `2 (A). If Σ + A ⊂ A, or equivalently Σ + A = A, then for each n ∈ Σ the set { eA,n+a | a ∈ A } is orthonormal and hence there is an isometry TnA on `2 (A) such A A A that TnA eA,a = eA,n+a . Straightforward calculations show that Tm+n = Tm Tn , A and that the range projections pairwise commute. The map n 7→ Tn is then an isometric representation of Σ on `2 (A) with commuting range projections, which we denote by T A . In particular, T N and T Σ are isometric representations. Note that if S is the unilateral shift on `2 (N) then TnN = S n . Remark 3.2. If Σ and Γ are numerical semigroups, and A is a non-empty subset of N such that Σ + A = A and Γ + A = A, then we shall denote both of the associated isometric representations of Σ and Γ from Example 3.1 by T A without reference to the particular numerical semigroup. The context will make clear which representation is intended. Remark 3.3. Let Σ be a numerical semigroup. The subsemigroup Σ of the additive group Z is the positive cone for the partial order ≤ on Z defined by n ≤ m if and only if m − n ∈ Σ. If Σ 6= N then the pair (Z, Σ) is not quasi-lattice ordered in the sense of Nica [7]. Indeed, the subset {C(Σ) + 1, C(Σ) + 2} of Z has an upper bound in Σ, however does not have a least upper bound in Σ. So the general theory of [7] and [5] does not apply in our situation.

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Consider an isometric representation V : Σ → B(H) of a numerical semigroup Σ with commuting range projections. If s, m ∈ Σ and m − s ∈ Σ then Vs∗ Vm = Vm−s ∗ and Vm∗ Vs = Vm−s . If m − s ∈ / Σ and m − s > 0, however, then we cannot expect to cancel words such as Vs∗ Vm in this manner. Therefore words in terms of V may appear quite complicated, however with the assumption of commuting range projections there are certain words that are also pairwise commuting projections. Lemma 3.4. Let Σ be a numerical semigroup and V : Σ → B(H) be an isometric representation of Σ on a Hilbert space H with commuting range projections. Then Vs∗ Vm Vm∗ Vs is a projection for s, m ∈ Σ, and commutes with every other such projection. Proof. Since Vs∗ Vm Vm∗ Vs is self-adjoint and (Vs∗ Vm Vm∗ Vs )2 = Vs∗ (Vm Vm∗ )(Vs Vs∗ )Vm Vm∗ Vs = Vs∗ Vs Vs∗ Vm Vm∗ Vm Vm∗ Vs = Vs∗ Vm Vm∗ Vs , Vs∗ Vm Vm∗ Vs is a projection. Further, (Vs∗ Vm Vm∗ Vs )(Vt∗ Vn Vn∗ Vt ) ∗ = Vs∗ Vm Vm∗ Vs Vt∗ (Vs∗ Vs )Vn Vn∗ (Vs∗ Vs )Vt = Vs∗ (Vm Vm∗ )(Vs Vs∗ )Vt∗ Vs+n Vs+n Vs+t ∗ ∗ ∗ ∗ = Vs∗ (Vt∗ Vt )Vm Vm∗ Vt∗ Vs+n Vs+n Vs+t = Vt+s (Vt+m Vt+m )(Vs+n Vs+n )Vs+t ∗ ∗ ∗ = Vt+s Vs+n Vs+n Vt+m Vt+m Vs+t (Vs∗ Vs ) = Vt∗ Vn Vn∗ Vs∗ Vt (Vm Vm∗ )(Vs Vs∗ )Vs

= Vt∗ Vn Vn∗ Vs∗ Vt Vs Vs∗ Vm Vm∗ Vs = (Vt∗ Vn Vn∗ Vt )(Vs∗ Vm Vm∗ Vs ), so any two such projections commute.

4. The collection A(Σ) Throughout this section we let Σ be a numerical semigroup with minimal system of generators {m1 , . . . , mr }, where mi < mj if i < j. In this section we shall associate to Σ a certain collection A(Σ) of subsets of N, such that each A ∈ A(Σ) gives an isometric representation T A as in Example 3.1. Definition 4.1. We define A(Σ) to be the collection of all subsets A of N such that Σ + A = A and Σ ⊂ A. Remark 4.2. If A is a subset of N such that Σ + A = A then the condition Σ ⊂ A is equivalent to 0 ∈ A. Remark 4.3. We trivially have that Σ and N are in A(Σ). For A ∈ A(Σ) we can write A = Σ ∪ (A \ Σ), so A is determined by the subset A \ Σ of N \ Σ. It must be noted, however, that not every subset of N \ Σ, when adjoined to Σ, will give a set A such that Σ + A = A. As an example, consider the numerical semigroup h3, 4i = N \ {1, 2, 5}, for which N \ h3, 4i = {1, 2, 5}, and define A := h3, 4i ∪ {2}. Then {2} is a subset of N \ h3, 4i, however h3, 4i + A = N \ {1} is not a subset of A since 5 ∈ / A.

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Theorem 4.4. A(Σ) is a finite collection such that: 1. For each A ∈ A(Σ) we have an isometric representation T A ; 2. Whenever C is a non-empty subset of N such that Σ + C = C there exists A ∈ A(Σ) such that T C is unitarily equivalent to T A ; 3. If A, B ∈ A(Σ) and A 6= B then T A is not unitarily equivalent to T B . Remark 4.5. Consider the equivalence relation on the collection of all non-empty subsets A of N satisfying Σ + A = A such that A is related to B if and only if the isometric representations T A and T B are unitarily equivalent. Then, by Theorem 4.4, A(Σ) consists of precisely one element A from each equivalence class, namely the unique element with Σ ⊂ A. Before proving Theorem 4.4 we shall establish some notation and technical results. As motivation, for A ∈ A(Σ) we decompose `2 (A) as a direct sum 2 L∞ A n A ∗n A n+1 A ∗ n+1 − (Tm ) (Tm ) (` (A)), (4.1) n=0 (Tm1 ) (Tm1 ) 1 1 A ∗ n+1 A n+1 A ∗n A n ) is the projection of `2 (A) onto the ) (Tm ) − (Tm ) (Tm where (Tm 1 1 1 1 A n+1 2 A n 2 orthogonal complement of (Tm1 ) (` (A)) in (Tm ) (` (A)). For m ∈ A and 1 A ∗ n+1 A n+1 A ∗n A n eA,m is equal to eA,m if n is the (Tm1 ) n ∈ N, (Tm1 ) (Tm1 ) − (Tm1 ) largest integer such that m − nm1 ∈ A (equivalently, m − nm1 is the smallest integer in A which is congruent to m modulo m1 ), and is zero otherwise. Denoting by α the smallest positive integer in Σ such that m1 and α are relatively prime, we apply this analysis to (TαA )i eA,0 = eA,iα , for 0 ≤ i ≤ m1 − 1, thereby identifying the subspace of the decomposition (4.1) to which the image of eA,0 under (TαA )i belongs. These properties will allow us to characterise the representation T A , and distinguish between distinct representations of this form. This leads us to the following definition.

Definition 4.6. Let α be the smallest positive integer in Σ such that m1 and α are relatively prime. For each A ∈ A(Σ) and 0 ≤ i ≤ m1 − 1 let bA,i be the smallest element of A such that bA,i ≡ iα (mod m1 ), and let qA,i ∈ N be the unique integer such that bA,i = iα − qA,i m1 . Remark 4.7. If Σ has a minimal system of generators {m1 , m2 } with m1 < m2 then α = m2 , a fact we use in Section 6. Remark 4.8. For the set { bA,i | 0 ≤ i ≤ m1 − 1 }, the residue classes modulo m1 of distinct elements are distinct, and the collection of such classes is a partition of Z. Lemma 4.9. Let A ∈ A(Σ). Each a ∈ A can be written uniquely in the form a = nm1 + bA,j for some n ∈ N and 0 ≤ j ≤ m1 − 1. Proof. As a ≡ jα (mod m1 ) for some 0 ≤ j ≤ m1 − 1 it follows by the definition of bA,j that a ≡ bA,j (mod m1 ) and a ≥ bA,j , hence there exists n ∈ N such that a = nm1 + bA,j . For uniqueness, if we can also write a = n0 m1 + bA,i with n0 ∈ N and 0 ≤ i ≤ m1 − 1 then bA,i ≡ bA,j (mod m1 ), so i = j and the result follows.

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The following proposition gives an indication of the significance of the bA,i . A ∗ Proposition 4.10. If A ∈ A(Σ) then ker((Tm ) ) = span{ eA,bA,i | 0 ≤ i ≤ m1 −1 }. 1 A ∗ Proof. The inclusion span{ eA,bA,i | 0 ≤ i ≤ m1 − 1 } ⊂ ker((Tm ) ) follows from 1 the observation that if 0 ≤ i ≤ m1 − 1 then the definition of bA,i as the smallest A ∗ element of A such that bA,i ≡ iα (mod m1 ) gives (Tm ) eA,bA,i = 0. For the reverse 1 inclusion consider a ∈ A. By Lemma 4.9 there exist n ∈ N and 0 ≤ j ≤ m1 − 1 A ∗ such that a = nm1 + bA,j . If n > 0 then (Tm ) eA,a = eA,(n−1)m1 +bA,j , and if 1 A ∗ A ∗ A ∗ ) ) is contained in n = 0 then (Tm1 ) eA,a = (Tm1 ) eA,bA,j = 0. Therefore ker((Tm 1 span{ eA,bA,i | 0 ≤ i ≤ m1 − 1 }.

Lemma 4.11. If A, B ∈ A(Σ) with A 6= B then there exists 1 ≤ k ≤ m1 − 1 such that bA,k 6= bB,j for all 0 ≤ j ≤ m1 − 1. Proof. Suppose bA,i ∈ { bB,j | 0 ≤ j ≤ m1 − 1 } for 1 ≤ i ≤ m1 − 1, and note that we then have { bA,i | 0 ≤ i ≤ m1 − 1 } = { bB,j | 0 ≤ j ≤ m1 − 1 }. If a ∈ A then, by Lemma 4.9, a = nm1 + bA,i for some n ∈ N and 0 ≤ i ≤ m1 − 1, and since bA,i = bB,j for some 0 ≤ j ≤ m1 − 1 we have a = nm1 + bB,j ∈ B. So A is a subset of B. A similar argument shows that B is a subset of A. Proposition 4.12. Let A ∈ A(Σ). Define the polynomial pA in 4 non-commuting indeterminates over C as Qm1 −1 i qA,i pA (v, w, x, y) := i=0 yv (1 − vw)wqA,i xi . A ∗ A ) , TαA , (TαA )∗ ) is the projection of `2 (A) onto span{eA,0 }, and , (Tm Then pA (Tm 1 1 B B ∗ for B ∈ A(Σ) with B 6= A, pA (Tm , (Tm ) , TαB , (TαB )∗ ) is zero on `2 (B). 1 1

Proof. Let C ∈ A(Σ). For simplicity of notation we shall denote by QC i the factor C C ∗ C C C ∗ ∗ C (Tiα ) TqA,i m1 1 − Tm1 (Tm1 ) (TqA,i m1 ) Tiα C ∗ C ) , TαC , (TαC )∗ ), for each 0 ≤ i ≤ m1 − 1. Recalling that the range , (Tm of pA (Tm 1 1 projections of T C pairwise commute, we observe that each QC i is a projection. Further, it follows from Lemma 3.4 that the factors QC i pairwise commute. So C C ∗ pA (Tm , (Tm ) , TαC , (TαC )∗ ) is a product of commuting projections, hence is itself 1 1 a projection. For each 0 ≤ i ≤ m1 − 1 and c ∈ C, recalling that bA,i = iα − qA,i m1 , we have ( eC,c if bA,i + c ∈ C and bA,i + c − m1 ∈ / C, C Qi eC,c = (4.2) 0 otherwise.

If c ∈ C and c > 0 then, choosing 0 ≤ k ≤ m1 − 1 with −c ≡ bA,k (mod m1 ), we have bA,k + c = qm1 for some integer q ≥ 1, hence both bA,k + c and bA,k + c − m1 are in C, so QC k eC,c = 0 by (4.2). It follows that the range of the projection C C ∗ pA (Tm , (T ) , TαC , (TαC )∗ ) is contained in span{eC,0 }. m 1 1 A A ∗ The definition of bA,i and (4.2) give pA (Tm , (Tm ) , TαA , (TαA )∗ )eA,0 = eA,0 , 1 1 so the range of this projection is as claimed.

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B B ∗ It remains to show that pA (Tm , (Tm ) , TαB , (TαB )∗ )eB,0 = 0 for B 6= A. By 1 1 Lemma 4.11 there exists 1 ≤ k ≤ m1 −1 such that bA,k 6= bB,j for all 0 ≤ j ≤ m1 −1. If bA,k ∈ B then by Lemma 4.9 there exist n ∈ N and 0 ≤ j ≤ m1 − 1 such that bA,k = nm1 + bB,j . Since bA,k 6= bB,j , n ≥ 1, so bA,k − m1 ∈ B. In any case, it B B ∗ B B ∗ follows from (4.2) that QB k eB,0 = 0, hence pA (Tm1 , (Tm1 ) , Tα , (Tα ) ) = 0.

Proof of Theorem 4.4. For A ∈ A(Σ) we can write A = Σ ∪ (A \ Σ), so A is determined by the subset A\Σ of N\Σ. Since N\Σ is finite by Proposition 2.1, there can only be finitely many distinct such sets A, hence A(Σ) is a finite collection. If A ∈ A(Σ) then, since Σ + A = A, we have an isometric representation T A as in Example 3.1. Let C be a non-empty subset of N such that Σ + C = C, let m be the smallest element of C, and define A := { c − m | c ∈ C }. Since Σ + C = C, Σ + A = A. Further, 0 ∈ A, so Σ ⊂ A. Hence A ∈ A(Σ). The representations T C and T A are unitarily equivalent since the map φ : `2 (C) → `2 (A) determined by φ(eC,c ) = eA,c−m for c ∈ C is a unitary isomorphism satisfying φTnC = TnA φ for n ∈ Σ. Finally, let A, B ∈ A(Σ) with A 6= B, and take pA as in Proposition 4.12. B ∗ B A ∗ A ) , TαB , (TαB )∗ ) , (Tm ) , TαA , (TαA )∗ ) is non-zero, whereas pA (Tm , (Tm Then pA (Tm 1 1 1 1 is zero, and it follows that T A and T B are not unitarily equivalent. From a practical point of view, one may wish to determine the collection A(Σ), and Proposition 4.13 shows that the sets in A(Σ) can be obtained by considering the subsets of the finite set N \ Σ. Proposition 4.13. Denote by Λ(Σ) the collection of all subsets Λ of N \ Σ such that {m1 , . . . , mr } + Λ ⊂ Σ ∪ Λ. The mapping Λ 7→ Σ ∪ Λ is a bijection from Λ(Σ) onto A(Σ), with inverse mapping A 7→ A \ Σ. Proof. It is enough to show the mappings are well-defined. Let Λ ∈ Λ(Σ). Since {m1 , . . . , mr } + Λ ⊂ Σ ∪ Λ, Σ + Λ ⊂ Σ ∪ Λ, so Σ + (Σ ∪ Λ) ⊂ Σ ∪ Λ. Further, Σ ⊂ Σ ∪ Λ, so we have Σ ∪ Λ ∈ A(Σ). For the inverse, if A ∈ A(Σ) then, since Σ + A ⊂ A, {m1 , . . . , mr } + (A \ Σ) is a subset of Σ ∪ (A \ Σ), so A \ Σ ∈ Λ(Σ).

5. The decomposition theorem In this section we state and prove our main theorem. The proof follows easily from the corresponding result for the special case of numerical semigroups with a minimal system of generators consisting of two elements. We also state the special case in this section, however due to the length of its proof we defer the proof until Section 6. Given isometric representations V and W of a semigroup P on Hilbert spaces HV and HW , we say that V is a multiple of W if there are a Hilbert space K and a unitary isomorphism U : HV → HW ⊗ K such that U Vp U ∗ = Wp ⊗ 1 for p ∈ P .

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For the representations T A of Example 3.1 we can identify the tensor product `2 (A) ⊗ K with `2 (A, K), and we move freely between these realisations. We can now state our main theorem. Theorem 5.1. Let V : Σ → B(H) be an isometric representation of a numerical semigroup Σ on a Hilbert space H with commuting L range projections. Then there is a unique direct-sum decomposition H = HU ⊕ ( A∈A(Σ) HA ) such that HU and HA for A ∈ A(Σ) are reducing for V , such that V |HU consists of unitary operators, and such that V |HA is a multiple of T A for A ∈ A(Σ). Proposition 5.2 is the special case of Theorem 5.1 for numerical semigroups with a minimal system of generators consisting of two elements. Proposition 5.2. Suppose Σ is a numerical semigroup with a minimal system of generators consisting of two elements, and let V : Σ → B(H) be an isometric representation of Σ on a Hilbert space H with commutingL range projections. Then there is a unique direct-sum decomposition H = HU ⊕ ( A∈A(Σ) HA ) such that HU and HA for A ∈ A(Σ) are reducing for V , such that V |HU consists of unitary operators, and such that V |HA is a multiple of T A for A ∈ A(Σ). The key to extending Proposition 5.2 to Theorem 5.1 is the next lemma. We first consider a few preliminary remarks for two numerical semigroups Γ and Σ with Γ ⊂ Σ. We have A(Σ) ⊂ A(Γ). Indeed, if A ∈ A(Σ) then Γ + A ⊂ Σ + A = A and Γ ⊂ Σ ⊂ A, so A ∈ A(Γ). We say that an isometric representation V : Σ → B(H) extends an isometric representation W : Γ → B(H) if W = V |Γ . Recalling our convention from Remark 3.2, for A ∈ A(Σ) ⊂ A(Γ) we shall denote both of the associated representations of Σ and Γ from Example 3.1 by T A . Lemma 5.3. Suppose that Γ and Σ are numerical semigroups with Γ ⊂ Σ, suppose that W : Γ → B(H) is an isometric representation on a Hilbert space H which extends to an isometric representation VL: Σ → B(H), and suppose that there is a direct-sum decomposition H = HU ⊕ ( A∈A(Γ) HA ) such that HU and HA for A ∈ A(Γ) are reducing for W , such that W |HU consists of unitary operators, and such that W |HA is a multiple of T A for A ∈ A(Γ). Then HA = {0} for A ∈ A(Γ) \ A(Σ), HU and HA for A ∈ A(Σ) are reducing for V , V |HU consists of unitary operators, and V |HA is a multiple of T A for A ∈ A(Σ). Proof. Throughout the proof we let c := C(Γ) + 1, so that t ∈ Γ ⊂ Σ whenever t ≥ c. Fix A ∈ A(Γ) \ A(Σ). We show that HA = {0}. Note that Σ + A is not a subset of A, for if Σ + A ⊂ A then, since 0 ∈ A, Σ ⊂ A, and hence A ∈ A(Σ). Choose s ∈ Σ and a ∈ A such that s + a ∈ / A. Suppose HA 6= {0}. Then there exist a non-zero Hilbert space K and a unitary isomorphism U : HA → `2 (A) ⊗ K

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such that Wt |HA = U ∗ (TtA ⊗ 1)U for t ∈ Γ. Choose a non-zero vector k in K. Since A s+a∈ / A, (TcA )∗ Tc+s eA,a = (TcA )∗ eA,c+s+a = 0, so Vs (U ∗ (eA,a ⊗ k)) = Vc∗ Vc+s (U ∗ (eA,a ⊗ k)) = Wc∗ Wc+s (U ∗ (eA,a ⊗ k)) A = U ∗ ((TcA )∗ Tc+s ⊗ 1)U (U ∗ (eA,a ⊗ k)) A = U ∗ ((TcA )∗ Tc+s eA,a ⊗ k) = 0.

Since eA,a ⊗ k is a non-zero vector in `2 (A) ⊗ K, this implies that Vs is not an isometry. Therefore we must have HA = {0}. We now show that the subspaces HU and HA for A ∈ A(Σ) reduce V . Indeed, if m ∈ Σ then Vm = Vc∗ Vc+m = Wc∗ Wc+m and ∗ ∗ Vm∗ = Vc+m Vc = Wc+m Wc , so any subspace of H that is reducing for W is also reducing for V , and the claim follows. To see that V |HU consists of unitaries, let m ∈ Σ and h ∈ HU . Then, as W |HU consists of unitaries, ∗ ∗ Vm Vm∗ h = Vc∗ Vc+m Vc+m Vc h = Wc∗ Wc+m Wc+m Wc h = Wc∗ 1HU Wc h = h,

hence Vm |HU is unitary. It remains to show that, for A ∈ A(Σ), V |HA is a multiple of T A . Since A ∈ A(Γ), W |HA is a multiple of T A , so there exist a Hilbert space K and a unitary isomorphism U : HA → `2 (A) ⊗ K such that U Wt U ∗ = TtA ⊗ 1 for t ∈ Γ. If m ∈ Σ then U Vm U ∗ = U Vc∗ Vc+m U ∗ = U Wc∗ Wc+m U ∗ = (U Wc∗ U ∗ )(U Wc+m U ∗ ) A A = ((TcA )∗ ⊗ 1)(Tc+m ⊗ 1) = Tm ⊗ 1.

Therefore V |HA is a multiple of T A .

Proof of Theorem 5.1. We begin with the case Σ = N. Applying the version of the Wold decomposition stated in [8, Proposition 3.2] to V1 , there is a reducing subspace HU for V1 with V1 |HU unitary, and, denoting by S the unilateral shift ⊥ on `2 (N), there is a subspace H0 and a unitary isomorphism φ : HU → `2 (N) ⊗ H0 ∗ ⊥ such that φV1 φ = SL ⊗ 1. Then, with HN := HU and noting that A(Σ) = {N}, we have H = HU ⊕ ( A∈A(Σ) HA ). Since Vn = V1n , for n ∈ N, and HU and HN reduce V1 , HU and HN reduce V . The representation V |HU consists of unitary operators since each Vn |HU = (V1 |HU )n is unitary. For n ∈ N, TnN = S n , so φVn φ∗ = φV1n φ∗ = S n ⊗ 1 = TnN ⊗ 1, hence V |HN is a multiple of T N . The 0 0 decomposition is unique, for if there is another H = HU ⊕ HN as in the theorem 0 0 then applying the Wold decomposition to V1 gives HU = HU and HN = HN . The only numerical semigroup with a minimal system of generators consisting of one element is N, so we now assume for the remainder of the proof that Σ has a minimal system of generators consisting of at least two elements. Let Γ be the

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numerical semigroup hC(Σ) + 1, C(Σ) + 2i, which is a subsemigroup of Σ, and let W : Γ → B(H) be the restriction of V to Γ. Note that the minimal system of generators of Γ is {C(Σ)+1, C(Σ)+2}. By Proposition 5.2 there is a unique directL sum decomposition H = HU ⊕ ( A∈A(Γ) HA ) such that HU and HA for A ∈ A(Γ) are reducing for W , such that W |HU consists of unitary operators, and such that W |HA is a multiple of T A for A ∈ A(Γ). Since W extends Lto the representation V : Σ → B(H), by Lemma 5.3 we have that H = HU ⊕ ( A∈A(Σ) HA ), HU and HA for A ∈ A(Σ) are reducing for V , V |HU consists of unitary operators, and V |HA is a multiple of T A for A ∈ A(Σ). L 0 0 For uniqueness, suppose H = HU ⊕ ( A∈A(Σ) HA ) is another decomposition 0

0

such that HU and HA for A ∈ A(Σ) are reducing for V , such that V |H0 consists U of unitary operators, and such that V |H0 is a multiple of T A for A ∈ A(Σ). If A

0

we define HA := {0} for A ∈ A(Γ) \ A(Σ) then we have another decomposition L 0 0 0 0 H = HU ⊕ ( A∈A(Γ) HA ) such that HU and HA for A ∈ A(Γ) are reducing for W , such that W |H0 consists of unitary operators, and such that W |H0 is a multiple U

A

0

of T A for A ∈ A(Γ), so by the uniqueness in Proposition 5.2 we have HU = HU 0 and HA = HA for A ∈ A(Γ), and the required uniqueness follows.

6. Proof of Proposition 5.2 We now turn to the proof of Proposition 5.2. In this section we shall assume that our numerical semigroup Σ has a minimal system of generators consisting of two elements {m1 , m2 } where m1 < m2 (note that we must have m1 ≥ 2). Our basic strategy in proving Proposition 5.2 is analogous to that in proving Theorem 3.1 of [8]. The representation V is determined by the two isometries Vm1 and Vm2 . We shall apply the version of the Wold decomposition stated in [8, Proposition 3.2] to the isometry Vm1 , and analyse the interaction of Vm2 with this decomposition. As motivation for our argument, for A ∈ A(Σ) we apply the Wold decomposition to A Tm . For such an isometry we have HU = {0}, and it follows from Proposition 4.10 1 ⊥

A ∗ A ) ) as the direct (`2 (A)) = ker((Tm that we have a decomposition of H0 = Tm 1 1 sum, recalling from Definition 4.6 that bA,i = im2 − qA,i m1 , Lm1 −1 Lm1 −1 A ∗qA,i A i H0 = i=0 span{eA,bA,i } = i=0 (Tm1 ) (Tm2 ) (span{eA,0 }). (6.1)

By Lemma 4.9 each a ∈ A can be expressed uniquely in the form a = nm1 + bA,j for some n ∈ N and 0 ≤ j ≤ m1 − 1, so sending eA,a 7→ enj gives a unitary isomorphism of `2 (A) onto `2 (N × {0, . . . , m1 − 1}) which carries T A into the isometric representation determined on f ∈ `2 (N × {0, . . . , m1 − 1}) by ( 0 if n = 0, A (Tm1 f )nj = (6.2) f(n−1)j if n ≥ 1

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and

A (Tm f )nj 2

 f(n−qA,j +qA,j−1 )(j−1)    0 =  f(n−m2 +qA,m1 −1 )(m1 −1)    0

if if if if

n ≥ qA,j − qA,j−1 and j ≥ 1, n < qA,j − qA,j−1 and j ≥ 1, n ≥ m2 − qA,m1 −1 and j = 0, n < m2 − qA,m1 −1 and j = 0,

(6.3)

where we have used that qA,m1 −1 ≤ m2 , and qA,j−1 ≤ qA,j for 1 ≤ j ≤ m1 − 1 (we do not prove these two results here, but note that they follow from the more general results in Lemmas 6.7 and 6.8). Note that, with respect to the unitary isomorphism sending eA,a 7→ enj , span{ enj | 0 ≤ j ≤ m1 − 1 } 2

is the subspace of ` (N × {0, . . . , m1 − 1}) corresponding to the subspace 2 A n A ∗n A n+1 A ∗ n+1 (Tm ) (Tm ) − (Tm ) (Tm ) (` (A)) 1 1 1 1 of `2 (A) in the decomposition (4.1). Viewing the representation T A in the manner described by (6.2) and (6.3) is rather instructive. Example 6.1. Consider the numerical semigroup h3, 4i = N\{1, 2, 5}. The collection A(h3, 4i) consists of the five sets A1 := h3, 4i, A2 := N, A3 := h3, 4i ∪ {1, 5}, A4 := h3, 4i ∪ {2, 5} and A5 := h3, 4i ∪ {5}. For   f02 f12 f22 f32 · · · f = f01 f11 f21 f31 · · · ∈ `2 (N × {0, 1, 2}), f00 f10 f20 f30 · · · we have from (6.2) that  0 f02 T3Ai f = 0 f01 0 f00

f12 f11 f10

f22 f21 f20

 ··· · · · ···

for 1 ≤ i ≤ 5, and from (6.3) that   f01 f11 f21 f31 f41 f51 · · · T4A1 f = f00 f10 f20 f30 f40 f50 · · · , 0 0 0 0 f02 f12 · · ·    0 f01 f11 f21 · · · f01 f11 f21 f31 A2 A T4 f = 0 f00 f10 f20 · · · , T4 3 f =  0 f00 f10 f20 0 0 f02 f12 · · · 0 0 0 f02   0 0 f01 f11 · · · T4A4 f = f00 f10 f20 f30 · · · 0 0 f02 f12 · · · and   0 f01 f11 f21 f31 · · · T4A5 f = f00 f10 f20 f30 f40 · · · . 0 0 0 f02 f12 · · ·

f41 f30 f12

 ··· · · · , ···

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We now begin the proof of Proposition 5.2. Applying the Wold decomposition to the isometry Vm1 gives a reducing subspace T∞ HU := n=0 Vmn1 (H) such that Vm1 |HU is unitary, and, since Vmm21 = Vmm12 , the isometry Vm2 and every other Vnm1 +pm2 = Vmn1 Vmp 2 are also unitary on HU . The Wold decomposition also ⊥ says that the complement HU of HU in H can be identified with `2 (N, H0 ) for ⊥

H0 := Vm1 (H) = ker Vm1 Vm∗ 1 . Our goal is to decompose H0 into subspaces where V behaves like each T A , and we begin by identifying the subspaces KA , for A ∈ A(Σ), of H0 consisting of vectors which behave under Vmi 2 as the vector e00 ∈ `2 (N × {0, . . . , m1 − 1}) A i does under (Tm ) for 1 ≤ i ≤ m1 − 1. The crucial observation that we make is 2 A i A qA,i A qA,i that (Tm2 ) e00 = eqA,i i = (Tm ) e0i belongs to the range of (Tm ) and is 1 1 A qA,i +1 orthogonal to the range of (Tm1 ) . With these observations in mind, we define for A ∈ A(Σ) q

q

A,i KA := { h ∈ H0 | Vmi 2 h ∈ VmA,i 1 (H) Vm1

+1

(H) for 1 ≤ i ≤ m1 − 1 }.

With the notation Pn := Vmn1 Vm∗n1 − Vmn+1 Vm∗n+1 , which is the projection of H onto 1 1 n+1 the orthogonal complement of Vm1 (H) in Vmn1 (H), we may write KA = { h ∈ H0 | Vmi 2 h ∈ PqA,i (H) for 1 ≤ i ≤ m1 − 1 } and H0 = P0 (H). In order to decompose H0 into subspaces where the representation V behaves like each T A we shall determine the projections of H onto the subspaces of H0 A i A ∗qA,i ) (span{eA,0 }) of ) (Tm corresponding to the subspaces span{eA,bA,i } = (Tm 2 1 the direct-sum decompositions in (6.1), observing that the subspace KA of H0 corresponds to the subspace span{eA,0 } of (6.1). Recall that, for m ∈ A and A n A ∗n A n+1 A ∗ n+1 n ∈ N, (Tm ) (T ) − (T ) (T ) e A,m is equal to eA,m if n is the m1 m1 m1 1 largest integer such that m−nm1 ∈ A, and is zero otherwise. Applying this analysis A j ) eA,bA,i = eA,jm2 +bA,i , for 0 ≤ j ≤ m1 − 1, identifies the subspace of the to (Tm 2 A j decomposition (4.1) to which the image of eA,bA,i under (Tm ) belongs. This leads 2 us to the following definition. Definition 6.2. For each A ∈ A(Σ) and 0 ≤ i, j ≤ m1 − 1 define rA,i,j ∈ N to be the unique integer such that jm2 + bA,i − rA,i,j m1 is the smallest integer in the set { jm2 + bA,i − rm1 | r ∈ N } ∩ A. Remark 6.3. For 0 ≤ j ≤ m1 − 1 we have rA,0,j = qA,j . Indeed, if i = 0 then, since bA,0 = 0, we have that jm2 + bA,i − rA,i,j m1 = jm2 − rA,0,j m1 is the smallest integer in the set { jm2 − rm1 | r ∈ N } ∩ A, hence by the definition of bA,j it follows that bA,j = jm2 − rA,0,j m1 , so rA,0,j = qA,j . The following lemma provides a useful expression for rA,i,j .

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Lemma 6.4. Fix A ∈ A(Σ) and 0 ≤ i, j ≤ m1 − 1. Choose 0 ≤ k ≤ m1 − 1 such that i + j ≡ k (mod m1 ). Then rA,i,j = qA,k − qA,i + (m2 /m1 )(i + j − k). Proof. For simplicity of notation define s := qA,k − qA,i + (m2 /m1 )(i + j − k). Since the definition of rA,i,j implies jm2 + bA,i − rA,i,j m1 ∈ A, we have bA,k + (s − rA,i,j )m1 = jm2 + bA,i − rA,i,j m1 ∈ A. So, as bA,k is the smallest element of A such that bA,k ≡ km2 (mod m1 ), we must have rA,i,j ≤ s. In particular, s ≥ 0. Further, jm2 + bA,i − sm1 = bA,k ∈ A, so by the definition of rA,i,j we must have rA,i,j = s. We are now in a position to describe the projections of H onto the relevant subspaces of H0 . Note that, for s ∈ Σ and m ∈ N, Vs∗ Pm Vs is self-adjoint and (Vs∗ Pm Vs )2 = Vs∗ Pm (Vs Vs∗ )Pm Vs = Vs∗ (Vs Vs∗ )Pm Pm Vs = Vs∗ Pm Vs , so Vs∗ Pm Vs is a projection. Furthermore, it follows from Lemma 3.4 that such projections pairwise commute. Qm1 −1 ∗j Proposition 6.5. Let A ∈ A(Σ) and 0 ≤ i ≤ m1 − 1. Then j=0 (Vm2 PrA,i,j Vmj 2 ) ∗qA,i i is the projection of H onto Vm1 Vm2 (KA ). Qm1 −1 ∗j Proof. Each j=0 (Vm2 PrA,i,j Vmj 2 ) is a product of commuting projections and hence is a projection, so it suffices to show that the ranges of these projections are as claimed. We begin with the case i = 0. For 0 ≤ j ≤ m1 − 1, Vmj 2 (KA ) ⊂ PqA,j (H) by the definition of KA , hence Vm∗j2 PqA,j Vmj 2 is the identity on KA . It follows that Qm1 −1 ∗j Qm1 −1 ∗j KA = j=0 (Vm2 PqA,j Vmj 2 )(KA ) ⊂ j=0 (Vm2 PqA,j Vmj 2 )(H). Qm1 −1 ∗j For the reverse inclusion we show that the elements of j=0 (Vm2 PqA,j Vmj 2 )(H) satisfy the defining conditions of KA . Indeed, Qm1 −1 ∗j Qm1 −1 ∗j j j j=0 (Vm2 PqA,j Vm2 )(H) = P0 j=1 (Vm2 PqA,j Vm2 )(H) ⊂ H0 , and, for 1 ≤ k ≤ m1 − 1, Qm1 −1 ∗j Qm1 −1 ∗j j PqA,k Vmk 2 P ) = PqA,k Vmk 2 (Vm∗k2 Vmk 2 ) j=0 (Vm2 PqA,j Vmj 2 ) (V V m2 qA,j m2 j=0 Qm1 −1 ∗j = PqA,k (Vmk 2 Vm∗k2 )Vmk 2 j=0 (Vm2 PqA,j Vmj 2 ) Qm1 −1 ∗j = Vmk 2 Vm∗k2 PqA,k Vmk 2 j=0 (Vm2 PqA,j Vmj 2 ) Qm1 −1 ∗j = Vmk 2 j=0 (Vm2 PqA,j Vmj 2 ), Qm1 −1 ∗j (Vm2 PqA,j Vmj 2 ) is contained in KA . so the range of j=0 ∗q Suppose now that i ≥ 1, and consider the partial isometry Vm1A,i Vmi 2 . Since Qm1 −1 ∗j KA = j=0 (Vm2 PqA,j Vmj 2 )(H)

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and (Vm∗q1A,i Vmi 2 )∗ Vm∗q1A,i Vmi 2

Qm1 −1

(Vm∗j2 PqA,j Vmj 2 ) Qm1 −1 ∗j (Vm2 PqA,j Vmj 2 ) = Vm∗i2 VmqA,i Vm∗q1A,i Vmi 2 Vm∗i2 PqA,i Vmi 2 j=0 1 Qm1 −1 ∗j = (Vm∗i2 VmqA,i Vm∗q1A,i (Vmi 2 Vm∗i2 )PqA,i Vmi 2 ) j=0 (Vm2 PqA,j Vmj 2 ) 1 Qm1 −1 ∗j = Vm∗i2 (VmqA,i Vm∗q1A,i PqA,i )Vmi 2 j=0 (Vm2 PqA,j Vmj 2 ) 1 Q m1 −1 (Vm∗j2 PqA,j Vmj 2 ) = Vm∗i2 PqA,i Vmi 2 j=0 Qm1 −1 ∗j = j=0 (Vm2 PqA,j Vmj 2 ),

∗q

j=0

∗q

(Vm1A,i Vmi 2 )∗ Vm1A,i Vmi 2 is the identity on KA . So KA is a closed subspace of the ∗q ∗q range of (Vm1A,i Vmi 2 )∗ Vm1A,i Vmi 2 , and hence by [8, Lemma 3.4] the projection of H ∗qA,i i onto Vm1 Vm2 (KA ) is ∗q

Vm1A,i Vmi 2

Qm1 −1 j=0

q (Vm∗j2 PqA,j Vmj 2 ) Vm∗i2 VmA,i 1 .

(6.4)

Now, for a fixed 0 ≤ j ≤ m1 − 1 choose 0 ≤ k ≤ m1 − 1 with k ≡ j − i (mod m1 ), then i + k ≡ j (mod m1 ) and rA,i,k = qA,j − qA,i + (m2 /m1 )(i + k − j) by Lemma 6.4, so Vm∗q1A,i Vmi 2 (Vm∗j2 PqA,j Vmj 2 ) = Vm∗q1A,i (Vm∗j2 Vmj 2 )Vmi 2 Vm∗j2 PqA,j Vmj 2 = Vm∗q1A,i Vm∗j2 Vmi 2 (Vmj 2 Vm∗j2 )PqA,j Vmj 2 = Vm∗q1A,i Vm∗j2 Vmi 2 PqA,j Vmj 2 = (Vm∗q1A,i VmqA,i )Vm∗q1A,i Vm∗j2 Vmi 2 PqA,j (Vm∗i2 Vmi 2 )Vmj 2 1 = Vm∗q1A,i (VmqA,i Vm∗q1A,i )(Vm∗j2 Vmi 2 PqA,j Vm∗i2 Vmj 2 )Vmi 2 1 = Vm∗q1A,i (Vm∗j2 Vmi 2 PqA,j Vm∗i2 Vmj 2 )(VmqA,i Vm∗q1A,i )Vmi 2 1 Vm∗q1A,i Vmi 2 = Vm∗q1A,i Vm∗j2 (Vm∗k2 Vmk 2 )Vmi 2 PqA,j Vm∗i2 (Vm∗k2 Vmk 2 )Vmj 2 VmqA,i 1 = Vm∗k2 Vm∗q1A,i Vm∗j2 Vmk 2 Vmi 2 PqA,j Vm∗i2 Vm∗k2 Vmj 2 VmqA,i Vmk 2 Vm∗q1A,i Vmi 2 1 = Vm∗k2 (Vm∗q1A,i Vm∗j2 Vmk 2 Vmi 2 VmqA,j )P0 (Vm∗q1A,j Vm∗i2 Vm∗k2 Vmj 2 VmqA,i )Vmk 2 Vm∗q1A,i Vmi 2 1 1 r

∗r

= Vm∗k2 VmA,i,k P0 Vm1A,i,k Vmk 2 Vm∗q1A,i Vmi 2 1 = (Vm∗k2 PrA,i,k Vmk 2 )Vm∗q1A,i Vmi 2 . Applying this result, for each 0 ≤ j ≤ m1 − 1, the projection (6.4) becomes Qm1 −1 j=0

∗q q (Vm∗j2 PrA,i,j Vmj 2 ) Vm1A,i Vmi 2 Vm∗i2 VmA,i 1 .

(6.5)

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By Lemma 6.4, rA,i,m1 −i = m2 − qA,i , so 1 −i (Vm∗m PrA,i,m1 −i Vmm21 −i )Vm∗q1A,i Vmi 2 Vm∗i2 VmqA,i 2 1 1 −i 1 −i 1 −i = Vm∗m PrA,i,m1 −i Vmm21 −i Vm∗q1A,i (Vm∗m Vmm21 −i )Vmi 2 Vm∗i2 (Vm∗m Vmm21 −i )VmqA,i 2 2 2 1 1 −i 1 −i 1 = Vm∗m PrA,i,m1 −i Vmm21 −i Vm∗m (Vm∗q1A,i Vmm21 Vm∗m VmqA,i )Vmm21 −i 2 2 2 1

r

∗r

1 −i 1 −i 1 −i = Vm∗m PrA,i,m1 −i (Vmm21 −i Vm∗m )VmA,i,m Vm1A,i,m1 −i Vmm21 −i 1 2 2

r

∗r

1 −i 1 −i (PrA,i,m1 −i VmA,i,m Vm1A,i,m1 −i )Vmm21 −i = Vm∗m 1 2 1 −i = Vm∗m PrA,i,m1 −i Vmm21 −i , 2

and then (6.5) becomes Q ∗m −i ∗q q m1 −1 ∗j j m1 −i 1 (V P V ) (V P V ) Vm1A,i Vmi 2 Vm∗i2 VmA,i rA,i,m1 −i m2 1 m2 rA,i,j m2 m2 j=0 ∗m −i Qm1 −1 ∗j j m1 −i 1 PrA,i,m1 −i Vm2 ) = j=0 (Vm2 PrA,i,j Vm2 ) (Vm2 Qm1 −1 ∗j = j=0 (Vm2 PrA,i,j Vmj 2 ), ∗q

which is the projection of H onto Vm1A,i Vmi 2 (KA ) as claimed.

Remark 6.6. Let A ∈ A(Σ), and recall that pA is the polynomial as defined in Proposition 4.12. Then, setting i = 0 in Proposition 6.5, we see that Qm1 −1 ∗j pA (Vm1 , Vm∗ 1 , Vm2 , Vm∗ 2 ) = j=0 (Vm2 PqA,j Vmj 2 ) is the projection of H onto KA . Our next step is the decomposition of H0 . For this we need some technical lemmas. Lemma 6.7. Let A ∈ A(Σ) and 0 ≤ i, j ≤ m1 − 1. Then rA,i,j ≤ m2 . Proof. Since j < m1 , and the definition of rA,i,j implies jm2 + bA,i − rA,i,j m1 ∈ A, we have bA,i + (m2 − rA,i,j )m1 = (m1 − j)m2 + (jm2 + bA,i − rA,i,j m1 ) ∈ A. Thus, since bA,i is the smallest element of A such that bA,i ≡ im2 (mod m1 ), we must have rA,i,j ≤ m2 . Lemma 6.8. Let A ∈ A(Σ) and 0 ≤ i ≤ m1 − 1. If 0 ≤ j < k ≤ m1 − 1 then rA,i,j ≤ rA,i,k . Proof. Since the definition of rA,i,j implies jm2 + bA,i − rA,i,j m1 ∈ A, we have km2 + bA,i − rA,i,j m1 = (k − j)m2 + (jm2 + bA,i − rA,i,j m1 ) ∈ A, so the definition of rA,i,k implies rA,i,j ≤ rA,i,k .

Lemma 6.9. Let A ∈ A(Σ) and 0 ≤ i, j ≤ m1 − 1. Choose 0 ≤ k ≤ m1 − 1 such that i + j ≡ k (mod m1 ). Then jm2 + bA,i − rA,i,j m1 = bA,k .

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Proof. Using Lemma 6.4 we have jm2 + bA,i − rA,i,j m1 = jm2 + bA,i − (qA,k − qA,i + (m2 /m1 )(i + j − k))m1 = km2 − qA,k m1 = bA,k .

Lemma 6.10. For integers 0 ≤ pj ≤ m2 , 1 ≤ j ≤ m1 − 1, where pj ≤ pk whenever j < k, there exist unique A ∈ A(Σ) and 0 ≤ i ≤ m1 − 1 such that pj = rA,i,j for 1 ≤ j ≤ m1 − 1. Proof. Our aim is to find A ∈ A(Σ) and 0 ≤ i ≤ m1 −1 so that, for 1 ≤ j ≤ m1 −1, jm2 + bA,i − pj m1 is the smallest integer in the set { jm2 + bA,i − rm1 | r ∈ N } ∩ A, and then the definition of rA,i,j will imply that pj = rA,i,j as required. Our strategy is to use the integers pj to find the integer bA,i , then construct A, and then find i. To motivate the proof, fix A ∈ A(Σ) and 0 ≤ i ≤ m1 − 1. If 0 ≤ j ≤ m1 − 1 then, choosing 0 ≤ k ≤ m1 −1 with i+j ≡ k (mod m1 ), jm2 +bA,i −rA,i,j m1 = bA,k by Lemma 6.9. In particular, jm2 +bA,i −rA,i,j m1 ≥ 0 for 0 ≤ j ≤ m1 −1. Moreover, if 1 ≤ i ≤ m1 −1 then (m1 −i)m2 +bA,i −rA,i,m1 −i m1 = bA,0 = 0, and if i = 0 then trivially bA,i = 0. It follows that bA,i = − min{ jm2 − rA,i,j m1 | 0 ≤ j ≤ m1 − 1 }. With this motivation we begin by constructing the required A ∈ A(Σ). Define p0 := 0 and define c0 := − min{ jm2 − pj m1 | 0 ≤ j ≤ m1 − 1 }. For 1 ≤ j ≤ m1 − 1 define cj := jm2 + c0 − pj m1 . Now, for 0 ≤ j ≤ m1 − 1 we have by the definition of c0 that −c0 ≤ jm2 − pj m1 , so cj = jm2 + c0 − pj m1 ≥ 0. Further, if c0 6= 0 then c0 = −(km2 − pk m1 ) for some 1 ≤ k ≤ m1 − 1, so ck = km2 + c0 − pk m1 = 0, thus in any case some element of { cj | 0 ≤ j ≤ m1 − 1 } is equal to zero. We now define A := { cj + nm1 | 0 ≤ j ≤ m1 − 1 and n ∈ N }. We will show that A ∈ A(Σ). Since cj ≥ 0 for 0 ≤ j ≤ m1 − 1 we have A ⊂ N. Note that if we have Σ + A = A then, since 0 ∈ A, Σ ⊂ A. So it suffices to show Σ + A = A, or equivalently Σ + A ⊂ A, and, since {m1 , m2 } generates Σ, the definition of A shows that it is enough to prove that m2 + cj ∈ A for 0 ≤ j ≤ m1 − 1. We consider two cases. If 0 ≤ j ≤ m1 − 2, then m2 + cj = m2 + (jm2 + c0 − pj m1 ) = (j + 1)m2 + c0 − pj m1 = cj+1 + (pj+1 − pj )m1 , which is in A as pj+1 ≥ pj . If j = m1 − 1 then m2 + cj = m2 + (jm2 + c0 − pj m1 ) = c0 + (m2 − pm1 −1 )m1 , which is in A as m2 ≥ pm1 −1 . We now want to show that there exists 0 ≤ i ≤ m1 − 1 such that pj = rA,i,j for 1 ≤ j ≤ m1 − 1. Note that if cj ≡ ck (mod m1 ) for some 0 ≤ j, k ≤ m1 − 1 then, since cj − ck = (j − k)m2 − (pj − pk )m1 , m1 | (j − k)m2 , so as m1 and m2 are relatively prime we must have j = k. It follows that the smallest element of A which is congruent to cj modulo m1 is cj itself, for each 0 ≤ j ≤ m1 − 1. So, choosing 0 ≤ i ≤ m1 − 1 such that c0 ≡ im2 (mod m1 ), c0 is the smallest element of A which is congruent to im2 modulo m1 , hence by the definition of bA,i we have c0 = bA,i . Moreover, for 1 ≤ j ≤ m1 − 1 we have that cj = jm2 + bA,i − pj m1

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is the smallest integer in the set { jm2 + bA,i − rm1 | r ∈ N } ∩ A, hence, by the definition of rA,i,j , we have pj = rA,i,j . For uniqueness, suppose there also exist B ∈ A(Σ) and 0 ≤ k ≤ m1 − 1 such that pj = rB,k,j for 1 ≤ j ≤ m1 − 1. We may assume without loss of generality that bB,k ≥ bA,i . We will show that we must have bB,k = bA,i . If k = 0 then bA,i ≤ bB,k = 0, so bA,i = 0 = bB,k . Suppose k ≥ 1. Then, by Lemma 6.9, (m1 − k)m2 + bB,k − rB,k,m1 −k m1 = bB,0 = 0, and, choosing 0 ≤ t ≤ m1 − 1 such that i + (m1 − k) ≡ t (mod m1 ), Lemma 6.9 also gives (m1 − k)m2 + bA,i − rA,i,m1 −k m1 = bA,t , hence (m1 − k)m2 + bB,k − rB,k,m1 −k m1 = 0 ≤ bA,t = (m1 − k)m2 + bA,i − rA,i,m1 −k m1 , which implies, since rB,k,m1 −k = rA,i,m1 −k , that bB,k ≤ bA,i . Hence bB,k = bA,i . We now show that A = B. Let 0 ≤ j ≤ m1 − 1, and choose 0 ≤ s ≤ m1 − 1 such that s ≡ j − k (mod m1 ). Then s + k ≡ j (mod m1 ), and by Lemma 6.9 we have sm2 + bB,k − rB,k,s m1 = bB,j . Further, choosing 0 ≤ r ≤ m1 − 1 such that i + s ≡ r (mod m1 ) we have sm2 + bA,i − rA,i,s m1 = bA,r by Lemma 6.9. Therefore, bB,j = sm2 + bB,k − rB,k,s m1 = sm2 + bA,i − rA,i,s m1 = bA,r . It follows that { bB,j | 0 ≤ j ≤ m1 − 1 } ⊂ { bA,r | 0 ≤ r ≤ m1 − 1 }. So Lemma 4.11 implies that A = B. Finally, since A = B, we have bA,i = bA,k , which implies that m1 | (i − k)m2 , so as m1 and m2 are relatively prime we must have i = k. Proposition 6.11. There is a direct-sum decomposition L Lm1 −1 ∗qA,i i H0 = A∈A(Σ) Vm2 (KA ) . i=0 Vm1 Remark 6.12. The projection of H onto the summand Qm1 −1 ∗j j j=0 (Vm2 PrA,i,j Vm2 ) by Proposition 6.5.

∗q Vm1A,i Vmi 2 (KA )

(6.6) of (6.6) is

Proof of Proposition 6.11. For 1 ≤ i ≤ m1 − 1, 1 −i 1 −i 2 +1 1 Vmi 2 P0 = Vm∗ 1 Vm∗m P0 = Vm∗m Vm∗ 1 P0 = 0, Vm∗m Vmi 2 P0 = Vm∗ 1 Vm∗m 2 2 2 1 Pm Pm so pi 2=0 Ppi Vmi 2 P0 = Vmi 2 P0 , hence P0 = pi 2=0 Vm∗i2 Ppi Vmi 2 P0 . Therefore, Qm1 −1 Pm2 ∗i i P0 = i=1 ( pi =0 Vm2 Ppi Vm2 ) P0 Pm2 Pm2 Qm1 −1 ∗j j = (6.7) j=1 (Vm2 Ppj Vm2 ) P0 . pm −1 =0 · · · p1 =0 1

If i, j, m, n ∈ N, i < j, and m > n then, since Vmi 2 Vm∗i2 commutes with Pm , and since Vm∗m Pn = 0, 1 (Vm∗i2 Pm Vmi 2 )(Vm∗j2 Pn Vmj 2 ) = Vm∗i2 Pm (Vmi 2 Vm∗i2 )Vm∗j−i Pn Vmj 2 = Vm∗i2 Pm Vm∗j−i Pn Vmj 2 2 2 = Vm∗i2 Vmm1 P0 Vm∗j−i Vm∗m Pn Vmj 2 = 0, 2 1

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Qm1 −1 ∗j so if a summand j=1 (Vm2 Ppj Vmj 2 ) of (6.7) is a non-zero projection then pj ≤ pk whenever j < k. Further, for integers 0 ≤ pj ≤ m2 , 1 ≤ j ≤ m1 − 1, where pj ≤ pk whenever j < k, Lemma 6.10 ensures the existence of unique A ∈ A(Σ) and 0 ≤ i ≤ m1 − 1 such that pj = rA,i,j for 1 ≤ j ≤ m1 − 1. Moreover, for any A ∈ A(Σ) and 0 ≤ i ≤ m1 − 1, it follows from Lemmas 6.7 and 6.8 that 0 ≤ rA,i,j ≤ m2 for 1 ≤ j ≤ m1 − 1, where rA,i,j ≤ rA,i,k whenever j < k. So (6.7) becomes P Pm1 −1 Qm1 −1 ∗j j P0 = A∈A(Σ) i=0 j=1 (Vm2 PrA,i,j Vm2 ) P0 P Pm1 −1 Qm1 −1 ∗j j = A∈A(Σ) i=0 j=1 (Vm2 PrA,i,j Vm2 ) P0 P Pm1 −1 Qm1 −1 ∗j j (6.8) = A∈A(Σ) i=0 j=0 (Vm2 PrA,i,j Vm2 ). Qm1 −1 ∗j By (6.8), the sum of the projections j=0 (Vm2 PrA,i,j Vmj 2 ) is the projection P0 , Qm1 −1 ∗j so the projections j=0 (Vm2 PrA,i,j Vmj 2 ) have mutually orthogonal ranges, and we have a direct-sum decomposition L Lm1 −1 Qm1 −1 ∗j j H0 = P0 (H) = A∈A(Σ) i=0 j=0 (Vm2 PrA,i,j Vm2 )(H) . Qm1 −1 ∗j ∗q Further, Proposition 6.5 gives j=0 (Vm2 PrA,i,j Vmj 2 )(H) = Vm1A,i Vmi 2 (KA ). Applying each isometry Vmm1 to the decomposition (6.6) of H0 = P0 (H) gives decompositions L Lm1 −1 m ∗qA,i i Pm (H) = Vmm1 (H0 ) = A∈A(Σ) Vm2 (KA ) , i=0 Vm1 Vm1 and, since the subspaces Pm (H) themselves give a direct-sum decomposition of ⊥ , we have HU L∞ L Lm1 −1 m ∗qA,i i H = HU ⊕ Vm2 (KA ) . m=0 A∈A(Σ) i=0 Vm1 Vm1 So defining, for A ∈ A(Σ), L∞ Lm1 −1 m ∗qA,i i HA := m=0 Vm2 (KA ) , i=0 Vm1 Vm1

(6.9)

we have the decomposition L H = HU ⊕ ( A∈A(Σ) HA ). L 0 0 This decomposition is unique, for given a decomposition H = HU ⊕ ( A∈A(Σ) HA ) 0

0

as in Proposition 5.2 this process will yield HU = HU and HA = HA for A ∈ A(Σ). We now show that the subspaces HU and HA , for A ∈ A(Σ), of H have the required properties. Proposition 6.13. The subspaces HU and HA , for A ∈ A(Σ), of H are reducing for V .

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1 2 Proof. Since {m1 , m2 } generates Σ and Vm∗ 2 = Vm∗m Vmm21 −1 = Vm∗m Vmm21 −1 , to 2 1 prove that a subspace of H is reducing for V it is enough to show that the subspace is invariant under Vm1 , Vm∗ 1 , and Vm2 . Each of our subspaces is clearly invariant T∞ T∞ under Vm1 . Since HU = n=0 Vmn1 (H) = n=1 Vmn1 (H), it is invariant under Vm∗ 1 , and, since Vm2 (Vmn1 (H)) = Vmn1 Vm2 (H) ⊂ Vmn1 (H), it is also invariant under Vm2 . Fix A ∈ A(Σ) and consider HA . We have S∞ Sm1 −1 m ∗qA,i i S∞ Sm1 −1 m ∗qA,i i Vm2 (KA ) = m=0 i=0 Vm1 Vm1 Vm2 (KA ) Vm∗ 1 m=1 i=0 Vm1 Vm1 ⊂ HA .

q

∗q

A,i (H) by the definition Further, Vmi 2 (KA ) is contained in PqA,i (H) = VmA,i 1 P0 Vm1 ∗qA,i i of KA , so Vm1 Vm2 (KA ) is contained in P0 (H) = H0 , hence Sm1 −1 ∗qA,i i Vm2 (KA ) = {0}. Vm∗ 1 i=0 Vm1

So HA is invariant under Vm∗ 1 . In showing that HA is invariant under Vm2 we will freely use the fact that q ∗qA,i for 1 ≤ i ≤ m1 −1. Vmi 2 (KA ) is contained in the range of the projection VmA,i 1 Vm1 Now, we first calculate Vm2 Vmm1 Vm∗q1A,i Vmi 2 (KA ) = Vmm1 Vm2 Vm∗q1A,i Vmi 2 (KA ) = Vmm1 (Vm∗q1A,i VmqA,i )Vm2 Vm∗q1A,i Vmi 2 (KA ) 1 = Vmm1 Vm∗q1A,i Vm2 (VmqA,i Vm∗q1A,i Vmi 2 (KA )) 1 = Vmm1 Vm∗q1A,i Vm2 Vmi 2 (KA ) = Vmm1 Vm∗q1A,i Vmi+1 (KA ), 2

(6.10)

and then consider the two cases i < m1 − 1 and i = m1 − 1. If i < m1 − 1 then, since qA,i ≤ qA,i+1 by Lemma 6.8, we have from (6.10) Vm2 Vmm1 Vm∗q1A,i Vmi 2 (KA ) = Vmm1 Vm∗q1A,i Vmi+1 (KA ) 2 = Vmm1 Vm∗q1A,i (VmqA,i+1 Vm∗q1A,i+1 Vmi+1 (KA )) 1 2 −qA,i ∗qA,i+1 i+1 = Vmm1 VmqA,i+1 Vm1 Vm2 (KA ) 1 A,i+1 −qA,i = Vmm+q Vm∗q1A,i+1 Vmi+1 (KA ) ⊂ HA . 1 2

If i = m1 − 1 then, since qA,i ≤ m2 by Lemma 6.7, we have from (6.10) Vm2 Vmm1 Vm∗q1A,i Vmi 2 (KA ) = Vmm1 Vm∗q1A,i Vmi+1 (KA ) = Vmm1 Vm∗q1A,i Vmm21 (KA ) 2 = Vmm1 Vm∗q1A,i Vmm12 (KA ) = Vmm1 Vmm12 −qA,i (KA ) 2 −qA,i = Vmm+m (KA ) ⊂ HA . 1

We next show that V |HA is equivalent to T A ⊗1KA for each A ∈ A(Σ). We identify A A `2 (A) ⊗ KA with `2 (N × {0, . . . , m1 − 1}, KA ) so that Tm ⊗ 1KA and Tm ⊗ 1KA 1 2 2 on ` (N × {0, . . . , m1 − 1}, KA ) are given by the same formulas (6.2) and (6.3) as

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Qm1 −1 ∗j A A j Tm and Tm . Denote by QA 00 the projection j=0 (Vm2 PqA,j Vm2 ) of H onto KA 1 2 as in Proposition 6.5, and define WA : HA → `2 (N × {0, . . . , m1 − 1}, KA ) by ∗j qA,j ∗n (WA h)nj = QA 00 Vm2 Vm1 Vm1 h.

It follows from the direct-sum decomposition (6.9) of HA that WA is a unitary isomorphism of HA onto `2 (N × {0, . . . , m1 − 1}, KA ). Proposition 6.14. We have WA (V |HA )WA∗ = T A ⊗ 1KA for A ∈ A(Σ). Proof. It suffices to prove that WA (Vs |HA ) = (TsA ⊗ 1)WA for s = m1 and s = m2 . j Let h ∈ HA , n ∈ N, and 0 ≤ j ≤ m1 − 1. Since Vmj 2 QA 00 (H) = Vm2 (KA ) is qA,j +1 orthogonal to Vm1 (H), ∗j qA,j +1 A (WA Vm1 h)0j = QA h = 0 = ((Tm ⊗ 1)WA h)0j . 00 Vm2 Vm1 1

For n ≥ 1, A ∗j qA,j ∗n−1 ∗j qA,j ∗n h = (WA h)(n−1)j (WA Vm1 h)nj = QA 00 Vm2 Vm1 Vm1 Vm1 h = Q00 Vm2 Vm1 Vm1 A = ((Tm ⊗ 1)WA h)nj . 1 A Before proving the equality WA (Vm2 |HA ) = (Tm ⊗ 1)WA we will establish 2 A ∗i ∗m qA,i that (WA k)mi = Q00 Vm2 Vm1 Vm1 k for k ∈ HA , m ∈ N, and 0 ≤ i ≤ m1 − 1, a result we will use freely throughout the remainder of this proof. Indeed, ∗i qA,i ∗m (WA k)mi = QA 00 Vm2 Vm1 Vm1 k ∗qA,i qA,i ∗i i ∗i qA,i ∗m Vm1 )k = (QA 00 (Vm2 PqA,i Vm2 ))Vm2 Vm1 Vm1 (Vm1 ∗i i ∗i qA,i ∗qA,i = QA )Vm∗m VmqA,i k 00 Vm2 PqA,i (Vm2 Vm2 )(Vm1 Vm1 1 1 ∗i qA,i ∗qA,i = QA )Vmi 2 Vm∗i2 Vm∗m VmqA,i k 00 Vm2 (PqA,i Vm1 Vm1 1 1 ∗i i ∗i ∗m qA,i A ∗i ∗m qA,i = (QA 00 (Vm2 PqA,i Vm2 ))Vm2 Vm1 Vm1 k = Q00 Vm2 Vm1 Vm1 k. A ⊗ 1)WA h by considering four We now prove the equality WA Vm2 h = (Tm 2 cases. If n ≥ qA,j − qA,j−1 and j ≥ 1 then, noting that qA,j − qA,j−1 ≥ 0 by Lemma 6.8, ∗j ∗n qA,j A ∗j−1 ∗n ∗ qA,j (WA Vm2 h)nj = QA 00 Vm2 Vm1 Vm1 Vm2 h = Q00 Vm2 Vm1 (Vm2 Vm2 )Vm1 h ∗j−1 ∗n qA,j −qA,j−1 qA,j−1 = QA Vm1 )h 00 Vm2 Vm1 (Vm1 ∗j−1 ∗n−qA,j +qA,j−1 qA,j−1 = QA Vm1 h = (WA h)(n−qA,j +qA,j−1 )(j−1) 00 Vm2 Vm1 A = ((Tm ⊗ 1)WA h)nj . 2 j−1 If n < qA,j − qA,j−1 and j ≥ 1 then, since Vmj−1 QA 00 (H) = Vm2 (KA ) is orthogonal 2 qA,j−1 +1 to Vm1 (H) and since qA,j − n ≥ qA,j−1 + 1, Vmj−1 QA 00 (H) is orthogonal to 2 qA,j −n Vm1 (H), so ∗j ∗n qA,j A ∗j−1 ∗n ∗ qA,j (WA Vm2 h)nj = QA 00 Vm2 Vm1 Vm1 Vm2 h = Q00 Vm2 Vm1 (Vm2 Vm2 )Vm1 h ∗j−1 qA,j −n A = QA h = 0 = ((Tm ⊗ 1)WA h)nj . 00 Vm2 Vm1 2

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If n ≥ m2 − qA,m1 −1 and j = 0 then, noting that m2 − qA,m1 −1 ≥ 0 by Lemma 6.7, A ∗n ∗m1 −1 m1 −1 ∗n Vm2 )Vm2 h (WA Vm2 h)nj = QA 00 Vm1 Vm2 h = Q00 Vm1 (Vm2 m −qA,m1 −1

∗m1 −1 ∗n m2 ∗m1 −1 ∗n = QA Vm1 Vm1 h = QA Vm1 (Vm12 00 Vm2 00 Vm2 ∗n−m2 +qA,m1 −1

∗m1 −1 = QA Vm1 00 Vm2

q

1 −1 VmA,m )h 1

q

1 −1 VmA,m h 1

A = (WA h)(n−m2 +qA,m1 −1 )(m1 −1) = ((Tm ⊗ 1)WA h)nj . 2 m1 −1 If n < m2 − qA,m1 −1 and j = 0 then, since Vmm21 −1 QA (KA ) is 00 (H) = Vm2 qA,m1 −1 +1 orthogonal to Vm1 (H) and since m2 − n ≥ qA,m1 −1 + 1, Vmm21 −1 QA 00 (H) is m2 −n orthogonal to Vm1 (H), so ∗n A ∗n ∗m1 −1 m1 −1 (WA Vm2 h)nj = QA Vm2 )Vm2 h 00 Vm1 Vm2 h = Q00 Vm1 (Vm2 ∗m1 −1 ∗n m2 ∗m1 −1 m2 −n = QA Vm1 Vm1 h = QA Vm1 h = 0 00 Vm2 00 Vm2 A = ((Tm ⊗ 1)WA h)nj . 2

7. The C ∗ -algebra of Σ Throughout this section we let Σ be a numerical semigroup with minimal system of generators {m1 , . . . , mr }, where mi < mj if i < j. We denote by C ∗ (Σ) the unital C ∗ -algebra generated by an isometric representation v : Σ → C ∗ (Σ) with commuting range projections which is universal for such representations: for every isometric representation V : Σ → B in a unital C ∗ -algebra B with commuting range projections there is a unique homomorphism πV : C ∗ (Σ) → B such that V = πV ◦ v. The two main results of this section are generalisations of the corresponding results in [8, Section 4] for the numerical semigroup N \ {1}. Namely, we describe a condition on an isometric representation V : Σ → B(H) on a Hilbert space H with commuting range projections which ensures that πV is faithful, and give a concrete description of C ∗ (Σ) in terms of the usual Toeplitz algebra. We recall from Definition 4.6 that α is the smallest positive integer in Σ such that m1 and α are relatively prime. Theorem 7.1. Let V : Σ → B(H) be an isometric representation on a Hilbert space H with commuting range projections. Then the representation πV of C ∗ (Σ) is faithful if and only if Qm1 −1 ∗i qA,i ∗qA,i q +1 ∗qA,i +1 − VmA,i Vm1 )Vαi 6= 0 for every A ∈ A(Σ). (7.1) 1 i=0 Vα (Vm1 Vm1 Remark 7.2. Condition (7.1) is equivalent to requiring that each subspace HA in the decomposition of Theorem 5.1 is non-zero. To see this, first consider the case Σ = N. Then (7.1) says that 1 − V1 V1∗ 6= 0 (equivalently each Vn , n ≥ 1, is non-unitary). Theorem 5.1 in conjunction with the ⊥ observation that the range of the projection 1 − V1 V1∗ is contained in HN = HU ∗ shows that 1 − V1 V1 6= 0 if and only if HN 6= {0}.

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Now suppose Σ 6= N, so that 2 ≤ m1 < α, and consider the numerical semigroup hm1 , αi, which has the minimal system of generators {m1 , α}. Since hm1 , αi ⊂ Σ we may consider the restriction V |hm1 ,αi : hm1 , αi → B(H) of V to hm1 , αi. Applying Theorem 5.1 (or Proposition 5.2) to the representation V |hm1 ,αi L gives a decomposition H = HU ⊕ ( A∈A(hm1 ,αi) HA ). By the uniqueness of this decomposition, each HA is the same subspace as in (6.9). So Qm1 −1 ∗i qA,i ∗qA,i q +1 ∗qA,i +1 − VmA,i Vm1 )Vαi (7.2) 1 i=0 Vα (Vm1 Vm1 is the projection onto the subspace KA of HA by Proposition 6.5. In particular, (7.2) is non-zero if and only if HA is non-zero. LSince V extends V |hm1 ,αi , we have, by Lemma 5.3, a decomposition H = HU ⊕ ( A∈A(Σ) HA ) for V , which is unique by Theorem 5.1, and the claim follows. Remark 7.3. Theorem 7.1 implies in particular that πLA∈A(Σ) T A is faithful. Proof of Theorem 7.1. Recall that pA is the polynomial in Proposition 4.12, and note that Qm1 −1 ∗i qA,i ∗qA,i q +1 ∗qA,i +1 − VmA,i Vm1 )Vαi = pA (Vm1 , Vm∗ 1 , Vα , Vα∗ ). 1 i=0 Vα (Vm1 Vm1 Since ∗ A A ∗ πT A pA (vm1 , vm , vα , vα∗ ) = pA (Tm , (Tm ) , TαA , (TαA )∗ ) 1 1 1 ∗ , vα , vα∗ ) is nonis the projection onto span{eA,0 } by Proposition 4.12, pA (vm1 , vm 1 ∗ zero in C (Σ). So, if πV is faithful then ∗ pA (Vm1 , Vm∗ 1 , Vα , Vα∗ ) = πV pA (vm1 , vm , vα , vα∗ ) 6= 0 1 for A ∈ A(Σ), hence (7.1) holds. Now suppose that V satisfies L condition (7.1). Then each HA is non-zero in the decomposition H = HU ⊕ ( A∈A(Σ) HA ) of Theorem 5.1. Write VU := V |HU , ∗ VA := V |HA for L A ∈ A(Σ), and fix a ∈ C (Σ). We can check on generators that πV = πVU ⊕ ( A∈A(Σ) πVA ), hence kπV (a)k = max {kπVU (a)k} ∪ { kπVA (a)k | A ∈ A(Σ) } . (7.3) Since VA is equivalent to T A ⊗ 1, and since we can check on generators that πT A ⊗1 = πT A ⊗ 1, we have that πVA is equivalent to πT A ⊗ 1. Since each HA is non-zero, we then have kπVA (a)k = kπT A (a)k, so (7.3) implies kπV (a)k = max {kπVU (a)k} ∪ { kπT A (a)k | A ∈ A(Σ) } . (7.4) Define U1 := ((VU )C(Σ)+1 )∗ (VU )C(Σ)+2 , which is unitary, and denote by S the unilateral shift on `2 (N). For n ∈ Σ we have (U1 )n = ((VU )C(Σ)+1 )∗n ((VU )C(Σ)+2 )n = (VU )n and S n = TnN , so the operator πVU (a)⊕πT N (a) belongs to the C ∗ -algebra generated by U1 ⊕ S. Hence the Lemma on page 724 of [1] implies that kπVU (a)k ≤ kπT N (a)k. Thus (7.4) implies kπV (a)k = max{ kπT A (a)k | A ∈ A(Σ) }.

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Since every C ∗ -algebra has a faithful representation and every representation of C (Σ) has the form πW for some isometric representation W , there is a faithful representation of the form πW . Applying Theorem 5.1 to W , we then have that each HA is non-zero by the first part of the proof. We can then deduce from the argument of the previous paragraph that ∗

kak = kπW (a)k = max{ kπT A (a)k | A ∈ A(Σ) } = kπV (a)k, which, since a is an arbitrary element of C ∗ (Σ), implies that πV is faithful.

We can view the Toeplitz algebra T either as the C ∗ -subalgebra of B(`2 (N)) generated by the unilateral shift, or as the C ∗ -subalgebra of B(H 2 (T)) generated by the Toeplitz operators Tφ with symbol φ ∈ C(T). In either realisation, the algebra K of compact operators is an ideal of T , and the quotient T /K is naturally isomorphic to C(T). In the proof of Theorem 7.4 we realise T as a subalgebra of B(`2 (N)). We denote by q : T → T /K the quotient map. Theorem 7.4. C ∗ (Σ) is isomorphic to L L C := { A∈A(Σ) RA ∈ A∈A(Σ) T | q(RN ) = q(RA ) for A ∈ A(Σ) }. We use the following lemma to prove this theorem. Lemma 7.5. Let A ∈ A(Σ) and denote A = { an | n ∈ N } where an < an+1 for n ∈ N. Let UA : `2 (N) → `2 (A) be the unitary isomorphism determined by UA eN,n = eA,an for n ∈ N. Then UA∗ TpA UA − TpN is a finite-rank operator on `2 (N) for every p ∈ Σ. Proof. If A = N then the result is trivial, so suppose A 6= N. Then we must have Σ 6= N, so C(Σ) ≥ 1. If aC(Σ) = C(Σ) then A = N, so we must have aC(Σ) ≥ C(Σ) + 1. Thus, for n ∈ N, n + aC(Σ) = an+C(Σ) . On the other hand, if 0 ≤ q < C(Σ) then, for n ∈ N \ {0}, n + aq is not necessarily equal to an+q . Therefore, using the notation h ⊗ k for the rank-one operator g 7→ ( g | k )h on `2 (N), PC(Σ)−1 ∗ UA∗ TpA UA − TpN = i=0 (UA eA,p+ai − eN,p+i ) ⊗ eN,i . Proof of Theorem 7.4. For L each A ∈ A(Σ) take UA as in L Lemma 7.5, and define the mapping ψ : C ∗ (Σ) → A∈A(Σ) B(`2 (N)) by ψ(a) := A∈A(Σ) UA∗ πT A (a)UA . ∗ We L claim that ψ isLan isomorphism of C (Σ) onto C. It is injective because A∈A(Σ) πT A = π A∈A(Σ) T A is faithful by Theorem 7.1. Since the operators πT N (vp ) = TpN are all powers of the unilateral shift, and Lemma 7.5 implies that UA∗ πT A (vp )UA = UA∗ TpA UA differs from TpN by a finite-rank operator, the range of ψ is contained in C. So it remains to prove that every element of C is in the range of ψ. L Let A∈A(Σ) (RN + KA ) ∈ C, where the KA are compact operators with N N ∗ KN = 0. Since (TC(Σ)+1 )∗ TC(Σ)+2 = πT N (vC(Σ)+1 vC(Σ)+2 ) is the unilateral shift,

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πT N maps C ∗ (Σ) onto T . Thus there exists a ∈ C ∗ (Σ) such that πT N (a) = RN , and then RN + KA = UA∗ πT A (a)UA + (πT N (a) − UA∗ πT A (a)UA ) + KA , which we may write as UA∗ πT A (a)UA + LA , where LA is compact, since it follows ∗ A from Lemma 7.5 that πT N (a) L − UA πT (a)UA is compact. So it suffices to show that, for each B ∈ A(Σ), R A∈A(Σ) A is in the range of ψ where RA = 0 for A 6= B and RB is compact, and for this it is enough to show that this is true when RB is a rank-one operator eN,i ⊗ eN,j . Therefore fix B ∈ A(Σ) and i, j ∈ N. Denote B = { bn | n ∈ N } where bn < bn+1 for n ∈ N. Recall that pB is the polynomial in Proposition 4.12. To simplify notation, for an isometric representation V of Σ in a C ∗ -algebra we denote pB (Vm1 , Vm∗ 1 , Vα , Vα∗ ) by pB (V ). Now, consider ∗ ∗ vC(Σ)+1+bi pB (v)vC(Σ)+1+b v ) ψ(vC(Σ)+1 j C(Σ)+1 L ∗ A ∗ A A A A = A∈A(Σ) UA (TC(Σ)+1 ) TC(Σ)+1+bi pB (T )(TC(Σ)+1+b )∗ TC(Σ)+1 UA . j

(7.5)

If A 6= B then, since pB (T A ) = 0 by Proposition 4.12, the summand of (7.5) corresponding to A is zero. Further, since pB (T B ) is the projection onto span{eB,0 } by Proposition 4.12, the summand of (7.5) corresponding to B satisfies B B B B UB∗ (TC(Σ)+1 )∗ TC(Σ)+1+b p (T B )(TC(Σ)+1+b )∗ TC(Σ)+1 UB eN,n i B j ( eN,i if n = j, = = (eN,i ⊗ eN,j )eN,n 0 if n 6= j and the result follows.

Acknowledgment The research presented in this article forms part of the author’s Ph.D. thesis, which was supervised by Professor Iain Raeburn. The author thanks Iain Raeburn for many helpful discussions and suggestions.

References [1] L. A. Coburn, The C ∗-algebra generated by an isometry, Bull. Amer. Math. Soc. 73 (1967), 722–726. [2] R. G. Douglas, On the C ∗-algebra of a one-parameter semigroup of isometries, Acta Math. 128 (1972), 143–152. [3] S. Y. Jang, Uniqueness property of C ∗-algebras like the Toeplitz algebra, Trends Math. 6 (2003), 29–32. [4] S. Y. Jang, Generalised Toeplitz algebra of a certain non-amenable semigroup, Bull. Korean Math. Soc. 43 (2006), 333–341. [5] M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabelian groups, J. Funct. Anal. 139 (1996), 415–440.

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[6] G. J. Murphy, Ordered groups and Toeplitz algebras, J. Operator Theory 18 (1987), 303–326. [7] A. Nica, C ∗-algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory 27 (1992), 17–52. [8] I. Raeburn and S. T. Vittadello, The isometric representation theory of a perforated semigroup, J. Operator Theory, to appear. [9] J. C. Rosales and P. A. Garc´ıa-S´ anchez, Finitely Generated Commutative Monoids, Nova Science, New York, 1999. Sean T. Vittadello School of Mathematics and Applied Statistics University of Wollongong NSW 2522 Australia e-mail: [email protected] Submitted: September 14, 2008.