Integr. Equ. Oper. Theory 66 (2010), 1–20 DOI 10.1007/s00020-009-1738-2 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Weyl Type Theorems for Left and Right Polaroid Operators Pietro Aiena, Elvis Aponte and Edixon Balzan Abstract. A bounded operator defined on a Banach space is said to be polaroid if every isolated point of the spectrum is a pole of the resolvent. In this paper we consider the two related notions of left and right polaroid, and explore them together with the condition of being a-polaroid. Moreover, the equivalences of Weyl type theorems and generalized Weyl type theorems are investigated for left and a-polaroid operators. As a consequence, we obtain a general framework which allows us to derive in a unified way many recent results, concerning Weyl type theorems (generalized or not) for important classes of operators. Mathematics Subject Classification (2010). Primary 47A10, 47A11. Secondary 47A53, 47A55. Keywords. Key words and phrases: Localized SVEP, semi B-Brower operators, left and right Drazin invertibility, Weyl’s theorem, property (w).
1. Introduction and preliminaries Let L(X) be the algebra of all bounded linear operators acting on an infinite dimensional complex Banach space X and if T ∈ L(X) let us denote by α(T ) the dimension of the kernel ker T and by β(T ) the codimension of the range T (X). Recall that the operator T ∈ L(X) is said to be upper semi-Fredholm, T ∈ Φ+ (X), if α(T ) < ∞ and the range T (X) is closed, while T ∈ L(X) is said to be lower semi-Fredholm, T ∈ Φ− (X), if β(T ) < ∞. If either T is upper or lower semi-Fredholm then T is said to be a semi-Fredholm operator, while if T is both upper and lower semi-Fredholm then T is said to be a Fredholm operator. If T is semi-Fredholm then the index of T is defined by ind (T ) := α(T ) − β(T ). A bounded operator T ∈ L(X) is said to be a Weyl operator, T ∈ W (X), if T is a Fredholm operator having index 0. The The first author was supported by ex-60 2008, Fondi Universit´ a di Palermo.
2
P. Aiena, E. Aponte and E. Balzan
IEOT
classes of upper semi-Weyl’s and lower semi-Weyl’s operators are defined, respectively: W+ (X) := {T ∈ Φ+ (X) : ind T ≤ 0}, W− (X) := {T ∈ Φ− (X) : ind T ≥ 0}. Clearly, W (X) = W+ (X) ∩ W− (X). The Weyl spectrum and the upper semiWeyl spectrum are defined, respectively, by / W (X)}. σw (T ) := {λ ∈ C : λI − T ∈ and σuw (T ) := {λ ∈ C : λI − T ∈ / W+ (X)}. The ascent of an operator T ∈ L(X) is defined as the smallest nonnegative integer p := p(T ) such that ker T p = ker T p+1 . If such integer does not exist we put p(T ) = ∞. Analogously, the descent of T is defined as the smallest non-negative integer q := q(T ) such that T q (X) = T q+1 (X), and if such integer does not exist we put q(T ) = ∞. It is well-known that if p(T ) and q(T ) are both finite then p(T ) = q(T ), see [1, Theorem 3.3]. Moreover, if λ ∈ C then 0 < p(λI − T ) = q(λI − T ) < ∞ if and only if λ is a pole of the resolvent of T . In this case λ is an eigenvalue of T and an isolated point of the spectrum σ(T ), see [28, Prop. 50.2]. A bounded operator T ∈ L(X) is said to be Browder (resp. upper semi-Browder, lower semi-Browder) if T ∈ Φ(X) and p(T ) = q(T ) < ∞ (resp. T ∈ Φ+ (X) and p(T ) < ∞, T ∈ Φ− (X) and q(T ) < ∞). Denote by B(X), B+ (X) and B− (X) the classes of Browder operators, upper semi-Browder operators and lower semiBrowder operators, respectively. Clearly, B(X) ⊆ W (X), B+ (X) ⊆ W+ (X) and B− (X) ⊆ W− (X). Let σb (T ) := {λ ∈ C : λI − T is not Browder} denote the Browder spectrum and let σub (T ) denote the upper semi-Browder spectrum of T , defined as σub (T ) := {λ ∈ C : λI − T is not upper semi-Browder}. Clearly, σw (T ) ⊆ σb (T ) and σuw (T ) ⊆ σub (T ). Lemma 1.1. If T ∈ L(X) and p = p(T ) < ∞ then the following statements are equivalent: (i) There exists n ≥ p + 1 such that T n (X) is closed; (ii) T n (X) is closed for all n ≥ p. Proof. Define ci (T ) := dim(ker T i / ker T i+1 ). Clearly, p = p(T ) < ∞ entails that ci (T ) = 0 for all i ≥ p, so ki (T ) := ci (T ) − ci+1 (T ) = 0 for all i ≥ p. The equivalence then easily follows from [33, Lemma 12]. The concept of semi-Fredholm operators has been generalized by Berkani ([16], [17]) in the following way: for every T ∈ L(X) and a nonnegative integer n let us denote by T[n] the restriction of T to T n (X) viewed as a map from the space T n (X) into itself (we set T[0] = T ). T ∈ L(X) is said to be
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
3
semi B-Fredholm (resp. B-Fredholm, upper semi B-Fredholm, lower semi BFredholm,) if for some integer n ≥ 0 the range T n (X) is closed and T[n] is a semi-Fredholm operator (resp. Fredholm, upper semi-Fredholm, lower semiFredholm). In this case T[m] is a semi-Fredholm operator for all m ≥ n ([17]). This enables one to define the index of a semi B-Fredholm as ind T = ind T[n] . A bounded operator T ∈ L(X) is said to be B-Weyl (respectively, upper semi B-Weyl, lower semi B-Weyl) if for some integer n ≥ 0 T n (X) is closed and T[n] is Weyl (respectively, upper semi-Weyl, lower semi-Weyl). The classes of operators previously defined generate the B-Weyl spectrum σbw (T ), the upper B-Weyl spectrum σusbw (T ), and the lower B-Weyl spectrum σlsbw (T ). Analogously, a bounded operator T ∈ L(X) is said to be B-Browder (respectively, (respectively, upper semi B-Browder, lower semi B-Browder) if for some integer n ≥ 0 T n (X) is closed and T[n] is Browder (respectively, upper semi-Browder, lower semi-Browder). The B-Browder spectrum is denoted by σbb (T ), the upper semi B-Browder spectrum by σusbb (T ) and the lower semi B-Browder spectrum by σlsbb (T ). This note also deals with the concept of Drazin invertibility which has been introduced in a more abstract setting than operator theory, see [25]. In the case of the Banach algebra L(X), T ∈ L(X) is said to be Drazin invertible (with a finite index) if and only if p(T ) = q(T ) < ∞ and this is equivalent to saying that T = T0 ⊕ T1 , where T0 is nilpotent and T1 is invertible, see [31, Corollary 2.2] and [29, Prop. A]. Every B-Fredholm operator T admits on a Banach space the representation T = T0 ⊕ T1 , where T0 is nilpotent and T1 is Fredholm [18], so every Drazin invertible operator is B-Fredholm. Drazin invertibility for bounded operators suggests the following definitions. Definition 1.2. T ∈ L(X) is said to be left Drazin invertible if p := p(T ) < ∞ and T p+1 (X) is closed, while T ∈ L(X) is said to be right Drazin invertible if q := q(T ) < ∞ and T q (X) is closed. Clearly, T ∈ L(X) is both right and left Drazin invertible if and only if T is Drazin invertible. In fact, if 0 < p := p(T ) = q(T ) then T p (X) = T p+1 (X) is the kernel of the spectral projection associated with the spectral set {0}, see [28, Prop. 50.2]. The left Drazin spectrum is then defined as σld (T ) := {λ ∈ C : λI − T is not left Drazin invertible}, the right Drazin spectrum is defined as σrd (T ) := {λ ∈ C : λI − T is not right Drazin invertible}, and the Drazin spectrum is defined as σd (T ) := {λ ∈ C : λI − T is not Drazin invertible}. Obviously, σd (T ) = σld (T ) ∪ σrd (T ). Theorem 1.3. [5] For every T ∈ L(X) we have σusbb (T ) = σld (T ),
σlsbb (T ) = σrd (T ),
σbb (T ) = σd (T ).
4
P. Aiena, E. Aponte and E. Balzan
IEOT
Define ∆(T ) := {n ∈ N : m ≥ n, m ∈ N ⇒ T n (X) ∩ ker T ⊆ T m (X) ∩ ker T }. The degree of stable iteration is defined as dis(T ) := inf ∆(T ) if ∆(T ) = ∅, while dis(T ) = ∞ if ∆(T ) = ∅. The following class of operators has been introduced by Labrousse [30] Definition 1.4. T ∈ L(X) is said to be quasi-Fredholm of degree d, if there exists d ∈ N such that: (a) dis(T ) = d, (b) T n (X) is a closed subspace of X for each n ≥ d, (c) T (X) + ker T d is a closed subspace of X. It should be noted that by Proposition 2.5 of [17] every semi B-Fredholm operator is quasi-Fredholm, in particular every left o right Drazin invertible operator is quasi-Fredholm. The analytic core of T ∈ L(X) is defined K(T ) := {x ∈ X : there exist c > 0 and a sequence (xn )n≥1 ⊆ X such that T x1 = x, T xn+1 = xn for all K(T ), and n ∈ N, and ||xn || ≤ cn ||x||for all n ∈ N}. Note that T (K(T )) = ∞ K(T ) is contained in the hyper-range of T defined by T ∞ (X) := n=0 T n (X), see [1, Chapter 1] for details. A bounded operator T ∈ L(X) is said to besemi-regular if T (X) is ∞ closed and N ∞ (T ) ⊆ T ∞ (X), where N ∞ (T ) := n=1 ker T n denotes the hyper-kernel of T . In the sequel we need the following result: Theorem 1.5. If T ∈ L(X) is quasi-Fredholm then there exists ε > 0 such that N ∞ (λI − T ) ⊆ (λI − T )∞ (X) for all 0 < |λ| < ε. If T is semi B-Fredholm then λI − T is semi-regular in a suitable punctured open disc centered at 0. Proof. Observe first that if T is quasi-Fredholm of degree d then T n (X) is closed for all n ≥ d, so T ∞ (X) is closed. Furthermore, by Theorem 3.4 of [27] the restriction T |T ∞(X) is onto, so T (T ∞(X)) = T ∞ (X). Let T0 := T |T ∞(X). Clearly, T0 is onto and hence λI − T is onto for all |λ| < ε, where ε := γ(T0 ) is the minimal modulus of T0 , see [1, Lemma 1.30]. Therefore, (λI − T )(T ∞ (X)) = T ∞ (X) for all |λ| < ε. Since T ∞ (X) is closed, by [1, Theorem 1.22] it then follows that T ∞ (X) ⊆ K(λI − T ) ⊆ (λI − T )∞ (X) for all |λ| < ε. By part (ii) of Theorem 1.3 of [1] we have N ∞ (λI − T ) ⊆ T ∞ (X) for all λ = 0, so we conclude that N ∞ (λI − T ) ⊆ (λI − T )∞ (X) for all 0 < |λ| < ε, and the first assertion is proved. To show the second assertion, suppose that T is semi B-Fredholm. Then there exists an open disc D centered at 0 such that λI − T is semi-Fredholm for all λ ∈ D \ {0} (this follows as a particular case of a result proved in [17, Corollary 3.2] for operators having topological uniform descent for n ≥ d). Since semi-Fredholm operators have closed range then the last assertion easily follows.
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
5
2. Left and right polaroid operators The first part concerns the basic properties of left and right polaroid operators. In the sequel by T we shall denote the dual of T ∈ L(X). Denote by M ⊥ the annihilator of M ⊆ X, while by ⊥ N we denote the pre-annihilator of N ⊆ X . Theorem 2.1. For every T ∈ L(X) the following equivalences hold: (i) T is left Drazin invertible ⇔ T is right Drazin invertible. (ii) T is right Drazin invertible ⇔ T is left Drazin invertible. (iii) T is Drazin invertible if and only if T is Drazin invertible. Proof. (i) Suppose that T is left Drazin invertible. Then p := p(T ) < ∞ and T p+1 (X) is closed, and hence also T p+1 (X ) is closed. By Lemma 1.1 p T p (X) is closed, and consequently also T (X ) is closed. From the equality p p+1 ker T = ker T and from the classical closed range theorem we then deduce that T (X ) = [ker T p ]⊥ = [ker T p+1 ]⊥ = T p
p+1
(X ).
This shows that T has finite descent q := q(T ) ≤ p and since T q (X ) = p T (X ) is closed it then follows that T is right Drazin invertible. Conversely, suppose that T is right Drazin invertible. Then q := q(T ) q q q+1 < ∞ and T (X ) is closed. From the equality T (X ) = T (X ) and from the closed range theorem we then obtain: ker T q =⊥ [T (X )] =⊥ [T q
q+1
(X )] = ker T q+1 ,
and hence p := p(T ) ≤ q. Since T (X ) is closed then also T q+1 (X) is closed and by Lemma 1.1 it then follows that T p+1 (X) is closed, so T is a left Drazin invertible. (ii) This may be proved in a similar way of part (i). (iii) Obviously, since both left and right Drazin invertibility entails Drazin invertibility. q+1
Recall that T ∈ L(X) is said to be bounded below if T is injective with closed range. The classical approximate point spectrum is defined by σa (T ) := {λ ∈ C : λI − T is not bounded below}. If σs (T ) denotes the surjectivity spectrum it is well known that σa (T ) = σs (T ) and σs (T ) = σa (T ) for all T ∈ L(X). Definition 2.2. Let T ∈ L(X), X a Banach space. If λI − T is left Drazin invertible and λ ∈ σa (T ) then λ is said to be a left pole of the resolvent of T . If λI − T is right Drazin invertible and λ ∈ σs (T ) then λ is said to be a right pole of the resolvent of T . It is easily seen that λ is a pole of T if and only if λ is both a left and a right pole of T . In fact, if λ is a pole of T then λI − T is Drazin invertible, so λI − T is both left and right Drazin invertible. Moreover, λ is both a left
6
P. Aiena, E. Aponte and E. Balzan
IEOT
and a right pole of T , since the condition 0 < p(λI − T ) = q(λI − T ) < ∞ entails that λ ∈ σa (T ) as well as λ ∈ σs (T ). The converse is clear. Remark 2.3. Suppose that for a linear operator T we have α(T ) < ∞. Then α(T n ) < ∞ for all n ∈ N. This may be easily seen by an inductive argument. Suppose that dim ker T n < ∞. Since T (ker T n+1 ) ⊆ ker T n then the restriction T0 := T | ker T n+1 : ker T n+1 → ker T n has kernel equal to ker T so the canonical mapping Tˆ : ker T n+1 / ker T → ker T n is injective. Therefore we have dim ker T n+1 / ker T ≤ dim ker T n < ∞, and since dim ker T < ∞ we then conclude that dim ker T n+1 < ∞. Analogously, if β(T ) < ∞ then β(T n ) < ∞ for all n ∈ N. In fact, suppose β(T n ) < ∞. Since the map T˜ : T n (X)/T n+1(X) → T n+1 (X)/T n+2(X), defined by T˜ (z + T n+1 (X)) = T z + T n+2 (X),
z ∈ T n (X),
is onto, then dim T n+1 (X)/T n+2 (X) ≤ dim T n (X)/T n+1 (X). This easily implies that β(T n+1 ) < ∞. Lemma 2.4. Let T ∈ L(X). Then we have: (i) T is upper semi B-Fredholm and α(T ) < ∞ if and only if T ∈ Φ+ (X). (ii) T is lower semi B-Fredholm and β(T ) < ∞ if and only T ∈ Φ− (X). Proof. (i) If T is upper semi B-Fredholm then there exists n ∈ N such that T n (X) is closed and T[n] is upper semi-Fredholm. Since α(T ) < ∞ then α(T n ) < ∞, hence T n is upper semi Fredholm. From the classical Fredholm theory then also T is upper semi-Fredholm. The converse is obvious. Part (ii) may be proved in a similar way. Definition 2.5. A bounded operator T ∈ L(X) is said to be left polaroid if every isolated point of σa (T ) is a left pole of the resolvent of T . T ∈ L(X) is said to be right polaroid if every isolated point of σs (T ) is a right pole of the resolvent of T . T ∈ L(X) is said to be polaroid if every isolated point of σ(T ) is a pole of the resolvent of T . Theorem 2.6. If T ∈ L(X) is both left and right polaroid then T is polaroid. Proof. If iso σ(T ) = ∅ there is nothing to prove. Suppose then λ ∈ iso σ(T ) = ∅. Since the boundary the spectrum is contained in σa (T ), see [1, Theorem 2.42], then λ ∈ iso σa (T ), so λ is a left pole and hence p(λI − T ) < ∞. On the other hand, λ ∈ σs (T ), (otherwise we have 0 = q(λI − T ) = p(λI − T ) and hence λ ∈ / σ(T )). Therefore, λ ∈ iso σs (T ) and since T is right polaroid then q(λI − T ) < ∞, from which conclude that λ is a pole of the resolvent of T . The following example shows that the converse of Theorem 2.6, in general, does not hold.
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
7
Example 2.7. Let R denote the right shift on 2 (N) defined by R(x1 , x2 , . . . ) := (0, x1 , x2 , . . . )
(xn ) ∈ 2 (N),
and let Q be the weighted left shift defined by Q(x1 , x2 , . . . ) := (x2 /2, x3 /3, . . . ) (xn ) ∈ 2 (N). Q is a quasi-nilpotent operator, σ(R) = D(0, 1), where D(0, 1) denotes the closed unit disc of C, and σa (R) = Γ, where Γ is the unit circle of C. Moreover, if en := (0, ..., 0, 1, 0...), where 1 is the n-th term, then en+1 ∈ ker Qn+1 while / ker Qn for every n ∈ N, so p(Q) = ∞. en+1 ∈ Define T := R ⊕ Q on X := 2 (N) ⊕ 2 (N). Clearly, σ(T ) = D(0, 1), and σa (T ) = Γ ∪ {0}. We have p(T ) = p(R) + p(Q) = ∞, so 0 is not a left pole. Therefore, T is polaroid, since iso σ(T ) = ∅, but not left polaroid. It is easily seen that the dual T is polaroid but not right polaroid, since q(T ) = ∞. The concept of left and right polaroid are dual each other: Theorem 2.8. If T ∈ L(X) then the following equivalences hold: (i) T is left polaroid if and only if T is right polaroid. (ii) T is right polaroid if and only if T is left polaroid. (iii) T is polaroid if and only if T is polaroid. Proof. (i) Suppose that T is left polaroid. If iso σs (T ) = ∅ there is nothing to prove. Suppose that λ ∈ iso σs (T ). Then λ is an isolated point of σa (T ), hence, by assumption, λI − T is left Drazin invertible. By Theorem 2.1 then λI − T is right Drazin invertible, so λ is a right pole of the resolvent of T . Conversely, suppose T right polaroid and let λ be an isolated point of σa (T ). Then λ ∈ iso σs (T ) and hence is a right pole of the resolvent of T . Therefore, λI − T is right Drazin invertible so, by Theorem 2.1, λI − T is left Drazin invertible. Thus, λ is a left pole of the resolvent of T . (ii) The proof is similar to that of part (i). (iii) It is well-known that σ(T ) = σ(T ) and that λ is a pole of the resolvent of T if and only if λ is a pole of the resolvent of T . Definition 2.9. A bounded operator T ∈ L(X) is said to be a-polaroid if every λ ∈ iso σa (T ) is a pole of the resolvent of T . Trivially, T a-polaroid ⇒ T left polaroid.
(1)
Moreover, iso (T ) ⊆ σa (T ) for every T ∈ L(X), since the boundary of σ(T ) is contained in σa (T ), from which we easily obtain: T a-polaroid ⇒ T polaroid.
(2)
The following example provides an operator that is left polaroid but not a-polaroid.
8
P. Aiena, E. Aponte and E. Balzan
IEOT
Example 2.10. Let R ∈ 2 (N) be the unilateral right shift defined as R(x1 , x2 , . . . ) := (0, x1 , x2 , · · · ) for all (xn ) ∈ 2 (N), and U (x1 , x2 , . . . ) := (0, x2 , x3 , · · · ) for all (xn ) ∈ 2 (N). If T := R ⊕ U then σ(T ) = D(0, 1), so iso σ(T ) = ∅. Moreover, σa (T ) = Γ∪{0}, Γ the unit circle, so iso σa (T ) = {0}. Since R is injective and p(U ) = 1 it then follows that p(T ) = p(R) + p(U ) = 1. Furthermore, T ∈ Φ+ (X) and hence T 2 ∈ Φ+ (X), so that T 2 (X) is closed. Therefore 0 is a left pole and hence T is left polaroid. On the other hand q(R) = ∞, so that q(T ) = q(R) + q(U ) = ∞, so T is not a-polaroid. Note that T is also polaroid. In the case of Hilbert space operators T ∈ L(H) instead of the dual T it is more appropriate to consider the Hilbert adjoint T ∗ . By means of the classical Fr´echet- Riesz representation theorem we know that if U is the conjugate-linear isometry that associates to each y ∈ H the linear form x → x, y then λI − T ∗ = (λI − T )∗ = U −1 (λI − T ) U.
(3)
This obviously implies that σa (T ∗ ) = σa (T ) and σs (T ∗ ) = σs (T ). Theorem 2.11. If T ∈ L(H), H a Hilbert space, then the following equivalences hold: (i) T is left polaroid if and only if T ∗ is right polaroid. (ii) T is right polaroid if and only if T ∗ is left polaroid. (iii) T is polaroid if and only if T ∗ is polaroid. Proof. (i) Suppose that T is left polaroid. If λ ∈ iso σs (T ∗ ) then λ ∈ iso σs (T ) = iso σa (T ), so λ is a left pole of T , hence p := p(λI − T ) < ∞ and (λI − T )p+1 (H) is closed. From ker (λI − T )p = ker (λI − T )p+1 we see that (λI − T ∗ )p (H) = [ker (λI − T )p ]⊥ = [ker (λI − T )p+1 ]⊥ = (λI − T ∗ )p+1 (H), where by N ⊥ we denote the orthogonal of N ⊆ H. Therefore q := q(λI − T ∗) ≤ p < ∞ and consequently (λI − T ∗ )q (H) = (λI − T ∗ )p (H) is closed, by Lemma 1.1. Thus λ is a right pole of T ∗ and T ∗ is right polaroid. Conversely, suppose that T ∗ is right polaroid and let λ ∈ iso σa (T ). Then λ ∈ iso σs (T ) = iso σs (T ∗ ), thus λ is a right pole of T ∗ . Consequently, q := q(λI − T ∗ ) < ∞ and (λI − T ∗ )q (H) = (λI − T ∗ )q+1 (H) is closed. Since ker(λI − T )q+1 = [(λI − T ∗ )q+1 (H)]⊥ = [(λI − T ∗ )q (H)]⊥ = ker(λI − T )q it then follows that p := p(λI − T ) ≤ q < ∞. By Lemma 1.1 (λI − T )p+1 (H) is closed, so λ is a left pole of T . (ii) This may be proved in a similar way of part (i). (iii) See Theorem 2.5 of [8].
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
9
The following property has relevant role in local spectral theory, see the recent monographs by Laursen and Neumann [32] and [1]. Definition 2.12. Let X be a complex Banach space and T ∈ L(X). The operator T is said to have the single valued extension property at λ0 ∈ C (abbreviated SVEP at λ0 ), if for every open disc D of λ0 , the only analytic function f : D → X which satisfies the equation (λI − T )f (λ) = 0 for all λ ∈ D is the function f ≡ 0. An operator T ∈ L(X) is said to have SVEP if T has SVEP at every point λ ∈ C. Evidently, T ∈ L(X) has SVEP at every isolated point of the spectrum. We also have p(λI − T ) < ∞ ⇒ T has SVEP at λ,
(4)
and dually
q(λI − T ) < ∞ ⇒ T has SVEP at λ, (5) see [1, Theorem 3.8]. Furthermore, from definition of localized SVEP it easily seen that σa (T ) does not cluster at λ ⇒ T has SVEP at λ,
(6)
σs (T ) does not cluster at λ ⇒ T has SVEP at λ.
(7)
and dually, The quasi-nilpotent part of T ∈ L(X) is defined as the set 1
H0 (T ) := {x ∈ X : lim T nx n = 0}. n→∞
Clearly, ker T n ⊆ H0 (T ) for every n ∈ N. Moreover, T is quasi-nilpotent if and only if H0 (λI − T ) = X, see Theorem 1.68 of [1]. Note that H0 (T ) generally is not closed and ([1, Theorem 2.31 ] H0 (λI − T ) closed ⇒ T has SVEP at λ.
(8)
Remark 2.13. All the implications (4), (5), (6), (7) and (8) are equivalences whenever λI − T is a quasi-Fredholm operator, in particular whenever λI − T is a semi B-Fredholm operator, see [3]. Theorem 2.14. Suppose that T ∈ L(X). Then the following assertions hold: (i) If T has SVEP then the properties of being polaroid, a-polaroid and left polaroid for T are all equivalent. (ii) If T has SVEP then the properties of being polaroid, a-polaroid and left polaroid for T are all equivalent. Proof. (i) Suppose that T has SVEP. Then σ(T ) = σa (T ), see Corollary 2.44 of [1], hence T is polaroid precisely when T is a-polaroid. By (1) the apolaroid condition for T entails that T is left polaroid. Conversely, if T is left polaroid and λ ∈ iso σ(T ) then λ ∈ iso σa (T ), again because the boundary of the spectrum is contained in σa (T ), hence λ is a left pole, so that then p(λI − T ) < ∞. On the other hand, since λI − T is left Drazin invertible, in
10
P. Aiena, E. Aponte and E. Balzan
IEOT
particular quasi-Fredholm the SVEP of T at λ entails that q(λI − T ) < ∞, see [3, Theorem 2.11], and consequently λ is a pole of the resolvent of T . Therefore T is polaroid. (ii) If T has SVEP, by Corollary 2.44 of [1] then σ(T ) = σs (T ) and hence σ(T ) = σa (T ), so T is polaroid if and only if T is a-polaroid. Again from (1) the a-polaroid condition for T entails that T is left polaroid. Suppose now that T is left polaroid. If λ ∈ iso σ(T ) = iso σ(T ) = iso σa (T ) then λI − T is left Drazin invertible, hence by Theorem 2.1 λI − T is right Drazin invertible, so that q(λI−T ) < ∞. On the other hand, λI−T is quasi-Fredholm so,by Theorem [3, 2.7], the SVEP for T at λ entails, that p(λI − T ) < ∞. Hence λ is a pole of T so that T is polaroid, or equivalently T is polaroid.
3. Weyl’s type theorem In this section we show the equivalence of some Weyl type theorems in the case that T is left polaroid or a-polaroid. If T ∈ L(X) define E(T ) := {λ ∈ iso σ(T ) : 0 < α(λI − T )}, and E a (T ) := {λ ∈ iso σa (T ) : 0 < α(λI − T )}. Evidently, E(T ) ⊆ E a (T ) for every T ∈ L(X). Define π00 (T ) : {λ ∈ iso σ(T ) : 0 < α(λI − T ) < ∞}, and a π00 (T ) : {λ ∈ iso σa (T ) : 0 < α(λI − T ) < ∞}.
Let p00 (T ) := σ(T ) \ σb (T ), i.e. p00 (T ) is the set of all poles of the resolvent of T having finite rank. Clearly, for every T ∈ L(X) we have a (T ). p00 (T ) ⊆ π00 (T ) ⊆ π00
(9)
It should be noted that the condition p00 (T ) = π00 (T ) is equivalent to saying that there exists p := p(λI − T ) ∈ N such that H0 (λI − T ) = ker (λI − T )p
for all λ ∈ π00 (T ),
(10)
see [7, Theorem 2.2]). Also the condition of being polaroid may be characterized by means of the quasi-nilpotent part: T is polaroid if and only if there exists p := p(λI − T ) ∈ N such that H0 (λI − T ) = ker (λI − T )p
for all λ ∈ iso σ(T ),
(11)
[4, Theorem 2.9]. Trivially, for every polaroid operator p00 (T ) = π00 (T ). a Lemma 3.1. If T is a-polaroid then p00 (T ) = π00 (T ) = π00 (T ) and E(T ) = a E (T ).
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
11
a Proof. To show the first equalities it suffices, by (9), to prove that π00 (T ) ⊆ a p00 (T ). Let λ ∈ π00 (T ). Then λ is isolated in σa (T ) and hence the a-polaroid condition entails that p(λI − T ) = q(λI − T ) < ∞. Moreover, α(λI − T ) < ∞, so by Theorem 3.4 of [1] we have β(λI − T ) < ∞, hence λ ∈ p00 (T ). To show the equality E(T ) = E a (T ) it suffices to prove the inclusion E a (T ) ⊆ E(T ). If λ ∈ E a (T ) then λ ∈ iso σa (T ) and hence is a pole of T , so λ is an isolated point of σ(T ).
We give now the definition of Weyl’s theorem and of some of its variants. The symbols here used could generate a certain confusion, but these are the most used in literature. Definition 3.2. A bounded operator T ∈ L(X) is said to satisfy Weyl’s theorem, in symbol (W ), if σ(T ) \ σw (T ) = π00 (T ). T ∈ L(X) is said to satisfy a a-Weyl’s theorem, in symbol (aW ), if σa (T ) \ σuw (T ) = π00 (T ). T ∈ L(X) is said to satisfy property (w), if σa (T ) \ σuw (T ) = π00 (T ). Weyl’s theorem for T entails Browder’s theorem for T , i.e. σw (T ) = σb (T ). Note that Browder’s theorem for T and Browder’s theorem for T ∗ are equivalent, since σw (T ) = σw (T ∗ ) and σb (T ) = σb (T ∗ ). Furthermore, by [2, Theorem 3.1], (W ) holds for T ⇔ Browder’s theorem holds for T and p00 (T ) = π00 (T ). Either a-Weyl’s theorem or property (w) entails Weyl’s theorem. Property (w) and a-Weyl’s theorem are independent, see [10]. The following result gives a very simple and useful framework for establishing Weyl’s theorem for several classes of operators: Theorem 3.3. It T ∈ L(X) is polaroid and either T or T has SVEP then both T and T satisfy Weyl’s theorem. Proof. The SVEP of either T or T entails Browder’s theorem for T , or equivalently Browder’s theorem for T . The polaroid condition for T entails that p00 (T ) = π00 (T ), so Weyl’s theorem holds for T . If T is polaroid then T is polaroid and hence p00 (T ) = π00 (T ), so Weyl’s theorem also holds for T . Definition 3.4. A bounded operator T ∈ L(X) is said to satisfy generalized Weyl’s theorem, in symbol (gW ), if σ(T )\σbw (T ) = E(T ). T ∈ L(X) is said to satisfy generalized a-Weyl’s theorem, in symbol (gaW ), if σa (T ) \ σubw (T ) = E a (T ). T ∈ L(X) is said to satisfy generalized property (w), in symbol (gw), if σa (T ) \ σubw (T ) = E(T ). For a bounded operator T ∈ L(X), define Πa00 (T )
:= σa (T )\ σubb (T ) = {λ ∈ σa (T ) : λI − T is upper semi B-Browder}.
Since σubb (T ) = σld (T ), by Theorem 1.3, it is clear that Πa00 (T ) is the set of all left poles of the resolvent. Lemma 3.5. If T ∈ L(X) then Πa00 (T ) ⊆ E a (T ).
12
P. Aiena, E. Aponte and E. Balzan
IEOT
Proof. If λ ∈ Πa00 (T ) then λ ∈ σa (T ), p(λI −T ) < ∞ and λI −T is left Drazin invertible. The condition p(λI − T ) < ∞ entails, by Remark 2.13, that σa (T ) does not cluster at λ. On the other hand we have 0 < α(λI − T ), otherwise if were α(λI − T ) = 0 then p(λI − T ) = 0 and from definition of left Drazin invertibility (λI − T )(X) would be closed, hence λ ∈ / σa (T ), a contradiction. Recall that T ∈ L(X) is said to satisfy generalized Browder’s theorem if σbb (T ) = σbw (T ), while T ∈ L(X) is said to satisfy generalized a-Browder’s theorem if σubb (T ) = σubw (T ). Browder’s Theorem and generalized Browder’s theorem are equivalent, and analogously a-Browder’s theorem and generalized a-Browder’s theorem are equivalent, see [5] or [14]. Generalized a-Weyl’s theorem, as well as generalized property (w), entails generalized a-Browder’s theorem. Theorem 3.6. T ∈ L(X) satisfies (gaW ) if and only if T satisfies generalized a-Browder’s theorem and E a (T ) = Πa00 (T ). In particular, every left polaroid operator which has SVEP satisfies (gaW ). Proof. The first assertion has been proved in [9]. The second assertion is clear: for every left polaroid operator we have E a (T ) = Πa00 (T ). The SVEP for T entails Browder’s theorem, or equivalently generalized Browder’s theorem. It should be noted that if T or T (or T ∗ , for Hilbert space operators) has SVEP then a-Browder’s theorem holds for T , or, equivalently, generalized a-Browder’s theorem, holds for T , see [6]. In the following diagram we resume the relationships between all Weyl type theorems: (gw) ⇒ (w) ⇒ (W ) (gaW ) ⇒ (aW ) ⇒ (W ), see [15, Theorem 2.3], [10] and [19]. Generalized property (w) and generalized a-Weyl’s theorem are also independent, see [15]. Furthermore, (gw) ⇒ (gW ) ⇒ (W ) (gaW ) ⇒ (gW ) ⇒ (W ), see [15] and [19]. The converse of all these implications in general does not hold. However, we have: Theorem 3.7. Let T ∈ L(X). Then we have: (i) If T is left-polaroid, then (aW ) and (gaW ) for T are equivalent. If T is right-polaroid then (aW ) and (gaW ) for T are equivalent. (ii) If T is polaroid, then (W ), and (gW ) for T are equivalent. Analogously, (W ), and (gW ) for T are equivalent.
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
13
Proof. (i) As already observed, generalized property (gaW ) entails property (aW ) without any assumption on T . Suppose now that property (aW ) holds for T , i.e. σa (T ) \ σuw (T ) = a (T ). We have to prove that σa (T ) \ σubw (T ) = E a (T ). We show first that π00 the inclusion σa (T ) \ σubw (T ) ⊆ E a (T ) holds without any assumption on T . Let λ ∈ σa (T ) \ σubw (T ). We can suppose that λ = 0. Therefore, 0 ∈ σa (T ) and T is upper semi B-Fredholm with index less or equal than 0. By [17, Corollary 3.2] then there exists ε > 0 such that µI − T ∈ W+ (X) for all 0 < |µ| < ε. We claim that T has SVEP at every µ. If µ ∈ / σa (T ) this is obvious. Suppose a (T ), so µ is an isolated point that µ ∈ σa (T ). Then µ ∈ σa (T ) \ σuw (T ) = π00 of σa (T ) and hence T has SVEP at µ. The following argument shows that T has SVEP at 0. Let f : D0 → X be an analytic function defined on an open disc D0 centered at 0 for which the equation (λI − T )f (λ) = 0 for all λ ∈ D0 . Take 0 = µ ∈ D0 and let D1 be an open disc centered at µ contained in D0 . The SVEP of T at µ implies that f ≡ 0 on D1 and hence, from the the identity theorem for analytic functions, it then follows that f ≡ 0 on D0 , so T has SVEP at 0. But T is upper semi B-Fredholm, so, by Theorem 2.7 of [3], 0 ∈ iso σa (T ). Suppose now that α(T ) = 0. By Lemma 3.5 then T ∈ Φ+ (X), so T (X) is closed and, consequently, 0 ∈ / σa (T ), a contradiction. Therefore α(T ) > 0, from which we conclude 0 ∈ E a (T ) and hence σa (T ) \ σubw (T ) ⊆ E a (T ). Suppose now that T is left polaroid and let λ ∈ E a (T ). Then λ is an isolated point of σa (T ), and hence by the left polaroid condition λ is a left pole of T . In particular, λI − T is left Drazin invertible or equivalently, by Theorem 1.3, an upper semi B-Browder operator. Since σubw (T ) ⊆ σubb (T ) we then have λ ∈ σa (T ) \ σubb (T ) ⊆ σa (T ) \ σubw (T ). Therefore, generalized a-Weyl’s theorem holds for T . The assertion concerning right-polaroid operators is obvious by Theorem 2.8. (ii) We have only to show that Weyl’s theorem entails generalized Weyl’s theorem. Suppose first that λ0 ∈ E(T ). Since T is polaroid then λ0 is a pole of T , hence 0 < p(λ0 I − T ) = q(λ0 I − T ) < ∞. Therefore, λ0 I − T is Drazin invertible or equivalently, by Theorem 1.3, λ0 I − T is B-Browder and hence B-Weyl. Consequently, λ0 ∈ σ(T ) \ σbw (T ) and hence E(T ) ⊆ σ(T ) \ σbw (T ). Conversely, assume that λ0 ∈ σ(T ) \ σbw (T ). Then λ0 I − T is B-Weyl and hence, again by [17, Corollary 3.2], there exists ε > 0 such that λI − T is Weyl for all 0 < |λ − λ0 | < ε. By Theorem 1.5 we know that λI − T is semi-regular in a punctured open disc centered at λ0 , so we can assume that ker (λI − T ) ⊆ N ∞ (λI − T ) ⊆ (λI − T )∞ (X) for all 0 < |λ − λ0 | < ε. As observed above Weyl’s theorem for T entails Browder’s theorem for T , i.e. σw (T ) = σb (T ). Therefore, λI − T is Browder for all 0 < |λ − λ0 | < ε and, consequently, p(λI − T ) = q(λI − T ) < ∞. By Lemma 3.2 of [1] we then have ker (λI − T ) = ker (λI − T ) ∩ (λI − T )∞ (X) = {0},
14
P. Aiena, E. Aponte and E. Balzan
IEOT
thus α(λI − T ) = 0 and since λI − T is Weyl we then conclude that also β(λI − T ) = 0, so λI − T is invertible for all 0 < |λ − λ0 | and hence λ0 ∈ iso σ(T ). To show that λ0 ∈ E(T ) it remains to prove that α(λ0 I − T ) > 0. Suppose that α(λ0 I − T ) = 0. Since λ0 I − T is B-Weyl then, by Lemma 2.4, λ0 I − T is Weyl and since α(λ0 I − T ) = 0 it then follows that λ0 I − T is invertible, a contradiction since λ0 ∈ σ(T ). Therefore, λ0 ∈ E(T ), so generalized Weyl’s theorem holds for T . The last assertion is clear: T is also polaroid. Corollary 3.8. If T ∈ L(X) is a-polaroid then (aW ), (gaW ), (w), (gw) for T are equivalent. Proof. Every a-polaroid operator is left polaroid so, by part (i) of Theorem 3.7, (aW ) and (gaW ) are equivalent. Property (w) and (aW ) are equiva alent, since by Lemma 3.1 we have π00 (T ) = π00 (T ). By Lemma 3.1 we also a have E(T ) = E (T ), from which it easily follows that (gaW ) and (gw) are equivalent. In the following example we show that the result of part (ii) of Theorem 3.7 does not hold if we replace the condition of being a-polaroid by the weaker conditions of being left polaroid or polaroid. Example 3.9. Let R and U be defined as in Example 2.10. As observed before T := R ⊕ U is both left polaroid and polaroid. Moreover, σa (T ) = Γ ∪ {0} and iso σ(T ) = π00 (T ) = ∅, so σa (T ) \ σuw (T ) = {0} = π00 (T ), i.e. T does a not satisfy property (w). On the other hand, we have π00 (T ) = {0}, hence T satisfies a-Weyl’s theorem. In the following result we show that if T is polaroid and T has SVEP, (respectively, T has SVEP), we can say much more: all Weyl type theorems, generalized or not, are equivalent and hold for T , (respectively, for T ). Theorem 3.10. Let T ∈ L(X) be polaroid. Then we have (i) If T has SVEP then (W ), (aW ), (w), (gW ), (gaW ) and (gw) hold for T . Moreover, T satisfies (gW ). (ii) If T has SVEP then (W ), (aW ), (w), (gW ), (gaW ) and (gw) hold for T . Moreover, T satisfies (gW ). Proof. (i) T satisfies (W ) by Theorem 3.3. The first statement is then proved if we show that (W ) is equivalent to each one of the other Weyl type theorems for T , generalized or not. Since T has SVEP (W ) and (aW ) for T are equivalent, by part (i) of [10, Theorem 2.16]. By Theorem 2.14 T is a-polaroid hence, by Theorem 3.8, (aW ), (gaW ), (w), (gw) for T are equivalent. Finally, by part (ii) of Theorem 3.7, (W ) and (gW ) for T are equivalent. By Theorem 3.3 T satisfies (W ) and since T is polaroid then, by part (ii) of Theorem 3.7, (gW ) holds for T . (ii) T satisfies (W ) by Theorem 3.3, so it suffices to prove that (W ) is equivalent to each one of the other Weyl type theorems, generalized or not, for T . Since, by Theorem 2.14, T is a-polaroid from Theorem 3.8 it then
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
15
follows that (aW ), (gaW ), (w), (gw) are equivalent for T . The SVEP for T entails by part (ii) of [10, Theorem 2.16] that (W ) and (aW ) are equivalent for T , while (W ) and (gW ) for T are equivalent by part (ii) of Theorem 3.7. By Theorem 3.3 T satisfies (W ) and since T is polaroid this is equivalent to (gW ) for T , by part (ii) of Theorem 3.7. Let H(σ(T )) denote the set of all analytic functions defined on an open neighborhood of σ(T ) define, by the classical functional calculus, f (T ) for every f ∈ H(σ(T )). Lemma 3.11. Suppose that f ∈ H(σ(T )) is non constant on each of the components of its domain. If T is left polaroid (respectively, right polaroid, polaroid), then f (T ) is left polaroid (respectively, right polaroid, polaroid). Proof. Let λ0 ∈ iso σa (f (T )). We have to show that λ0 is a left pole of f (T ). Since σa (T ) satisfies the spectral mapping theorem we have λ0 ∈ iso f (σa (T )). We show that λ0 ∈ f (iso σa (T )). Let µ0 ∈ σa (T ) be such that f (µ0 ) = λ0 . Denote by Ω the open and connected component of the domain of f which contains µ0 . Suppose that µ0 is not isolated in σa (T ). Then there exists a sequence (µn ) ⊂ σa (T ) of distinct scalars such that µn → µ0 . Clearly, for n sufficiently large µn ∈ Ω and since K := {µ0 , µ1 , µ2 , . . . } is compact subset of Ω, the classical principle of isolated zeros of analytic functions says to us that f may assume the value λ0 = f (µ0 ) only a finite number of points of K, so for n sufficiently large f (µn ) = f (µ0 ) = λ0 , and since f (µn ) → f (µ0 ) = λ0 it then follows that λ0 is not an isolated point of f (σa (T )), a contradiction. Hence λ0 = f (µ0 ), with µ0 ∈ iso σa (T ). Since T is left polaroid then µ0 is a left pole of T and by [11, Theorem 2.9] it then follows that λ0 is a left pole of f (T ), which proves that f (T ) is left polaroid. The proofs for right polaroid and polaroid operators are analogous, just use the spectral mapping theorems for σs (T ) and σ(T ), respectively, and [11, Theorem 2.9]. The result of Theorem 3.10 may be considerably extended as follows Theorem 3.12. Let T ∈ L(X) be polaroid an suppose that f ∈ H(σ(T )) is not constant on each of the components of its domain. Then we have (i) If T has SVEP then (W ), (aW ), (w), (gW ), (gaW ) and (gw) hold for f (T ). (ii) If T has SVEP then (W ), (aW ), (w), (gW ), (gaW ) and (gw) hold for f (T ). Proof. (i) If T ∗ has SVEP then f (T ) = f (T ) has SVEP, see [1, Theorem 2.40]. Moreover, T is left polaroid by Theorem 2.14, so f (T ) is left polaroid by Lemma 3.11. Again by Theorem 2.14, the SVEP of f (T ) entails that f (T ) is polaroid, hence Theorem 3.10 applies to f (T ). (ii) Argue as in the proof of part (i), just replace T with T .
16
P. Aiena, E. Aponte and E. Balzan
IEOT
Remark 3.13. Obviously, in the case of Hilbert space operators, the condition T has SVEP in Theorem 3.12 may be replaced by the SVEP of the Hilbert adjoint T ∗ . Theorem 3.14. Let T ∈ L(X) . Then we have (i) If T ∈ L(X) is left-polaroid and has SVEP, f ∈ H(σ(T )) is not constant on each of the components of its domain, then (aW ) holds for f (T ), or equivalently (gaW ) holds for f (T ). (ii) If T ∈ L(X) is polaroid and has SVEP, f ∈ H(σ(T )) is not constant on each of the components of its domain, then (W ) holds for f (T ), or equivalently (gW ) holds for f (T ). Proof. (i) If T is left polaroid then f (T ) is left polaroid, by Lemma 3.11. By [1, Theorem 2.40] f (T ) has SVEP, hence Theorem 3.6 applies to f (T ). The equivalence of (aW ) and (gaW ) follows from Theorem 3.7. (ii) If T is polaroid then f (T ) is polaroid and has SVEP, so Theorem 3.3 applies to f (T ). The equivalence of (W ) and (gW ) follows from Theorem 3.7.
4. Some applications Weyl type theorems, in their classical and more recently in their generalized form, have been studied by a large number of authors. The results of the previous sections give us an unifying theoretical framework for establishing all Weyl type theorems for a large number of the commonly considered classes of operators. It should be noted that for these classes of operators, Weyl type theorems, or their generalized versions, have been proved in several papers. The classes of polaroid operators introduced in the previous sections are rather large. In the sequel we list some, by no means all, of these classes of operators. In the following H always denotes a Hilbert space. (a) A bounded operator T ∈ L(X) is said to belong to the class H(p) if there exists a natural p := p(λ) such that: H0 (λI − T ) = ker (λI − T )p
for all λ ∈ C.
(12)
From the implication (8) we see that every operator T which belongs to the class H(p) has SVEP. Moreover, from (11) it follows that every H(p) operator T is polaroid. Consequently, by Theorem 3.14, f (T ) satisfies Weyl’s theorem, or equivalently, (gW ) for every f ∈ H(σ(T )) which is not constant on each of the components of its domain. Note that, by [34, Theorem 3.4], T ∈ H(p) if and only if there exists a f ∈ H(σ(T )), non constant on each of the components of its domain, such that f (T ) is H(p) and this is equivalent to saying that f (T ) is H(p) for all f ∈ H(σ(T )). Consequently, if T is algebraically H(p) (i.e. there exists a non trivial polynomial h such that h(T ) is H(p)) then T is H(p), so (W ) and (gW ) hold for f (T ). The class H(p) has been introduced by Oudghiri in [34] and in [12] this
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
17
class of operators has been studied for p := p(λ) = 1 for all λ ∈ C. Property H(p) is satisfied by every generalized scalar operator, and in particular for p-hyponormal, log-hyponormal or M-hyponormal operators on Hilbert spaces, see [34]. Therefore, algebraically p-hyponormal or algebraically M hyponormal operators are H(p). (W ) or (gW ) for f (T ), where T is algebraically M -hyponormal, have been proved in different papers [23] and [24], respectively. In [20, Theorem 3.3] it is shown that if T ∈ L(H) is such that T ∗ is p-hyponormal or M - hyponormal then (gaW ) holds for f (T ) for all f ∈ H(σ(T )). Since T ∗ is H(p), hence polaroid (or, equivalently, T is polaroid), then, by Theorem 3.12 and Remark 3.13, all Weyl’s theorems (generalized or not) hold (and are equivalent!) for f (T ). Finally, Theorem 3.14 subsumes also [21, Theorem 3.4]: if T is analytically hyponormal (i.e. there exists h ∈ H(σ(T )) for which h(T ) is hyponormal) then Weyl’s theorem holds for f (T ). Clearly, since analytically hyponormal operators are H(p), then (W ) and (gW ) are equivalent for f (T ), (b) A bounded operator T ∈ L(X) on a Banach space X is said to be paranormal if T x2 ≤ T 2 xx holds for all x ∈ X. Every paranormal operator on a Hilbert space has SVEP [7]. An operator T ∈ L(H) for which there exists a complex nonconstant polynomial h such that h(T ) is paranormal is said to be algebraically paranormal. Every algebraic paranormal operator T defined on a Hilbert space is polaroid, see [7]. Moreover, the SVEP for h(T ) entails the SVEP for T , see [1, Theorem 2.40]. (gW ) for f (T ), T algebraically paranormal, has been proved in [35, Theorem 3.1] and [23, Theorem 4.14], but this is immediate from Theorem 3.14 and (gW ) for f (T ) is equivalent to (W ). In [35, Theorem 3.2] it has been also proved that (gaW ) holds for T if T ∗ is algebraically paranormal. This result is clear from Theorem 3.10, and in particular (gaW ) for T is equivalent to any Weyl type theorem, generalized or not. Furthermore, Theorem 3.12 and Remark 3.13 extend to f (T ) all Weyl type theorems. Similar considerations may be done for analytically paranormal operators on Hilbert spaces studied in [22]. Analytically paranormal operators T are polaroid and have SVEP, again by [1, Theorem 2.40], so that (W ) holds for f (T ), for f ∈ H(σ(T )) non constant on each of the components of its domain, and this is equivalent to saying that (gW ) holds for f (T ). (gW ) for f (T ), T analytically paranormal, has been proved [22, Theorem 3.1, part (a)]. (gaW ) for f (T ∗ ), T analytically paranormal, was proved in [22, Theorem 3.1, part (b)]. This is clear by Theorem 3.12 and Remark 3.13 and, again, all Weyl type theorems, generalized or not, are equivalent for f (T ). Also the result of [22, Theorem 3.2, part] easily follows from Theorem 3.12 and Remark 3.13: If T ∗ ∈ L(H) is analytically paranormal then all Weyl type theorems, generalized or not, are satisfied by f (T ) and all of them are equivalent. (c) A bounded operator T ∈ L(H) is a class A operator if |T |2 ≤ |T 2 |, and T is said to be a quasi-class A operator if T ∗ |T |2 T ≤ T ∗ |T 2 |T. If
18
P. Aiena, E. Aponte and E. Balzan
IEOT
T ∈ L(H) is algebraically quasi-class operator (i.e. there exists a nonconstant polynomial h such that h(T ) is a quasi-class A operator), then T is polaroid ([13, Lemma 2.3]) and since h(T ) has SVEP ([26, Lemma 1.5], also T has SVEP. Consequently, by Theorem 3.14 f (T ) satisfies (W ), or equivalently (gW ). If T ∗ is an algebraically quasi-class operator then T ∗ , or equivalently T , is polaroid and the SVEP for T ∗ entails, by Theorem 3.12 and Remark 3.13, that all Weyl’s theorems (generalized or not) hold (and are equivalent!) for f (T ). Therefore Theorem 3.12 subsumes and extends [13, Theorem 2.4 and Theorem 3.3]. (d) Every multiplier T of a semi-simple commutative Banach algebra A, (see [1, Chapter 4] for definitions and details) is H(1), see [12], in particular every convolution Tµ operator of L1 (G), L1 (G) the group algebra of a locally compact Abelian group G is H(1). Therefore, T is polaroid and has SVEP so that (W ), or equivalently (gW ) holds for f (T ). If A is regular and Tauberian, this is the case for instance of the group algebra L1 (G), G a compact Abelian group, then σ(T ) = σa (T ) for every multiplier T , see Corollary 5.88 of [1]. Therefore, if A is regular and Tauberian every multiplier T is a-polaroid. In particular, every convolution operator Tµ on L1 (G) whenever G is compact, is a-polaroid. Consequently, by Theorem 3.10, (W ), (aW ), (w), (gW ), (gaW ) and (gw) for f (T ) are equivalent and f (T ) satisfies all of them. Another example of a-polaroid operator is given by a multiplier T of a Banach algebra A with an orthogonal basis. In fact also in this case σ(T ) = σa (T ), see Theorem 4.46 of [1], so that Theorem 3.10 applies to T .
References [1] P. Aiena Fredholm and local spectral theory, with application to multipliers. Kluwer Acad. Publishers (2004). [2] P. Aiena. Classes of Operators Satisfying a-Weyl’s theorem. Studia Math. 169 (2005), 105-122. [3] P. Aiena Quasi Fredholm operators and localized SVEP, Acta Sci. Math. (Szeged), 73 (2007), 251-263. [4] P. Aiena, M. T. Biondi, F. Villaf˜ ane Property (w) and perturbations III J. Math. Anal. Appl. 353 (2009), 205-214. [5] P. Aiena, M. T. Biondi, C. Carpintero On Drazin invertibility, Proc. Amer. Math. Soc. 136, (2008), 2839-2848. [6] P. Aiena, C. Carpintero, E. Rosas. Some characterization of operators satisfying a-Browder theorem. J. Math. Anal. Appl. 311, (2005), 530-544. [7] P. Aiena, J. R. Guillen Weyl’s theorem for perturbations of paranormal operators. Proc. Amer. Math. Soc.35, (2007), 2433-2442. [8] P. Aiena, J. Guillen, P. Pe˜ na Property (w) for perturbation of polaroid operators. Linear Alg. and Appl. 424 (2008), 1791-1802. [9] P. Aiena, T. L. Miller On generalized a-Browder’s theorem. Studia Math. 180, 3, (2007), 285300.
Vol. 66 (2010)
Weyl Type Theorems for Left and Right Polaroid Operators
19
[10] P. Aiena, P. Pe˜ na A variation on Weyl’s theorem. J. Math. Anal. Appl. 324 (2006), 566-579. [11] P. Aiena, J. E. Sanabria On left and right poles of the resolvent. Acta Sci. Math. 74 (2008),669-687. [12] P. Aiena, F. Villaf˜ ane Weyl’s theorem for some classes of operators. Int. Equa. Oper. Theory 53, (2005), 453-466. [13] J. An, Y. M. Han Weyl’s theorem for algebraically Quasi-class A operators. Int. Equa. Oper. Theory 62, (2008), 1-10. [14] M. Amouch, H. Zguitti On the equivalence of Browder’s and generalized Browder’s theorem. Glasgow Math. Jour. 48, (2006), 179-185. [15] M. Berkani, M. Amouch On the property (gw). Mediterr. J. Math. 5 (2008), no. 3, 371–378. [16] M. Berkani On a class of quasi-Fredholm operators. Int. Equa. Oper. Theory 34 (1), (1999), 244-249. [17] M. Berkani, M. Sarih On semi B-Fredholm operators, Glasgow Math. J. 43 (2001), 457-465. [18] M. Berkani Index of B-Fredholm operators and generalization of a Weyl’s theorem, Proc. Amer. Math. Soc., vol. 130, 6, (2001), 1717-1723. [19] M. Berkani, J. J. Koliha Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 359–376. [20] X. Cao, M. Guo, B. Meng Weyl type theorems for p-hyponormal and M hyponormal operators, Studia Math. 163 (2) (2004), 177-187. [21] X. Cao Weyl’s type theorem for analytically hyponormal operators, Linear Alg. and Appl. 405 (2005), 229-238. [22] X. Cao Topological uniform descent and Weyl’s type theorem, Linear Alg. and Appl. 420 (2007), 175-182. [23] R. E. Curto, Y. M. Han Generalized Browder’s and Weyl’s theorems for Banach spaces operators, J. Math. Anal. Appl. 2 (2007), 1424-1442. [24] R. E. Curto, Y. M. Han Weyl’s theorem for algebraically paranormal operators, Integ. Equa. Oper. Theory 50, (2004), No.2, 169-196. [25] M. P. Drazin Pseudoinverse in associative rings and semigroups. Amer. Math. Monthly 65, (1958), 506-514. [26] B.P. Duggal, I. H. Jeon, I. H. Kim On Weyl’s theorem for quasi-class A operators J. Korean Math. Soc. 43, (2006), 899-909. [27] S. Grabiner Uniform ascent and descent of bounded operators J. Math. Soc. Japan 34 (1982), 317-337. [28] H. Heuser Functional Analysis, Marcel Dekker, New York 1982. [29] J. J. Koliha Isolated spectral points Proc. Amer. Math. Soc. 124 (1996), 34173424. [30] J.P. Labrousse Les op´erateurs quasi-Fredholm., Rend. Circ. Mat. Palermo, XXIX 2, (1980) [31] D. C. Lay Spectral analysis using ascent, descent, nullity and defect. Math. Ann. 184 (1970), 197-214. [32] K. B. Laursen, M. M. Neumann Introduction to local spectral theory., Clarendon Press, Oxford 2000.
20
P. Aiena, E. Aponte and E. Balzan
IEOT
[33] M. Mbekhta, V. M¨ uller On the axiomatic theory of the spectrum II. Studia Math. 119 (1996), 129-147. [34] M. Oudghiri Weyl’s and Browder’s theorem for operators satisfying the SVEP Studia Math. 163, 1, (2004), 85-101. [35] H. Zguitti A note on generalized Weyl’s theorem, J. Math. Anal. Appl. 316 (1) (2006), 373-381. Pietro Aiena Dipartimento di Metodi e Modelli Matematici Facolt` a di Ingegneria Universit` a di Palermo 90128 Palermo Italy e-mail:
[email protected] Elvis Aponte Departamento de Matem´ aticas Facult´ ad de Ciencias UCLA Barquisimeto Venezuela e-mail:
[email protected] Edixon Balzan Departamento de Matem´ aticas Facult´ ad de Ciencias Universidad del Zulia Maracaibo Venezuela e-mail:
[email protected] Submitted: May 20, 2009.
Integr. Equ. Oper. Theory 66 (2010), 21–40 DOI 10.1007/s00020-009-1736-4 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Weighted Composition Operators from H ∞ to the Bloch Space of a Bounded Homogeneous Domain Robert F. Allen and Flavia Colonna Abstract. Let D be a bounded homogeneous domain in Cn . In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space H ∞ (D) into the Bloch space of D. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if D is a bounded symmetric domain, the bounded multiplication operators from H ∞ (D) to the Bloch space of D are the operators whose symbol is bounded. Mathematics Subject Classification (2010). Primary 47B38; Secondary 32A18, 30D45. Keywords. Weighted composition operators, Bloch space, homogeneous domains.
1. Introduction Let X be a Banach space of holomorphic functions on a bounded domain D in Cn . For ψ a complex-valued holomorphic function on D and ϕ a holomorphic self-map of D, we define a linear operator Wψ,ϕ on X, called the weighted composition operator with multiplicative symbol ψ and composition symbol ϕ, by Wψ,ϕ f = ψ(f ◦ ϕ), f ∈ X. Setting Mψ f = ψf and Cϕ f = f ◦ ϕ, we may write Wψ,ϕ = Mψ Cϕ . Then Mψ is called multiplication operator with symbol ψ and Cϕ is called composition operator with symbol ϕ. The study of the weighted composition operators on the Bloch space began with the work of Ohno and Zhao in [19] where the operators from the Bloch space of the open unit disk D into itself were considered. In higher dimensions, these operators on the Bloch space have been studied by Chen,
22
R. F. Allen and F. Colonna
IEOT
Stevi´c and Zhou in [6] (which was motivated by papers and [21] and [7]), and by the authors in [4]. For related work, see [27]. In [18], Ohno investigated the weighted composition operators between the Bloch space of D and the Hardy space H ∞ (D) of bounded analytic functions on D. Characterizations of the boundedness and the compactness of the weighted composition operators from the Bloch space to H ∞ were given by Hosokawa, Izuchi and Ohno in [14] in the one-dimensional case, and by Li and Stevi´c in the case of the unit ball [17]. In [22], Stevi´c determined the norm of the bounded weighted composition operators from the Bloch space and the little Bloch space to the weighted Hardy space Hµ∞ of the unit ball (where µ is a weight). The weighted composition operators from a larger class of spaces known as the α-Bloch spaces to H ∞ on the polydisk where studied by Li and Stevi´c in [16]. In [4], we characterized the boundedness, determined the operator norm, and gave a sufficient condition for compactness in the case of a general bounded homogeneous domain in Cn . The study of the weighted composition operators from H ∞ to the Bloch space in several variables was carried out by Li and Stevi´c when the ambient space is the unit polydisk [15]. For the case of the unit ball, the study of these operators from H ∞ into the α-Bloch spaces was carried out by Li and Stevi´c in [17] and Zhang and Chen in [26]. In this paper, we analyze the weighted composition operators from H ∞ into the Bloch space on a bounded homogeneous domain in Cn . In Section 2, we give the background on the bounded homogeneous domains in Cn and special class of such domains that have a canonical representation (up to biholomorphic transformation) due to Cartan [5], the bounded symmetric domains. We then review the notion of the Bloch space of a bounded homogeneous domain [12], [23], and of a subspace we refer to as the ∗-little Bloch space. We also recall the definition of little Bloch space on a bounded symmetric domain [24]. In Section 3, in the environment of a general bounded homogeneous domain, we characterize the bounded weighted composition operators from H ∞ into the Bloch space and into the ∗-little Bloch space, thereby extending the results of Ohno [18] in the one-dimensional case, of Li and Stevi´c for the polydisk [15] and for the unit ball [17], and of Zhang and Chen for the unit ball [26]. We also give estimates on the operator norm. In Section 4, we describe sufficient conditions for the compactness of a weighted composition operator from H ∞ to either the Bloch space or the ∗-little Bloch space of a bounded homogeneous domain. We conjecture these conditions to be necessary, and prove the necessity when the domain is the unit polydisk and the unit ball. In the latter setting, we obtain a result equivalent to a special case of Theorem 2 in [26]. Furthermore, we show that compactness and boundedness are equivalent when the operator maps H ∞ (Bn ) into B0 (Bn ), a result not observed in [26].
Vol. 66 (2010)
Weighted Composition Operators
23
In Section 5, we show that the bounded multiplication operators from H ∞ into the Bloch space (respectively, the ∗-little Bloch space) of a bounded symmetric domain are precisely those whose symbol is bounded (respectively, in the ∗-little Bloch space and bounded). Furthermore, we obtain operator norm estimates in terms of the Bergman constant of the domain. We then discuss the boundedness of the composition operators on a bounded homogeneous domain and establish operator norm estimates. While composition operators from H ∞ to the α-Bloch spaces of the polydisk have been studied by Stevi´c in [21], to our knowledge, there are no results in the literature for the multiplication or the composition operators between H ∞ and the Bloch spaces on general bounded homogeneous or symmetric domains. Finally, in Section 6, we show there are no isometries amongst the multiplication or composition operators from the Hardy space H ∞ into the Bloch space when the domain is the unit disk. We also show that there are no isometric weighted composition operators from the Bloch space into H ∞ if the ambient space is the unit polydisk. We conjecture that, likewise, there are no isometric weighted composition operators from H ∞ to the Bloch space.
2. Preliminaries Let D be a domain in Cn . We denote by H(D) the set of holomorphic functions from D into C, and by Aut(D) the set of biholomorphic maps of D. The space H ∞ (D) of bounded holomorphic functions on D is a Banach algebra equipped with norm ||f ||∞ = supz∈D |f (z)|. A domain D is homogeneous if Aut(D) acts transitively on D. Every homogeneous domain is equipped with a canonical metric invariant under the action of Aut(D), called the Bergman metric [13]. A domain D is symmetric at z0 ∈ D if there exists an involution ϕ ∈ Aut(D) for which z0 is an isolated fixed point. A domain that is symmetric at every point is called symmetric. A bounded symmetric domain is homogeneous and a bounded homogeneous domain that is symmetric at one point is symmetric [13]. The unit ball Bn = {z = (z1 , . . . , zn ) ∈ Cn : ||z|| = 2
n
|zk |2 < 1}
k=1
and the unit polydisk Dn = {(z1 , . . . , zn ) ∈ Cn : |zk | < 1, k = 1, . . . , n} are bounded symmetric domains since they are homogeneous and symmetric at the origin via the map z → −z. While bounded homogeneous domains in dimensions 2 and 3 are symmetric, there are examples of bounded homogeneous domains in dimensions greater than 3 which are not symmetric [20]. In [5], Cartan showed that every bounded symmetric domain in Cn is biholomorphically equivalent to a finite product of irreducible bounded symmetric domains, unique up to order. He classified the irreducible bounded
24
R. F. Allen and F. Colonna
IEOT
symmetric domains into six classes, four of which are known as the Cartan classical domains, and the other two, each consisting of a single domain, are known as the exceptional domains. A bounded symmetric domain written as such a product is said to be in standard form. The Cartan classical domains are defined as: RI = {Z ∈ Mm,n (C) : Im − ZZ ∗ > 0}, for m ≥ n ≥ 1, RII = {Z ∈ Mn (C) : Z = Z T , In − ZZ ∗ > 0}, for n ≥ 2, RIII = {Z ∈ Mn (C) : Z = −Z T , In − ZZ ∗ > 0}, for n ≥ 5, RIV = {z = (z1 , . . . , zn ) ∈ Cn : A > 0, ||z|| < 1} , for n ≥ 5, where Mm,n (C) denotes the set of m × n matrices with complex entries, Mn (C) = Mn,n (C), Z T and Z ∗ are the transpose and the adjoint of Z, 2 respectively, and A = | zk2 |2 + 1 − 2 ||z|| . See [11] for a description of the exceptional domains. The notion of Bloch function in higher dimensions was first introduced by Hahn in [12]. In [23] and [24], Timoney studied in depth the Bloch functions on a bounded homogeneous domain. In this paper, we conform to his definition and notation, as follows. Let D be a bounded homogeneous domain. For z ∈ D and f ∈ H(D), define |∇(f )(z)u| , Qf (z) = sup 1/2 n H z (u, u) u∈C \{0} where ∇(f )(z) is the gradient of f at z, for u = (u1 , . . . , un ), ∇(f )(z)u =
n ∂f (z)uk , ∂zk
k=1
and Hz is the Bergman metric on D at z. The Bergman metric for the unit ball Bn is defined as n + 1 (1 − ||z||2 ) u, v + u, z z, v
, · 2 2 (1 − ||z|| )2 where u, v ∈ Cn , z ∈ Bn , and u, v = uj vj . For the unit polydisk Dn , the Bergman metric is defined as Hz (u, v) =
Hz (u, v) =
n
uj vj
j=1
(1 − |zj | )2
2
,
where u, v ∈ Cn and z ∈ Dn . The Bloch space B(D) on a bounded homogeneous domain D is the set of all functions f ∈ H(D) for which βf = sup Qf (z) < ∞. z∈D
Timoney proved that B(D) is a Banach space under the norm ||f ||B = |f (z0 )| + βf ,
Vol. 66 (2010)
Weighted Composition Operators
25
where z0 is some fixed point in D [23]. For convenience, we shall assume throughout that 0 ∈ D and choose z0 = 0. The ∗-little Bloch space of D is the subspace of B(D) defined as B0∗ (D) = f ∈ B(D) : lim∗ Qf (z) = 0 , z→∂ D
∗
where ∂ D is the distinguished boundary of D. Timoney defined the little Bloch space B0 (D) of a bounded symmetric domain D to be the closure of the polynomials in B(D). Its elements f satisfy the condition lim Qf (z) = 0.
z→∂ ∗ D
If D is the unit ball, then ∂D = ∂ ∗ D and thus B0 (D) = B0∗ (D). When D = Bn , B0 (D) is a proper subspace of B0∗ (D), and B0∗ (D) is a non-separable subspace of B(D) [24]. In [23], Timoney proved that the space H ∞ (D) of bounded holomorphic functions on a bounded homogeneous domain D is a subspace of B(D) and for each f ∈ H ∞ (D), ||f ||B ≤ |f (0)|+c ||f ||∞ where c is a constant depending only on the domain D. In [8], Cohen and the second author defined the Bloch constant of a bounded homogeneous domain D in Cn as cD = sup{Qf (z) : f ∈ H ∞ (D), ||f ||∞ ≤ 1, z ∈ D}. The precise value of the Bloch constant was calculated for each Cartan classical domain to be the reciprocal of the inner radius of the domain with respect to the Bergman metric, that is, 1 . cD = inf ξ∈∂D H0 (ξ, ξ)1/2 In [25], Zhang determined the Bloch constant for the two exceptional domains RV of dimension 16 and RV I of dimension 27. The value of cD for each irreducible bounded symmetric domain D is shown in Theorem 5.2. By Theorem 3 of [8], extended to include the exceptional domains, if D = D1 × · · · × Dk is a bounded symmetric domain in standard form, then cD = max cDj .
(1)
1≤j≤k
Furthermore, it was shown that there exist polynomial functions f (hence in the little Bloch space of D) such that ||f ||∞ ≤ 1 and cD = Qf (0). Let D be a bounded homogeneous domain. In [2], we showed that a function f ∈ H(D) is Bloch if and only if there exists c > 0 such that |f (z) − f (w)| ≤ cρ(z, w), for each z, w ∈ D, where ρ is the Bergman distance on D. Furthermore, the Bloch seminorm of f is precisely the infimum of all such constants c. As a consequence, we obtain that for a Bloch function f on D, |f (z) − f (w)| ≤ ||f ||B ρ(z, w), for each z, w ∈ D.
(2)
26
R. F. Allen and F. Colonna
IEOT
3. Boundedness From now on, unless specified otherwise, we shall assume D is a bounded homogeneous domain, ψ ∈ H(D), and ϕ = (ϕ1 , . . . , ϕn ) is a holomorphic self-map of D. Define θψ,ϕ = supz∈D |ψ(z)|θϕ (z), where θϕ (z) = sup{Qf ◦ϕ (z) : f ∈ H ∞ (D), ||f ||∞ ≤ 1}. For z ∈ D, denote by Jϕ(z) the Jacobian matrix of ϕ at z, that is, the ∂ϕ matrix whose (j, k)-entry is ∂zkj (z). Furthermore, define Bϕ (z) =
sup u∈Cn \{0}
Hϕ(z) (Jϕ(z)u, Jϕ(z)u)1/2 . Hz (u, u)1/2
The quantity Bϕ = supz∈D Bϕ (z) is bounded above by a constant dependent only on D [23]. By the invariance of the Bergman metric under the action of Aut(D), if ϕ ∈ Aut(D), then for each z ∈ D, u ∈ Cn , Hϕ(z) (Jϕ(z)u, Jϕ(z)u) = Hz (u, u). For f ∈ B(D) and z ∈ D, Qf ◦ϕ (z) ≤ Bϕ (z)Qf (ϕ(z)). Taking the supremum over all functions f ∈ H ∞ (D) with ||f ||∞ ≤ 1, we obtain (3) θϕ (z) ≤ cD Bϕ (z) for each z ∈ D. Theorem 3.1. (a) The weighted composition operator Wψ,ϕ : H ∞ (D) → B(D) is bounded if and only if ψ ∈ B(D) and θψ,ϕ is finite. (b) Wψ,ϕ : H ∞ (D) → B0∗ (D) is bounded if and only if ψ ∈ B0∗ (D), θψ,ϕ is finite, and lim |ψ(z)|θϕ (z) = 0.
z→∂ ∗ D
(4)
Furthermore, if Wψ,ϕ is bounded as an operator into B(D) or B0∗ (D), then the norm of Wψ,ϕ satisfies the estimates ||ψ||B ≤ ||Wψ,ϕ || ≤ ||ψ||B + θψ,ϕ .
(5)
Proof. (a) Assume first Wψ,ϕ is bounded. Then ||ψ||B = ||Wψ,ϕ 1||B , thus ψ ∈ B(D) and the lower estimate of (5) holds. For each f ∈ H ∞ (D) with ||f ||∞ ≤ 1, and each z ∈ D, we have ∇(ψ(f ◦ ϕ))(z) = ψ(z)∇(f ◦ ϕ)(z) + f (ϕ(z))∇(ψ)(z)
(6)
so that |ψ(z)| Qf ◦ϕ (z) ≤ ||Wψ,ϕ f ||B + Qψ (z) ≤ ||Wψ,ϕ || + ||ψ||B . Thus, taking the supremum over all such functions f and over all z ∈ D, we see that θψ,ϕ is finite.
Vol. 66 (2010)
Weighted Composition Operators
27
Conversely, assume ψ ∈ B(D) and θψ,ϕ is finite. Then for f ∈ H ∞ (D), with f ∞ ≤ 1, we have Qψ(f ◦ϕ) (z) ≤ |ψ(z)|Qf ◦ϕ (z) + |f (ϕ(z))|Qψ (z) ≤ θψ,ϕ + βψ . Thus, Wψ,ϕ f ∈ B(D) and ||ψ(f ◦ ϕ)||B ≤ |ψ(0)f (ϕ(0))| + θψ,ϕ + βψ ≤ ||ψ||B + θψ,ϕ , proving the boundedness of Wψ,ϕ and the upper estimate of (5). (b) From part (a), it suffices to show that if Wψ,ϕ is bounded, then (4) holds and, conversely, if ψ ∈ B0∗ (D), θψ,ϕ is finite, and (4) holds, then for each f ∈ B0∗ (D), ψ(f ◦ ϕ) ∈ B0∗ (D). Assume Wψ,ϕ is bounded. Then ψ = Wψ,ϕ 1 ∈ B0∗ (D) and, for each f ∈ H ∞ (D) and each z ∈ D, from (6) we have |ψ(z)|Qf ◦ϕ (z) ≤ Qψ(f ◦ϕ) (z) + Qψ (z) ||f ||∞ . Taking the limit as z → ∂ ∗ D, we obtain |ψ(z)|Qf ◦ϕ (z) → 0. Since this holds for each f ∈ H ∞ (D), we deduce (4). Next assume ψ ∈ B0∗ (D), θψ,ϕ is finite, and (4) holds. Then for each nonzero function f ∈ H ∞ (D) and each z ∈ D, using the function f˜ = f / ||f ||∞ , we see that Qf ◦ϕ (z) ≤ θϕ (z) ||f ||∞ . Thus, for each f ∈ H ∞ (D) and each z ∈ D, we obtain Qψ(f ◦ϕ) (z) ≤
Qψ (z) f ||∞ + |ψ(z)|Qf ◦ϕ (z)
≤
(Qψ (z) + |ψ(z)|θϕ (z)) ||f ||∞ .
Taking the limit as z → ∂ ∗ D, we conclude that ψ(f ◦ ϕ) ∈ B0∗ (D).
In the special case of D = Bn , the finiteness of θψ,ϕ follows from (4) since ∂ ∗ Bn = ∂Bn . From this, we obtain the following characterization of the bounded weighted composition operators from H ∞ (Bn ) into B0 (Bn ). Corollary 3.2. Wψ,ϕ : H ∞ (Bn ) → B0 (Bn ) is bounded if and only if ψ ∈ B0 (Bn ) and lim |ψ(z)| θϕ (z) = 0.
||z||→1
In the case of D = D, we can improve the lower bound on the operator norm of Wψ,ϕ . Lemma 3.3. Let ψ ∈ H(D) and ϕ a holomorphic self-map of D. Then θψ,ϕ = sup |ψ(z)| z∈D
(1 − |z| ) |ϕ (z)| 2
1 − |ϕ(z)|2
.
(7)
28
R. F. Allen and F. Colonna
IEOT
Proof. Let f ∈ H ∞ (D) such that ||f ||∞ ≤ 1. Since βf ≤ ||f ||∞ , for all z ∈ D, Qf ◦ϕ (z) = (1 − |z|2 ) |f (ϕ(z))| |ϕ (z)| = (1 − |ϕ(z)| ) |f (ϕ(z))| 2
≤ βf ≤
(1 − |z| ) |ϕ (z)| 2
2
1 − |ϕ(z)|
(1 − |z| ) |ϕ (z)| 2
1 − |ϕ(z)|2
(1 − |z| ) |ϕ (z)| 2
2
1 − |ϕ(z)|
.
Thus θϕ (z) ≤
(1 − |z| ) |ϕ (z)| 2
2
1 − |ϕ(z)|
.
On the other hand, fixing z ∈ D, the function f (w) =
ϕ(z) − w 1 − ϕ(z)w
, w ∈ D,
is an automorphism of D, and thus ||f ||∞ = 1. Moreover, Qf ◦ϕ (z) = and so θϕ (z) =
(1−|z|2 )|ϕ (z)| , 1−|ϕ(z)|2
(1 − |z|2 ) |ϕ (z)| 2
1 − |ϕ(z)|
(8)
which yields (7).
Theorem 3.4. Let Wψ,ϕ be bounded from H ∞ (D) to B(D) or B0 (D). Then max{||ψ||B , θψ,ϕ } ≤ ||Wψ,ϕ || ≤ ||ψ||B + θψ,ϕ . Proof. By Theorem 3.1, It suffices to show that θψ,ϕ ≤ ||Wψ,ϕ ||. Fix λ ∈ D and for z ∈ D, let ϕ(λ) − z . fλ (z) = 1 − ϕ(λ)z Then ||fλ ||∞ = 1 and fλ (ϕ(λ)) = 0. From (8), we obtain ||Wψ,ϕ || ≥ ||Wψ,ϕ fλ ||B ≥ sup(1 − |z| ) |ψ (z)fλ (ϕ(z)) + ψ(z)(fλ ◦ ϕ) (z)| 2
z∈D
≥ |ψ(λ)|Qfλ ◦ϕ (λ) (1 − |λ| ) |ψ(λ)| |ϕ (λ)| 2
=
1 − |ϕ(λ)|
2
.
Taking the supremum over all λ ∈ D, we get ||Wψ,ϕ || ≥ θψ,ϕ .
Vol. 66 (2010)
Weighted Composition Operators
29
4. Compactness The following lemma will be used to prove a sufficient condition for the compactness of Wψ,ϕ . Lemma 4.1. Wψ,ϕ is compact from H ∞ (D) into B(D) if and only if for every bounded sequence {fk } in H ∞ (D) converging to 0 locally uniformly in D, ||ψ(fk ◦ ϕ)||B → 0 as k → ∞. Proof. Assume that Wψ,ϕ : H ∞ (D) → B(D) is compact. Let {fk } be a bounded sequence in H ∞ (D) which converges to 0 locally uniformly in D. By rescaling fk , we may assume ||fk ||∞ ≤ 1 for all k ∈ N. We need to show that ||ψ(fk ◦ ϕ)||B → 0 as k → ∞. Since Wψ,ϕ is compact, the sequence {ψ(fk ◦ϕ)} has a subsequence (which for convenience we reindex as the original sequence) converging in the Bloch norm to some function f ∈ B(D). We are going to show that f is identically 0 by proving that ψ(fk ◦ ϕ) → 0 locally uniformly. Fix z0 ∈ D and, subtracting from the elements of this sequence the value of f at z0 , we may assume f (z0 ) = 0. Then ψ(z0 )fk (ϕ(z0 )) → 0 as k → ∞. For z ∈ D, using (2), we find |ψ(z)fk (ϕ(z)) − f (z)| ≤ |ψ(z)fk (ϕ(z)) − f (z) − (ψ(z0 )fk (ϕ(z0 )) − f (z0 ))| + |ψ(z0 )fk (ϕ(z0 ))| ≤ ||ψ(fk ◦ ϕ) − f ||B ρ(z, z0 ) + |ψ(z0 )fk (ϕ(z0 ))| . The right-hand side converges to 0 locally uniformly as k → ∞ since ψ(fk ◦ ϕ) − f → 0 in B(D). On the other hand, ψ(fk ◦ ϕ) → 0 locally uniformly, so f is identically 0. Conversely, suppose that whenever {gk } is a bounded sequence in H ∞ (D) converging to 0 locally uniformly in D, ||ψ(gk ◦ ϕ)||B → 0 as k → ∞. To prove the compactness of Wψ,ϕ , it suffices to show that if {fk } is a sequence in H ∞ (D) with ||fk ||∞ ≤ 1 for all k ∈ N, there exists a subsequence {fkj } such that ψ(fkj ◦ ϕ) converges in norm in B(D). Fix z0 ∈ D. Replacing fk by fk − fk (z0 ), we may assume that fk (z0 ) = 0 for all k ∈ N. Since {fk } is uniformly bounded on D, by Montel’s theorem some subsequence {fkj } converges locally uniformly to some holomorphic function f on D such that ||f ||∞ ≤ 1. Then, letting gkj = fkj − f , we obtain a bounded ∞ sequence converging to 0 locally uniformly in D. By the hypoth in H (D) esis, ψ(gkj ◦ ϕ)B → 0 as j → ∞. Therefore, ψ(fkj ◦ ϕ) converges in norm to ψ(f ◦ ϕ), completing the proof. Theorem 4.2. (a) Let ψ ∈ B(D). Then Wψ,ϕ : H ∞ (D) → B(D) is compact if lim
ϕ(z)→∂D
Qψ (z) = 0, and
lim
ϕ(z)→∂D
|ψ(z)| θϕ (z) = 0.
(9)
(b) Let ψ ∈ B0∗ (D). Then Wψ,ϕ : H ∞ (D) → B0∗ (D) is compact if lim Qψ (z) = 0, and
z→∂D
lim |ψ(z)| θϕ (z) = 0.
z→∂D
(10)
30
R. F. Allen and F. Colonna
IEOT
Proof. Assume conditions (9) both hold. By Lemma 4.1, to prove that Wψ,ϕ is compact from H ∞ (D) into B(D) it suffices to show that for any bounded sequence {fk } such that ||fk ||∞ ≤ 1 and fk → 0 locally uniformly in D, ||ψ(fk ◦ ϕ)||B → 0 as k → ∞. Let {fk } be such a sequence and fix ε > 0. Then, there exists r > 0 such that for all k ∈ N, |ψ(z)| Qfk ◦ϕ (z) < 2ε and Qψ (z) < 2ε whenever ρ(ϕ(z), ∂D) ≤ r. Thus, if ρ(ϕ(z), ∂D) ≤ r, then Qψ(fk ◦ϕ) (z) ≤ |ψ(z)| Qfk ◦ϕ (z) + Qψ (z) < ε. On the other hand, since fk → 0 locally uniformly in D, |fk (ϕ(z))| → 0 uniformly on the compact set Er = {z ∈ D : ρ(ϕ(z), ∂D) ≥ r}. Thus, ∇fk (ϕ(z)) approaches the zero vector, and hence Qfk ◦ϕ (z) → 0, uniformly on Er as k → ∞. Consequently, recalling that ψ ∈ B(D), we see that for all k sufficiently large, Qψ(fk ◦ϕ) (z) < ε for all z ∈ D. Furthermore, |ψ(0)fk (ϕ(0))| → 0 as k → ∞, so ||ψ(fk ◦ ϕ)||B → 0, completing the proof of (a). The proof of part (b) is analogous. Remark 1. If D is not the unit ball, the multiplicative symbol of the weighted composition operators satisfying conditions (10) reduces to a constant, and hence the compact weighted composition operators of this type have composition component which is compact. The case when D = Bn is discussed in Corollary 4.3. We conjecture that under boundedness assumptions, conditions (9) and (10) are necessary as well. Conjecture. If D is a bounded homogeneous domain, then the bounded operator Wψ,ϕ : H ∞ (D) → B(D) is compact if and only if lim
ϕ(z)→∂D
Qψ (z) = 0, and
lim
ϕ(z)→∂D
|ψ(z)| θϕ (z) = 0.
4.1. Compactness from H ∞ (Bn ) to B0 (Bn ) or B(Bn ) We begin this section by extending Theorem 4 in [18] to the unit ball. Corollary 4.3. Let ϕ be a holomorphic self-map of Bn and ψ ∈ H(Bn ). Then the following are equivalent: (a) Wψ,ϕ : H ∞ (Bn ) → B0 (Bn ) is bounded. (b) Wψ,ϕ : H ∞ (Bn ) → B0 (Bn ) is compact. (c) ψ ∈ B0 (Bn ) and lim |ψ(z)| θϕ (z) = 0. ||z||→1
Proof. The implication (b) =⇒ (a) is obvious. The implication (a) =⇒ (c) follows from Corollary 3.2. Finally, (c) =⇒ (b) follows from part (b) of Theorem 4.2. We now prove the above conjecture in the case of the unit ball. The following result is equivalent to a characterization of the compactness obtained by Zhang and Chen ([26], Theorem 2).
Vol. 66 (2010)
Weighted Composition Operators
31
Theorem 4.4. Let ϕ be a holomorphic self-map of Bn , and ψ ∈ H(Bn ). Then Wψ,ϕ : H ∞ (Bn ) → B(Bn ) is compact if and only if it is bounded, lim
Qψ (z) = 0, and
(11)
lim
|ψ(z)| θϕ (z) = 0.
(12)
||ϕ(z)||→1 ||ϕ(z)||→1
Proof. If Wψ,ϕ is bounded and conditions (11) and (12) hold, then ψ ∈ B(Bn ), so by Theorem 4.2, Wψ,ϕ is compact. Conversely, suppose Wψ,ϕ is compact. Then Wψ,ϕ is bounded and by Theorem 2 in [26], condition (11) holds, and lim
||ϕ(z)||→1
|ψ(z)| Bϕ (z) = 0.
Condition (12) now follows from the inequality θϕ (z) ≤ Bϕ (z) for each z ∈ Bn . 4.2. Compactness from H ∞ (Dn ) to B(Dn ) In [15], Li and Stevi´c characterized the compact weighted composition operators from H ∞ (Dn ) into B(Dn ) in the following result. Theorem 4.5 ([15], Theorem 1.2). Let ϕ = (ϕ1 , . . . , ϕn ) be a holomorphic selfmap of Dn and ψ(z) a holomorphic function on Dn . Then Wψ,ϕ : H ∞ (Dn ) → B(Dn ) is compact if and only if the following conditions are satisfied: (a) Wψ,ϕ : H ∞ (Dn ) → B(Dn ) is bounded; n 2 ∂ψ (b) lim (1 − |zk | ) (z) = 0; ∂zk ϕ(z)→∂Dn k=1 n ∂ϕj 1 − |zk |2 (c) lim n |ψ(z)| (z) ∂zk 1 − |ϕ (z)|2 = 0. ϕ(z)→∂D j k,j=1 We will prove the conjecture posed in the previous section in the setting of the polydisk Dn . To do this, we need the following lemma. Lemma 4.6. Let ϕ = (ϕ1 , . . . , ϕn ) be a holomorphic self-map of Dn . For z ∈ Dn , n ∂ϕj 1 − |zk |2 |ψ(z)| θϕ (z) ≤ |ψ(z)| ∂zk (z) 1 − |ϕ (z)|2 . j j,k=1
32
R. F. Allen and F. Colonna
IEOT
Proof. Observe that by (1.2) of [9], for all z ∈ Dn , |ψ(z)| θϕ (z) ≤ |ψ(z)| Bϕ (z)
⎛ 2 ⎞1/2 n n 2 ∂ϕ | )w (1 − |z j k k ⎠ = |ψ(z)| max ⎝ (z) 2 ||w||=1 ∂zk 1 − |ϕj (z)| j=1 k=1
⎛
n 2 ⎞1/2 n ∂ϕj (1 − |zk |2 ) |wk | ⎠ ≤ |ψ(z)| max ⎝ ∂zk (z) 1 − |ϕ (z)|2 ||w||=1 j=1
j
k=1
n ∂ϕj 1 − |zk |2 ≤ |ψ(z)| ∂zk (z) 1 − |ϕ (z)|2 . j
k,j=1
Theorem 4.7. Let ϕ be a holomorphic self-map of Dn and ψ ∈ B(Dn ). Then Wψ,ϕ : H ∞ (Dn ) → B(Dn ) is compact if and only if lim
ϕ(z)→∂Dn
Qψ (z) = 0 and
lim
ϕ(z)→∂Dn
|ψ(z)| θϕ (z) = 0.
(13)
Proof. By Theorem 4.2, it suffices to show that if Wψ,ϕ is compact from H ∞ (Dn ) into B(Dn ), then conditions (13) hold. First, observe that by Theorem 3.3 of [9], for z ∈ Dn ,
∂ψ ∂ψ (z), . . . , (1 − |zn |2 ) (z) Qψ (z) = (1 − |z1 |2 ) ∂z1 ∂zn n 2 ∂ψ ≤ (1 − |zk | ) (z) . ∂zk k=1
Since Wψ,ϕ is compact, Theorem 4.5(b) implies that lim
ϕ(z)→∂Dn
Qψ (z) = 0.
In addition, by Lemma 4.6, we have n ∂ϕj 1 − |zk |2 (z) |ψ(z)| θϕ (z) ≤ |ψ(z)| ∂zk 1 − |ϕ (z)|2 j k,j=1
for all z ∈ Dn . Thus, by part (c) of Theorem 4.5, we obtain lim
ϕ(z)→∂Dn
completing the proof.
|ψ(z)| θϕ (z) = 0,
5. Component Operators In this section, we look at the issues of boundedness and compactness of the multiplication and the composition operators separately.
Vol. 66 (2010)
Weighted Composition Operators
33
5.1. Multiplication Operators from H ∞ into the Bloch Space Let us now consider the implications of Theorem 3.1 for the case when ϕ is the identity and D is a bounded symmetric domain. Theorem 5.1. Let D be a bounded symmetric domain in standard form and let ψ ∈ H(D). Then (a) Mψ : H ∞ (D) → B(D) is bounded if and only if ψ ∈ H ∞ (D). (b) Mψ : H ∞ (D) → B0∗ (D) is bounded if and only if ψ ∈ B0∗ (D)∩H ∞ (D). Furthermore, if Mψ is bounded as an operator from H ∞ (D) into the Bloch space or the ∗-little Bloch space, then max{||ψ||B , cD ||ψ||∞ } ≤ ||Mψ || ≤ ||ψ||B + cD ||ψ||∞ , where cD is the Bloch constant of D. Proof. Suppose first D is an irreducible domain. In [8] and [25], it was shown that there exists a polynomial p on D and ξ ∈ ∂D such that p(0) = 0, ||p||∞ = 1, |∇p(0)ξ| = 1, and Qp (0) =
1 = cD . H0 (ξ, ξ)1/2
(14)
If D = D1 × · · · × Dk , a product of irreducible domains, then by (1), there exists j = 1, . . . , k such that cD = cDj , and so there exists a polynomial pj on Dj and ξj ∈ ∂Dj such that ||pj ||∞ = 1, |∇(pj )(0)ξj | = 1, and Qpj (0) = cD . Then letting p be the polynomial on D such that p(z) = pj (zj ) (where zj denotes the component of z in Dj ), and ξ the vector whose component in Dj equals ξj and whose components in each irreducible factor other than Dj are 0, we obtain a function on D with supremum norm 1 and a vector in ∂D such that |∇(p)(0)ξ| = |∇(pj )(0)ξj | = 1, p(0) = 0, and Qp (0) = Qpj (0) = cD . Fix a ∈ D and let S ∈ Aut(D) be such that S(a) = 0. Since the Jacobian matrix JS(a) is invertible, there exists a nonzero v ∈ Cn such that JS(a)v = ξ. Composing p with S, we obtain a function g ∈ B0∗ (D) such that ||g||∞ = 1 and Qg (a) = cD . In particular, θψ,id = sup |ψ(a)| sup{Qf (a) : ||f ||∞ ≤ 1} = cD ||ψ||∞ . a∈D
Moreover, by the boundedness of Mψ , the invariance of the Bergman metric under the action of Aut(D), recalling that |∇p(0)ξ| = 1 and using (14), we obtain |∇(ψg)(a)u| ||Mψ g||B ≥ Qψg (a) = sup 1/2 u∈Cn \{0} Ha (u, u) =
sup
≥ =
|ψ(a)∇(p)(0)JS(a)u|
H0 (JS(a)u, JS(a)u)1/2 |∇(p)(0)ξ| |ψ(a)| = |ψ(a)|Qp (0) H0 (ξ, ξ)1/2 |ψ(a)|cD .
u∈Cn \{0}
34
R. F. Allen and F. Colonna
IEOT
Taking the supremum over all a ∈ D, we deduce ||Mψ || ≥ cD ||ψ||∞ . The result now follows at once from Theorem 3.1. Theorem 5.2 ([8],[25]). If D is an irreducible bounded symmetric domain in Cn , then ⎧ 2 ⎪ if D is of type RI , ⎪ m+n ⎪ ⎪ ⎪ ⎪ 2 ⎪ if D is of type RII , ⎪ n+1 ⎪ ⎪ ⎨√ 1 if D is of type RIII , cD = n−1 ⎪ 2 ⎪ if D is of type RIV , ⎪ n ⎪ ⎪ ⎪ 1 ⎪ √ ⎪ if D = RV , ⎪ ⎪ ⎩1 6 if D = RV I . 3 Recalling that the Bloch seminorm of a bounded analytic function is no greater than its supremum norm and observing from Theorem 5.2 that cD ≤ 1 for any bounded symmetric domain D in standard form, and the only domains D for which cD = 1 are those which contain the unit disk as a factor, we obtain the following result. Corollary 5.3. Let D be a bounded symmetric domain in standard form with D as a factor and let ψ ∈ H(D). If Mψ is bounded from H ∞ (D) into either B(D) or B(D)0∗ , then max{||ψ||B , ||ψ||∞ } ≤ ||Mψ || ≤ ||ψ||B + ||ψ||∞ . Furthermore, if ψ(0) = 0, then ||ψ||∞ ≤ ||Mψ || ≤ ||ψ||B + ||ψ||∞ . We now characterize the compact multiplication operators from H ∞ to B when the underlying space is the ball or the polydisk. Theorem 5.4. For D = Bn or Dn , the following statements are equivalent: (a) Mψ : H ∞ (D) → B(D) is compact. (b) ψ is identically zero. Proof. (a) =⇒ (b): By Theorems 4.4 and 4.7, the compactness of Mψ implies that lim |ψ(z)| sup{Qf (z) : ||f ||∞ ≤ 1} = 0. z→∂D
Since sup{Qf (z) : ||f ||∞ ≤ 1} = cD , it follows that lim |ψ(z)| = 0, hence ψ z→∂D
is identically 0. (b) =⇒ (a) is obvious.
Using Corollary 4.3, we deduce that there are no nontrivial bounded multiplication operators from H ∞ (Bn ) to B0 (Bn ). Corollary 5.5. The following statements are equivalent: (a) Mψ : H ∞ (Bn ) → B0 (Bn ) is bounded. (b) Mψ : H ∞ (Bn ) → B0 (Bn ) is compact.
Vol. 66 (2010)
Weighted Composition Operators
35
(c) ψ is identically zero. We next look at the case when ψ is identically 1, that is, the weighted composition operator reduces to the composition operator from H ∞ (D) into B(D). 5.2. Composition Operators from H ∞ into the Bloch Space From (3) we deduce that for any holomorphic self-map of a bounded homogeneous domain D, the supremum θϕ of θϕ (z), over all z ∈ D, is finite. Indeed, θϕ ≤ cD Bϕ . Thus, Theorem 3.1 yields the following result. Corollary 5.6. Let D be a bounded homogeneous domain and let ϕ be a holomorphic self-map of D. Then (a) Cϕ : H ∞ (D) → B(D) is bounded. (b) Cϕ : H ∞ (D) → B0∗ (D) is bounded if and only if lim θϕ (z) = 0.
z→∂ ∗ D
Furthermore, if Cϕ is bounded as an operator into B(D) or B0∗ (D), then 1 ≤ ||Cϕ || ≤ 1 + θϕ . Remark 2. By Corollary 4.3, all bounded composition operators from H ∞ (D) to B0 (D) are compact and the corresponding symbol ϕ must satisfy the condition 2 (1 − |z| ) |ϕ (z)| = 0. (15) lim 2 |z|→1 1 − |ϕ(z)| Besides the symbols whose range is relatively compact in D, examples of b , symbols satisfying (15) include the functions of the form ϕ(z) = 1−λz 2 where |λ| = 1 and 0 < b < 1.
6. Isometries 6.1. Isometric multiplication operators In [1] and [3], we proved that the only isometric multiplication operators from the Bloch space of the unit disk or of a bounded symmetric domain that does not have the disk as a factor to itself are those induced by constant functions of modulus one. We now show that there are no isometric multiplication operators from H ∞ (D) into the Bloch space B(D) (and hence into the little Bloch space as well). Lemma 6.1. If Mψ : H ∞ (D) → B(D) is an isometry, then ψ(0) = 0. Proof. Arguing by contradiction, assume ψ(0) = 0. Since Mψ is an isometry, ||ψ||B = βψ = ||Mψ 1||B = 1. For a ∈ D define the automorphism of D a−z , z ∈ D. La (z) = 1 − az
36
R. F. Allen and F. Colonna
IEOT
Then, La ∈ H ∞ (D) with ||La ||∞ = 1. Again, since Mψ is an isometry, we obtain ||ψLa ||B = ||La ||∞ = 1. Noting that |ψ(a)| = (1 − |a| ) |(ψLa ) (a)| ≤ βψLa = ||ψLa ||B = 1, 2
taking the supremum over all a ∈ D, it follows that ||ψ||∞ ≤ 1. Since βψ ≤ ||ψ||∞ , we have 1 = βψ ≤ ||ψ||∞ ≤ 1. Thus 1 = ||ψ||∞ = ||Mψ (ψ)||B = βψ2 .
(16)
On the other hand, by the Schwarz-Pick lemma, we get βψ2 = 2 sup(1 − |z|2 ) |ψ(z)| |ψ (z)| ≤ 2 sup |ψ(z)| (1 − |ψ(z)|2 ) z∈D
z∈D
4 ≤ 2 max (x − x ) = √ < 1, x∈[0,1] 3 3 3
which contradicts (16).
Theorem 6.2. There are no isometric multiplication operators Mψ from H ∞ (D) to B(D). Proof. Assume Mψ is an isometry from H ∞ (D) into B(D). By Lemma 6.1, the symbol ψ cannot fix the origin. Since the identity function has supremum norm 1, the function f defined by f (z) = zψ(z) has Bloch semi-norm 1. Thus, by Theorem 2.1 of [9], either f is a rotation or the zeros of f form an infinite sequence {zk } satisfying the condition lim sup(1 − |zk |2 )|f (zk )| = 1.
(17)
k→∞
If f is a rotation, then ψ is a constant of modulus 1. Observe that constants cannot induce isometric multiplication operators since there exist functions in H ∞ (D) with supremum norm 1 which fix the origin and have Bloch seminorm strictly less than 1 (e.g. the function z → z 2 ). Thus, ψ cannot be a constant of modulus 1. If f is not a rotation, then ψ has the same non-zero zeros of f and |zk | → 1 as k → ∞. Since f (zk ) = zk ψ (zk ) + ψ(zk ) = zk ψ (zk ), by (17) we obtain 1 = lim sup(1 − |zk |2 )|zk ψ (zk )| = lim sup(1 − |zk |2 )|ψ (zk )|. k→∞
k→∞
Hence βψ = 1. On the other hand, since ψ does not fix the origin, 1 = ||ψ||B = |ψ(0)| + βψ > 1, which yields a contradiction. Therefore, no isometric multiplication operators from H ∞ (D) into B(D) can exist. 6.2. Isometric composition operators In [10] and [2], it was shown that there is a large class of isometric composition operators on the Bloch space of the disk and more generally on bounded homogeneous domains that have the disk as a factor. We shall now prove that there are no isometries among the composition operators between the Hardy space H ∞ and the Bloch space of the disk.
Vol. 66 (2010)
Weighted Composition Operators
37
Lemma 6.3. If ϕ is an analytic self-map of D inducing an isometric composition operator, then ϕ(0) = 0. Proof. Assume Cϕ is an isometry from H ∞ (D) into B(D). Then letting f be the identity, we have |ϕ(0)| + βϕ = ||ϕ||B = ||Cϕ f ||B = ||f ||∞ = 1.
(18)
Furthermore, letting, for z ∈ D, g+ (z) =
1+z 1−z , and g− (z) = , 2 2
we see that ||g+ ||∞ = ||g− ||∞ = 1. Thus ||Cϕ g+ ||B = ||Cϕ g− ||B = 1, and hence |1 + ϕ(0)| + βϕ = 2 = |1 − ϕ(0)| + βϕ . This, combined with (18), yields |1 + ϕ(0)| = 1 + |ϕ(0)| = |1 − ϕ(0)|, which implies ϕ(0) = 0. Theorem 6.4. There are no isometric composition operators from H ∞ (D) to B(D). Proof. Suppose Cϕ is an isometry from H ∞ (D) to B(D). Consider the function f (z) = z 2 for z ∈ D. Since Cϕ is an isometry, we have ||Cϕ f ||B = ||f ||∞ = 1. On the other hand, arguing as in the proof of Lemma 6.1, by Lemma 6.3 and the Schwarz-Pick Lemma, we obtain ||Cϕ f ||B
=
sup 2(1 − |z| ) |ϕ(z)| |ϕ (z)| 2
z∈D
≤
4 2 sup 2 |ϕ(z)| (1 − |ϕ(z)| ) ≤ √ < 1, 3 3 z∈D
reaching a contradiction.
6.3. Isometric weighted composition operators from B to H ∞ in the polydisk Let D be a bounded homogeneous domain. In [4], we characterized the bounded weighted composition operators from B(D) to H ∞ (D). In particular, for D = Dn , we obtained a characterization of boundedness equivalent to the following result proved by Li and Stevi´c in [15]. Theorem 6.5 ([15]). Wψ,ϕ : B(Dn ) → H ∞ (Dn ) is bounded if and only if ψ ∈ H ∞ (Dn ) and sup |ψ(z)|
z∈D
n k=1
log
2 < ∞. 1 − |ϕj (z)|
(19)
As a consequence we obtain the following result. Theorem 6.6. There are no isometric weighted composition operators from B(Dn ) to H ∞ (Dn ).
38
R. F. Allen and F. Colonna
IEOT
Proof. Assume Wψ,ϕ : B(Dn ) → H ∞ (Dn ) is an isometry. Then ||ψ||∞ = ||Wψ,ϕ 1||∞ = 1 and, fixing j = 1, . . . , n, ||ψϕj ||∞ = ||Wψ,ϕ pj ||∞ = ||pj ||B = 1, where pj is the z → zj . Thus, there exists a sequence {z (m) } in projection n (m) (m) D such that ψ(z )ϕj (z ) → 1 as m → ∞. Since both ψ and ϕj map Dn into D, it follows that ψ(z (m) ) → 1 and ϕj (z (m) ) → 1 as m → ∞. On the other hand, by the boundedness of Wψ,ϕ , (19) implies that ψ(z (m) ) → 0, a contradiction. 6.4. Final remarks By Corollary 4.3, the bounded weighted composition operators from H ∞ (Bn ) to B0 (Bn ) are necessarily compact, so there exist no isometric weighted composition operators from H ∞ (Bn ) to B0 (Bn ). We have not been able to prove or disprove the existence of isometric weighted composition operators from H ∞ to the Bloch space, even in the case of the unit disk. We conjecture that there are no isometric weighted composition operators from H ∞ (D) to B(D) for any bounded homogeneous domain D. Acknowledgments We wish to thank the referees for their thoughtful comments and helpful suggestions, and for bringing to our attention important references.
References [1] R. F. Allen and F. Colonna, Isometries and spectra of multiplication operators on the Bloch space, Bull. Austral. Math. Soc. 79 (2009), 147–160. [2] R. F. Allen and F. Colonna, On the isometric composition operators on the Bloch space in Cn , J. Math. Anal. Appl. 355 (2009), 675–688. [3] R. F. Allen and F. Colonna, Multiplication operators on the Bloch space of bounded homogeneous domains, Comput. Methods Function Theory 9(2) (2009), 679–693. [4] R. F. Allen and F. Colonna, Weighted composition operators on the Bloch space of a bounded homogeneous domain, Proc. IWOTA (to appear). [5] E. Cartan, Sur les domains bourn´ es de l’espace de n variable complexes (French), Abh. Math. Sem. Univ. Hamburg 11 (1935), 116–162. [6] R. Chen, S. Stevi´c and Z. Zhou, Weighted composition operators between Bloch type spaces in the polydisk, Mat. Sb. (to appear). [7] D. Clahane, S. Stevi´c and Z. Zhou, On composition operators between Bloch type spaces in polydisc, arXiv:math\0507339v1.
Vol. 66 (2010)
Weighted Composition Operators
39
[8] J. M. Cohen and F. Colonna, Bounded holomorphic functions on bounded symmetric domains, Trans. Amer. Math. Soc. 343 (1994), 135–156. [9] J. M. Cohen and F. Colonna, Isometric composition operators on the Bloch space in the polydisk, Contemp. Math., 454 (2008), 9–21. [10] F. Colonna, Characterisation of the isometric composition operators on the Bloch space, Bull. Austral. Math. Soc. 72 (2005), 283–290. [11] D. Drucker, Exception Lie algebras and the structure of Hermitian symmetric spaces, Mem. Amer. Math. Soc. 208 (1978), 1–207. [12] K. T. Hahn, Holomorphic mappings of the hyperbolic space in the complex Euclidean space and the Bloch theorem, Canad. J. Math 27 (1975), 446–458. [13] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York (1962). [14] T. Hosokawa, K. Izuchi and S. Ohno, Topological structure of the space of weighted composition operators on H ∞ , Integr. Equ. Oper. Theory 53 (2005), 509–526. [15] S. Li and S. Stevi´c, Weighted composition operators from H ∞ to the Bloch space on the polydisc, Abstr. Appl. Anal. Vol. 2007, Article ID 48478, (2007), 13 pp. [16] S. Li and S. Stevi´c, Weighted composition operators from α-Bloch spaces to H ∞ on the polydisk, Numer. Funct. Anal. Optimization 28(7) (2007), 911–925. [17] S. Li and S. Stevi´c, Weighted composition operators between H ∞ and α-Bloch spaces in the unit ball, Taiwanese J. Math. 12 (2008), 1625–1639. [18] S. Ohno, Weighted composition operators between H ∞ and the Bloch space, Taiwanese J. Math. 5(3) (2001), 555563. [19] S. Ohno and R. Zhao, Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc 63 (2001), 177–185. ˇ [20] I. I. Pjatecki˘ı-Sapiro, On a problem proposed by E. Cartan (Russian), Dokl. Akad. Nauk SSSR 124 (1959), 272–273. [21] S. Stevi´c, Composition operators between H ∞ and the α-Bloch spaces on the polydisc, Z. Anal. Anwendungen, 25(4) (2006), 457–466. [22] S. Stevi´c, Norm of weighted composition operators from Bloch space to H ∞ on the unit ball, Ars. Combin. 88 (2008), 125–127. [23] R. M. Timoney, Bloch functions in several complex variables I, Bull. London Math. Soc. 12 (1980), 241–267. [24] R. M. Timoney, Bloch functions in several complex variables II, J. Reine Angew. Math. 319 (1980), 1–22. [25] G. Zhang, Bloch constants of bounded symmetric domains, Trans. Amer. Math. Soc. 349 (1997), 2941–2949. [26] M. Zhang and H. Chen, Weighted composition operators of H ∞ into α-Bloch spaces on the unit ball, Acta Math. Sinica (English) 25 (2009), 265–278. [27] Z. Zhou and R. Chen, Weighted composition operators from F (p, q, s) to Bloch type spaces on the unit ball (preprint) (http://arxiv.org/abs/math/0503614v9).
40
R. F. Allen and F. Colonna
Robert F. Allen Department of Mathematics University of Wisconsin–La Crosse La Crosse, WI 54601 USA e-mail:
[email protected] Flavia Colonna Department of Mathematical Sciences George Mason University Fairfax, VA 22030 USA e-mail:
[email protected] Submitted: March 24, 2009. Revised: September 1, 2009.
IEOT
Integr. Equ. Oper. Theory 66 (2010), 41–52 DOI 10.1007/s00020-009-1737-3 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
The Heat Equation for the Generalized Hermite and the Generalized Landau Operators Viorel Catan˘a
Abstract. We give a formula for the one-parameter strongly continuous λ ˜ semigroups e−tL and e−tA , t > 0 generated by the generalized Hermite λ operator L , λ ∈ R\{0} respectively by the generalized Landau operator ˜ These formula are derived by means of pseudo-differential operators A. of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L2 (R2n ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L2 estimate for the solutions of initial value problems for the heat equations governed by ˜ in terms of Lp norm, 1 ≤ p ≤ ∞ of the initial data Lλ respectively A, are given. Mathematics Subject Classification (2010). Primary 47G10, 47G30; Secondary 35S10. Keywords. Weyl transform, Fourier-Wigner transform, Wigner transform, one-parameter strongly continuous semigroup.
Following Wong’s point of view (see [6], by Wong), we give a formula for λ the one-parameter strongly continuous semigroup e−tL , t > 0, generated by the generalized Hermite operator Lλ , for a fixed λ ∈ R \ {0}, in terms of the Weyl transforms. Then we use it to obtain an L2 estimate for the solution of the initial value problem for the heat equation governed by Lλ , in terms of the Lp norm, 1 ≤ p ≤ ∞, of the initial data. Similar results have also been derived for the generalized Landau operator A˜ which was firstly introduced by M.A. de Gosson (see [3] by de Gosson) who has studied its spectral properties.
This work was completed with the support of our TEX-pert.
42
V. Catana˘
For a fixed λ ∈ R \ {0} let
IEOT
∂ ∂ , be the linear partial differential ∂z λ ∂ z¯λ
operators on R2 , given by ∂ ∂ ∂ − i|λ|1/2 , = |λ|−1/2 λ ∂z ∂x ∂y
and
∂ ∂ ∂ + i|λ|1/2 . = |λ|−1/2 λ ∂ z¯ ∂x ∂y
Then we define the linear partial differential operator Lλ on R2 by 1 Lλ = − (Z λ Z¯ λ + Z¯ λ Z λ ), (1) 2 where ∂ 1 + z¯λ , z¯λ = |λ|1/2 x − i|λ|−1/2 y Zλ = ∂z λ 2 and ∂ 1 Z¯ λ = λ − z λ , z λ = |λ|1/2 x + i|λ|−1/2 y. ∂ z¯ 2 In fact −Z¯ λ is the formal adjoint of Z λ and Lλ is an elliptic partial differential operator on R2 , given by ∂2 ∂ ∂2 1 ∂ Lλ = −|λ|−1 2 − |λ| 2 + (|λ|x2 + |λ|−1 y 2 ) − i x|λ| − y|λ|−1 . ∂x ∂y 4 ∂y ∂x (2) We call the partial differential operator defined by (1) or (2) the generalized Hermite operator. If λ ∈ R, |λ| = 1 in (2), then the operator Lλ becomes 1 the ordinary Hermite operator −∆ + (x2 + y 2 ) perturbed by the partial 4 ∂ ∂ −y . differential operator −iN , where N = x ∂y ∂x First we need some basic definitions (see the book [5], by Wong) and preliminary results. Let f, g be functions in the Schwartz space S(R) on R. Then we define the Fourier-Wigner transform V (f, g) of f and g by +∞ p p dy (3) V (f, g)(q, p) = (2π)−1/2 eiqy f y + g y− 2 2 −∞ for all q, p ∈ R. The Wigner transform W (f, g) of f and g is defined by W (f, g) = V (f, g)∧ , where ∧ is the Fourier transform. Now, let σ ∈ Lp (R2 ), 1 ≤ p ≤ ∞. Then we define Wσ f to be the temperate distribution on R by +∞ +∞ −1/2 σ(x, ξ)W (f, g)(x, ξ)dxdξ, (4) (Wσ f, g) = (2π) −∞
−∞
for all g ∈ S(R). We call Wσ the Weyl transform associated to the symbol σ. For k ∈ N, let ek be the Hermite function on R of order k defined by √ −x2 ek (x) = (2k · k! π)−1/2 e 2 Hk (x), where Hk is the Hermite polynomial of degree k. Now, for k ∈ N and a fixed λ ∈ R \ {0}, we introduce the generalized Hermite function on R, eλk , defined by eλk (x) = |λ|−1/4 ek (|λ|−1/2 x), for all
Vol. 66 (2010)
The Heat Equation for the Generalized Hermite
43
x ∈ R (see [1]). For j, k ∈ R and a fixed λ ∈ R \ {0} we define the generalized Hermite function eλj,k on R2 by eλj,k (x, y) = V (eλj , eλk )(x, y), for all x, y ∈ R. Theorem 1. {eλj ; j = 0, 1, 2, . . .} is an orthonormal basis for L2 (R). Proof. Firstly let us observe that (eλj , eλk ) = δjk , for all j, k = 0, 1, 2, . . . . Thus (eλj )j∈N is an orthonormal system. To prove that (eλj )j∈N is an orthonormal basis for L2 (R) we need to verify that for f ∈ L2 (R) such that (f, eλj ) = 0, for all j ∈ N, it follows f = 0. To this end we use the fact that (ej )j∈N is an orthonormal basis for L2 (R) and that (f, ejλ ) = (g, ej ), where f ∈ L2 (R) and g = |λ|1/4 f (|λ|1/2 .) ∈ L2 (R), j = 0, 1, 2, . . .. So, if (f, eλj ) = 0 for all j ∈ N, it follows that (g, ej ) = 0, for all j ∈ N. This implies that g = 0 and a fortiori f = 0. Thus the proof is complete. Theorem 2. {eλj,k ; j, k = 0, 1, 2, . . .} is an orthonormal basis for L2 (R2 ). The proof of Theorem 2 is based on Theorem 1 and is similar with the proof of Theorem 21.2 in the book [5] by Wong, so we omit this. The spectral analysis of the generalized Hermite operator Lλ on R2 is based on the following theorem. Theorem 3. Lλ eλj,k = (2k + 1)eλj,k , for all j and k in N . Proof. Following the same reasoning as in Theorem 22.1 in the book [5] by Wong we can prove that for all x and y ∈ R, (Z λ eλj,k )(x, y) = i(2k)1/2 eλj,k−1 (x, y), j = 0, 1 . . . , k = 1, 2, . . .
(5)
and (Z¯ λ eλj,k )(x, y) = i(2k + 2)1/2 eλj,k+1 (x, y), j = 0, 1, 2, . . . , k = 0, 1, 2, . . . . (6) Hence by (1), (5) and (6) the conclusion in Theorem 3 follows.
Remark. Theorem 3 says that for k = 0, 1, 2, . . . the number 2k+1 is an eigenvalue of the generalized Hermite operator Lλ , and the generalized Hermite functions eλj,k , j = 0, 1, 2, . . . on R2 are eigenfunctions of Lλ corresponding to the eigenvalue 2k + 1. λ
A consequence of Theorem 3 is a formula for Hermite semigroup e−tL , t > 0 which is given in the following theorem. Theorem 4. For f ∈ S(R2 ) and for all t > 0, λ
e−tL f = (2π)1/2
∞
e−(2k+1)t V (Wfˆeλk , eλk ),
k=0
where the convergence is uniform and absolute on R2 . Proof. Let f ∈ S(R2 ). Then for t > 0, we use the Theorem 3 to get λ
e−tL f =
∞ ∞ k=0 j=0
e−(2k+1)t (f, eλj,k )eλj,k ,
(7)
44
V. Catana˘
IEOT
where the series is convergent in L2 (R2 ) and is also uniform and absolute convergent on R2 . Now by the definitions of Fourier-Wigner transform and Wigner transform respectively and Plancherel’s theorem, f (z)V (eλj , eλk )(z)dz = fˆ(ζ)V (eλj , eλk )∧ (ζ)dζ (f, eλj,k ) = R2
R2
1/2
(Wfˆeλk , eλj ),
= (2π)
j, k = 0, 1, 2, . . .
(8)
2
Similarly, for j, k = 0, 1, 2, . . . and g ∈ S(R ), we get (eλj,k , g) = (g, eλj,k ) = (2π)1/2 (Wgˆ eλk , eλj ) = (2π)1/2 (eλj , Wgˆ eλk ).
(9)
So by the relations (7)-(9), Fubini’s theorem and Parseval’s identity, ∞ ∞ λ e−(2k+1)t (f, eλj,k )(eλj,k , g) (e−tL f, g) = k=0 j=0 ∞
e
= (2π)
= (2π)
k=0 ∞
−(2k+1)t
∞
(Wfˆeλk , eλj )(eλj , Wgˆ eλk )
j=0
e−(2k+1)t (Wfˆeλk , Wgˆ eλk ),
for
t > 0,
(10)
k=0
where the series is absolutely convergent on R. But, by the definition and proprieties of the Wigner transform and Plancherel’s theorem, (Wfˆeλk , Wgˆ eλk )
=
(Wgˆ eλk , Wfˆeλk )
= (2π)−1/2
R2
−1/2
= (2π)
R2
gˆ(z)W (eλk , Wfˆeλk )(z)dz
W (Wfˆeλk , eλk )(z)ˆ g(x)dz,
for k = 0, 1, 2, . . . . (11)
Thus, by the relations (10)-(11) and Fubini’s theorem, ∞ λ e−(2k+1)t V (Wfˆeλk , eλk )∧ (ζ)ˆ g (ζ)dζ (e−tL f, g) = (2π)1/2 = (2π)1/2 1/2
= (2π)
k=0 ∞
R2
e−(2k+1)t (V (Wfˆeλk , eλk ), g)
k=0 ∞
e
−(2k+1)t
V
(Wfˆeλk , eλk ), g
,
(12)
k=0
for all f, g ∈ S(R2 ) and t > 0. Thus by the relation (12), ∞ −tLλ 1/2 e f = (2π) e−(2k+1)t V (Wfˆeλk , eλk ),
(13)
k=0
for all f ∈ S(R2 ) and t > 0, where the uniform and absolute convergence of the series follows from the properties of the Fourier-Wigner transform and the following theorem (see the Theorem 14.3 in the book [5] by Wong).
Vol. 66 (2010)
The Heat Equation for the Generalized Hermite
45
Theorem 5. Let σ ∈ Lp (R2 ), 1 ≤ p ≤ 2. Then Wσˆ is a bounded linear operator from L2 (R2 ) into L2 (R) and ||Wσˆ ||∗ ≤ (2π)−1/p ||σ||Lp (R2 ) ,
(14)
where ||Wσˆ ||∗ is the operator norm of Wσˆ : L2 (R) → L2 (R). As an application of the formula for the Hermite semigroup given in Theorem 4 we can now prove the following theorem. λ
Theorem 6. For t > 0, the Hermite semigroup e−tL , initially defined on S(R2 ), can be extended to a unique bounded linear operator from Lp (R2 ) λ into L2 (R2 ) which we again denote by e−tL and λ
||e−tL f ||L2 (R2 ) ≤ (2π)1/2−1/p
1 ||f ||Lp (R2 ) , for all f ∈ Lp (R2 ), 1 ≤ p ≤ 2. 2sht
We need the following result, which is known as the Moyal identity and can be found in the book [5] by Wong. Theorem 7. For all f, g ∈ S(R), ||V (f, g)||L2 (R2 ) = ||f ||L2 (R) ||g||L2 (R) .
(15)
Proof of Theorem 6. Let f ∈ S(R2 ). Then by Theorem 4 and Theorem 7, and Minkowski’s inequality, λ
||e−tL f ||L2 (R2 ) ≤ (2π)1/2 = (2π)1/2 = (2π)1/2
∞ k=0 ∞ k=0 ∞
e−(2k+1)t ||V (Wfˆeλk , eλk )||L2 (R2 ) e−(2k+1)t ||Wfˆeλk ||L2 (R) ||eλk ||L2 (R) e−(2k+1)t ||Wfˆeλk ||L2 (R) , for t > 0.
(16)
k=0
So by the relation (16) and Theorem 5, we get for t > 0, λ
||e−tL f ||L2 (R2 ) ≤ (2π)1/2 ≤ (2π)1/2
∞ k=0 ∞
e−(2k+1)t ||Wfˆ||∗ ||eλk ||L2 (R) e−(2k+1)t (2π)−1/p ||f ||Lp (R2 )
k=0
= (2π)1/2−1/p
1 ||f ||Lp (R2 ) , 2sh t
(17)
for all f ∈ S(R2 ). Now, by the relation (17) and a density argument, the proof is complete.
46
V. Catana˘
IEOT
Remark. Theorem 6 gives an L2 estimate for the solution of the initial value problem ∂u (z, t) = (−Lλ u)(z, t), z ∈ R2 , t > 0 ∂t u(z, 0) = f (z), z ∈ R2 , in terms of the Lp norm of the initial data f , 1 ≤ p ≤ 2. 2
Theorem 8. Let Λ ∈ S(R2 ) be a function on R2 defined by Λ(z) = π −1 e−|z| , z ∈ R2 , let g ∈ Lp (R2 ), 1 ≤ p ≤ ∞, and let uλ be the solution of the initial value problem considered in Remark 2, with initial data (g ∗ Λ)V , where V is the inverse Fourier transform. Then ||uλ ||L2 (R2 ) ≤ (2π)1/2−1/p
1 ||g||Lp (R2 ) . 2sht
For the proof of Theorem 8, we need the following results (see Chapter 17 of the book [7] by Wong and Theorem 15.4 and Theorem 17.1 in the book [5] by Wong). Theorem 9. Let Λ : C → R, be a function defined by 2
Λ(z) = π −1 e−|z| , z ∈ C. Then for all F ∈ Lp (C), 1 ≤ p ≤ ∞, WF ∗Λ = LF , where LF is the localization operator on the Weyl-Heisenberg group with symbol F . Theorem 10. Let F ∈ Lp (C), 1 ≤ p ≤ ∞. Then ||LF ||∗ ≤ (2π)−1/p ||F ||Lp (C) ,
(18)
where LF is the localization operator on the Weyl-Heisenberg group with symbol F . The proof of Theorem 8 is the same as that of Theorem 6, if we note that, by Theorem 9, WFˆ = Wg∗Λ = Lg , and hence the estimate follows from Theorem 10. Remark. For λ ∈ R, |λ| = 1, we obtain from our statements the results obtained in [6], by Wong. Remark. The above results are valid for the Hermite operator Lλ on R2n . Now we want to give similar results as the preceding ones in the case of a class of generalized Landau operators. This class of operators has been firstly introduced and studied by de Gosson in [3]. To this end let us first recall the following definitions and results from Shubin [4] (Chapter 4). Let m0 , m1 and ρ be real numbers such that m0 ≤ m1 and 0 < ρ ≤ 1.
Vol. 66 (2010)
The Heat Equation for the Generalized Hermite
47
1 ,m0 We shall write a ∈ HΓm (R2n ), if a ∈ C ∞ (R2n ) and there exist ρ R, C0 , C1 ≥ 0, and for every α ∈ Nn , there exists Cα ≥ 0 such that the following estimates hold for |z| ≥ R
C0 |z|m0 ≤ |a(z)| ≤ C1 |z|m1 |∂zα a(z)| ≤ Cα |a(z)||z|−ρ|α| . 1 ,m0 (R2n ) we denote the class of operators A, given by the By HGm ρ 1 ,m0 (R2n ); this means that for every τ ∈ R, there exists τ -symbols aτ ∈ HΓm ρ 1 ,m0 aτ ∈ HΓm (R2n ) such that ρ ei(x−y)·ξ aτ ((1 − τ )x + τ y, ξ)u(y)dydξ, u ∈ S(Rn ). Au(x) = (2π)−n
R2n
1 , in particular, we observe that every Weyl operator 2 m1 ,m0 1 ,m0 with symbol a ∈ HΓρ (R2n ) is in HGm (R2n ). Indeed, by Proposition ρ 1 ,m0 (R2n ) is true for some τ ∈ R, 25.1 in Shubin [4] (Chapter 4), if aτ ∈ HΓm ρ 1 ,m0 (R2n ) is also then it is true for all τ ∈ R. So, the condition a ∈ HΓm ρ m1 ,m0 2n sufficient to write A ∈ HGρ (R ). The following result tell us about discretness of the spectrum of the operators of Shubin’s classes. If we choose τ =
1 ,m0 (R2n ), where Theorem 11. (see Theorem 26.3 in [4]). Let A ∈ HGm ρ m0 > 0. Suppose that A is formally self-adjoint, that is (Au, v) = (u, Av) for all u, v ∈ C0∞ (Rn ). Then A has discret spectrum in L2 (Rn ). More precisely, there exists an orthonormal basis of eigenfunctions ϕj ∈ S(Rn ), j = 0, 1, 2, . . . , with eigenvalues λj ∈ R such that |λj | → ∞ as j → ∞.
In order to define the Landau operators A˜ which have been introduced by de Gosson in [3] and to state the main result about these, we need some preliminary. Let ϕ ∈ S(Rn ), ||ϕ|| = 1 and let the mapping Wϕ : S(Rn ) → S(R2n ) defined by π n/2 1 ˜ W (u, ϕ) z , (19) Wϕ u(z) = 2 2 ˜ (u, ϕ) = (2π)−n/2 W (u, ϕ) and W (u, ϕ) is the Wigner transform of where W the pair (u, ϕ). Explicitely: −n/2 2i ξx e e−iξy u(y)ϕ(x ¯ − y)dy. (20) Wϕ u(z) = (2π) Rn
We can extend it by continuity into a linear mapping Wϕ : L2 (Rn ) → L2 (R2n ). The integral operator Wϕ defined by (20) has been introduced by de Gosson in [2] and it has been called the ”Wigner wave-packet transform” associated with ϕ.
48
V. Catana˘
IEOT
Let A = aw (x, D) be the Weyl operator corresponding to the symbol a ∈ C ∞ (R2n ) (or generally a ∈ S (R2n )) 1 (x + y), ξ u(y)dydξ. (21) Au(x) = (2π)−n ei(x−y)ξ a 2 R2n Then we define a ˜ ∈ C ∞ (R4n ) by a ˜(z, ζ) = a
1 z − Jζ , 2
(22)
where Jζ = (ζ2 , −ζ1 ) if ζ = (ζ1 , ζ2 ) and we consider the operator A˜ = a ˜w (z, −i∂z ). 2
(23)
2
Let us observe that if we take a(x, ξ) = x + ξ in (23) then the operator A˜ coincide with the ”Landau Hamiltonian” ˜ = −(∂x2 + ∂y2 ) + i(x∂y − y∂x ) + 1 (x2 + y 2 ) H 4 ˜ (see [3]). So, the operator A generalize the ”Landau Hamiltonian”. Then we will call the operator A˜ defined by (23) the generalized Landau operator in the same manner as it has been called by de Gosson in [3]. Now we can state the main result in the article [3], by de Gosson, concerning the generalized Landau operators. 1 ,m0 (R2n ). Then the Theorem 12. (see Theorem 6 in [3]). Let A ∈ HGm ρ following statements are true: (i) The operators A and A˜ have the same eigenvalues. If u is an eigenfunction of A corresponding to the eigenvalue λ, then Uϕ = Wϕ u is an eigenfunction of A˜ corresponding to the eigenvalue λ, for every ϕ, and we have Uϕ ∈ S(R2n ). (ii) Suppose that m0 > 0 and that A is formally self-adjoint. Then A˜ has discrete spectrum (λj )j≥0 and |λj | → ∞ as j → ∞. (iii) In the context of (ii) the eigenfunctions of A˜ are given by Φjk = Wϕj ϕk , where ϕj , j = 0, 1, 2, . . . are the eigenfunctions of A. (iv) Φjk ∈ S(R2n ) and Φjk , j, k = 0, 1, 2, . . . form an orthonormal basis of L2 (R2n ).
As a consequence of Theorem 12 and by aditional hypothesis we can give ˜ a formula for the Landau semigroup e−tA , t > 0 in the following theorem. Theorem 13. Let A˜ be a positive generalized Landau operator in the context of (ii) of Theorem 12. Suppose that its symbol a ˜ is a real function and its spectrum (λ ) is a faster increasing sequence (this means that the series j j≥0 −λj t e is convergent; for example λj > j, (∀)j ≥ j0 , for some j0 ∈ N). j≥0
Then, for f ∈ S(R2n ) and for all t > 0, ∞ ˜ e−tA f = (2π)n/2 e−λj t θ1/2 W (Wθ2 f ϕj , ϕj ), j=0
(24)
Vol. 66 (2010)
The Heat Equation for the Generalized Hermite
49
where the convergence is uniform and absolute on R2n and (θa f )(z) = f (az), for all a ∈ R \ {0}, z ∈ R2n . Proof. Let f ∈ S(R2n ). Then for t > 0, we use Theorem 12 ((ii), (iii)) to get ∞ ∞ ˜ −tA e f= e−λj t (f, Φjk )Φjk , (25) j=0 k=0
where the series is convergent in L2 (R2n ) and is also uniform and absolute convergent on R2n . Now by the definitions of Φjk and Weyl transform, π n/2 1 ˜ z dz f (z)W (ϕk , ϕj ) (f, Φjk ) = 2 2 2n R ˜ (ϕj , ϕk )(w)dw = (8π)n/2 f (2w)W R2n = 2n (θ2 f )(w)W (ϕj , ϕk )(w)dw R2n n/2
= (8π)
(Wθ2 f ϕj , ϕk ), j, k = 0, 1, 2, . . .
(26)
2n
Similarly, for j, k = 0, 1, 2, . . . and g ∈ S(R ), we get (Φjk , g) = (g, Φjk ) = (8π)n/2 (Wθ2 g ϕj , ϕk ) = (8π)n/2 (ϕk , Wθ2 g ϕj ).
(27)
Thus by (25)-(27) and Fubini’s theorem and Parseval’s identity ∞ ∞ ˜ (e−tA f, g) = e−λj t (f, Φjk )(Φjk , g) j=0 k=0
= (8π)n
∞
e−λj t
j=0
= (8π)n
∞
∞
(Wθ2 f ϕj , ϕk )(ϕk , Wθ2 g ϕj )
k=0
e−λj t (Wθ2 f ϕj , Wθ2 g ϕj ), for t > 0,
(28)
j=0
where the series is absolutely convergent on R. By the definition of Weyl transform, and by the appropriate changes of variables, we get (Wθ2 f ϕj , Wθ2 g ϕj ) = (Wθ2 g ϕj , Wθ2 f ϕj ) −n/2 = (2π) (θ2 g)(z)W (ϕj , Wθ2 f ϕj )(z)dz 2n R W (Wθ2 f ϕj , ϕj )(z)(θ2 g)(z)dz = (2π)−n/2 R2n = (32π)−n/2 W (Wθ2 f ϕj , ϕj ) (1/2w) g(w)dw R2n
−n/2
= (32π) 2n
(θ1/2 W (Wθ2 f ϕj , ϕj ), g),
for all f, g ∈ S(R ) and t > 0.
(29)
50
V. Catana˘
IEOT
Thus by (28), (29) and Fubini’s theorem ˜
(e−tA f, g) = (2π)n/2
∞
e−λj t (θ1/2 W (Wθ2 f ϕj , ϕj ), g)
j=0
⎛ = (2π)n/2 ⎝
∞
⎞ e−λj t θ1/2 W (Wθ2 f ϕj , ϕj ), g ⎠ ,
(30)
j=0
for all f, g ∈ S(R2n ) and t > 0. Thus by (30), ˜
e−tA f = (2π)n/2
∞
e−λj t θ1/2 W (Wθ2 f ϕj , ϕj ),
(31)
j=0
for all f ∈ S(R2n ) and t > 0, where the uniform and absolute convergence of the series on R2n follows from the properties of Wigner transform and the following theorem. Theorem 14. (see the book [5] by Wong). Let σ ∈ Lp∗ (R2n ), 2 ≤ p ≤ ∞. Then Wσ : L2 (Rn ) → L2 (Rn ) is a bounded linear operator and
σ ||Lp (R2n ) , ||Wσ ||∗ ≤ (2π)−n/p ||ˆ
σ ∈ Lp∗ (R2n )
(32)
2 ≤ p ≤ ∞, where Lp∗ (R2n ) = {f ∈ Lp (R2n ); fˆ ∈ Lp (R2n )}. Now we can give an application of the formula for the Landau semigroup which was proved in Theorem 13. To this end let us recall the following result which is known as the Moyal identity for the Wigner transform (see the book [5] by Wong). Theorem 15. W : S(Rn ) × S(Rn ) → S(R2n ) can be extended uniquely to a bilinear operator W : L2 (Rn ) × L2 (Rn ) → L2 (R2n ) such that ||W (f, g)||L2 (R2n ) = ||f ||L2 (Rn ) ||g||L2 (Rn ) , 2
(33)
n
for all f and g in L (R ). ˜
Theorem 16. For t > 0, the Landau semigroup e−tA initially defined on S(R2n ) can be extended to a unique bounded linear operator from Lp∗ (R2n ) ˜ into L2 (R2n ) which we again denote by e−tA and ˜
||e−tA f ||L2 (R2n ) ≤ 2(1/2−1/p)n π (1/2−1/p )n
∞
e−λj t ||f ||Lp∗ (R2n ) ,
2 ≤ p ≤ ∞,
j=0
(34) for all f ∈ Lp∗ (R2n ), where Lp∗ (R2n ) = {g ∈ Lp (R2n ); gˆ ∈ Lp (R2n )}, and g||Lp (R2n ) is a norm on Lp∗ (R2n ). ||g||Lp∗ (R2n ) = ||ˆ
Vol. 66 (2010)
The Heat Equation for the Generalized Hermite
51
Proof. Let f ∈ S(R2n ). Then by Theorem 13, Theorem 15, Minkowski’s inequality and Theorem 14, ˜
||e−tA f ||L2 (R2n ) ≤ (2π)n/2
∞
e−λj t ||θ1/2 W (Wθ2 f ϕj , ϕj )||L2 (R2n )
j=0
= (8π)n/2
∞
e−λj t ||W (Wθ2 f ϕj , ϕj )||L2 (R2n )
j=0
= (8π)n/2
∞
e−λj t ||Wθ2 f ϕj ||L2 (Rn ) ||ϕj ||L2 (Rn )
j=0
≤ (8π)n/2
∞
e−λj t ||Wθ2 f ||∗ ||ϕj ||L2 (Rn )
j=0
≤ (8π)n/2 (2π)−n/p
∞ j=0
=2
e−λj t ||θ 2 f ||Lp (R2n )
(1/2−1/p)n (1/2−1/p )n
π
∞
e−λj t ||f ||Lp∗ (R2n )
(35)
j=0
Now, by (35) and a density argument, the proof is complete.
Remark. Theorem 14 gives an L2 estimate for the solution of the initial value problem ∂u ˜ (z, t) = (−Au)(z, t), z ∈ R2n , t > 0 ∂t u(z, 0) = f (z), in terms of the
Lp∗
z ∈ R2n ,
(36)
norm of the initial data f , 2 ≤ p ≤ ∞.
Theorem 17. Let Λ ∈ S(R2n ) be a function on R2n defined by Λ(z) = 2 π −n e−|z| , z ∈ R2n , let g ∈ Lp (R2n ), 1 ≤ p ≤ ∞ and let u be the solution of the initial value problem (36), with initial data f = θ1/2 (g ∗ Λ). Then ||u||L2 (R2n ) ≤ 2(3/2−1/p)n π (1/2−1/p)n
∞
e−λj t ||g||Lp (R2n ) .
(37)
j=0
For the proof, we need the following results (see Theorem 15.4 and Theorem 17.1 in the book [5] by Wong and Chapter 17 of the book [7] by Wong). Theorem 18. Let Λ : Cn → R be a function defined by 2
Λ(z) = π −n e−|z| ,
z ∈ Cn .
Then for all F ∈ Lp (Cn ), 1 ≤ p ≤ ∞, WF ∗Λ = LF , where LF : L2 (Rn ) → L2 (Rn ) is the localization operator on the Weyl-Heisenberg group with symbol F .
52
V. Catana˘
IEOT
Theorem 19. Let F ∈ Lp (Cn ), 1 ≤ p ≤ ∞. Then ||LF ||∗ ≤ (2π)−n/p ||F ||Lp (Cn ) , 2
n
2
(38)
n
where LF : L (R ) → L (R ) is the localization operator on the WeylHeisenberg group with symbol F and ||LF ||∗ is its operator norm. Proof of Theorem 17. Now, we use the same reasoning as that of Theorem 16, noting that by Theorem 18, Wθ2 f = Wg∗Λ = Lg , and hence the estimate (37) follows from (38). Thus the proof is complete. Acknowledgement The author is grateful to the referee for some useful observations on the presentation of this paper.
References [1] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group. Studia Math. 56 (1976), 165-173. [2] M.A. de Gosson, Symplectic Geometry and Quantum Mechanics. Basel, Birkh¨ auser, 2006. [3] M.A. de Gosson, Spectral properties of a class of generalized Landau operators. Comm. Part. Diff. Equ., 33 (2008), 2096-2104. [4] M.A. Shubin, Pseudodifferential Operators and Spectral Theory. Berlin, Springer Verlag, 1987. [5] M.W. Wong, Weyl Transforms. Springer-Verlag, 1998. [6] M.W. Wong, The heat equation for the Hermite operator on the Heisenberg group. Hokkaido Math. Journal, 34 (2005), 393-404. [7] M.W. Wong, Wavelet Transforms and Localization Operators. Birkh¨ auser, 2002. Viorel Catan˘ a University Politehnica of Bucharest Department of Mathematics I Splaiul Independent¸ei 313 060042 Bucharest Romania e-mail: catana
[email protected] Submitted: April 9, 2009. Revised: July 10, 2009.
Integr. Equ. Oper. Theory 66 (2010), 53–77 DOI 10.1007/s00020-009-1734-6 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Toeplitz Operators on Generalized Bergman Spaces Kamthorn Chailuek and Brian C. Hall Abstract. We consider the weighted Bergman spaces HL2 (Bd, µλ ), where we set dµλ (z) = cλ (1 − |z|2 )λ dτ (z), with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces. Mathematics Subject Classification (2010). Primary 47B35; Secondary 32A36, 81S10. Keywords. Bergman space, Toeplitz operator, quantization, holomorphic Sobolev space, Berezin transform.
1. Introduction 1.1. Generalized Bergman spaces Let Bd denote the (open) unit ball in Cd and let τ denote the hyperbolic volume measure on Bd , given by dτ (z) = (1 − |z|2 )−(d+1) dz,
(1.1)
where dz denotes the 2d-dimensional Lebesgue measure. The measure τ is natural because it is invariant under all of the automorphisms (biholomorphic mappings) of Bd . For λ > 0, let µλ denote the measure dµλ (z) = cλ (1 − |z|2 )λ dτ (z), where cλ is a positive constant whose value will be specified shortly. Finally, let HL2 (Bd , µλ ) denote the (weighted) Bergman space, consisting of those holomorphic functions on Bd that are square-integrable with respect to µλ . Supported in part by a grant from Prince of Songkla University. Supported in part by NSF Grant DMS-0555862.
K. Chailuek and B. C. Hall
54
IEOT
(Often these are defined using the Lebesgue measure as the reference measure, but all the formulas look nicer if we use the hyperbolic volume measure instead.) These spaces carry a projective unitary representation of the group SU (d, 1). If λ > d, then the measure µλ is finite, so that all bounded holomorphic functions are square-integrable. For λ > d, we choose cλ so that µλ is a probability measure. Calculation shows that Γ(λ) , λ > d. (1.2) cλ = d π Γ(λ − d) (This differs from the value in Zhu’s book [33] by a factor of π d /d!, because Zhu uses normalized Lebesgue whereas we use un-normalized Lebesgue measure in (1.1).) On the other hand, if λ ≤ d, then µλ is an infinite measure. In this case, it is not hard to show that there are no nonzero holomorphic functions that are square-integrable with respect to µλ (no matter which nonzero value for cλ we choose). Although the holomorphic L2 space with respect to µλ is trivial (zero dimensional) when λ ≤ d, there are indications that life does not end at λ = d. First, the reproducing kernel for HL2 (Bd , µλ ) is given by 1 Kλ (z, w) = (1 − z · w) ¯ λ for λ > d. The reproducing kernel is defined by the property that it is antiholomorphic in w and satisfies Kλ (z, w)f (w) dµλ (w) = f (z) Bd
2
for all f ∈ HL (B , µλ ). Nothing unusual happens to Kλ as λ approaches d. In fact, Kλ (z, w) := (1 − z · w) ¯ −λ is a “positive definite reproducing kernel” for all λ > 0. Thus, it is possible to define a reproducing kernel Hilbert space for all λ > 0 that agrees with HL2 (Bd , µλ ) for λ > d. Second, in representation theory, one is sometimes led to consider spaces like HL2 (Bd , µλ ) but with λ < d. Consider, for example, the much-studied metaplectic representation of the connected double cover of SU (1, 1) ∼ = Sp(1, R). This representation is a direct sum of two irreducible representations, one of which can be realized in the Bergman space HL2 (B1 , µ3/2 ) and the other of which can be realized in (a suitably defined version of) the Bergman space HL2 (B1 , µ1/2 ). To be precise, we can say that the second summand of the metaplectic representation is realized in a Hilbert space of holomorphic functions having Kλ , λ = 1/2, as its reproducing kernel. See [14, Sect. 4.6]. Last, one often wants to consider the infinite-dimensional (d → ∞) limit of the spaces HL2 (Bd , µλ ). (See, for example, [25] and [23].) To do this, one wishes to embed each space HL2 (Bd , µλ ) isometrically into a space of functions on Bd+1 , as functions that are independent of zn+1 . It turns out that if one uses (as we do) hyperbolic volume measure as the reference measure, then the desired isometric embedding is achieved by embedding d
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
55
HL2 (Bd , µλ ) into HL2 (Bd+1 , µλ ). That is, if we use the same value of λ on Bd+1 as on Bd , then the norm of a function f (z1 , . . . , zd ) is the same whether we view it as a function on Bd or as a function on Bd+1 that is independent of zd+1 . (See, for example, Theorem 4, where the inner product of z m with z n is independent of d.) If, however, we keep λ constant as d tends to infinity, then we will eventually violate the condition λ > d. Although it is possible to describe the Bergman spaces for λ ≤ d as reproducing kernel Hilbert spaces, this is not the most convenient description for calculation. Instead, drawing on several inter-related results in the literature, we describe these spaces as “holomorphic Sobolev spaces,” also called Besov spaces. The inner product on these spaces, which we denote as H(Bd , λ), is an L2 inner product involving both the functions and derivatives of the functions. For λ > d, H(Bd , λ) is identical to HL2 (Bd , µλ ) (the same space of functions with the same inner product), but H(Bd , λ) is defined for all λ > 0. It is worth mentioning that in the borderline case λ = d, the space H(Bd , λ) can be identified with the Hardy space of holomorphic functions that are square-integrable over the boundary. To see this, note that the normalization constant cλ tends to zero as λ approaches d from above. Thus, the measure of any compact subset of Bd tends to zero as λ → d+ , meaning that most of the mass of µλ is concentrated near the boundary. As λ → d+ , µλ converges, in the weak-∗ topology on Bd , to the unique rotationally invariant probability measure on the boundary. Alternatively, we may observe that the formula for the inner product of monomials in H(Bd , d) (Theorem 4 with λ = d) is the same as in the Hardy space. 1.2. Toeplitz operators One important aspect of Bergman spaces is the theory of Toeplitz operators on them. If φ is a bounded measurable function, the we can define the Toeplitz operator Tφ on HL2 (Bd , µλ ) by Tφ f = Pλ (φf ), where Pλ is the orthogonal projection from L2 (Bd , µλ ) onto the holomorphic subspace. That is, Tφ consists of multiplying a holomorphic function by φ, followed by projection back into the holomorphic subspace. Of course, Tφ depends on λ, but we suppress this dependence in the notation. The function φ is called the (Toeplitz) symbol of the operator Tφ . The map sending φ to Tφ is known as the Berezin–Toeplitz quantization map and it (and various generalizations) have been much studied. See, for example, the early work of Berezin [5, 6], which was put into a general framework in [26, 27], along with [22, 8, 7, 10], to mention just a few works. The Berezin–Toeplitz quantization may be thought of as a generalization of the anti-Wick-ordered quantization on Cd (see [15]). When λ < d, the inner product on H(Bd , λ) is not an L2 inner product, and so the “multiply and project” definition of Tφ no longer makes sense. Our strategy is to find alternative formulas for computing Tφ in the case λ > d, with the hope that these formulas will continue to make sense (for certain classes of symbols φ) for λ ≤ d. Specifically, we will identify classes of symbols φ for which Tφ can be defined as:
56
K. Chailuek and B. C. Hall
IEOT
• A bounded operator on H(Bd , λ) (Section 4) • A Hilbert–Schmidt operator on H(Bd , λ) (Section 5). We also consider in Section 3 Toeplitz operators whose symbols are polynomials in z and z¯ and observe some unusual properties of such operators in the case λ < d.
2. H(Bd , λ) as a holomorphic Sobolev space In this section, we construct a Hilbert space of holomorphic functions on Bd with reproducing kernel (1 − z · w) ¯ −λ , for an arbitrary λ > 0. We denote this d space as H(B , λ). The inner product on this space is an L2 inner product with respect to the measure µλ+2n , where n is chosen so that λ + 2n > d. The inner product, however, involves not only the holomorphic functions but also their derivatives. That is, H(Bd , λ) is a sort of holomorphic Sobolev space (or Besov space) with respect to the measure µλ+2n . When λ > d, our space is identical to HL2 (Bd , µλ )—not just the same space of functions, but also the same inner product. When λ ≤ d, the Hilbert space H(Bd , λ), with the associated projective unitary action of SU (d, 1), is sometimes referred to as the analytic continuation (with respect to λ) of the holomorphic discrete series. Results in the same spirit as—and in some cases almost identical to— the results of this section have appeared in several earlier works, some of which treat arbitrary bounded symmetric domains and not just the ball in Cd . For example, in the case of the unit ball in Cd , Theorem 3.13 of [30] would presumably reduce to almost the same expression as in our Theorem 4, except that Yan has all the derivatives on one side, in which case the inner product has to be interpreted as a limit of integrals over a ball of radius 1 − ε. (Compare the formula for Dλk on p. 13 of [30] to the formula for A and B in Theorem 4.) See also [2, 4, 21, 31, 32]. Note, however, a number of these references give a construction that yields, for λ > d, the same space of functions as HL2 (Bd , µλ ) with a different but equivalent norm. Such an approach is not sufficient for our needs; we require the same inner product as well as the same space of functions. Although our results in this section are not really new, we include proofs to make the paper self-contained and to get the precise form of the results that we want. The integration-by-parts argument we use also serves to prepare for our definition of Toeplitz operators on H(Bd , λ) in Section 4. We ourselves were introduced to this sort of reasoning by the treatment in Folland’s book [14] of the disk model for the metaplectic representation. The paper [16] obtains results in the same spirit as those of this section, but in the context of a complex semisimple Lie group. We begin by showing that for λ > d, the space HL2 (Bd , µλ ) can be expressed as a subspace of HL2 (Bd , µλ+2n ), with a Sobolev-type norm, for
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
57
any positive integer n. Let N denote the “number operator,” defined by N=
d j=1
zj
∂ . ∂zj
This operator satisfies N z m = |m|z m for all multi-indices m. If f is holomorphic, then N f coincides with the “radial derivative” df (rz)/dr|r=1 . We use ¯ = d z¯j ∂/∂ z¯j . also the operator N j=1 A simple computation shows that ¯ N N 2 α 2 α+1 (1 − |z| ) = I − = I− (1 − |z| ) (1 − |z|2 )α+1 . (2.1) α+1 α+1 We will use (2.1) and the following integration by parts result, which will also be used in Section 4. Lemma 1. If λ > d and ψ is a continuously differentiable function for which ψ and N ψ are bounded, then N 2 λ−d−1 ψ(z)(1 − |z| ) dz = cλ+1 cλ I+ ψ (z)(1 − |z|2 )λ−d dz λ d d B B ¯ N = cλ+1 I+ ψ (z)(1 − |z|2 )λ−d dz. λ Bd Here dz is the 2d-dimensional Lebesgue measure on Bd . Proof. We start by applying (2.1) and then think of the integral over Bd as the limit as r approaches 1 of the integral over a ball of radius r < 1. On the ball of radius r, we write out ∂/∂zj in terms of ∂/∂xj and ∂/∂yj . For, say, the ∂/∂xj term we express the integral as a one-dimensional integral with respect to xj (with limits of integration depending on the other variables) followed by an integral with respect to the other variables. We then use ordinary integration by parts in the xj integral, and similarly for the ∂/∂yj term. The integration by parts will yield a boundary term involving zj ψ(z)(1− |z|2 )λ−d ; this boundary term will vanish as r tends to 1, because we assume λ > d. In the nonboundary term, the operator N applied to (1 − |z|2 )λ−d d will turn into the operator − j=1 ∂/∂zj ◦ zj = −(dI + N ) applied to ψ. Computing from (1.2) that cλ /cλ+1 = (λ − d)/λ, we may simplify and let r tend to 1 to obtain the desired result involving N. The same reasoning gives ¯ as well. the result involving N We now state the key result, obtained from (2.1) and Lemma 1, relating the inner product in HL2 (Bd , µλ ) to the inner product in HL2 (Bd , µλ+1 ) (compare [14, p. 215] in the case d = 1).
58
K. Chailuek and B. C. Hall
IEOT
Proposition 2. Suppose that λ > d and f and g are holomorphic functions on Bd for which f, g, N f, and N g are all bounded. Then N f, g L2 (Bd ,µλ ) = f, I + g λ L2 (Bd ,µλ+1 ) (2.2) N = I+ . f, g λ L2 (Bd ,µλ+1 ) Proof. Recalling the formula (1.1) for the measure τ, we apply Lemma 1 with ψ(z) = f (z)g(z) with f and g holomorphic. Observing that N (f¯g) = f¯N g ¯ (f¯g) = (N f )g gives the second gives the first equality and observing that N equality. Now, a general function in HL2 (Bd , µλ ) is not bounded. Indeed, the pointwise bounds on elements of HL2 (Bd , µλ ), coming from the reproducing kernel, are not sufficient to give a direct proof of the vanishing of the boundary terms in the integration by parts in Proposition 2. Nevertheless, (2.2) does hold for all f and g in HL2 (Bd , µλ ), provided that one interprets the inner product as the limit as r approaches 1 of integration over a ball of radius r. (See [14, p. 215] or [30, Thm. 3.13].) We are going to iterate (2.2) to obtain an expression for the inner product on HL2 (Bd , µλ ) involving equal numbers of derivatives on f and g. This leads to the following result. Theorem 3. Fix λ > d and a non-negative integer n. Then a holomorphic function f on Bd belongs to HL2 (Bd , µλ ) if and only if N l f belongs to HL2 (Bd , µλ+2n ) for 0 ≤ l ≤ n. Furthermore, f, g HL2 (Bd ,µλ ) = Af, Bg HL2 (Bd ,µλ+2n )
(2.3)
for all f, g ∈ HL2 (Bd , µλ ), where N N N A= I+ I+ ··· I + λ+n λ+n+1 λ + 2n − 1 N N N I+ ··· I + . B= I+ λ λ+1 λ+n−1 Let us make a few remarks about this result before turning to the proof. Let σ = λ + 2n. It is not hard to see that N k f belongs to HL2 (Bd , µσ ) for 0 ≤ k ≤ n if and only if all the partial derivatives of f up to order n belong to HL2 (Bd , µµ ), so we may describe this condition as “f has n derivatives in HL2 (Bd , µσ ).” This condition then implies that f belongs to HL2 (Bd , µσ−2n ), 2 which in turn means that f (z)/(1 − |z| )n belongs to L2 (Bd , µσ ). Since 1/(1 − |z|2 )n blows up at the boundary of Bd , saying that f (z)/(1 − |z|2 )n belongs to L2 (Bd , µσ ) says that f (z) has better behavior at the boundary than a typical element of HL2 (Bd , µσ ). We may summarize this discussion by saying that each derivative that f ∈ HL2 (Bd , µσ ) has in HL2 (Bd , µσ ) results, roughly speaking, in an improvement by a factor of (1 − |z|2 ) in the behavior of f near the boundary.
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
59
This improvement is also reflected in the pointwise bounds on f coming from the reproducing kernel. If f has n derivatives in HL2 (Bd , µσ ), then f belongs to HL2 (Bd , µσ−2n ), which means that f satisfies the pointwise bounds 1/2
|f (z)| ≤ f L2 (Bd ,µσ−2n ) (Kσ−2n (z, z))
σ2 −n 1 = f L2 (Bd ,µσ−2n ) . 2 1 − |z|
(2.4)
2
These bounds are better by a factor of (1 − |z| )n than the bounds on a typical element of HL2 (Bd , µσ ). See also [16] for another setting in which the existence of derivatives in a holomorphic L2 space can be related in a precise way to improved pointwise behavior of the functions. The results of the two previous paragraphs were derived under the assumption that λ = σ − 2n > d. However, Theorem 4 will show that (2.4) still holds under the assumption λ = σ − 2n > 0. Proof. If f and g are polynomials, then (2.3) follows from iteration of Proposition 2. Note that N is a non-negative operator on polynomials, because the monomials form an orthogonal basis of eigenvectors with non-negative eigenvalues. It is well known and easily verified that for any f in HL2 (Bd , µλ ), the partial sums of the Taylor series of f converge to f in norm. We can therefore choose polynomials fj converging in norm to f . If we apply (2.3) with f = g = (fj − fk ) and expand out the expressions for A and B, then the positivity of N will force each of the terms on the right-hand side to tend to zero. In particular, N l fj is a Cauchy sequences in HL2 (Bd , µλ+2n ), for all 0 ≤ l ≤ n. It is easily seen that the limit of this sequence is N l f ; for holomorphic functions, L2 convergence implies locally uniform convergence of the derivatives to the corresponding derivatives of the limit function. This shows that N l f is in HL2 (Bd , µλ+2n ). For any f, g ∈ HL2 (Bd , µλ ), choose sequences fj and gk of polynomials converging to f, g. Since N l fj and N l gj converge to N l f and N l g, respectively, plugging fj and gj into (2.3) and taking a limit gives (2.3) in general. In the other direction, suppose that N l f belongs to HL2 (Bd , µλ+2n ) for all 0 ≤ l ≤ n. Let fj denote the jth partial sum of the Taylor series of f . Then since N z m = |m|z m for all multi-indices m, the functions N l fj form the partial sums of a Taylor series converging to N l fj , and so these must be the partial sums of the Taylor series of N l f. Thus, for each l, we have that N l fj converges to N l f in HL2 (Bd , µλ+2n ). If we then apply (2.3) with f = g = fj − fk , convergence of each N l fj implies that all the terms on the right-hand side tend to zero. We conclude that fj is a Cauchy sequence in HL2 (Bd , µλ ), which converges to some fˆ. But L2 convergence of holomorphic functions implies pointwise convergence, so the limit in HL2 (Bd , µλ ) (i.e., fˆ) coincides with the limit in HL2 (Bd , µλ+2n ) (i.e., f ). This shows that f is in HL2 (Bd , µλ ).
K. Chailuek and B. C. Hall
60
IEOT
Now, when λ ≤ d, Proposition 2.2 no longer holds. This is because the boundary terms, which involve (1 − |z|2 )λ−d , no longer vanish. This failure of equality is actually a good thing, because if we take f = g, then |f (z)|2 (1 − |z|2 )λ dτ (z) = +∞ cλ Bd
for all nonzero holomorphic functions, no matter what positive value we assign to cλ . (Recall that when λ > d, cλ is chosen to make µλ a probability measure, but this prescription does not make sense for λ ≤ d.) Although the left-hand side of (2.2) is infinite when f = g and λ ≤ d, the right-hand side is finite if λ + 1 > d and, say, f is a polynomial. More generally, for any λ ≤ d, we can choose n big enough such that λ + 2n > d. We then take the right-hand side of (2.3) as a definition. Theorem 4. For all λ > 0, choose a non-negative integer n so that λ + 2n > d and define
H(Bd , λ) = f ∈ H(Bd ) N k f ∈ HL2 (Bd , µλ+2n ), 0 ≤ k ≤ n . Then the formula f, g λ = Af, Bg HL2 (Bd ,µλ+2n ) where
N N N I+ I+ ··· I + λ+n λ+n+1 λ + 2n − 1 N N N B= I+ I+ ··· I + λ λ+1 λ+n−1 A=
defines an inner product on H(Bd , λ) and H(Bd , λ) is complete with respect to this inner product. The monomials z m form an orthogonal basis for H(Bd , λ) and for all multi-indices l and m we have l m m!Γ(λ) . z , z λ = δl,m Γ(λ + |m|) Furthermore, H(Bd , λ) has a reproducing kernel given by Kλ (z, w) =
1 . (1 − z · w) ¯ λ
Using power series, it is easily seen that for any holomorphic function f, if N n f belongs to HL2 (Bd , µλ+2n ), then N k f belongs to HL2 (Bd , µλ+2n ) for 0 ≤ k < n. Note that the reproducing kernel and the inner product of the monomials are independent of n. Thus, we obtain the same space of functions with the same inner product, no matter which n we use, so long as λ + 2n > d. From the reproducing kernel we obtain the pointwise bounds given by 2 2 2 |f (z)| ≤ f λ (1 − |z| )−λ .
Toeplitz Operators on Generalized Bergman Spaces
Vol. 66 (2010)
61
Proof. Using a power series argument, it is easily seen that if f and N k f 2 d k belong to HL (B , µλ+2n ), then f, N f L2 (Bd ,µ ≥ 0. From this, we λ+2n ) obtain positivity of the inner product ·, · λ . If fj is a Cauchy sequence in H(Bd , λ), then positivity of the coefficients in the expressions for A and B imply that for 0 ≤ k ≤ n, N k fj is a Cauchy sequence in HL2 (Bd , µλ+2n ), which converges (as in the proof of Theorem 3) to N k f. This shows that N k f is in HL2 (Bd , µλ+2n ) for each 0 ≤ k ≤ n, and so f ∈ H(Bd , λ). Further, convergence of each N k fj to N k f implies that fj converges to f in H(Bd , λ). To compute the inner product of two monomials in H(Bd , λ), we apply the definition. Since N z m = |m|z m , we obtain l m z ,z λ λ + |m| λ + 1 + |m| λ + 2n − 1 + |m| m!Γ(λ + 2n) = δl,m ··· λ λ+1 λ + 2n − 1 Γ(λ + 2n + |m|) m!Γ(λ) = δl,m , Γ(λ + |m|) where we have used the known formula for the inner product of monomials in HL2 (Bd , µλ+2n ) (e.g., [33]). Completeness of the monomials holds in H(Bd , λ) for essentially the same reason it holds in the ordinary Bergman spaces. For f ∈ HL2 (Bd , µλ ), expand f in a Taylor series and then consider z m , f λ . Each term in the inner product is an integral over Bd with respect to µλ+2n , and each of these integrals can be computed as the limit as r tends to 1 of integrals over a ball of radius r < 1. On the ball of radius r, we may interchange the integral with the sum in the Taylor series. But distinct monomials are orthogonal not just over Bd but also over the ball of radius r, as is easily verified. The upshot of all of this is that z m , f λ is a nonzero multiple of the mth Taylor coefficient of f. Thus if z m , f λ = 0 for all m, f is identically zero. Finally, we address the reproducing kernel. Although one can use essentially the same argument as in the case λ > d, using the orthogonal basis of monomials and a binomial expansion (see the proof of Theorem 12), it is more enlightening to relate the reproducing kernel in H(Bd , λ) to that in HL2 (Bd , µλ+2n ). We require some elementary properties of the operators A and B; since the monomials form an orthogonal basis of eigenvectors for these operators, these properties are easily obtained. We need that A is self-adjoint on its natural domain and that A and B have bounded inverses. be the unique element of HL2 (Bd , µλ+2n ) for which Let χλ+2n z λ+2n χz , f L2 (Bd ,µ = f (z) ) λ+2n
2
for all f in HL (B , µλ+2n ). Explicitly, χλ+2n (w) = (1 − z¯ · w)−(λ+2n) . (This z is Theorem 2.2 of [33] with our λ corresponding to n + α + 1 in [33].) Now, a simple calculation shows that d
(I + N/a)(1 − z¯ · w)−a = (1 − z¯ · w)−(a+1) ,
(2.5)
62
K. Chailuek and B. C. Hall
IEOT
where N acts on the w variable with z fixed. From this, we see that N k χλ+2n z is a bounded function for each fixed z ∈ Bd and k ∈ N, so that χλ+2n is in z H(Bd , λ). For any f ∈ H(Bd , λ) we compute that f, (AB)−1 χλ+2n = Af, B(AB)−1 χλ+2n z z λ L2 (Bd ,µλ+2n) λ+2n = f, χz = f (z). L2 (Bd ,µ λ+2n)
d
This shows that the reproducing kernel for H(B , λ) is given by Kλ (z, w) = [(AB)−1 χλ+2n ](w). Using (2.5) repeatedly gives the desired result. z We conclude this section with a simple lemma that will be useful in Section 5. Lemma 5. For all λ1 , λ2 > 0, if f is in H(Bd , λ1 ) and g is in H(Bd , λ2 ) then f g is in H(Bd , λ1 + λ2 ). Proof. If, say, λ1 > d, then we have the following simple argument: 2 2 2 2
f g λ1 +λ2 = cλ1 +λ2 |f (z)| |g(z)| (1 − |z| )λ1 +λ2 dτ (z) Bd 2 2 2 2 ≤ cλ1 +λ2 g λ2 |f (z)| (1 − |z| )−λ2 (1 − |z| )λ1 +λ2 dτ (z) Bd
cλ +λ 2 2 = 1 2 f λ1 g λ2 . cλ1 Unfortunately, cλ1 +λ2 /cλ1 tends to infinity as λ1 approaches d from above, so we cannot expect this simple inequality to hold for λ1 < d. For any λ1 , λ2 > 0, choose n so that λ1 + n > d and λ2 + n > d. Then f g belongs to H(Bd , λ1 + λ2 ) provided that N n (f g) belongs to HL2 (Bd , λ1 + λ2 + 2n). But n n n N (f g) = (2.6) N k f N n−k g. k k=0
Using Theorem 4, it is easy to see that if f belongs to H(Bd , λ1 ) then N k f belongs to H(Bd , λ1 + 2k). Thus,
k
N f (z) 2 ≤ ak (1 − |z|2 )−(λ1 +2k) . Now, for each term in (2.6) with k ≤ n/2, we then obtain the following norm estimate:
k
N f (z)N n−k g(z) 2 (1 − |z|2 )λ1 +λ2 +2n dτ (z) cλ1 +λ2 +2n d B
2
n−k 2
N ≤ cλ1 +λ2 +2n ak g(z) (1 − |z| )λ2 +2n−2k dτ (z). (2.7) Bd
Since k ≤ n/2, we have λ2 + 2n − 2k ≥ λ2 + n > d. We are assuming that g is in H(Bd , λ2 ), so that N n−k g is in H(Bd , λ2 + 2n − 2k), which coincides with HL2 (Bd , µλ2 +2n−2k ). Thus, under our assumptions on f and g, each term in
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
63
(2.6) with k ≤ n/2 belongs to HL2 (Bd , λ1 + λ2 + 2n). A similar argument with the roles of f and g reversed takes care of the terms with k ≥ n/2.
3. Toeplitz operators with polynomial symbols In this section, we will consider our first examples of Toeplitz operators on generalized Bergman spaces, those whose symbols are (not necessarily holomorphic) polynomials. Such examples are sufficient to see some interesting new phenomena, that is, properties of ordinary Toeplitz operator that fail when extended to these generalized Bergman spaces. The definition of Toeplitz operators for the case of polynomial symbols is consistent with the definition we use in Section 4 for a larger class of symbols. For λ > d, we define the Toeplitz operator Tφ by Tφ f = Pλ (φf ) for all f in HL2 (Bd , µλ ) and all bounded measurable functions φ. Recall that Pλ is the orthogonal projection from L2 (Bd , τ ) onto the holomorphic subspace. Because Pλ is a self-adjoint operator on L2 (Bd , µλ ), the matrix entries of Tφ may be calculated as f1 , Tφ f2 HL2 (Bd ,µλ ) = f1 , φf2 L2 (Bd ,µλ ) ,
λ > d,
(3.1)
for all f1 , f2 ∈ HL2 (Bd , µλ ). From this formula, it is easy to see that Tφ¯ = (Tφ )∗ . If ψ is a bounded holomorphic function and φ is any bounded measurable function, then it is easy to see that Tφψ = Tφ Mψ . Thus, for any two multiindices m and n, we have Tz¯m zn = (Mzm )∗ (Mzn ).
(3.2)
We will take (3.2) as a definition for 0 < λ ≤ d. Our first task, then, is to show that Mzn is a bounded operator on HL2 (Bd , µλ ) for all λ > 0. Proposition 6. For all λ > 0 and all multi-indices n, the multiplication operator Mzn is a bounded operator on H(Bd , λ). Thus, for any polynomial φ, the Toeplitz operator Tφ defined in (3.2) is a bounded operator on H(Bd , λ). Proof. The result is a is a special case of a result of Arazy and Zhang [3] and also of the results of Section 4, but it is easy to give a direct proof. It suffices to show that Mzj is bounded for each j. Since Mzj preserves the orthogonality of the monomials, we obtain m Mzj = sup zj z λ = sup mj + 1 .
z m λ m m |m| + λ
Note that mj ≤ |m| with equality when mk = 0 for k = j. Thus the supremum is finite and is easily seen to have the value of 1 if λ ≥ 1 and 1/λ if λ < 1.
K. Chailuek and B. C. Hall
64
IEOT
We now record some standard properties of Toeplitz operators on (ordinary) Bergman spaces. These properties hold for Toeplitz operators (defined by the “multiply and project” recipe) on any holomorphic L2 space. We will show that these properties do not hold for Toeplitz operators with polynomial symbols on the generalized Bergman spaces H(Bd , λ), λ < d. Proposition 7. For λ > d and φ(z) bounded, the Toeplitz operator Tφ on the space HL2 (Bd , dµλ ), which is defined by Tφ f = Pλ (φf ), has the following properties. 1. Tφ ≤ supz |φ(z)| 2. If φ(z) ≥ 0 for all z, then Tφ is a positive operator. Both of these properties fail when λ < d. In fact, for λ < d, there is no constant C such that Tφ ≤ C supz |φ(z)| for all polynomials φ. As we remarked in the introduction, when λ = d, the space H(Bd , λ) may be identified with the Hardy space. Thus Properties 1 and 2 in the proposition still hold when λ = d, if, say, φ is continuous up to the boundary of Bd (or otherwise has a reasonable extension to the closure of Bd ). Proof. When λ > d, the projection operator Pλ has norm 1 and the multiplication operator Mφ has norm equal to supz |φ(z)| as an operator on L2 (Bd , µλ ). Thus, the restriction to HL2 (Bd , µλ ) of Pλ Mφ has norm at most supz |φ(z)|. Meanwhile, if φ is non-negative, then from (3.1) we see that f, Tφ f ≥ 0 for all f ∈ HL2 (Bd , µλ ). Let us now assume that 0 < λ < d. Computing on the orthogonal basis in Theorem 4, it is a simple exercise to show that Tz¯j zj (z m ) =
Γ(λ + |m|) (m + ej )! 1 + mj m zm = z . m! Γ(λ + |m| + 1) λ + |m|
(3.3)
If we take φ(z) = |z|2 , then summing (3.3) on j gives Tφ z m =
d + |m| m z . λ + |m|
Since λ < d, this shows that Tφ > 1, even though |φ(z)| < 1 for all z ∈ Bd . Thus, Property 1 fails for λ < d. (From this calculation it easily follows that 2 if φ(z) = (1 − |z| )/(λ − d), then Tφ is the bounded operator (λI + N )−1 , for all λ = d.) For the second property, we let ψ(z) = 1 − φ(z) = 1 − |z|2 which is positive. From the above calculation we obtain d + |m| Tψ z m , z m Hλ = z m 2Hλ −
z m 2H(Bd ,λ) , λ + |m| which is negative if 0 < λ < d.
Toeplitz Operators on Generalized Bergman Spaces
Vol. 66 (2010)
65
We now show that there is no constant C such that Tφ ≤ C supz |φ(z)|. Consider k
d 2 k 2 φk (z) := (|z| ) = |zi | i=1
k! k! (|z1 |2 )i1 (|z2 |2 )i2 · · · (|zd |2 )id = = zi zi. i! i! |i|=k
|i|=k
Computing on the orthogonal basis in Theorem 4 we obtain k! k! i!Γ(λ) k!Γ(λ) (Tzi zi 1) = 1=I 1, T φk 1 = i! i! Γ(λ + k) Γ(λ + k) |i|=k
|i|=k
where 1 is the constant function. Here, I is the of multi-indices i of number length d such that |i| = k, which is equal to k+d−1 . Thus d−1 T φk 1 =
k−1 d+j (d + k − 1) · · · (d) (k + d − 1)! Γ(λ) 1= 1= 1. (d − 1)! Γ(λ + k) (λ + k − 1) · · · (λ) λ+j j=0
. Since d > λ, the terms d−λ λ+j ∞ d+j are positive diverges. This implies j=0 λ+j = ∞. Since supz |φk (z)| = 1 for all k, there is no a constant C such that Tφ ≤ C supz |φ(z)|. k−1
d+j j=0 λ+j = ∞ d−λ and j=0 λ+j
Consider
k−1 j=0 1 +
d−λ λ+j
Remark 8. For λ < d, there does not exist any positive measure ν on Bd such that f λ = f L2 (Bd ,ν) for all f in H(Bd , λ). If such a ν did exist, then the argument in the first part of the proof of Proposition 7 would show that Properties 1 and 2 in the proposition hold.
4. Bounded Toeplitz operators In this section, we will consider a class of symbols φ for which we will be able to define a Toeplitz operator Tφ as a bounded operator on H(Bd , λ) for all λ > 0. Our definition of Tφ will agree (for the relevant class of symbols) with the usual “multiply and project” definition for λ > d. In light of the examples in the previous section, we cannot expect boundedness of φ to be sufficient to define Tφ as a bounded operator. Instead, we will consider functions φ for which φ and a certain number of derivatives of φ are bounded. Our strategy is to use integration by parts to give an alternative expression for the matrix entries of a Toeplitz operator with sufficiently regular symbol, in the case λ > d. We then take this expression as our definition of Toeplitz operator in the case 0 < λ ≤ d. Theorem 9. Assume λ > d and fix a positive integer n. Let φ be a function ¯ k N l φ is bounded that is 2n times continuously differentiable and for which N
K. Chailuek and B. C. Hall
66
for all 0 ≤ k, l ≤ n. Then
f, Tφ g HL2 (Bd ,µλ ) = cλ+2n
C
Bd
IEOT
λ+2n f (z)φ(z)g(z) 1 − |z|2 dτ (z)
for all f, g ∈ HL2 (Bd , µλ ), where C is the operator given by ¯ ¯ N N N N C= I+ ··· I + I+ ··· I + . (4.1) λ + 2n − 1 λ+n λ+n−1 λ Thus, there exist constants Ajklm (depending on n and λ) such that f, Tφ g HL2 (Bd ,µλ ) =
n
k l m ¯ N φ N g 2 d Ajklm N j f, N L (B ,µ
λ+2n )
. (4.2)
j,k,l,m=1
Proof. Assume at first that f and g are polynomials, so that f and g and all of their derivatives are bounded. We use (3.1) and apply the first equality in Lemma 1 with ψ = f¯φg. We then apply the first equality in the lemma again with ψ = (I +N/λ)[f¯φg]. We continue on in this fashion until we have applied the first equality in Lemma 1 n times and the second equality n times. This establishes the desired equality in the case that f and g are polynomials. For general f and g in HL2 (Bd , µλ ), we approximate by sequences fa and ga of polynomials. From Theorem 3 we can see that convergence of fa and ga in HL2 (Bd , µλ ) implies convergence of N j fa and N k ga to N j f and N k g, so that applying (4.2) to fa and ga and taking a limit establishes the desired result for f and g. Definition 10. Assume 0 < λ ≤ d and fix a positive integer n such that λ + 2n > d. Let φ be a function that is 2n times continuously differentiable ¯ k N l φ is bounded for all 0 ≤ k, l ≤ n. Then we define the and for which N Toeplitz operator Tφ to be the unique bounded operator on H(Bd , λ) whose matrix entries are given by λ+2n f, Tφ g H(Bd ,λ) = cλ+2n C f (z)φ(z)g(z) 1 − |z|2 dz, (4.3) Bd
where C is given by (4.1). Note that from Theorem 4, N j f and N m g belong to L2 (Bd , µλ+2n ) for all 0 ≤ j, m ≤ n, for all f and g in HL2 (Bd , µλ ). Furthermore, N j f 2 d and N m g 2 d are bounded by constants times L (B ,µλ+2n )
L (B ,µλ+2n )
f λ and g λ , respectively. Thus, the right-hand side of (4.3) is a continuous sesquilinear form on H(Bd , λ), which means that there is a unique bounded operator Tφ whose matrix entries are given by (4.3). If λ = d, then (as discussed in the introduction) the Hilbert space H(Bd , λ) is the Hardy space of holomorphic functions that are square-integrable over the boundary. In that case, the Toeplitz operator Tφ will be the zero operator whenever φ is identically zero on the boundary of Bd . If λ = d− 1, d− 2, . . . , then the inner product on H(Bd , λ) can be related to the inner product on the Hardy space. It is not hard to see that in these cases,
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
67
Tφ will be the zero operator if φ and enough of its derivatives vanish on the boundary of Bd . Let us consider the case in which φ(z) = ψ1 (z)ψ2 (z), where ψ1 and ψ2 are holomorphic functions such that the function and the first n derivatives are bounded. Then when applying C to f (z)φ(z)g(z), all the N -factors go ¯ -factors go onto f (z)ψ1 (z). Reonto the expression ψ2 (z)g(z) and all the N calling from Theorem 4 the formula for the inner product on H(Bd , λ), we see that f, Tφ g HL2 (Bd ,µλ ) = ψ1 f, ψ2 g H(Bd ,λ) , as expected. This means that in this case, Tψ¯1 ψ2 = (Mψ1 )∗ (Mψ2 ), as in the case λ > d. In particular, Definition 10 agrees with the definition we used in Section 3 in the case that φ is a polynomial in z and z¯.
5. Hilbert–Schmidt Toeplitz operators 5.1. Statement of results In this section, we will give sufficient conditions under which a Toeplitz operator Tφ can be defined as a Hilbert–Schmidt operator on H(Bd , λ). Specifically, if φ belongs to L2 (Bd , τ ) then Tφ can be defined as a Hilbert–Schmidt operator, provided that λ > d/2. Meanwhile, if φ belongs to L1 (Bd , τ ), then Tφ can be defined as a Hilbert–Schmidt operator for all λ > 0. In both cases, we define Tφ in such a way that for all bounded functions f and g in H(Bd , λ), we have f, Tφ g λ = cλ
Bd
f (z)φ(z)g(z)(1 − |z|2 )λ dτ (z),
(5.1)
where cλ is defined by cλ = Γ(λ)/(π d Γ(λ − d)). This expression is identical to (3.1) in the case λ > d. The value of cλ should be interpreted as 0 when λ−d = 0, −1, −2, . . .. This means that for φ in L2 (Bd , τ ) or L1 (Bd , τ ) (but not for other classes of symbols!), Tφ is the zero operator when λ = d, d − 1, . . . . This strange phenomenon is discussed in the next subsection. Note that we are not claiming Tφ = 0 for arbitrary symbols when λ = d, d − 1, . . . , but only for symbols that are integrable or square-integrable with respect to the hyperbolic volume measure τ. Such functions must have reasonable rapid decay (in an average sense) near the boundary of Bd . In the case φ ∈ L2 (Bd , τ ), the restriction λ > d/2 is easy to explain: the function (1 − |z|2 )λ belongs to L2 (Bd , τ ) if and only if λ > d/2. Thus, if f and g are bounded and φ is in L2 (Bd , τ ), then (5.1) is absolutely convergent for λ > d/2. In this subsection, we state our results; in the next subsection, we discuss some unusual properties of Tφ for λ < d; and in the last subsection of this section we give the proofs. We begin by considering symbols φ in L2 (Bd , τ ).
68
K. Chailuek and B. C. Hall
IEOT
Theorem 11. Fix λ > d/2 and let cλ = Γ(λ)/(π d Γ(λ − d)). (We interpret cλ to be zero if λ is an integer and λ ≤ d.) Then the operator Aλ given by λ (1 − |z|2 )(1 − |w|2 ) φ(w) dτ (w) Aλ φ(z) = c2λ ¯)(1 − w ¯ · z) Bd (1 − w · z is a bounded operator from L2 (Bd , τ ) to itself. Theorem 12. Fix λ > d/2. Then for each φ ∈ L2 (Bd , τ ), there is a unique Hilbert–Schmidt operator on H(Bd , λ), denoted Tφ , with the property that f, Tφ g λ = cλ f (z)φ(z)g(z)(1 − |z|2 )λ dτ (z) (5.2) Bd
for all bounded holomorphic functions f and g in H(Bd , λ). The Hilbert– Schmidt norm of Tφ is given by 2
Tφ HS = φ, Aλ φ L2 (Bd ,τ ) . If λ > d and φ ∈ L2 (Bd , τ ) ∩ L∞ (Bd , τ ), then the definition of Tφ in Theorem 12 agrees with the “multiply and project” definition; compare (3.1). Applying Lemma 5 with λ1 = λ2 = λ and λ > d/2, we see that for all 2 f and g in H(Bd , λ), the function z → f (z)g(z)(1 − |z| )λ is in L2 (Bd , τ ). This means that the integral on the right-hand side of (5.2) is absolutely convergent for all f, g ∈ H(Bd , λ). It is then not hard to show that (5.2) holds for all f, g ∈ H(Bd , λ). The operator Aλ coincides, up to a constant, with the Berezin transform. Let χλz (w) := Kλ (z, w) be the coherent state at the point z, which satisfies f (z) = χλz , f λ for all f ∈ H(Bd , λ). Then one standard definition of the Berezin transform Bλ is λ χz , Tφ χλz λ . Bλ φ = χλz , χλz λ The function Bλ φ may be thought of as the Wick-ordered symbol of Tφ , where Tφ is thought of as the anti-Wick-ordered quantization of φ. Using the formula (Theorem 4) for the reproducing kernel along with (5.2), we see χλz (w) is a bounded function of w for each fixed that Aλ = cλ Bλ . (Note that d λ λ z ∈ B and that χz , χz λ = Kλ (z, z).) Note that τ is an infinite measure, which means that if φ is in L2 (Bd , τ ) 1 or L (Bd , τ ), then φ must tend to zero at the boundary of Bd , at least in an average sense. This decay of φ is what allows (5.2) to be a convergent integral. If, for example, we want to take φ(z) ≡ 1, then we cannot use (5.2) to define Tφ , but must instead use the definition from Section 3 or Section 4. Note also that the space of Hilbert–Schmidt operators on H(Bd , λ) may be viewed as the quantum counterpart of L2 (Bd , τ ). It is thus natural to investigate the question of when the Berezin–Toeplitz quantization maps L2 (Bd , τ ) into the Hilbert–Schmidt operators. We now show that if one considers a symbol φ in L1 (Bd , τ ), then one obtains a Hilbert–Schmidt Toeplitz operator Tφ for all λ > 0.
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
69
Theorem 13. Fix λ > 0 and let cλ be as in Theorem 12. Then for each φ ∈ L1 (Bd , τ ), there exists a unique Hilbert–Schmidt operator on H(Bd , λ), denoted Tφ , with the property that f (z)φ(z)g(z)(1 − |z|2 )λ dτ (z) (5.3) f, Tφ g λ = cλ Bd
for all bounded holomorphic functions f and g in H(Bd , λ). The Hilbert– Schmidt norm of Tφ satisfies
Tφ HS ≤ cλ φ L1 (Bd ,τ ) . Using the pointwise bounds on elements of H(Bd , λ) coming from the reproducing kernel, we see immediately that for all f, g ∈ H(Bd , λ), the function z → f (z)g(z)(1−|z|2 )λ is bounded. It is then not hard to show that (5.3) holds for all f, g ∈ H(Bd , λ). We have already remarked that the definition of Tφ given in this section agrees with the “multiply and project” definition when λ > d (and φ is bounded). It is also easy to see that the definition of Tφ given in this section agrees with the one in Section 4, when φ falls under the hypotheses of both Definition 10 and either Theorem 12 or Theorem 13. For some positive integer n, consider the set of λ’s for which λ + 2n > d and λ > d/2, i.e., λ > ¯ l φ is max(d−2n, d/2). Now suppose that φ belongs to L2 (Bd , τ ) and that N k N bounded for all 0 ≤ k, l ≤ n. It is easy to see that the matrix entries f, Tφ g λ depend real-analytically on λ for fixed polynomials f and g, whether Tφ is defined by Definition 10 or by Theorem 12. For λ > d, the two matrix entries agree because both definitions of Tφ agree with the “multiply and project” definition. The matrix entries therefore must agree for all λ > max(d − 2n, d/2). Since polynomials are dense in H(Bd , λ) and both definitions of Tφ give bounded operators, the two definitions of Tφ agree. The same reasoning shows agreement of Definition 10 and Theorem 13. 5.2. Discussion Before proceeding on with the proof, let us make a few remarks about the way we are defining Toeplitz operators in this section. For λ > d, cλ is the normalization constant that makes the measure µλ a probability measure, which can be computed to have the value Γ(λ)/(π d Γ(λ − d)). For λ ≤ d, although the measure (1 − |z|2 )λ dτ (z) is an infinite measure, we simply use the same formula for cλ in terms of the gamma function. We understand this to mean that cλ = 0 whenever λ is an integer in the range (0, d]. It also means that cλ is negative when d − 1 < λ < d and when d − 3 < λ < d − 2, etc. In the cases where cλ = 0, we have that Tφ = 0 for all φ in L1 (Bd , τ ) 2 or L (Bd , τ ). This first occurs when λ = d. Recall that for λ = d, the space H(Bd , λ) can be identified with the Hardy space of holomorphic functions square-integrable over the boundary. Meanwhile, having φ being integrable or square-integrable with respect to τ means that φ tends to zero (in an average sense) at the boundary, in which case it is reasonable that Tφ should be zero as an operator on the Hardy space. For other integer values of λ ≤ d, the
K. Chailuek and B. C. Hall
70
IEOT
inner product on H(Bd , λ) can be expressed using the methods of Section 2 in terms of integration over the boundary, but involving the functions and their derivatives. In that case, we expect Tφ to be zero if φ has sufficiently rapid decay at the boundary, and it is reasonable to think that having φ in L1 or L2 with respect to τ constitutes sufficiently rapid decay. Note, however, that the conclusion that Tφ = 0 when cλ = 0 applies only when φ is in L1 or L2 ; for other classes of symbols, such as polynomials, Tφ is not necessarily zero. For example, Tzm is equal to Mzm , which is certainly a nonzero operator on H(Bd , λ), for all λ > 0. Meanwhile, if cλ < 0, then we have the curious situation that if φ is positive and in L1 or L2 with respect to τ, then the operator Tφ is actually a negative operator. This is merely a dramatic example of a phenomenon we have already noted: for λ < d, non-negative symbols do not necessarily give rise to non-negative Toeplitz operators. Again, though, the conclusion that Tφ is negative for φ positive applies only when φ belongs to L1 or L2 . For example, the constant function 1 always maps to the (positive!) identity operator, regardless of the value of λ.
5.3. Proofs As motivation, we begin by computing the Hilbert–Schmidt norm of Toeplitz operators in the case λ > d. For any bounded measurable φ, we extend the Toeplitz operator Tφ to all of L2 (Bd , µλ ) by making it zero on the orthogonal complement of the holomorphic subspace. This extension is given by the formula Pλ Mφ Pλ . Then the Hilbert–Schmidt norm of the operator Tφ on HL2 (Bd , µλ ) is the same as the Hilbert–Schmidt norm of the operator Pλ Mφ Pλ on L2 (Bd , µλ ). Since Pλ is computed as integration against the reproducing kernel, we may compute that Pλ Mφ Pλ f (z) = where
Bd
Kφ (z, w)f (w) dµλ (w),
Kφ (z, w) =
Bd
K(z, u)φ(u)K(u, w) dµλ (u).
If we can show that Kφ is in L2 (Bd × Bd , µλ × µλ ), then it will follow by a standard result that Tφ is Hilbert–Schmidt, with Hilbert–Schmidt norm equal to the L2 norm of Kφ . For sufficiently nice φ, we can compute the L2 norm of Kφ by rearranging the order of integration and using twice the repro ducing identity K(z, w)K(w, u) dµλ (w) = K(z, u). (This identity reflects that Pλ2 = Pλ .) This yields Bd ×Bd
2
|Kφ (z, w)| dµλ (z) dµλ = φ, Aφ L2 (Bd ,τ ) ,
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
71
where Aλ is the integral operator given by 2 |K(z, w)| (1 − |z|2 )λ (1 − |w|2 )λ φ(w) dτ (w) Aλ φ(z) = c2λ Bd
=
c2λ
Bd
(1 − |z|2 )(1 − |w|2 ) (1 − w ¯ · z)(1 − z¯ · w)
λ φ(w) dτ (w).
(5.4)
In the case d/2 < λ ≤ d, it no longer makes sense to express Tφ as Pλ Mφ Pλ . Nevertheless, we can consider an operator Aλ defined by (5.4). Our goal is to show that for all λ > d/2, (1) Aλ is a bounded operator on L2 (Bd , τ ) and (2) if we define Tφ by (5.1), then the Hilbert–Schmidt norm of Tφ is given by φ, Aλ φ L2 (Bd ,τ ) . We will obtain similar results for all λ > 0 if φ ∈ L1 (Bd , τ ). Proof of Theorem 11. We give two proofs of this result; the first generalizes more easily to other bounded symmetric domains, whereas the second relates Aλ to the Laplacian for Bd (compare [13]). First Proof. We let λ (1 − |z|2 )(1 − |w|2 ) Fλ (z, w) = c2λ ; (1 − w ¯ · z)(1 − z¯ · w) i.e., Fλ is the integral kernel of the operator Aλ . A key property of Fλ is its invariance under automorphisms: Fλ (ψ(z), ψ(w)) = Fλ (z, w) for each automorphism (biholomorphism) ψ of Bd and all z, w ∈ Bd . To establish the invariance of Fλ , let (5.5) fλ (z) = c2λ (1 − |z|2 )λ . According to Lemma 1.2 of [33], Fλ (z, w) = fλ (φw (z)), where φw is an automorphism of Bd taking 0 to w and satisfying φ2w = I. Now, if ψ is any automorphism, the classification of automorphisms (Theorem 1.4 of [33]) implies that ψ ◦ φw = φψ(w) ◦ U for some unitary matrix U. From this we can obtain φψ(w) = U ◦ φw ◦ ψ −1 , and so fλ (φψ(w) (ψ(z))) = fλ (U (φw (ψ −1 (ψ(z)))) = fλ (φw (z)), i.e., Fλ (ψ(z), ψ(w)) = Fλ (z, w). The invariance of Fλ under automorphisms means that Aλ φ can be thought of as a convolution (over the automorphism group P SU (d, 1)) of φ with the function fλ . What this means is that Aλ φ(z) = fλ (gh−1 · 0)φ(h · 0) dh, G
where g ∈ G is chosen so that g · 0 = z. Here G = P SU (d, 1) is the group of automorphisms of Bd (given by fractional linear transformations) and dh is an appropriately normalized Haar measure on G. Furthermore, L2 (Bd , τ ) can be identified with the right-K-invariant subspace of L2 (G, dg), where K := U (d) is the stabilizer of 0. If λ > d, then fλ is in L1 (Bd , τ ), in which case it is easy to prove that Aλ is bounded; see, for example, Theorem 2.4 in [5]. This argument does not
72
K. Chailuek and B. C. Hall
IEOT
work if λ ≤ d. Nevertheless, if λ > d/2, an easy computation shows that fλ belongs to L2 (Bd , τ ) and also to Lp (Bd , τ ) for some p < 2. We could at this point appeal to a general result known as the Kunze–Stein phenomenon [24]. The result states that on connected semisimple Lie groups G with finite center (including P SU (d, 1)), convolution with a function in Lp (G, dg), p < 2, is a bounded operator from L2 (G, dg) to itself. (See [11] for a proof in this generality.) However, the proof of this result is simpler in the case we are considering, where the function in Lp (G, dg) is bi-K-invariant and the other function is right-K-invariant. (In our case, the function in Lp (G, dg) is the function g → fλ (g · 0) and the function in L2 (G, dg) is g → φ(g · 0).) Using the Helgason Fourier transform along with its behavior under convolution with a bi-K-invariant function ([19, Lemma III.1.4]), we need only show that the spherical Fourier transform of fλ is bounded. (Helgason proves Lemma III.1.4 under the assumption that the functions are continuous and of compact support, but the proof also applies more generally.) Meanwhile, standard estimates show that for every ε > 0, the spherical functions are in L2+ε (G/K), with L2+ε (G/K) norm bounded independent of the spherical function. (Specifically, in the notation of [18, Sect. IV.4], for all λ ∈ a∗ , we have |φλ (g)| ≤ φ0 (g), and estimates on φ0 (e.g., [1, Prop. 2.2.12]) show that φ0 is in L2+ε for all ε > 0.) Choosing ε so that 1/p + 1/(2 + ε) = 1 establishes the desired boundedness. Second proof. If cλ = 0 (i.e., if λ ∈ Z and λ ≤ d), then there is nothing to prove. Thus we assume cλ is nonzero, in which case cλ+1 is also nonzero. The invariance of Fλ under automorphisms together with the square-integrability of the function (1 − |z|2 )λ for λ > d/2 show that the integral defining Aλ f (z) is absolutely convergent for all z. We introduce the (hyperbolic) Laplacian ∆ for Bd , given by ∆ = (1 − |z|2 )
d
(δjk − z¯j zk )
j,k=1
∂2 . ∂ z¯j ∂zk
(5.6)
(This is a negative operator.) This operator commutes with the automorphisms of Bd . It is known (e.g., [28]) that ∆ is an unbounded self-adjoint operator on L2 (Bd , τ ), on the domain consisting of those f ’s in L2 (Bd , τ ) for which ∆f in the distribution sense belongs to L2 (Bd , τ ). In particular, if f ∈ L2 (Bd , τ ) is C 2 and ∆f in the ordinary sense belongs to L2 (Bd , τ ), then f ∈ Dom(∆). We now claim that ∆z Fλ (z, w) = λ(λ − d)(Fλ (z, w) − Fλ+1 (z, w)),
(5.7)
where ∆z indicates that ∆ is acting on the variable z with w fixed. Since ∆ commutes with automorphisms, it again suffices to check this when w = 0, in which case it is a straightforward algebraic calculation. Suppose, then, that φ is a C ∞ function of compact support. In that case, we are free to differentiate
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
73
under the integral to obtain ∆Aλ φ = λ(λ − d)Aλ φ − λ(λ − d)Aλ+1 φ. 2
(5.8)
d
Now, the invariance of Fλ tells us that L (B , τ ) norm of Fλ (z, w) as a function of z is finite for all w and independent of w. Putting the L2 norm inside the integral then shows that Aλ φ and Aλ+1 φ are in L2 (Bd , τ ). This shows that Aλ φ is in Dom(∆). Furthermore, the condition λ > d/2 implies that λ(λ − d/2) > −d2 /4. It is known that the L2 spectrum of ∆ is (−∞, −d2 /4]. For general symmetric space of the noncompact type, the 2 L2 spectrum of the Laplacian is (−∞, − ρ ], where ρ is half the sum of the positive (restricted) roots for G/K, counted with their multiplicity. In our case, there is one positive root α with multiplicity (2d − 2) and another positive root 2α with multiplicity 1. (See the entry for “A IV” in Table VI of Chapter X of [17].) Thus, ρ = dα. It remains only to check that if the metric is normalized so that the Laplacian comes out as in (5.6), then α 2 = 1/4. This is a straightforward but unilluminating computation, which we omit. Since λ(λ − d) is in the resolvent set of ∆, we may rewrite (5.8) as Aλ φ = −λ(λ − d)[∆ − λ(λ − d)I]−1 Aλ+1 φ. Suppose now that λ + 1 > d, so that (as remarked above) Aλ+1 is bounded. Since [∆ − cI]−1 is a bounded operator for all c in the resolvent of ∆, we see that Aλ has a bounded extension from Cc∞ (Bd ) to L2 (Bd , τ ). Since the integral computing Aλ φ(z) is a continuous linear functional on L2 (Bd , τ ) (integration against an element of L2 (Bd , τ )), it is easily seen that this bounded extension coincides with the original definition of Aλ . The above argument shows that Aλ is bounded if λ > d/2 and λ+1 > d. Iteration of the argument then shows boundedness for all λ > d/2. Proof of Theorem 12. We wish to show that for all λ > d/2, if φ is in L2 (Bd , τ ), then there is a unique Hilbert–Schmidt operator Tφ with matrix entries given in (5.1) for all polynomials, and furthermore, Tφ 2HS = φ, Aλ φ λ . At the beginning of this section, we had an calculation of Tφ in terms of Aλ , but this argument relied on writing Tφ as Pλ Mφ Pλ , which does not make sense for λ ≤ d. We work with an orthonormal basis for H(Bd , λ) consisting of normalized monomials, namely, Γ(λ + |m|) em (z) = z m , m!Γ(λ) for each multi-index m. Then we want to establish the existence of a Hilbert– Schmidt operator whose matrix entries in this basis are given by el (z)φ(z)em (z)(1 − |z|2 )λ dτ (z). (5.9) alm := cλ Bd
There will exist a unique such operator provided that
l,m
|alm |2 < ∞.
K. Chailuek and B. C. Hall
74
IEOT
If we assume, for the moment, that Fubini’s Theorem applies, we obtain Γ(λ + |l|) Γ(λ + |m|) l l m m 2 2 z¯ w z w |alm | = cλ ¯ l!Γ(λ) m!Γ(λ) d d B B l,m
l,m
× φ(z)φ(w)(1 − |z|2 )λ (1 − |w|2 )λ dτ (z) dτ (w),
(5.10)
where l and m range over all multi-indices of length d. We now apply the binomial series ∞ 1 λ+k−1 k = r k (1 − r)λ k=0
for r ∈ C with |r| < 1, where λ+k−1 Γ(λ + k) . = k k!Γ(λ) (This is the so-called negative binomial series.) We apply this with r = z ¯ w j j , and we then apply the (finite) multinomial series to the compuj tation of (¯ z · w)k . The result is that Γ(λ + |l|) 1 z¯l wl = , (5.11) l!Γ(λ) (1 − z¯ · w)λ l
where the sum is over all multi-indices l. Applying this result, (5.10) becomes 2 |alm | = φ, Aλ φ λ , (5.12) l,m
which is what we want to show. Assume at first that φ is “nice,” say, continuous and supported in a ball of radius r < 1. This ball has finite measure and φ is bounded on it. Thus, if we put absolute values inside the sum and integral on the right-hand side of (5.10), finiteness of the result follows from the absolute convergence of the series (5.11). Thus, Fubini’s Theorem applies in this case. Now for a general φ ∈ L2 (Bd , τ ), choose φj converging to φ with φj “nice.” Then (5.12) tells us that Tφj is a Cauchy sequence in the space of Hilbert–Schmidt operators, which therefore converges in the Hilbert–Schmidt norm to some operator T. The matrix entries of Tφj in the basis {em } are by construction given by the integral in (5.9). The matrix entries of T are the limit of the matrix entries of Tφj , hence also given by (5.9), because el and em are bounded and (1 − |z|2 )λ belongs to L2 (Bd , τ ) for λ > d/2. We can now establish that (5.2) in Theorem 12 holds for all bounded holomorphic functions f and g in H(Bd , λ) by approximating these functions by polynomials. Proof of Theorem 13. In the proof of Theorem 12, we did not use the assumption λ > d/2 until the step in which we approximated arbitrary functions in L2 (Bd , τ ) by “nice” functions. In particular, if φ is nice, then (5.9) makes
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
75
sense for all λ > 0, and (5.12) still holds. Now, since Fλ (z, w) = fλ (φw (z)), where fλ is given by (5.5), we see that |Fλ (z, w)| ≤ c2λ for all z, w ∈ Bd . Thus, 2
φ, Aλ φ λ ≤ c2λ φ L1 (Bd ,τ ) for all nice φ. An easy approximation argument then establishes the existence of a Hilbert–Schmidt operator with the desired matrix entries for all φ ∈ L1 (Bd , τ ), with the desired estimate on the Hilbert–Schmidt norm.
Acknowledgment The authors thank M. Engliˇs for pointing out to them several useful references and B. Driver for useful suggestions regarding the results in Section 4. This article is an expansion of the Ph.D. thesis of the first author, written under the supervision of the second author. We also thank the referee for helpful comments and corrections.
References [1] J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces. Geom. Funct. Anal. 9 (1999), 1035-1091. [2] J. Arazy, Integral formulas for the invariant inner products in spaces of analytic functions on the unit ball. Function spaces (Edwardsville, IL, 1990), 9–23, Lecture Notes in Pure and Appl. Math., 136, Dekker, New York, 1992. [3] J. Arazy and G. Zhang, Homogeneous multiplication operators on bounded symmetric domains. J. Funct. Anal. 202 (2003), 44–66. [4] F. Beatrous Jr. and J. Burbea, Holomorphic Sobolev spaces on the ball. Dissertationes Math. (Rozprawy Mat.) 276 (1989), 60 pp. [5] F. A. Berezin, Quantization. Math. USSR Izvestija, 8 (1974), 1109-1165. [6] F. A. Berezin, Quantization in complex symmetric spaces. Math. USSR Izvestija 9 (1976), 341-379. [7] M. Bordemann, E. Meinrenken, and M. Schlichenmaier, Toeplitz quantization of K¨ ahler manifolds and gl(N ), N → ∞ limits. Comm. Math. Phys. 165 (1994), 281–296. [8] D. Borthwick, A. Lesniewski, and H. Upmeier, Nonperturbative deformation quantization of Cartan domains. J. Funct. Anal. 113 (1993), 153–176. [9] D. Borthwick, T. Paul, and A. Uribe, Legendrian distributions with applications to relative Poincar´e series. Invent. Math. 122 (1995), 359–402. [10] L. A. Coburn, Deformation estimates for the Berezin-Toeplitz quantization. Comm. Math. Phys. 149 (1992), 415–424. [11] M. Cowling, The Kunze–Stein phenomenon. Ann. Math. 107 (1978), 209-234. [12] P. Duren and A. Schuster, Bergman spaces. Mathematical surveys and monographs; no.100, American Mathematical Society, 2004.
76
K. Chailuek and B. C. Hall
IEOT
[13] M. Engliˇs, Berezin transform and the Laplace-Beltrami operator. Algebra i Analiz 7 (1995), 176–195; translation in St. Petersburg Math. J. 7 (1996), 633–647. [14] G. Folland, Harmonic analysis in phase space. Princeton University Press, 1989. [15] B. C. Hall, Holomorphic methods in analysis and mathematical physics. In: First Summer School in Analysis and Mathematical Physics (S. P´erez-Esteva and C. Villegas-Blas, Eds.), 1–59, Contemp. Math., 260, Amer. Math. Soc., 2000. [16] B. C. Hall and W. Lewkeeratiyutkul, Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform. J. Funct. Anal. 217 (2004), 192–220. [17] S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Academic Press, 1978. [18] S. Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, corrected reprint of the 1984 edition. Amer. Math. Soc., 2000. [19] S. Helgason, Geometric analysis on symmetric spaces. Amer. Math. Soc., 1994. [20] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces. Springer-Verlag, 2000. [21] H. T. Kaptano˘ glu, Besov spaces and Bergman projections on the ball. C. R. Math. Acad. Sci. Paris 335 (2002), 729–732. [22] S. Klimek and A. Lesniewski, Quantum Riemann surfaces I. The unit disc. Comm. Math. Phys., 46 (1976) 103-122. [23] A. Konechny, S. G. Rajeev, and O. T. Turgut, Classical mechanics on Grassmannian and disc. In: Geometry, integrability and quantization (Varna, 2000), 181–207, Coral Press Sci. Publ., Sofia, 2001. [24] R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the 2 × 2 unimodular group. Amer. J. Math. 82 (1960), 1-62. [25] S. G. Rajeev and O. T. Turgut, Geometric quantization and two-dimensional QCD. Comm. Math. Phys. 192 (1998), 493–517. [26] J. H. Rawnsley, Coherent states and K¨ ahler manifolds. Quart. J. Math. Oxford Ser. (2) 28 (1977), 403–415. [27] J. H. Rawnsley, M. Cahen, S. Gutt, Quantization of K¨ ahler manifolds. I. Geometric interpretation of Berezin’s quantization. J. Geom. Phys. 7 (1990), 45–62. [28] R. Strichartz, Harmonic analysis as spectral theory of Laplacians. J. Funct. Anal. 87 (1989), 51–148. [29] A. Unterberger and H. Upmeier The Berezin transform and invariant differential operators. Comm. Math. Phys, 164 (1994), 563-597. [30] Z. Yan, Invariant differential operators and holomorphic function spaces. J. Lie Theory 10 (2000), 31 pp. [31] R. Zhao and K. H. Zhu, Theory of Bergman spaces in the unit ball of Cn . Preprint, arxiv.org/abs/math/0611093.
Vol. 66 (2010)
Toeplitz Operators on Generalized Bergman Spaces
77
[32] K. Zhu, Holomorphic Besov spaces on bounded symmetric domains. Quart. J. Math. Oxford Ser. (2) 46 (1995), 239–256. [33] K. Zhu, Spaces of holomorphic functions in the unit ball. Springer-Verlag, 2004. Kamthorn Chailuek Department of Mathematics Prince of Songkla University Hatyai Songkhla 90112 Thailand e-mail:
[email protected] Brian C. Hall Department of Mathematics University of Notre Dame 255 Hurley Building Notre Dame, IN 46556-4618 USA e-mail:
[email protected] Submitted: January 21, 2009. Revised: July 10, 2009.
Integr. Equ. Oper. Theory 66 (2010), 79–112 DOI 10.1007/s00020-009-1733-7 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Interpolation of Banach Lattices and Factorization of p-Convex and q-Concave Operators Yves Raynaud and Pedro Tradacete ˇ Abstract. We extend a result of Sestakov to compare the complex inon-Lozanovskii’s construction terpolation method [X0 , X1 ]θ with Calder´ X01−θ X1θ , in the context of abstract Banach lattices. This allows us to prove that an operator between Banach lattices T : E → F which is p-convex and q-concave, factors, for any θ ∈ (0, 1), as T = T2 T1 , where p q )-convex and T1 is ( 1−θ )-concave. T2 is ( θ+(1−θ)p Mathematics Subject Classification (2010). Primary 47B60; Secondary 46B42, 46B70. Keywords. Interpolation of Banach lattices, method of Calder´ on-Lozanovskii, factorization, p-convex operator, q-concave operator.
1. Introduction In [10], J. L. Krivine showed that the composition T2 T1 of a p-convex operator T1 : X → E and a p-concave operator T2 : E → Y , where X, Y are Banach spaces and E is a Banach lattice, factors always through a space Lp (µ). Motivated by this fact, in this note we study factorization properties of pconvex and q-concave operators. More precisely, we consider the following question: if an operator between Banach lattices T : E → F is both p-convex and q-concave, does it necessarily factor as T = T2 T1 where T2 is p-convex and T1 q-concave? Note that such a product is always p-convex and q-concave, hence we are interested in a converse statement. In general, the answer to this question is negative (see Examples 4 and 5). However, we show that for every θ ∈ (0, 1), the operator T can be
The second author was partially supported by grants MICINN MTM2008-02652, AP-20044841, Grupo UCM 910346 and Santander/Complutense PR34/07-15837.
80
Y. Raynaud and P. Tradacete
IEOT
p q written as T = T2 T1 where T2 is ( θ+(1−θ)p )-convex and T1 is ( 1−θ )-concave (see Theorem 15). To prove this fact, we exhibit first a canonical way in which a p-convex (respectively q-concave) operator factors through a p-convex (resp. q-concave) Banach lattice. Afterwards, we present some interpolation results regarding the complex interpolation method and the Calder´ on-Lozanovskii construction for Banach lattices. In particular, we prove a comparison theorem between these two constructions that had been apparently known in the literature only in the case of Banach lattices of measurable functions. ˇ Thus, we extend this comparison theorem due to Sestakov (see [19]) to the more general setting of compatible pairs of Banach lattices which need not be function spaces (that is, ideals in the space of measurable functions on some measure space). This will constitute a key ingredient in our proof of the main factorization result.
The problem of factoring an operator through p-convex and q-concave operators had also been considered, although in a quite different manner, by S. Reisner in [17]; in particular, Theorem 1 was essentially proved in [17, Sec. 2, Lemma 6]. Moreover, this author showed that for fixed p, q, the class of operators between Banach spaces T : E → F such that the composition with the canonical inclusion jF : F → F ∗∗ factors as jF T = U V with V pconvex and U q-concave, forms an operator ideal. However, his approach to an analogous statement of Theorem 1 for p-convex operators is not satisfactory for our interests, because in [17] this is only considered as a dual fact to that of factoring q-concave operators, and, as we will show in Section 3, these factorizations do not behave in an entirely dual way. Moreover, from Theorem 1 we can only get that for a p-convex operator T : E → F , the bi-adjoint T ∗∗ : E ∗∗ → F ∗∗ factors through a p-convex Banach lattice, which suffices for the purposes in [17], but are not enough to prove our main result on factorization (Theorem 15). We mention that our proofs of Theorems 1 and 3 have been inspired in fact by the work of P. Meyer-Nieberg in [14] on factorization of cone p-summing and p-majorizing operators (see also [15, 2.8]). Then we realized that some of the main ideas of our work were already present in the paper [17]. The organization of the paper goes as follows. Section 2 contains the proofs of the basic factorizations for p-convex (resp. q-concave) operators. It is also shown that these constructions can be equivalently obtained by means of maximality properties of factorization diagrams. The next section, Section 3, is devoted to the study of the duality relation between the factorization spaces for p-convex and q-concave operators. Next, Section 4 is ˇ mainly devoted to the proof of the extension of Sestakov’s result to compatible pairs of Banach lattices. Then, in Section 5 we prove the main theorem on factorization of operators which are both p-convex and q-concave. Here ˇ the extension of Sestakov’s result is used for interpolating operators which are not necessarily positive (at the difference of the situation in [17]) between Banach lattices which are perhaps not representable as ideal function spaces. In this section we show also how some examples can be used to see that in
Vol. 66 (2010)
Interpolation and Factorization of Operators
81
general the factorization cannot be improved much further. Finally, in Section 6 we show the connection between the constructions of the first section and the factorization theorem of Krivine. We refer the reader to [11], [15] and [18] for any unexplained terminology on Banach lattice theory, and to [4] and [9] for those of interpolation theory.
2. Two basic constructions Let E be a Banach lattice and X a Banach space. Recall that an operator T : E → X is q-concave for 1 ≤ q ≤ ∞, if there exists a constant M < ∞ so that 1q 1q n n q q , T xi ≤ M |xi | if 1 ≤ q < ∞, i=1
or
i=1
n |xi | max T xi ≤ M , 1≤i≤n
if q = ∞,
i=1
for every choice of vectors (xi )ni=1 in E (cf. [11, 1.d]). The smallest possible value of M is denoted by M(q) (T ). Similarly, an operator T : X → E is p-convex for 1 ≤ p ≤ ∞, if there exists a constant M < ∞ such that p1 p1 n n p p |T xi | xi , if 1 ≤ p < ∞, ≤M i=1
or
i=1
n |T xi | max xi , ≤ M 1≤i≤n
if p = ∞,
i=1
for every choice of vectors (xi )ni=1 in X. The smallest possible value of M is denoted by M (p) (T ). Recall that a Banach lattice is q-concave (resp. pconvex) whenever the identity operator is q-concave (resp. p-convex). The following result was essentially proved in [17, Sec. 2, Lemma 6]. However, we include a similar proof for completeness, since we will be using the explicit construction throughout. Theorem 1. Let E be a Banach lattice, X a Banach space and 1 ≤ q ≤ ∞. An operator T : E → X is q-concave if and only if there exist a q-concave Banach lattice V , a positive operator φ : E → V (in fact, a lattice homomorphism with dense image), and another operator S : V → X such that T = Sφ. E@ @@ @@ φ @@
T
V
/X > ~ ~~ ~ ~ ~~ S
82
Y. Raynaud and P. Tradacete
IEOT
Proof. Let us suppose q < ∞. The proof for the case q = ∞ is trivial because every Banach lattice is ∞-concave. However, the precise construction carried out here for q < ∞ has its analogue for q = ∞. For the “if” part, let (xi )ni=1 in E. Since V is q-concave and φ is positive, by [11, Prop. 1.d.9] we have n 1q 1 n q q , T xi q ≤ SM(q) (IV )φ |x | i i=1
i=1
which yields that T is q-concave. Now, for the other implication, given x ∈ E, let us consider q1 1q n n ρ(x) = sup T xi q : |xi |q ≤ |x| . i=1
i=1
If M(q) (T ) denotes the q-concavity constant of T , then for (xi )ni=1 in E, we have n q1 1q n q q . T xi ≤ M(q) (T ) |xi | i=1
i=1
In particular, for all x ∈ E
T x ≤ ρ(x) ≤ M(q) (T )x. Moreover, ρ is a lattice semi-norm on E. Indeed, for any x ∈ E and λ ≥ 0 it is clear that ρ(λx) = λρ(x). In order to prove the triangle inequality, let x, y ∈ E and z = |x| + |y|, and denote Iz ⊂ E the ideal generated by z in E, which is identified with a space C(K) in which z corresponds to the function identically one [18, II.7]. Now, for every ε > 0 let z1 , . . . , zn ∈ E n
1 |zi |q q ≤ |z| and such that i=1
ρ(z) ≤
n
T zi q
1q + ε.
i=1
Since x, y ∈ Iz , they correspond to functions f, g ∈ C(K) such that |f (t)| + |g(t)| = 1 for every t ∈ K. Similarly, zi corresponds to hi ∈ C(K) with n
1 q q ≤ 1 for every t ∈ K. Hence we can consider i=1 |hi (t)| fi (t) = hi (t)f (t), gi (t) = hi (t)g(t), which belong to C(K) and satisfy 1q 1q n n q q |fi (t)| ≤ |f (t)| and |gi (t)| ≤ |g(t)|. i=1
i=1
This means that we can consider (xi )ni=1 and (yi )ni=1 such that q1 q1 n n q q |xi | ≤ |x| and |yi | ≤ |y| i=1
i=1
Vol. 66 (2010)
Interpolation and Factorization of Operators
83
in E, with xi + yi = zi . Thus, it follows that ρ(x + y) ≤ ρ(x) + ρ(y) + ε, and since this holds for every ε > 0, the triangle inequality is proved. n q1 n q |xi | ≤ |y|, it Now, if |y| ≤ |x|, then for any (xi )i=1 such that i=1 1q 1q n n |xi |q ≤ |x|, hence for any such {xi }ni=1 , T xi q ≤ holds that i=1
i=1
ρ(x). This implies that ρ(y) ≤ ρ(x). Let V denote the Banach lattice obtained by completing E/ρ−1 (0) with the norm induced by ρ. Let φ denote the quotient map from E to E/ρ−1 (0), seen as a map to V . Now, for x ∈ E let us define S(φ(x)) = T (x). Since T x ≤ ρ(x), S is well defined and extends to a bounded operator S : V → X, such that T = Sφ. Now, let (xi )ni=1 in E. For every ε > 0 and for every i = 1, . . . , n there
1 ki i q q i exist {yji }kj=1 in E such that ≤ |xi | and j=1 |yj | ρ(xi )q = sup
k
T yj q :
k
j=1
|yj |q
1q
j=1
ki εq ≤ |xi | ≤ T yji q + , n j=1
for every i = 1, . . . , n. Therefore, we have n i=1
q
ρ(xi )
1q
≤ρ
n
|xi |
q
1q + ε,
i=1
and since this holds for every ε > 0, the normed lattice E/ρ−1 (0) is q-concave; hence, the same holds for its completion V . Since the lattice V constructed in the proof depends on the operator T : E → X and q, we will denote it by VT,q whenever needed. Similarly we will denote ρT for the expression defining the norm of VT,q . Remark 1. Note that VT,q has q-concavity constant one. In particular if E is q-concave and T = idE is the identity, then VT,q is the usual lattice renorming of E with q-concavity constant one. Remark 2. In [8], it was proved that an order weakly compact operator T : E → Y (i.e. T [−x, x] is relatively weakly compact for every x ∈ E+ ) always factors through an order continuous Banach lattice F . The Banach lattice F is constructed by means of the expression xF = sup{T y : |y| ≤ |x|}, for x ∈ E, which yields a Banach lattice in the usual way. Notice that if T : E → Y is q-concave, which implies being order weakly compact, then i xF ≤ ρT (x), hence we can consider a natural map VT,q → F such that we
84
Y. Raynaud and P. Tradacete
IEOT
can factor T as follows:
φ
VT,q
/Y O
T
E
T
/F
i
Moreover, F coincides with VT,∞ , so in a sense the previous Theorem is an extension of [8, Thm. I.2]. The factorization given in Theorem 1 is in a certain sense maximal, as the following Proposition shows. Proposition 2. Let T : E → X be a q-concave operator. Suppose that T factors through a q-concave Banach lattice V with factors A : E → V and B : V → X, such that A is a lattice homomorphism whose image is dense in V , and T = B ◦ A. Then there is a lattice homomorphism u : V → VT,q such that φ = u ◦ A and S ◦ u = B. T / E =NNN p7 @ X p == NNN p B ppp == NNAN ppp == NNN p p NN' ppp == == V S φ == == == u = VT,q
Proof. Let us define for x ∈ E, u(A(x)) = φ(x). Notice that, since A is a n 1q |xi |q ≤ |x|, we lattice homomorphism, for {xi }ni=1 in E, such that i=1
have n i=1
T xi
q
q1 =
n
BAxi
q
1q
≤ B
i=1
n
Axi
q
1q
i=1
1q n q ≤ BM(q) (IV ) |A(x )| i i=1
n q1 q = BM(q) (IV )A |xi | i=1
≤ BM(q) (IV )A(x). Therefore, u(A(x)) = φ(x) = ρT (x) ≤ BM(q) (IV )A(x). Since A has dense image, the preceding inequality implies that u can be extended to a bounded operator u : V → V(T,q) , which is clearly a lattice homomorphism and satisfies the required properties.
Vol. 66 (2010)
Interpolation and Factorization of Operators
85
There is an analogous version of Theorem 1 for p-convex operators, which could be considered, in a sense, as a predual construction to that given in Theorem 1 (see Section 3). Theorem 3. Let E be a Banach lattice, X a Banach space and 1 ≤ p ≤ ∞. An operator T : X → E is p-convex if and only if there exist a p-convex Banach lattice W , a positive operator (in fact, an injective interval preserving lattice homomorphism) ϕ : W → E and another operator R : X → W such that T = ϕR. T /E XB > BB } BB }} } B }} ϕ R BB }} W Proof. Let us suppose p < ∞. The proof for the case p = ∞ is analogous, with the usual changes. As in the proof of Theorem 1, [11, 1.d.9] yields one implication. For the non-trivial one, let T : X → E be p-convex. Let us consider the following set p1 k k |T xi |p , where xi p ≤ 1 and k ∈ N}. S = {y ∈ E : |y| ≤ i=1
i=1
We can consider the Minkowski functional defined by its closure S in E zW = inf{λ > 0 : z ∈ λS}. Clearly S is solid, and since T is p-convex, it is also a bounded set of E. Let us consider the space W = {z ∈ E : zW < ∞}. We claim that for any k
1/p z1 , . . . , zn in W , it holds that |zi |p belongs to W and i=1
p1 k p |z | i
≤
W
i=1
p1 n zi pW . i=1
Indeed, given z1 , . . . , zn in W , for every ε > 0 and for every i = 1, . . . , n there exist λi with zi ∈ λi S, such that εp p p λi ≤ inf µ : zi ∈ µS + , n for each i = 1, . . . , n. This means that for every i = 1, . . . , n, and for every δ > 0 there exists yiδ in E with zi − yiδ E ≤ δ, and m p1 i,δ i,δ p δ |T xj | , |yi | ≤ j=1
where
mi,δ {xi,δ j }j=1
satisfy m i,δ j=1
p xi,δ j
p1
≤ λi ,
86
Y. Raynaud and P. Tradacete
IEOT
for each i = 1, . . . , n, and each δ > 0. Now, for each δ > 0 let p1 n δ p wδ = |yi | . i=1
Notice that n n p1 p1 n p δ p |z | − w ≤ |z − y | ≤ zi − yiδ E ≤ nδ. i δ i i E
i=1
E
i=1
Moreover, note that for every δ > 0, wδ belongs to |wδ | =
n
|yiδ |p
p1
≤
i,δ n m
i=1
and
Hence,
n
|zi |
p
p1
i=1
n
p1 |zi |p i=1
p xi,δ j
∈
n
λpi i=1
≤
W n i=1
λpi
p1 S. Indeed,
i=1 p |T xi,δ j |
p1 ,
n
λpi
p1 .
i=1
p1
= inf{µ > 0 : ≤
p1
i=1 j=1
n
i=1 j=1
i,δ n m
i=1
S. Therefore, it follows that n
|zi |
p
p1
∈ µS} ≤
i=1
n
p
1
ε inf µp : zi ∈ µS + n
p
λpi
p1
i=1
≤
n
zi pW
p1
+ ε.
i=1
Since this holds for every ε > 0, we finally have n 1 p1 n p p p |z | ≤ z . i i W i=1
W
i=1
It follows that the Minkowski functional .W is a norm on W . Indeed since S is bounded, xW = 0 implies x = 0. Moreover if x, y ∈ W are |x| |y| xW yW , v = y , α = xW non zero, set u = x +yW , β = xW +yW . Since W W uW = vW = 1, α, β ≥ 0 and α + β = 1 we have x + yW ≤ |x| + |y| W = (xW + yW )αu + βvW ≤ (xW + yW )(αup + βv p )1/p W 1/p
≤ (xW + yW )(αupW + βvp )W = xW + yW Therefore, (W, · W ) is a p-convex normed lattice. We claim that W is complete, and hence a p-convex Banach lattice. Indeed, let (wi )∞ i=1 be a Cauchy sequence in W . Since for every z ∈ E it holds that zE ≤ M (p) (T )zW ,
Vol. 66 (2010)
Interpolation and Factorization of Operators
87
it follows that (wi )∞ i=1 is also a Cauchy sequence in E. Let w ∈ E be its limit. Notice that since wi are bounded in W , there exists some λ < ∞ such that wi ∈ λS for every i = 1, 2, . . . and since S is closed in E, we must have w ∈ λS. Thus, w belongs to W , and we will show that (wi )∞ i=1 converges to w also in W . To this end, let ε > 0. Since (wi )∞ is a Cauchy sequence, there i=1 exists N such that wi − wj ∈ εS when i, j ≥ N . Thus, if i ≥ N we can write w − wi = (w − wj ) + (wj − wi ) for every j ∈ N, and letting j → ∞ we obtain that w − wi ∈ εS. This shows that wi → w in W , and hence W is complete, as claimed. Clearly, by the definition of W we have T xW ≤ xX for every x ∈ X. Moreover, as noticed above it also holds that zE ≤ M (p) (T )zW for each z ∈ E, therefore the formal inclusion ϕ : W → E is clearly an injective interval preserving lattice homomorphism, and we have the following diagram T /E XB > BB } BB }} } B } R BB }} ϕ .} W
where R is defined by Rx = T x for x ∈ X. This finishes the proof.
As with the Banach lattice constructed in Theorem 1, we will denote by WT,p the Banach lattice obtained in the proof of Theorem 3. Remark 3. The operator ϕ : WT,p → E constructed in the proof is an injective, interval preserving lattice homomorphism. Moreover, it satisfies that the image of the closed unit ball ϕ(BWT ,p ) is a closed set in E. This let us introduce the class C consisting of operators T : E → F between Banach lattices which are injective, interval preserving lattice homomorphisms, such that the image of the closed unit ball T (BE ) is closed in F . The importance of this class will become clear next. Remark 4. Note that if T : X → E is p-convex, then it is also p -convex for every 1 ≤ p ≤ p. Hence, if we consider the factorization spaces WT,p and WT,p it always holds that WT,p → WT,p , with norm smaller than or equal to one. Indeed, this follows from the following two facts. First, the set p1 k k p S = {y ∈ E : |y| ≤ |T xi | , with xi p ≤ 1} i=1
i=1
88
Y. Raynaud and P. Tradacete
IEOT
can be equivalently described by p1 k k p S = {y ∈ E : |y| ≤ ai |T wi | , with wi ≤ 1, ai ≥ 0, ai = 1}. i=1
i=1
Furthermore, for 1 ≤ p ≤ p, and ai ≥ 0 with
k
ai = 1 it always holds that
i=1
k
ai |T wi |p
1 p
≤
k
i=1
ai |T wi |p
p1 .
i=1
Hence the unit ball of WT,p is contained in that of WT,p . Remark 5. WT,p has p-convexity constant equal to one. If E is already pconvex and T : E → E is the identity then WT,p is a renorming of E with p-convexity constant one. As for Proposition 2, the construction of Theorem 3 is in a sense minimal. Proposition 4. Let T : X → E be a p-convex operator, such that there exist and operators A : X → W and B : W →E a p-convex Banach lattice W with T = BA and B belonging to the class C. Then there exists an operator such that vR = A and Bv = ϕ. v : WT,p → W T / X >OOO oo7 @ E >> OOO o o B oo >> OOAO ooo >> OOO o o OO' >> ooo >> W ϕ O R >> >> >> v > WT,p
Proof. Let us define v. Let y ∈ WT,p with yWT ,p ≤ 1. By definition, there exists a sequence (yn )∞ n=1 in E such that yn → ϕ(y) in E, and for each n ∈ N, |yn | ≤ with
|T xni |p
p1
i=1
kn
n p i=1 xi X kn
≤ 1. Notice that since B is a lattice homomorphism
|T xni |p
p1
i=1
kn
=
kn i=1
p1
|BAxni |p
p1
=B
kn
|Axni |p
p1
,
i=1
. Hence, since B is interval preserving belongs to W
1 with |wn | ≤ kn |Axn |p p such that B(wn ) = yn . there exists wn ∈ W i i=1 where
kn i=1
|Axni |p
Vol. 66 (2010)
Interpolation and Factorization of Operators
89
is p-convex and kn xi p ≤ 1, for every n we have Notice that since W i=1 X kn
p1 |Axni |p wn W ≤ i=1
W
kn
p1 ≤ M (p) (IW Axni p ≤ M (p) (IW ) )A. i=1
under B is closed, and B(wn ) = yn Now, since the image of the unit ball of W with w ≤ M (p) (I )A such converge to ϕ(y) ∈ E, there exists w ∈ W W W that B(w) = ϕ(y). Moreover, this element is unique because B is injective. Let us define v(y) = w. is linear. It is clear, because of the injectivity of B, that v : WT,p → W Moreover, by the previous argument v is a bounded operator of norm less than or equal to M (p) (IW )A. It is clear by construction that B ◦ v = ϕ. Moreover, since B ◦ A = T = ϕ ◦ R = B ◦ v ◦ R and B is injective, we also get that A = v ◦ R as desired. This finishes the proof. Remark 6. Notice that the factorizations given in Theorems 1 and 3 also make sense in the more general context of quasi-Banach lattices and for pconvex or q-concave operators with p, q ∈ (0, ∞) (not necessarily p, q ≥ 1). It can be seen that in these cases, the factorization spaces are quasi-Banach lattices which need not be locally convex, except in the case when p ≥ 1.
3. Duality relations In this section we show the precise relation between the Banach lattices constructed in the proofs of Theorems 1 and 3. Namely we will prove the following Theorem 5. Let T : X → E be p-convex. By Theorem 3, T can be factored through WT,p ; moreover, since T ∗ : E ∗ → X ∗ is q-concave for 1p + 1q = 1 (see [11, Prop. 1.d.4]), T ∗ can also be factored through VT ∗ ,q . It holds that: 1. VT ∗ ,q is lattice isometric to a sublattice of (WT,p )∗ , 2. WT,p is lattice isometric to a sublattice of (VT ∗ ,q )∗ . Moreover, under this identifications VT ∗ ,q is always an ideal in (WT,p )∗ , and if E is order continuous WT,p is an ideal of (VT ∗ ,q )∗ . We need some preliminary lemmas first. ∗ . Lemma 6. Let E be a Banach lattice, x, y ∈ E+ , x ∧ y = 0, and z ∗ ∈ E+ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ There exist u , v in E+ such that z = u + v , u ∧ v = 0 and ∗
z , x = u∗ , x
z ∗ , y = v ∗ , y
Proof. By [15, Lemma 1.4.3], there exist ∗
z (x), u = z ∗ , u
z ∗ (x), u = 0 ∗
z (y), u = z ∗ , u
z ∗ (y), u = 0
∗ z ∗ (x) and z ∗ (y) in E+ such that
for all u ∈ Ex for all u ∈ {x}⊥ for all u ∈ Ey for all u ∈ {y}⊥
90
Y. Raynaud and P. Tradacete
IEOT
where Ex denotes the principal ideal generated by x in E, and {x}⊥ denotes the orthogonal complement of x (i.e. {x}⊥ = {u ∈ E : u ∧ x = 0}). Moreover, without loss of generality we can assume that z ∗ (x), z ∗ (y) ≤ ∗ z (simply consider z ∗ (x) ∧ z ∗ and z ∗ (y) ∧ z ∗ ), and that z ∗ (x) ∧ z ∗ (y) = 0 (consider z ∗ (x) − z ∗ (x) ∧ z ∗ (y) and z ∗ (y) − z ∗ (x) ∧ z ∗ (y)). Let then P be the band projection onto the band generated by z ∗ (x) in the Dedekind complete Banach lattice E ∗ and Q be the complementary band projection. Then set u∗ = P z ∗ , v ∗ = Qz ∗ . ∗ Lemma 7. Let E be a Banach lattice. For any z ∗ ∈ E+ , and x1 , . . . , xn ∈ E+ , n ∗ , such that z ∗ = x∗i , x∗i ∧ x∗j = 0 for i = j, and there exist x∗1 , . . . , x∗n in E+ i=1
z ∗ ,
n
xi =
i=1
n
x∗i , xi .
i=1
Proof. Given x, y ∈ E+ , Lemma 6 applied to x − x ∧ y and y − x ∧ y yields the result for n = 2. An easy induction on n completes the proof. Recall that given a Banach space X, the polar of a set A in X is the set A0 = {x∗ ∈ X ∗ : | x∗ , x| ≤ 1, ∀x ∈ A}. Similarly, for a set B in X ∗ , the prepolar of B is the set B0 = {x ∈ X : | x∗ , x| ≤ 1, ∀x∗ ∈ B}. Lemma 8. Let T : X → E be p-convex, and let p1 k k p S := {y ∈ E : |y| ≤ |T xi | , with xi p ≤ 1}. i=1
i=1
Since T ∗ : E ∗ → X ∗ is q-concave (with 1p + 1q = 1), we can consider ρT ∗ , the seminorm which induces the norm on VT ∗ ,q (see Theorem 1). Hence, we can also consider the convex set U := {y ∗ ∈ E ∗ : ρT ∗ (y ∗ ) ≤ 1}. Then 0
S = U, where S denotes the closure of S in E. Proof. First of all, we claim that S ⊂ U0 . Indeed, let y ∈ E be such that |y| ≤
n
|T xi |
p
p1 with
i=1
For every y ∗ ∈ E ∗ such that ρT ∗ (y ∗ ) ≤ 1, we have:
n
xi p ≤ 1.
i=1
| y ∗ , y| n p1 |T xi |p ≤ |y ∗ |, i=1 n n ∗ = |y |, sup ai T xi : |ai |q ≤ 1 i=1 i=1 N n n ∗ m m q |y |, : = sup ai T xi |ai | ≤ 1, m = 1, . . . , N, N ∈ N . m=1
i=1
i=1
Where we have made use of [15, Cor. 1.3.4.ii)] in the last step.
Vol. 66 (2010)
Interpolation and Factorization of Operators
91
∗ N Now, by Lemma 7, there exist (ym )m=1 pairwise disjoint elements of N ∗ ∗ such that |y | = m=1 ym and N n N n ∗ m = y |y ∗ |, am T x , a T x i i . i m i
∗ E+
m=1
m=1
i=1
Therefore, setting zi∗ =
N
i=1
∗ am i ym , we have
m=1
q1 p1 N n n n ∗ ∗ q p ≤ |y ∗ |, am T x T z x . i i i i m=1
i=1
Note that, since
i=1
∗ N (ym )m=1 n
i=1
are pairwise disjoint we have that
|zi∗ |q
1q
N
≤
∗ ym = |y ∗ |.
m=1
i=1
Since ρT ∗ (y ∗ ) ≤ 1, this implies that any y ∗ with ρT ∗ (y ∗ ) ≤ 1,
n i=1
T ∗ zi∗ q
1q
≤ 1. Therefore, for
| y ∗ , y| N q q1 n n ∗ m ∗ m q ai y m : |ai | ≤ 1, m ≤ N, N ∈ N ≤ sup T ≤ 1.
i=1
m=1
i=1
This means that y ∈ U0 . Since U0 is closed, this proves that S ⊆ U0 as claimed. 0 0 Therefore, it follows that (U0 )0 ⊆ S . So in particular, U ⊆ S . 0 0 Let us prove now the converse inclusion (S ⊆ U ). Given y ∗ ∈ S , we ∗ ∗ ∗ want to show that ρT ∗ (y ) ≤ 1. To this end, let y1 , . . . , yk be elements in E ∗ , such that 1q k |yi∗ |q ≤ |y ∗ |. i=1 0
0
Notice that since S is solid, then so is S . In particular, |y ∗ | ∈ S whenever 0 y∗ ∈ S . k Now, for every ε > 0 there exist x1 , . . . , xk in X, such that xi p ≤ 1, i=1
and
k
T ∗ yi∗ q
1 q
k ≤ | T ∗ yi∗ , xi | + ε.
i=1
i=1
Moreover, by [11, Prop. 1.d.2] we have 1q p1 k k k k ∗ ∗ ∗ ∗ q p | T yi , xi | = | yi , T xi | ≤
|yi | , |T xi | ≤1 i=1
i=1
i=1
i=1
92
Y. Raynaud and P. Tradacete
IEOT
0
0
because |y ∗ | ∈ S . Therefore, ρT ∗ (y ∗ ) ≤ 1 for every y ∗ ∈ S . This finishes the proof. 0
Remark 7. Note that the equality S = U proved above, yields in particular that U is weak*-closed. Hence, by the bipolar theorem it also holds that S = U0 . Remark 8. Moreover, notice that Lemma 8 also provides an alternative proof of the fact that the norm · W defined by S in Theorem 3, is a p-convex lattice norm. Now we can give the proof of Theorem 5. Proof of Theorem 5. We stick to the notation of Theorems 1 and 3. Let us consider the inclusion ϕ : WT,p → E. Hence, we also have ϕ∗ : E ∗ → (WT,p )∗ . Notice that for every y ∗ ∈ E ∗ we have ϕ∗ (y ∗ )(WT ,p )∗ = sup{ ϕ∗ (y ∗ ), y : yWT ,p ≤ 1} = sup{ y ∗ , ϕ(y) : y ∈ S} 0
= inf{λ > 0 : y ∗ ∈ λS } = ρT ∗ (y ∗ ), by Lemma 8. Thus ker ϕ∗ ⊃ ρ−1 T (0), which allows us to define A:
−→ (WT,p )∗ E ∗ /ρ−1 T ∗ (0) −1 ∗ y + ρT ∗ (0) −→ ϕ∗ (y ∗ )
Moreover, A can be extended to an isometry from VT ∗ ,q into (WT,p )∗ . Furthermore, since the unit ball of WT,p is a solid subset of E, then ϕ + is interval preserving (i.e. ϕ([0, x]) = [0, ϕ(x)] for x ∈ WT,p ). Thus, ϕ∗ is a lattice homomorphism (cf. [1, Theorem 1.35]). Now, for v ∈ VT ∗ ,q , we can consider a sequence (yn∗ ) in E ∗ such that lim yn∗ + ρ−1 T ∗ (0) = v in VT ∗ ,q . Hence, n
we have
∗ ∗ A(|v|) = lim A(|yn∗ | + ρ−1 T ∗ (0)) = lim ϕ (|yn |) n
n
= lim |ϕ∗ (yn∗ )| = lim |A(yn∗ + ρ−1 T ∗ (0))|, n
n
which coincides with |A(v)|. Therefore, A is a lattice homomorphism, which implies that VT ∗ ,q is lattice isometric to a sublattice of (WT,p )∗ . In order to see that VT ∗ ,q is in fact an ideal of (WT,p )∗ , let y ∈ (WT,p )∗ with 0 ≤ y ≤ A(x) for some x ∈ VT ∗ ,q . Notice that x = lim φ(x∗n ) in VT ∗ ,q , where (x∗n ) belong to E ∗ . Thus, A(x) = lim A(φ(x∗n )) = lim ϕ∗ (xn ). If we denote yn = y ∧ ϕ∗ (xn ), then we clearly have that yn tends to y in (WT,p )∗ . Moreover, since ϕ is a lattice homomorphism, by [1, Thm. 1.35], it follows that ϕ∗ is interval preserving. Hence, since 0 ≤ yn ≤ ϕ∗ (x∗n ), for every n ∈ N, there exists u∗n ∈ [0, x∗n ], such that yn = ϕ∗ (u∗n ). Notice that ϕ∗ (u∗n )
Vol. 66 (2010)
Interpolation and Factorization of Operators
93
tends to y in (WT,p )∗ . In particular, we have ρT ∗ (u∗n −u∗m ) = ϕ∗ (u∗n −u∗m ) → 0 when n, m → ∞, which yields that φ(u∗n ) tends to some u∗ in VT ∗ ,q . By construction, we obtain that A(u∗ ) = y, which implies that A is interval preserving. This shows that VT ∗ ,q is an ideal of (WT,p )∗ , as claimed. On the other hand, we can also define a mapping B : WT,p → (VT ∗ ,q )∗ . Indeed, given y ∈ S and y ∗ ∈ E ∗ , since S = U0 , we have y ∗ , ϕ(y) ≤ ρT ∗ (y ∗ ). Therefore, for every y ∈ WT,p and y ∗ ∈ E ∗ we get y ∗ , ϕ(y) ≤ ∗ ρT ∗ (y ∗ )yWT ,p . Hence, there exists a unique element B(y) ∈ (E ∗ /ρ−1 T ∗ (0)) such that ∗
y ∗ + ρ−1 T ∗ (0), B(y) = y , ϕ(y) for every y ∗ ∈ E ∗ . Clearly, B(y) is a linear functional which is continuous for the norm in VT ∗ ,q , thus, it can be extended to an element of (VT ∗ ,q )∗ , with B(y)(VT ∗ ,q )∗ ≤ yWT ,p . Hence, B : WT,p → (VT ∗ ,q )∗ is a linear mapping which is bounded of norm ≤ 1. Moreover, for y ∈ WT,p we have B(y)(VT ∗ ,q )∗ = sup{ v, B(y) : vVT ∗ ,q ≤ 1}
= sup{ y ∗ , ϕ(y) : ρT ∗ (y ∗ ) ≤ 1}
which is the value of the Minkowski functional of U0 = S at ϕ(y). Hence, B(y)(VT ∗ ,q )∗ = inf{λ ≥ 0 : ϕ(y) ∈ λS} = yWT ,p . This means that B is an isometry. ∗ Moreover, for y ∗ ∈ E+ and every y ∈ WT,p we have −1 ∗ ∗ ∗
y ∗ + ρ−1 T ∗ (0), |B(y)| = sup{| x + ρT ∗ (0), B(y)| : |x | ≤ y }
= sup{| x∗ , ϕ(y)| : |x∗ | ≤ y ∗ }
= y ∗ , |ϕ(y)|, and since ϕ is a lattice homomorphism we have −1 ∗ ∗
y ∗ + ρ−1 T ∗ (0), |B(y)| = y , ϕ(|y|) = y + ρT ∗ (0), B(|y|). ∗ Since this holds for every y ∗ ∈ E+ , we have that |B(y)| = B(|y|). Therefore, B is a lattice homomorphism and the claimed result follows. To prove the last statement, let u ∈ (VT ∗ ,q )∗ such that 0 ≤ u ≤ B(y) for some y ∈ WT,p . We consider φ : E ∗ → VT ∗ ,q the operator induced by the quotient map. Since φ is positive, so is φ∗ : (VT ∗ ,q )∗ → E ∗∗ . It holds that
φ∗ (u) ≤ φ∗ (B(y)) = ϕ(y). Indeed, for every y ∗ ∈ E ∗ we have
φ∗ (B(y)), y ∗ = B(y), φ(y ∗ ) = ϕ(y), y ∗ . Hence, φ∗ (u) ∈ [0, ϕ(y)] in E ∗∗ . However, if E is order continuous and ϕ(y) belongs to E, then we have [0, ϕ(y)] ⊂ E. Moreover, since ϕ is interval preserving, there exists x ∈ [0, y] in WT,p , such that φ∗ (u) = ϕ(x). This implies that u = B(x), which means that WT,p is an ideal in (VT ∗ ,q )∗ .
94
Y. Raynaud and P. Tradacete
IEOT
Notice that the isometries A and B given in the proof of Theorem 5 may not be surjective, as the following examples show. Moreover, if E is not order continuous, WT,p may not be an ideal in (VT ∗ ,q )∗ . Example 1. Let T : L∞ (0, 1) → L1 (0, 1) denote the formal inclusion. Clearly, for every 1 ≤ p ≤ ∞, T is p-convex. First, notice that the set n n
p1 |T fi |p , fi pL∞ ≤ 1}, S = {f ∈ L1 (0, 1) : |f | ≤ i=1
i=1
satisfies that S = {f ∈ L1 (0, 1) : f L∞ ≤ 1}. This implies that WT,p = L∞ (0, 1). On the other hand, if we consider the adjoint operator T ∗ : L1 (0, 1)∗ → L∞ (0, 1)∗ , which is p -concave (for p1 + p1 = 1), then for f ∈ L∞ (0, 1) = L1 (0, 1)∗ we clearly have T ∗ f L∗∞ = f L1 . From here, it follows that the expression n n
1
1 p p ∗ p p ρT ∗ ,p (f ) = sup T fi L∗∞ : |fi | ≤ |f | , i=1
i=1
trivially satisfies f L1 ≤ ρT ∗ ,p (f ). While on the other hand, for f ∈ L∞ (0, 1) and (fi )ni=1 with 1/p n |fi |p ≤ |f | i=1
we have n
T ∗fi pL∗∞
1 p
=
i=1
n
fi pL1
1 p
n
1 p ≤ |fi |p
i=1
i=1
L1
≤ f L1 .
Thus, ρT ∗ ,p (f ) = f L1 , which implies that VT ∗ ,p = L1 (0, 1). Hence, the isometry A : VT ∗ ,p → (WT,p )∗ given in Theorem 5 is not surjective. Example 2. Let T : 1 → ∞ denote the formal inclusion. Clearly, T is ∞-convex. Moreover, it is easy to see that the set S = {y ∈ ∞ : |y| ≤
n
|yi |,
i=1
n
yi 1 ≤ 1},
i=1
satisfies S = Bc0 . Hence, WT,∞ = c0 . On the other 1-convex. It is
∗
hand, let T : ∗∞ well known that ∗∞
→ ∗1 be the adjoint operator, which is = ∗∗ 1 can be decomposed as
⊥ ∗∗ 1 = J(1 ) ⊕ J(1 ) ,
where J(1 ) denotes the canonical image of 1 in its bidual, and J(1 )⊥ its disjoint complement.
Vol. 66 (2010)
Interpolation and Factorization of Operators
95
Notice that every y ∈ J(1 )⊥ , viewed as an element of ∗∞ , satisfies y|c0 = 0. Indeed, for every n ∈ N, let en denote the sequence formed by zeros except 1 in the nth entry. For y ∈ J(1 )⊥ , by disjointness we have 0 = |y| ∧ J(en ), en = inf{ |y|, x + en , z : x, z ∈ + ∞ , x + z = en } = inf{λ |y|, en + 1 − λ : λ ∈ [0, 1]} = |y|, en , for every n ∈ N, which clearly implies y|c0 = 0. In particular, for y ∈ J(1 )⊥ we have T ∗ (y) = sup{ T ∗ (y), x : x ∈ 1 , x1 ≤ 1} = sup{ y, T x : x ∈ 1 , x1 ≤ 1} = 0, since T x ∈ c0 ⊂ ∞ for every x ∈ 1 . Therefore, for y ∈ J(1 )⊥ , since J(1 )⊥ is solid, we have n n ρT ∗ ,1 (y) = sup T ∗ yi ∞ : |yi | ≤ |y| = 0. i=1
i=1
While for y ∈ J(1 ) we have n n T ∗ yi ∞ : |yi | ≤ |y| = y1 . ρT ∗ (y) = sup i=1
i=1
Hence, VT ∗ ,1 = 1 , which implies that the isometry B : WT,∞ → (VT ∗ ,1 )∗ of Theorem 5 is not surjective. Example 3. Let T : 1 → c be defined by T (x1 , x2 , . . . , xn , . . .) = (x1 , x1 + n x2 , . . . , k=1 xk , . . .), where c denotes the space of convergent sequences of real numbers with the supremum norm. Clearly, T is positive and p-convex for every 1 ≤ p ≤ ∞. Notice that the set n n
1 p p |T yi | , yi p1 ≤ 1}, S = {y ∈ c : |y| ≤ i=1
i=1
contains the constant sequence equal to one, so since S is solid, S coincides with the closed unit ball of c. Hence, WT,p = c. Now, we can consider the adjoint operator T ∗ : c∗ → ∗1 , which is clearly q-concave for every 1 ≤ q ≤ ∞. Recall that c∗ can be identified with the space 1 (N) in the following way: for an element x = (x0 , x1 , . . .) in 1 (N) and another element y = (y1 , y2 , . . .) in c, we set ∞ xn yn .
x, y = x0 lim yn + n=1
Therefore, for a positive element x ∈ c∗ we have T ∗ x∗1
= sup{ T ∗ x, y : y1 ≤ 1} = sup{ x, T y : y1 ≤ 1} ≥ x, T e1 = ∞ n=0 xn = xc∗ .
96
Y. Raynaud and P. Tradacete
IEOT
Since T ≤ 1, it holds that T ∗ x∗1 = xc∗ for every positive x ∈ c∗ . This implies that n n
1q
1q T ∗xi q∗ : |xi |q ≤ |x| = xc∗ , ρT ∗ ,q (x) = sup 1
i=1
i=1 ∗
which yields that V =c . Notice that, in particular, the operator ϕ : c → c defined in Theorem 3 coincides with the identity on c, and the operator φ : c∗ → c∗ defined in Theorem 1 coincides as well with the identity on c∗ . Now, by the definition of the operator B : WT,p → (VT ∗ ,q )∗ in Theorem 5, it follows that for every y ∈ c and y ∗ ∈ c∗ we have T ∗ ,q
B(y), y ∗ = B(y), φ(y ∗ ) = ϕ(y), φ(y ∗ ) = y, y ∗ . Hence, B = J, where J : c → c∗∗ denotes the canonical inclusion of c into its bidual (= ∞ ). Now since c is not order continuous, it follows that B(c) is not an ideal in (VT ∗ ,q )∗ , and this shows that the last statement of Theorem 5 does not hold without the assumption of order continuity on E.
4. Interpolation of Banach lattices Throughout this section we will be using the complex interpolation method for Banach lattices, hence we need to consider complex Banach lattices. However, our final results, which are given in the next section, remain true for real Banach lattices by means of “complexifying” and considering the real part after the interpolation constructions. Notice that the results presented in the previous section work equally for both real or complex Banach lattices. We refer to [15, Section 2.2] for the notion of complex Banach lattice. Recall that a compatible pair of Banach spaces (X0 , X1 ) is a pair of Banach spaces X0 , X1 which are continuously included in a topological vector space X. In the context of Banach lattices, we will say that two Banach lattices X0 , X1 form a compatible pair of Banach lattices (X0 , X1 ) if there exists a complete Riesz space X, and inclusions ij : Xj → X which are continuous interval preserving lattice homomorphisms, for j = 0, 1. In this way, the space X0 + X1 = {x ∈ X : x = x0 + x1 , with x0 ∈ X0 , x1 ∈ X1 }, equipped with the norm x = inf{x0 X0 + x1 X1 : x = x0 + x1 } is a Banach lattice which contains X0 and X1 as (non-closed) ideals. Given a compatible pair of Banach lattices, (X0 , X1 ), for each θ ∈ [0, 1] we will consider three different constructions: 1. X01−θ X1θ denotes the space of elements x ∈ X0 + X1 such that |x| ≤ λ|x0 |1−θ |x1 |θ , for some λ > 0, x0 ∈ X0 and x1 ∈ X1 , with x0 X0 ≤ 1, x1 X1 ≤ 1. Notice that the expressions of the form |f |1−θ |g|θ can be defined in any
Vol. 66 (2010)
Interpolation and Factorization of Operators
Banach lattice by means of the functional calculus due to Krivine (see [11, pp. 40-43]). The norm in this space is given by xX 1−θ X θ 0
1
= inf{λ > 0 : |x| ≤ λ|x0 |1−θ |x1 |θ for some x0 X0 ≤ 1, x1 X1 ≤ 1}. 2. [X0 , X1 ]θ denotes the space of elements x ∈ X0 + X1 which can be represented as x = f (θ) for some f ∈ F(X0 , X1 ). Here F (X0 , X1 ) denotes the linear space of functions f (z) defined in the strip Π = {z ∈ C : z = x + iy, 0 ≤ x ≤ 1}, with values in the space X0 + X1 , such that • f (z) is continuous and bounded for the norm of X0 + X1 in Π, • f (z) is analytic for the norm of X0 + X1 in the interior of Π, • f (it) assumes values in X0 and is continuous and bounded for the norm of X0 , while f (1 + it) assumes values in X1 and is continuous and bounded for the norm of X1 . In F (X0 , X1 ) we can consider the norm f F (X0 ,X1 ) = max{sup f (it)X0 , sup f (1 + it)X1 }. t
t
The norm in [X0 , X1 ]θ is given by x[X0 ,X1 ]θ = inf{f F (X0,X1 ) : f (θ) = x}. 3. [X0 , X1 ]θ denotes the space of elements x ∈ X0 + X1 which can be represented as x = f (θ) for some f ∈ F (X0 , X1 ). Now F (X0 , X1 ) denotes the linear space of functions f (z) defined in the strip Π = {z ∈ C : z = x + iy, 0 ≤ x ≤ 1}, with values in the space X0 + X1 , such that • f (z)X0 +X1 ≤ c(1 + |z|) for some constant c > 0 and for every z ∈ Π, • f (z) is continuous in Π and analytic in the interior of Π for the norm of X0 + X1 , • f (it1 ) − f (it2 ) has values in X0 and f (1 + it1 ) − f (1 + it2 ) in X1 for any −∞ < t1 < t2 < ∞, endowed with the norm f F(X0 ,X1 ) f (it2 ) − f (it1 ) , sup f (1 + it2 ) − f (1 + it1 ) = max sup . t2 − t 1 t2 − t 1 t1 ,t2 X0 t1 ,t2 X1 The norm in [X0 , X1 ]θ is given by x[X0 ,X1 ]θ = inf{f F(X0 ,X1 ) : f (θ) = x}. These spaces are Banach lattices provided that (X0 , X1 ) is a compatible pair of Banach lattices. Moreover, [X0 , X1 ]θ and [X0 , X1 ]θ are always interpolation spaces, while X01−θ X1θ is an intermediate space between X0 and X1 which is an interpolation space under certain extra assumptions. We will refer to X01−θ X1θ as the Calder´ on-Lozanovskii construction for (X0 , X1 ). We refer to [6], [9], [12], and [13] for more information on these spaces.
97
98
Y. Raynaud and P. Tradacete
IEOT
ˇ Next theorem extends a result of Sestakov [19], which was originally proved only for the case of Banach lattices of measurable functions, showing how these constructions are related to each other. Theorem 9. Let X0 , X1 be a compatible pair of Banach lattices. For every θ ∈ (0, 1) it holds that [X0 , X1 ]θ = X0 ∩ X1
[X0 ,X1 ]θ
= X0 ∩ X1
X01−θ X1θ
,
with equality of norms. Before the proof of Theorem 9 we need the following. Lemma 10. Let F : Π → X0 + X1 be a function in F (X0 , X1 ) of the form F (z) = eδz
2
N
xj eλj z ,
j=1
where δ > 0, the λj are real, and xj ∈ X0 ∩ X1 . It holds that F (θ)X 1−θ X θ ≤ F F (X0 ,X1 ) . 0
1
Proof. Let F : Π → X0 ∩ X1 be a function in F (X0 , X1 ) of the form F (z) = eδz
2
N
xj eλj z ,
j=1
where δ > 0, the λj are real, and xj ∈ X0 ∩ X1 . Let x = N j=1 |xj |. We can consider the principal (non closed) ideal in X0 ∩ X1 generated by x, equipped with the norm that makes it isomorphic to a C(K) space for some compact K (i.e. y = inf{λ > 0 : |y| ≤ λx}, cf. [18, Chapter II. §7]). We clearly have inclusions C(K) → X0 ∩ X1 → X0 + X1 , which are bounded lattice homomorphisms. Moreover, since |xj | ≤ x, we have xj ∈ C(K), so we can consider F (ω, z) = eδz
2
N
xj (ω)eλj z ,
j=1
as a function of ω ∈ K, and z ∈ Π. For each z ∈ Π, F (·, z) belongs to C(K). Hence, applying [6, §9.4, ii)], for any ω ∈ K we have
1 |F (ω, θ)| ≤ 1−θ
+∞
1−θ +∞ θ 1 |F (ω, it)|µ0 (θ, t)dt |F (ω, 1 + it)|µ1 (θ, t)dt , θ
−∞
−∞
where µ0 and µ1 are the Poisson kernels for the strip Π, given by (see [6, §9.4]): µ0 (θ, t) =
e−πt sin πθ , sin2 πθ + [cos πθ − e−πt ]2
µ1 (θ, t) =
e−πt sin πθ . sin2 πθ + [cos πθ + e−πt ]2
Vol. 66 (2010)
Interpolation and Factorization of Operators
99
Hence setting 1 g(ω) = 1−θ
+∞
1 |F (ω, it)|µ0 (θ, t)dt and h(ω) = θ
−∞
+∞
|F (ω, 1 + it)|µ1 (θ, t)dt, −∞
we find that g and h belong to C(K). Indeed, for any ω1 , ω2 in K, we have 1 |g(ω1 ) − g(ω2 )| ≤ 1−θ 1 ≤ 1−θ 1 ≤ 1−θ ≤
N
+∞
|F (ω1 , it) − F (ω2 , it)|µ0 (θ, t)dt −∞ +∞ N
2 (xj (ω1 ) − xj (ω2 ))eiλj t e−δt µ0 (θ, t)dt
−∞ j=1 +∞ N
2
|xj (ω1 ) − xj (ω2 )|e−δt µ0 (θ, t)dt
−∞ j=1
|xj (ω1 ) − xj (ω2 )|,
j=1
since
+∞
µ0 (θ, t)dt = 1 − θ (see [6, §29.4]). This inequality together with the
−∞
fact that xj belongs to C(K) for j = 1, . . . , N , proves that g ∈ C(K). The proof for h is identical. Moreover, +∞ 1 gX0 = |F (ω, it)|µ (θ, t)dt 0 1 − θ X0 −∞
≤
1 1−θ
+∞
F (ω, it)X0 µ0 (θ, t)dt −∞
1 ≤ F F (X0 ,X1 ) 1−θ
+∞
µ0 (θ, t)dt −∞
= F F (X0 ,X1 ) , and similarly hX1 ≤ F F (X0 ,X1 ) . Since |F (θ)| ≤ g 1−θ hθ (in C(K), and thus in X0 + X1 since Krivine’s calculus is preserved under lattice homomorphisms), we have F (θ)X 1−θ X θ ≤ F F (X0 ,X1 ) . 0
And the proof is finished.
1
100
Y. Raynaud and P. Tradacete
IEOT
[X0 ,X1 ]θ
Proof of Theorem 9. If x is an element in X0 ∩ X1 , by the definition of the norm in [X0 , X1 ]θ , for every ε > 0, we can take F in F (X0 , X1 ), such that F (θ) = x and F F (X0 ,X1 ) ≤ x[X0 ,X1 ]θ + ε. By [9, Chapter IV, Thm. 1.1], we can consider a sequence (Fn )∞ n=1 in F (X0 , X1 ), of elements of the form eδz
2
N
xj eλj z ,
j=1
where xj ∈ X0 ∩ X1 and λj ∈ R, such that F − Fn F (X0 ,X1 ) → 0. Then we have Fn (θ) − x[X0 ,X1 ]θ = Fn (θ) − F (θ)[X0 ,X1 ]θ ≤ Fn − F F (X0 ,X1 ) → 0. By Lemma 10, for n, m ∈ N we have Fn (θ) − Fm (θ)X 1−θ X θ ≤ Fn − Fm F (X0 ,X1 ) → 0, 0
1
when n, m → ∞, and Fn (θ)X 1−θ X θ ≤ Fn F (X0 ,X1 ) → F F (X0 ,X1 ) ≤ x[X0 ,X1 ]θ + ε. 0
1
Therefore, Fn (θ) also converges to a limit in X01−θ X1θ of norm not exceeding x[X0 ,X1 ]θ + ε. However, since X01−θ X1θ and [X0 , X1 ]θ are both continuously embedded in X0 + X1 , it follows that x is also the limit of Fn (θ) for the norm of X01−θ X1θ . Hence, x ∈ X01−θ X1θ and xX 1−θ X θ ≤ x[X0 ,X1 ]θ + ε. 1 0 Since this is true for all ε > 0, we have xX 1−θ X θ ≤ x[X0 ,X1 ]θ . 0
We will show now that
1
X01−θ X1θ
⊂ [X0 , X1 ]θ and the inclusion mapping
X01−θ X1θ → [X0 , X1 ]θ is bounded with norm smaller than or equal to one. Indeed, let x ∈ X01−θ X1θ be such that xX 1−θ X θ ≤ 1. Then for every ε > 0 we have g ∈ X0+ , and 0
1
h ∈ X1+ such that gX0 ≤ 1, hX1 ≤ 1, and |x| ≤ (1 + ε)g 1−θ hθ in X0 + X1 . Now, if I denotes the (non closed) order ideal generated by g ∨ h in X0 +X1 , then I can be viewed as a space C(K) over some compact Hausdorff space K. Since |x| ≤ (1 + ε)g 1−θ hθ in X0 + X1 , we can consider f (t) =
x(t) , g 1−θ (t)hθ (t)
which is well defined for all t ∈ K such that g(t)h(t) = 0. This allows us to define f (t)g(t)1−z h(t)z if g(t)h(t) = 0, F (t, z) = 0 in any other case.
Vol. 66 (2010)
Interpolation and Factorization of Operators
101
Note that, since g, h ≤ g ∨ h, we have gC(K), hC(K) ≤ 1; hence, for every z ∈ Π, sup |F (t, z)| ≤ 1 + ε. t∈K ◦
Clearly, for z ∈ Π we can consider φ(z) ∈ C(K) defined by φ(z)(t) = F (t, z). It is routine to verify that the map ◦
φ : Π → C(K) is continuous. We claim that it is analytic. Indeed, note that for every t ∈ K ◦
fixed, F (t, ·) is analytic on Π. Hence, φ(z)(t) = F (t, z) =
1 2πi
γ
F (t, ξ) dξ ξ−z ◦
for every t ∈ K, and for any circumference γ of center z contained in Π. Since this identity is valid for every t ∈ K, we get φ(ξ) 1 dξ. φ(z) = 2πi γ ξ − z ◦
This means that φ : Π → C(K) is analytic. Now, let us define F (t, ξ)dξ, F1 (t, z) = γz 1 2
for t ∈ K and z ∈ Π, where γz is any continuous path joining ◦
and z, with ◦
all its points except possibly z inside Π. Note that since F is analytic in Π and sup|F (t, z)| ≤ 1 + ε, for all z ∈ Π, F1 is independent of the path γz , t∈K
so it is well defined. Therefore, we can define φ1 : Π → B(K), where B(K) denotes the bounded measurable functions on K, by φ(ξ)dξ, φ1 (z) = F1 (·, z) = γz ◦
◦
for z ∈ Π. Since φ : Π → C(K) is analytic, so is φ1 on Π, and clearly ◦
φ1 (Π) ⊆ C(K). Moreover, φ1 (z) − φ1 (z )C(K) ≤ (1 + ε)|z − z | ◦
◦
for z, z ∈ Π. Now, for any z in the border of Π, let zn ∈ Π be such that zn → z. Since φ1 (zn )−φ1 (zm )C(K) ≤ (1+ε)|zn −zm |, we get that φ1 (zn ) is a Cauchy sequence in C(K), hence convergent to some ψ ∈ C(K). In particular, for every t ∈ K, φ1 (zn )(t) → ψ(t) and since φ1 (zn )(t) = γz F (t, ξ)dξ we get n that ψ(t) = γz F (t, ξ)dξ. This implies that φ1 (Π) ⊆ C(K), and φ1 (z) − φ1 (z )C(K) ≤ (1 + ε)|z − z | for z, z ∈ Π. Thus φ1 : Π → C(K) is continuous.
102
Y. Raynaud and P. Tradacete
IEOT
Now, for u, v ∈ R, and for every α ∈ (0, 1) let γα be the path formed by the rectilinear segments L1 = [iu, α + iu], L2 = [α + iu, α + iv] and L3 = [α + iv, iv]. Hence, for every α ∈ (0, 1) and t ∈ K such that g(t)h(t) = 0 |F1 (t, iu) − F1 (t, iv)| ≤ |F (t, ξ)|dξ γα |F (t, ξ)|dξ + |F (t, ξ)|dξ + intL3 |F (t, ξ)|dξ = L1
L2
≤ α(1 + ε) + (1 + ε)g(t)1−α h(t)α |u − v| + α(1 + ε) ≤ (g(t)1−α h(t)α |u − v| + 2α)(1 + ε). Thus, letting α → 0+ , we get |F1 (t, iu) − F1 (t, iv)| ≤ (1 + ε)g(t) |u − v| for t ∈ K with g(t)h(t) = 0. Since the same inequality holds trivially if g(t)h(t) = 0, we have that |φ1 (iu) − φ1 (iv)| ≤ (1 + ε)g |u − v| in X0 . Analogously we have |φ1 (1 + iu) − φ1 (1 + iv)| ≤ (1 + ε)h |u − v| in X1 . Since X0 and X1 are order ideals, it clearly follows that 1 (1+iv)| ∈ X1 . X0 and |φ1 (1+iu)−φ |u−v| dφ1 Therefore, since dz z=θ = φ(θ)
|φ1 (iu)−φ1 (iv)| |u−v|
∈
= x, we get that x ∈ [X0 , X1 ]θ and
x[X0 ,X1 ]θ |φ1 (iu) − φ1 (iv)| |φ1 (1 + iu) − φ1 (1 + iv)| ≤ max sup , , sup |u − v| |u − v| u,v X0 u,v X1 which is smaller than 1 + ε. Since this holds for every ε > 0, we have proved that X01−θ X1θ → [X0 , X1 ]θ is continuous with norm smaller than or equal to one. In particular, we have an inclusion X0 ∩ X1
X01−θ X1θ
→ X0 ∩ X1
[X0 ,X1 ]θ
with norm smaller than one. Now, by [3], we have X0 ∩ X1
[X0 ,X1 ]θ
= X0 ∩ X1
[X0 ,X1 ]θ
with equality of norms. This proves the theorem.
,
Vol. 66 (2010)
Interpolation and Factorization of Operators
103
5. Factorization for operators which are both p-convex and q-concave In section 2, it was proved that every p-convex (resp. q-concave) operator factors in a nice way through a p-convex (resp. q-concave) Banach lattice. However, if the operator is both p-convex and q-concave, can this factorization be improved? It is well-known that if E is a q-concave Banach lattice and F a p-convex Banach lattice, then every operator T : E → F is both p-convex and q-concave. Moreover, if an operator T : E → F between Banach lattices, has a factorization of the following form E
T
ψ
φ
E1
/F O
R
/ F1
where φ and ψ are positive, E1 q-concave, and F1 p-convex, then T is both p-convex and q-concave [10]. Hence, the following question is natural: Can a p-convex and q-concave operator T : E → F factor always in this way? According to Theorems 1 and 3, this is true if the operator T : E → F can be written as T = T1 ◦ T2 , where T1 is p-convex, and T2 is q-concave. In fact, it turns out that the previous question is equivalent to the following one. If T : E → F is p-convex and q-concave, do there exist operators T1 and T2 , such that T = T1 ◦ T2 , where T1 is p-convex, and T2 is q-concave? In general, the answer to this question is negative, as the following examples show. Proposition 11. Let T : E → F be an operator from an ∞-convex Banach lattice (an AM -space) E to a q-concave Banach lattice F (q < ∞). If T can be factored as T = SR, with R q-concave and S ∞-convex, then T is compact. Proof. If T : E → F has such a factorization, then by Theorems 1 and 3 we must have T /F E O ϕ
φ
V
T1
/W
where V is q-concave, W an AM -space, and φ, ϕ lattice homomorphisms. Since φ and ϕ are positive and take values in q-concave Banach lattices, by [11, Prop. 1.d.9], they are q-concave operators. Moreover, since both operators are defined on AM -spaces, by [11, Theorem 1.d.10], φ and ϕ are q-absolutely summing. Therefore, T = ϕ ◦ (T1 ◦ φ) is a product of two q-absolutely summing operators, hence it is compact, because every q-absolutely summing operator is weakly compact and Dunford-Pettis (cf. [2, Cor. 8.2.15]).
104
Y. Raynaud and P. Tradacete
IEOT
Example 4. The formal inclusion T : C(0, 1) → Lq (0, 1) is q-concave and ∞-convex, but it does not factor as T = T1 ◦ T2 , with T1 ∞-convex, and T2 q-concave. Proof. Since T is positive, it is q-concave and ∞-convex by [11, Prop. 1.d.9]. However T is not compact since the closure in Lq (0, 1) of the unit ball of C(0, 1) contains the Rademacher functions. By duality, Proposition 11 immediately yields the following. Corollary 12. Let T : E → F be an operator from a p-convex Banach lattice E to a 1-concave Banach lattice (an AL-space) F . If T can be factored as T = SR, with R 1-concave and S p-convex, then T is compact. A different argument can be used to see that the formal inclusion i : Lp (0, 1) → Lq (0, 1) with 1 < q < p < ∞ (which is clearly p-convex and q-concave) does not factor as i = T2 T1 with T1 q-concave and T2 p-convex. First we need the following lemma: Lemma 13. Let 1 < q < p < ∞. There is no disjointness preserving nonzero operator T : Lq (0, 1) → Lp (0, 1). Proof. Assume f := T h = 0 for some h ∈ Lq (0, 1) with hq = 1. If U : Lq (0, 1) → Lq (0, 1) is a linear isometry such that U χ[0,1] = h, then S := T U is also disjointness preserving. For each n ∈ N, let us consider the partition {0, n1 , n2 , . . . , 1}. Notice that for each n ∈ N there must exist kn ≤ n such that f p S(χ[ kn −1 , kn ] )p ≥ 1/p . n n n Otherwise, by the fact that S is disjointness preserving, we would have n n n
p1 f p p1 f = S(χ[ k−1 , k ] ) = S(χ[ k−1 , k ] )p < = f , n n n n n k=1
k=1
which is clearly a contradiction. Hence, since χ[ k−1 , k ] q = n
n
1 n1/q
k=1
for every k = 1, . . . , n, we have
f p f p = 1/p−1/q χ[ kn −1 , kn ] q . n n n1/p n Therefore, since q < p, for n large enough we get a contradiction with the fact that S is bounded. S(χ[ kn −1 , kn ] )p ≥ n
n
Recall that given a Banach lattice E and a Banach space X, an operator T : E → X is called AM-compact if T [−x, x] is relatively compact for every positive x ∈ E. Theorem 14. If a lattice homomorphism T : Lp (0, 1) → Lq (0, 1) (q < p) can be factored as T = T2 T1 with T1 q-concave and T2 p-convex, then T is AM-compact.
Vol. 66 (2010)
Interpolation and Factorization of Operators
105
Proof. Suppose that we have Lp (0, 1) GG GG GG T1 GGG #
T
X
/ Lq (0, 1) ; ww ww w ww T2 ww
with T1 q-concave and T2 p-convex. Hence, by Theorems 1 and 3 we have Lp (0, 1)
T
ϕ
φ
V
/ Lq (0, 1) O
S
/W
where V is q-concave, W p-convex, and φ, ϕ are lattice homomorphisms. Now, by Krivine’s Theorem ([11, Theorem 1.d.11]) we can factor φ /V Lp (0, 1) = II { { II { II { II {{ φ1 I$ {{ φ2 Lq (µ)
/ Lq (0, 1) WD DD u: DD uu u u D ϕ1 DD uu ϕ ! uu 2 Lp (ν) ϕ
where φi and ϕi are still lattice homomorphisms. Therefore, we can consider the closure of φ1 (Lp (0, 1)) in Lq (µ), which is lattice isomorphic to some µ), and the quotient Lp (ν)/ ker(ϕ2 ) which is lattice isomorphic to Lp ( ν) Lq ( for certain measures µ and ν. Thus, we can consider the following diagrams: / Lq (µ) Lp (0, 1) II v; II v v II v v II v 1 I$ φ , vvv i µ) Lq ( φ1
/ Lq (0, 1) Lp (ν) AA AA u: u AA uu u A π AA uu uu ϕ2 Lp ( ν) ϕ2
µ) → Lp ( ν ) be defined by R = πϕ1 Sφ2 i. It follows that R Now, let R : Lq ( µ), we can consider (xn ) in is a lattice homomorphism. Indeed, given x ∈ Lq ( µ). Since T is a lattice homomorphism Lp (0, 1) such that φ1 (xn ) → x in Lq ( 2 is an injective lattice homomorT (|xn |) = |T xn | for every n, and since ϕ 1 (|xn |) = |Rφ 1 (xn )|, and by continuity and the fact phism we get that Rφ 1 is also a lattice homomorphism, we achieve R(|x|) = |R(x)|. that φ µ) and By considering the diffuse and atomic parts of the spaces Lq ( Lp ( ν ), we can decompose them as Lq (0, 1) ⊕ q and Lp (0, 1) ⊕ p (lattice isomorphically). Accordingly, every operator between them can be decomposed into four parts acting between each of the summands, that is R = R11 R12 with R21 R22 R11 : Lq (0, 1) → Lp (0, 1) R21 : Lq (0, 1) → p
R12 : q → Lp (0, 1) R22 : q → p .
106
Y. Raynaud and P. Tradacete
IEOT
Clearly, if R is a lattice homomorphism, so are Rij , and since the intervals in p and q are compact, we have that R12 , R21 and R22 are AM-compact. Finally, by Lemma 13 we see that R11 has to be the zero operator. This finishes the proof. Example 5. For 1 < q < p < ∞, the formal inclusion i : Lp (0, 1) → Lq (0, 1) cannot be factored as i = T2 T1 with T1 q-concave and T2 p-convex. Proof. Since i is positive, by [11, Prop. 1.d.9], i is q-concave and p-convex. Moreover, since i : Lp (0, 1) → Lq (0, 1) is a lattice homomorphism and it is not AM-compact (consider for instance the Rademacher functions), by Theorem 14, we conclude that it cannot be factored as i = T2 T1 with T1 q-concave and T2 p-convex. Despite these facts, as an application of the results of section 4, we have the following factorization for operators which are both p-convex and q-concave. Theorem 15. Let E and F be Banach lattices, and let T : E → F be both p-convex and q-concave. For every θ ∈ (0, 1) we can factor T in the following way /F O
T
E
ϕθ
φθ
Eθ
/ Fθ
Rθ
q where φθ and ϕθ are interval preserving lattice homomorphisms, Eθ is ( 1−θ )p concave, and Fθ is ( θ+(1−θ)p )-convex.
Before the proof, we need some lemmas first. Recall, that given a Banach space X, and 1 ≤ p < ∞, p (X) denotes the space of sequences (xn ) in X such that (xn X ) belongs to p . This is a Banach space with the norm ∞
p1 xn pX . (xn )p (X) = n=1
In order to keep a unified notation, for p = ∞, ∞ (X) will denote the space of sequences (xn ) of X such that (xn X ) belongs to c0 , equipped with the norm (xn )∞ (X) = sup xn X . Notice that this space is usually denoted c0 (X) in the literature. Analogously, given a Banach lattice E, and 1 ≤ p ≤ ∞, E(p ) denotes the completion of the space of eventually null sequences (xn ) of E under the norm ⎧
p1 ⎪ n ⎪ |xi |p E if 1 ≤ p < ∞, ⎨ sup n i=1 (xn )E(p ) = n ⎪ ⎪ if p = ∞. ⎩ sup |xi | n
i=1
E
Vol. 66 (2010)
Interpolation and Factorization of Operators
107
The following lemma consists of standard results. In the case of Banach lattices of measurable functions, this can be obtained from [5, Theorem 3], however, in general we cannot use the measurability tools and thus some functional calculus needs to be carried out. Lemma 16. Let (F, G) be a compatible pair of Banach lattices, let r, s ∈ θ [1, +∞] and θ ∈ (0, 1). For 1t = 1−θ r + s , we have: 1. r (F )1−θ s (G)θ = t (F 1−θ Gθ ), with equality of norms. r (F )1−θ s (G)θ
F 1−θ Gθ
2. r (F ) ∩ s (G) = t (F ∩ G ). 3. the inclusion F (nr )1−θ G(ns )θ → F 1−θ Gθ (nt ) is bounded of norm ≤ 1. 4. E(r )1−θ E(s )θ = E(t ), with equality of norms. We skip the proof of the lemma and proceed with the proof of the main result. Proof of Theorem 15. Since T is p-convex, it can be factored through a pconvex Banach lattice Z as in Theorem 3: E@ @@ @@ R @@
T
Z
/F ? ~ ~~ ~ ~ϕ ~~
where ϕ : Z → F is an injective interval preserving lattice homomorphism, and Rx = T x for all x ∈ E. Therefore, (Z, F ) can be considered as a compatible interpolation pair of Banach lattices, and we can interpolate T : E → F and R : E → Z by the complex method of interpolation (see [6]) with parameter θ, (thus, we complexify E and Z if they are not complex Banach lattices) and we get a Banach lattice Fθ = [(Z, F )]θ , and an operator Tθ : E → Fθ . Moreover, since ϕ is an inclusion, Fθ is also continuously included in F . Let us denote this inclusion by ϕθ : Fθ → F . p We claim that Fθ is pθ convex, with p1θ = pθ + 1−θ 1 , that is pθ = θ+(1−θ)p . Indeed, first notice that if Z is p-convex then F 1−θ Z θ is pθ -convex. This is because for any positive operator S it holds that S(|x0 |1−θ |x1 |θ ) ≤ (S|x0 |)1−θ (S|x1 |)θ . This implies that for any positive operator S acting simultaneously from X0 into Y0 and from X1 into Y1 the interpolated operator S : X01−θ X1θ → Y01−θ Y1θ is bounded. In our particular case, (see the discussion following [11, 1.d.3]) for every n ∈ N, we have operators Iˆn : n1 (F ) (x1 , . . . , xn )
−→ F (n1 ) −→ (x1 , . . . , xn )
Iˆn : np (Z) −→ Z(np ) (x1 , . . . , xn ) −→ (x1 , . . . , xn )
which are bounded uniformly on n ∈ N. Since they are clearly positive, by the previous remark the following operators are also uniformly bounded In : n1 (F )1−θ np (Z)θ −→ F (n1 )1−θ Z(np )θ (x1 , . . . , xn ) −→ (x1 , . . . , xn )
108
Y. Raynaud and P. Tradacete
IEOT
Using (1) and (3) of Lemma 16 we get that the operators pnθ (F 1−θ Gθ ) −→ n1 (F )1−θ np (Z)θ −→ F (n1 )1−θ Z(np )θ −→ F 1−θ Gθ (npθ ) −→ (x1 , . . . , xn )
(x1 , . . . , xn )
−→ (x1 , . . . , xn )
−→ (x1 , . . . , xn )
are also uniformly bounded on n. This means that F 1−θ Gθ is pθ -convex. Now, F 1−θ Z θ
by Theorem 9, Fθ = F ∩ Z , and since F ∩ Z is a sublattice of F 1−θ Z θ , Fθ is also pθ -convex. q )-concave. Indeed, since T : E → F is Now we claim that Tθ is ( 1−θ q-concave and R : E → Z is ∞-concave, the following maps are bounded: Tˇ : E(q ) (x1 , x2 , . . .)
ˇ : E(∞ ) R (x1 , x2 , . . .)
−→ q (F ) −→ (T x1 , T x2 , . . .)
−→ −→
∞ (Z) (Rx1 , Rx2 , . . .)
Therefore, the interpolated map Tˇθ : [(E(q ), E(∞ ))]θ → [(q (F ), ∞ (Z))]θ is also bounded (cf. [4] or [6, §4]). Note that by Theorem 9 and (4) of Lemma 16, we have [(E(q ), E(∞ ))]θ = E(q ) ∩ E(∞ )
E(q )1−θ E(∞ )θ
E(q )1−θ E(∞ )θ
= E(q ) where
1 qθ
=
θ ∞
+
1−θ q .
= E(qθ ),
And by Lemma 16, we have the identity q (F )1−θ ∞ (Z)θ
[(q (F ), ∞ (Z))]θ = q (F ) ∩ ∞ (Z) = qθ (F ∩ Z
F 1−θ Z θ
) = qθ (Fθ ),
with equality of norms. Therefore, the map Tˇθ : E(qθ ) → qθ (Fθ ) is bounded, q which means that Tθ is qθ -concave (qθ = 1−θ ). Hence, we can now apply Theorem 1 to Tθ : E → Fθ , and we get the factorization E@ @@ @@ @ φθ @@
Tθ
Eθ
/ Fθ > } }} } }} }} Rθ
through the qθ -concave Banach lattice Eθ . Therefore, T can be factorized as claimed. Remark 9. It is easy to see that in the case when the spaces E and F are real Banach lattices, after complexifying and making the previous argument, the obtained operators are all “complexified” operators, i.e. TC (x + iy) = T (x) + iT (y). Hence, by considering the restriction to the real part in each space, we obtain the same factorization result for real Banach lattices.
Vol. 66 (2010)
Interpolation and Factorization of Operators
109
6. Connections with Krivine’s theorem Recall the classical result proved in [10]: Given Banach spaces X, Y and a Banach lattice E, if T1 : X → E is p-convex and T2 : E → Y is p-concave, then T2 T1 factors through Lp (µ) for certain measure µ. We remark that the factorization Theorems 1 and 3 allow us to reduce Krivine’s theorem to the following purely lattice theoretical version: Lemma 17. If W , V are quasi-Banach lattices with W p-convex and V pconcave, then every lattice homomorphism h : W → V factors through some space Lp (µ), and the factors are lattice homomorphisms. Proof. We may assume by renorming that the p-convexity constant of W , resp. the q-concavity constant of V , are equal to one (see Rem. 1 and 5). Notice that by an elementary p-concavification/ convexification argument (see [11, pp. 53-54]), the proof of the Lemma reduces itself to the case p = 1 (this is because a lattice homomorphism h : W → V is bounded if and only if it is bounded between the p-convexifications h : W (p) → V (p) ). In this case, Krivine’s argument becomes transparent: indeed, let us consider F1 = {x ∈ W : xW < 1} and F2 = h−1 ({y ∈ V : y ≥ 0 and yV ≥ h}). Clearly, both sets are convex and satisfy F1 ∩ F2 = ∅. Hence, by HahnBanach’s Theorem we can find a functional f ∈ W ∗ such that f (x) ≤ 1 for each x ∈ F1 and f (x) ≥ 1 for each x ∈ F2 . Thus, f is positive and for x ∈ W 1 we have h h(x)V ≤ f (|x|) ≤ xW . This allows us to define a seminorm on W by x → f (|x|) which induces a lattice norm norm on the vector lattice W/{x ∈ W : f (|x|) = 0}, the completion of which (for this new norm) is, by Kakutani’s theorem [11, Theorem 1.b.2], isomorphic as normed lattice to a space L1 (µ) for a certain measure µ. Moreover, if π denotes the map W → L1 (µ) induced by the quotient map W → W/{x ∈ W : f (|x|) = 0}, we have 1 h(x)V ≤ π(x) ≤ xW . h This means that we can factor /V WE y< EE y EE yy E yy π EE " yy h L1 (µ) h
where π and h are lattice homomorphisms and h is defined so that h(π(x)) = h(x).
110
Y. Raynaud and P. Tradacete
IEOT
Now, let T1 : X → E be a p-convex operator and T2 : E → Y be p-concave. Using Theorems 1 and 3 we have XB BB BB B R1 BB
T2 /E /Y >} @@ > ~ @@ ~ }} ~ } @ ~ ~ }} ϕ φ @@ }} ~~ S2 W V T1
where W is p-convex, V p-concave, and ϕ, φ are lattice homomorphisms. This diagram shows clearly how Krivine’s theorem can be obtained from the previous Lemma. Remark 10. The same argument plus a standard application of Maurey’s Theorem [2, Theorem 7.1.2] yields that if T1 : X → E is p-convex and T2 : E → Y is q-concave, with p > q, then T2 T1 can be factored through the canonical inclusion i : Lp (µ) → Lq (µ) for a certain measure µ (this was essentially proved in [17, Sec. 2, Corollary 7] when Y is reflexive). In a similar direction, as another application of Theorem 9, we have the following result (compare with [17, Sec. 3, Proposition 2]). Proposition 18. Let T : X → E be p-convex and S : E → Y q-concave. For every θ ∈ (0, 1) we can factor ST through a Banach lattice Uθ which is p q pθ -convex and qθ -concave (with as usual pθ = p(1−θ)+θ and qθ = 1−θ ). Proof. By Theorem 3, we can factor T in the following way, where W is a p-convex Banach lattice and i a positive operator: T /E XB BB }> } BB } BB }} B . }}} i T W
Moreover, since S ◦ i : W → Y is q-concave, by Theorem 1 we have the lattice seminorm ρS◦i which is continuous with respect to the norm in W (ρS◦i (x) ≤ Mq (S ◦ i)xW ), and such that W/ρ−1 S◦i (0) with the norm that ρS◦i induces becomes a q-concave Banach lattice, such that S ◦ i factors through it. But, since W is p-convex and ρ−1 S◦i (0) is a closed ideal, it follows that W/ρ−1 (0) with its quotient norm is also p-convex. S◦i (0) with its quotient norm, and X1 = We can consider X0 = W/ρ−1 S◦i −1 W/ρ (0) (the completion under ρ ) with the norm induced by ρ . Note S◦i
S◦i
S◦i
that, for all y with ρS◦i (y) = 0, we have that ρS◦i (x) = ρS◦i (x + y) ≤ M(q) (S ◦ i)x + y. Thus, xX1 = ρS◦i (x) ≤ M(q) (S ◦ i) inf{x + y : ρS◦i (y) = 0} = M(q) (S ◦ i)xX0 ,
Vol. 66 (2010)
Interpolation and Factorization of Operators
111
which means that the inclusion X0 → X1 is bounded of norm less than or equal to ≤ M(q) . Therefore, we can interpolate X0 and X1 . Since X0 is pconvex and X1 is q-concave, by [16] we get that Uθ = X01−θ X1θ is pθ -convex and qθ -concave. The following diagram illustrates the situation: /: E uu u uu uu , uuu i φ
T
XB BB BB B BB T W
X0
/ X 1−θ X θ 1 0
S
/ X1 O / X? 1
/Y }> } }} }} }} S1
Remark 11. A similar result to Proposition 18 was also given in [17, Sec. 3, Proposition 2] for interval preserving lattice homomorphisms from a p-convex to a q-concave Banach lattice with essentially the same proof. The idea of using interpolation to produce this kind of factorization has been initiated both by S. Reisner in [17] and, in parallel, by T. Figiel in [7] using the real method of interpolation. Corollary 19. If T : E → E is p-convex and q-concave, then T 2 factors through a pθ -convex and qθ -concave Banach lattice. In particular, it factors through a super reflexive Banach lattice. Acknowledgment We thank Prof. F. L. Hern´ andez for bringing the reference [17] to our attention after that a first draft of the present paper was completed.
References [1] Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory. Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. [2] F. Albiac, and N. J. Kalton, Topics in Banach space theory. Graduate Texts in Mathematics, 233, Springer, New York, 2006. [3] J. Bergh, On the relation between the two complex methods of interpolation. Indiana Univ. Math. J. 28 (1979), no. 5, 775–778. [4] J. Bergh, and J. L¨ ofstr¨ om, Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, no. 223, Springer-Verlag, Berlin-New York, 1976. [5] A. V. Bukhvalov, Interpolation of linear operators in spaces of vector-valued functions and with mixed norm. (Russian) Sibirsk. Mat. Zh. 28 (1987), no. 1, i, 37–51. English translation: Siberian Math. J. 28 (1987), no. 1, 24–36. [6] A. P. Calder´ on, Intermediate spaces and interpolation, the complex method. Studia Math. 24 (1964), 113–190. [7] T. Figiel, Uniformly convex norms on Banach lattices. Studia Math. 68 (1980), 215–247.
112
Y. Raynaud and P. Tradacete
IEOT
[8] N. Ghoussoub, and W. B. Johnson, Factoring operators through Banach lattices not containing C(0, 1). Math. Z. 194 (1987), 153-171. [9] S. G. Kre˘ın, Yu. ¯I. Petun¯ın, and E. M. Sem¨enov, Interpolation of linear operators. Translations of Mathematical Monographs, 54, American Mathematical Society, 1982. [10] J. L. Krivine, Th´eor`emes de factorisation dans les espaces r´ eticul´es. (French) S´eminaire Maurey-Schwartz 1973–1974: Espaces Lp , applications radonifiantes ´ et g´eom´etrie des espaces de Banach, Exp. Nos. 22 et 23. Centre de Math., Ecole Polytech., Paris, 1974. [11] J. Lindenstrauss, and L. Tzafriri, Classical Banach Spaces II: Function Spaces. Springer-Verlag, Berlin-New York, 1979. ˇ 10 [12] G. Ya. Lozanovskii, Certain Banach lattices. (Russian) Sibirsk. Mat. Z. (1969), 584–599. ˇ 14 [13] G. Ya. Lozanovskii, Certain Banach lattices IV. (Russian) Sibirsk. Mat. Z. (1973), 140–155. [14] P. Meyer-Nieberg, Kegel p-absolutsummierende und p-beschr¨ ankende Operatoren. (German) Nederl. Akad. Wetensch. Indag. Math. 40 (1978), no. 4, 479– 490. [15] P. Meyer-Nieberg, Banach Lattices. Springer-Verlag, 1991. [16] G. Pisier, Some applications of the complex interpolation method to Banach lattices. J. Analyse Math. 35 (1979), 264–281. [17] S. Reisner, Operators which factor through convex Banach lattices. Canad. J. Math. 32 (1980), 1482–1500. [18] H. H. Schaefer, Banach lattices and positive operators. Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, 1974. ˇ [19] V. A. Sestakov, Complex interpolation in Banach spaces of measurable functions. (Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom. Vyp. 4 (1974), 64–68, 171. Yves Raynaud Institut de Math´ematiques de Jussieu CNRS and UPMC-Univ. Paris-06 case 186 75005 Paris France e-mail:
[email protected] Pedro Tradacete Departamento de An´ alisis Matem´ atico Universidad Complutense de Madrid 28040 Madrid Spain e-mail:
[email protected] Submitted: January 20, 2009. Revised: August 25, 2009.
Integr. Equ. Oper. Theory 66 (2010), 113–140 DOI 10.1007/s00020-009-1735-5 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Integral Equations, Dichotomy of Evolution Families on the Half-Line and Applications Adina Luminit¸a Sasu and Bogdan Sasu Abstract. The purpose of this paper is to provide a new, unified and complete study for uniform dichotomy and exponential dichotomy on the half-line. First we deduce conditions for the existence of uniform dichotomy, using classes of function spaces over R+ which are invariant under translations. After that, we obtain a classification of the main classes of function spaces over R+ , in order to deduce necessary and sufficient conditions for the existence of exponential dichotomy, emphasizing on the main technical qualitative properties of the underlying spaces. We motivate our approach by illustrative examples and show that the main hypotheses cannot be dropped. We provide optimal methods regarding the input space in the study of dichotomy and deduce as particular cases some interesting situations as well as several dichotomy results published in the past few years. Mathematics Subject Classification (2010). Primary 34D09; Secondary 34D05. Keywords. Uniform dichotomy; exponential dichotomy; evolution equation; integral equation; evolution family.
1. Introduction Exponential dichotomy is one of the most interesting asymptotic properties of evolution equations being in the front line of various research studies (see [1]–[4], [6]–[12], [14]–[16], [19]–[25]). For an evolution equation on a Banach space X, exponential dichotomy is described through the existence of a projections family {P (t)}, which determines a decomposition of the space X at every moment, into a direct sum of stable subspace and unstable subspace such that the norm of the projection onto the stable subspace of any orbit decays exponentially as t → ∞ and the norm of the projection onto the The work is supported by the Exploratory Research Grant PN II ID 1080 code 508/2009, director Professor Mihail Megan.
114
A. L. Sasu and B. Sasu
IEOT
unstable subspace of any orbit grows exponentially as t → ∞. This asymptotic property and its generalizations was at the center of intensive research studies, among which we mention the classical monographs of Massera and Sch¨ affer [12], Daleckii and Krein [8] and Coppel [7], the notable contributions from [1]–[4] and also the recent results obtained in [6], [9], [10], [11], [14]–[17], [19]–[25]. Let X be a Banach space and let U = {U (t, s)}t≥s,t,s∈J be an evolution family on X (see e.g. [21], Definition 2.1 and the present paper Definition 3.1). If J = R then we say that U is defined on the real line and if J = R+ then we say that U is defined on the half-line. In the study of the exponential dichotomy, there are two crucial differences between the real line and the half-line. The first one is that on the real line, if an evolution family is exponentially dichotomic with respect to a family of projections then the projections family is uniquely determined (see [6], [11], [21], [22], [25] and the references therein). In contrast, on the half-line an evolution family may be exponential dichotomic with respect to an infinite class of projection families (see e.g. [16], Example 1.1.1). The second aspect is related to the characterization of the exponential dichotomy in terms of the admissibility with respect to an associated integral equation: on the real line the admissibility concept relies on the unique solvability of the integral equation, while on the half-line the uniqueness requirement may be removed. These arguments and also many others led to distinct studies concerning these two cases: the real line on the one side (see e.g. [6], [11], [21], [22], [25]) and the half-line on the other side (see e.g. [10], [14], [15], [19], [23], [24]). If X is a Banach space and U = {U (t, s)}t≥s≥0 is an evolution family on X, then a pair of function spaces (O(R+ , X), I(R+ , X)) is said to be admissible for U if for every v ∈ I(R+ , X) there exists a continuous function f ∈ O(R+ , X) such that t (EU ) f (t) = U (t, s)f (s) + U (t, τ )v(τ ) dτ, ∀t ≥ s ≥ 0. s
I(R+ , X) is called the input space and O(R+ , X) is called the output space. Taking into account that the input-output techniques rely on the verification of the solvability of the equation (EU ) in the output space, for various input test functions, a natural concern is to obtain characterizations such that the input space is as small as possible, while the output space is as large as possible. For the case of evolution equations on the half-line, an interesting open problem is to classify the classes of function spaces where the input and the output space from the admissible pair would belong to, such that the admissibility provides the existence of dichotomy. For the case of evolution families defined on the real line, this question was answered in some cases in papers [21] and [25]. Another open problem on the half-line is to determine the main differences between the case of uniform dichotomy and the case of exponential dichotomy and to identify the corresponding classes of input and output spaces
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
115
in each case. These problems bring the subject on a new line of research and request an approach based on solid arguments from interpolation theory. In order to solve them, new methods must be provided at the interference of qualitative theory of evolution equations, dynamic control and interpolation theory. The main purpose of the present paper is to give complete answers to the above questions and to deduce new applications, emphasizing on the characteristic properties of the dichotomous families on the half-line. We propose a new, unified and complete study concerning the dichotomy of evolution families on the half-line using the input-output admissibility of pairs of function spaces. The paper is organized as follows: in the second section we present some basic properties of Banach function spaces arising from interpolation theory. We consider a general class Q(R+ ) of Banach function spaces which are invariant to translations, contain the continuous functions v with v(0) = 0 and compact support, and satisfy a simple integral property and we associate a subclass denoted V(R+ ), which contains the Banach function spaces in Q(R+ ) satisfying the ideal property. We also introduce three subclasses F (R+ ), L(R+ ), B(R+ ) (see Section 2) and discuss some properties that occur in the study of dichotomy on the half-line and establish the connections between the classes of function spaces considered throughout the paper. Next, our attention focuses on the property of uniform dichotomy using specific control tools. In several steps, we deduce the connections between the admissibility of the pair (O(R+ , X), I(R+ , X)) (see Definition 3.3), with I ∈ Q(R+ ) and O ∈ V(R+ ) and the uniform dichotomy of an evolution family U = {U (t, s)}t≥s≥0 . The main result of the third section shows that if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the initial subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X, then U is uniformly dichotomic. Naturally, the question arises under what kind of hypotheses the admissibility of the pair (O(R+ , X), I(R+ , X)), with I ∈ Q(R+ ) and O ∈ V(R+ ) implies the uniform exponential dichotomy of U = {U (t, s)}t≥s≥0 . The purpose of the fourth section is to answer this question. Using constructive methods, we analyze both cases O ∈ F(R+ ) and I ∈ L(R+ ), respectively. In these cases, we prove that if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X, then U is uniformly exponentially dichotomic. Moreover, if I ⊂ O and one of the spaces I or O belongs to B(R+ ), then U is uniformly exponentially dichotomic if and only if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) is closed and complemented in X. After that, by concrete examples we prove that the assumptions on the structure of the underlying function spaces are indeed necessary and we motivate our methods. We show that conditions “O ∈ F(R+ ) or I ∈ L(R+ )” cannot be dropped and that our context is the most general in this topic. The last section is also devoted to some optimization methods regarding the
116
A. L. Sasu and B. Sasu
IEOT
input space and to the connections between the main results of our paper and the previous contributions in this area. Finally, we present as particular cases many interesting situations among which we note several dichotomy results obtained in the past few years.
2. Banach function spaces: basic notations and preliminary results In this section, for the sake of clarity we recall some basic definitions and notations from the theory of Banach function spaces. Let M(R+ ) be the linear space of all Lebesgue measurable functions u : R+ → R, identifying the functions equal a.e. Definition 2.1. A linear subspace B of M(R+ ) is called a normed function space, if there is a mapping | · |B : B → R+ such that: (i) |u|B = 0 if and only if u = 0 a.e.; (ii) |αu|B = |α| |u|B , for all (α, u) ∈ R × B; (iii) |u + v|B ≤ |u|B + |v|B , for all u, v ∈ B; (iv) if u ∈ B, then |u| ∈ B; (v) if u, v ∈ B with |u(·)| ≤ |v(·)| a.e., then |u|B ≤ |v|B . If (B, | · |B ) is complete, then B is called a Banach function space. For A ⊂ R we denote by χA the characteristic function of the set A. Definition 2.2. A Banach function space (B, | · |B ) is said to be invariant to translations if for every u : R+ → R and every t > 0 we have that u ∈ B if and only if the function ut : R+ → R, ut (s) = u(s − t)χ[t,∞) (s) belongs to B and |ut |B = |u|B . Let Cc (R+ , R) be the linear space of all continuous functions v : R+ → R with compact support and let C0c (R+ , R) = {v ∈ Cc (R+ , R) : v(0) = 0}. Notation. We denote by Q(R+ ) the class of all Banach function spaces B, which are invariant to translations, C0c (R+ , R) ⊂ B and with the properties: (i) if q ∈ Cc (R, R) and u ∈ B then qu ∈ B; t (ii) for every t > 0 there is c(t) > 0 such that 0 |u(τ )| dτ ≤ c(t) |u|B , for all u ∈ B. Example. The space C00 (R+ , R) = {u : R+ → R : u continuous and lim u(t) = u(0) = 0} t→∞
with the norm |||u||| := sup ||u(t)||, C00 (R+ , R) is a Banach function space which belongs to Q(R+ ).
t≥0
Example. Let p ∈ [1, ∞). Then the space p L (R+ , R) = u ∈ M(R+ ) :
∞ 0
|u(τ )| dτ < ∞ p
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
with respect to the norm ||u||p = space which belongs to Q(R+ ).
∞ 0
|u(τ )|p dτ
1/p
117
is a Banach function
Example. Let L∞ (R+ , R) be the linear space of all essentially bounded functions u ∈ M(R+ ). With respect to the norm ||u||∞ := ess supt≥0 |u(t)|, L∞ (R+ , R) is a Banach function space which belongs to Q(R+ ). Example. (Orlicz spaces) Let ϕ : R+ → [0, ∞] be a non-decreasing leftcontinuous function, which is not identically zero or ∞ on (0, ∞). The Young t function associated with ϕ is defined by Yϕ (t) = 0 ϕ(s) ds. For every u ∈ ∞ M(R+ ) we define Mϕ (u) := 0 Yϕ (|u(s)|) ds. The set Oϕ of all u ∈ M(R+ ) with the property that there is k > 0 such that Mϕ (ku) < ∞ is easily checked to be a linear space. With respect to the norm |u|ϕ := inf{k > 0 : Mϕ (u/k) ≤ 1}, Oϕ is a Banach space, called the Orlicz space associated with ϕ. Proposition 2.3. If ϕ(1) < ∞, then Oϕ ∈ Q(R+ ). Proof. This follows using similar arguments with those in the proof of Proposition 3.4 in [21]. Remark 2.4. The Lp -spaces are particular cases of Orlicz spaces. Notation. In what follows, we denote by V(R+ ) the class of all Banach function spaces B ∈ Q(R+ ) with the properties: (i) Cc (R+ , R) ⊂ B; (ii) if |u(·)| ≤ |v(·)| a.e. and v ∈ B, then u ∈ B. Remark 2.5. If ϕ(1) < ∞, then Oϕ ∈ V(R+ ). Lemma 2.6. Let B ∈ V(R+ ). The following assertions hold: (i) χ[a,b) ∈ B, for all b > a ≥ 0; (ii) if un → u in B, then there is a subsequence (ukn ) such that ukn → u a.e. Proof. Since Cc (R+ , R) ⊂ B, the first assertion is obvious. For the second we refer to [18]. Lemma 2.7. Let B ∈ V(R+ ) and ν > 0. Then the function eν : R+ → R, eν (t) = e−νt belongs to B. Proof. We observe that ∞ ∞ eν (t) = e−νt χ[n,n+1) (t) ≤ e−νn χ[n,n+1) (t), n=0
∀t ≥ 0.
n=0
This implies that eν ∈ B and |eν |B ≤ |χ[0,1) |B /(1 − e−ν ). Lemma 2.8. Let B ∈ V(R+ ), u ∈ B and t > 0. Then the function u ˜t : R+ → R, u˜t (s) = u(s + t) belongs to B and |˜ ut |B ≤ |u|B .
118
A. L. Sasu and B. Sasu
IEOT
Proof. Let v : R+ → R, v(s) = u(s)χ[t,∞) (s). Since B ∈ V(R+ ), we have that v ∈ B and |v|B ≤ |u|B . Using the invariance to translations of B we deduce that u ˜t ∈ B and |˜ ut | = |v|B , which concludes the proof. Lemma 2.9. Let B ∈ V(R+ ). Then, for every u ∈ B t+1 sup |u(τ )| dτ < ∞. t≥0
t
1 Proof. Let c(1) > 0 be such that 0 |v(τ )| dτ ≤ c(1) |v|B , for all v ∈ B. Let now u ∈ B and t ≥ 0. Define u˜t : R+ → X, u ˜t (s) = u(s+t). From Lemma 2.8 t+1 1 ut |B ≤ |u|B . Then t |u(τ )| dτ = 0 |˜ ut (s)| ds ≤ we have that u ˜t ∈ B and |˜ t+1 c(1) |˜ ut |B ≤ c(1) |u|B . It follows that supt≥0 t |u(τ )| dτ ≤ c(1) |u|B , for all u ∈ B. Definition 2.10. Let B ∈ V(R+ ). The function FB : (0, ∞) → R+ , FB (t) = |χ[0,t) |B is called the fundamental function of the space B. Remark 2.11. The fundamental function FB is non-decreasing. Notation. We consider F (R+ ) - the space of all Banach function spaces B ∈ V(R+ ) with the property that sup FB (t) = ∞. t>0
Remark 2.12. If ϕ(t) ∈ (0, ∞), for all t ∈ (0, ∞), then the Orlicz space Oϕ belongs to F (R+ ) (see e.g. [13], Proposition 2.1.). Notation. Let L(R+ ) denote the class of all Banach function spaces B ∈ Q(R+ ) with the property that B \ L1 (R+ , R) = ∅. Remark 2.13. If B ∈ L(R+ ), then there exists α : R+ → R+ such that α ∈ B \ L1 (R+ , R). Notation. In what follows we denote by B(R+ ) the class of all Banach function spaces B ∈ V(R+ ) with the property that for every u ∈ B the function t+1 λu : R+ → R+ , λu (t) = t u(s) ds belongs to B. Lemma 2.14. If ϕ(1) < ∞, then the Orlicz space Oϕ belongs to B(R+ ). Proof. The Orlicz spaces are rearrangement invariant (see e.g. [5]). So they are interpolation spaces between L1 (R+ , R) and L∞ (R+ , R) (see e.g. [5], Theo t+1 rem 2.2, p. 106). Let A : L∞ (R+ , R) → L∞ (R+ , R), (A(u))(t) = t u(s) ds. Then A is a bounded linear operator with the property that the restriction A| : L1 (R+ , R) → L1 (R+ , R) is correctly defined and bounded. Then A(Oϕ ) ⊂ Oϕ , so Oϕ ∈ B(R+ ). Proposition 2.15. Let B ∈ B(R+ ) and ν > 0. Then for every u ∈ B the functions fu , gu : R+ → R given by t ∞ −ν(t−s) e u(s) ds and gu (t) = e−ν(s−t) u(s) ds fu (t) = 0
belong to B.
t
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
119
Proof. Let u ∈ B. Then |u| ∈ B. Since B ∈ B(R+ ), the function t+1 ˜ u : R+ → R+ , λ ˜u (t) = λ |u(s)| ds t
belongs to B. 1 Let c > 0 be such that 0 |v(τ )| dτ ≤ c |v|B , for all v ∈ B. For t ∈ [0, 1), we have that t 1 |u(s)| ds ≤ |u(s)| ds ≤ c |u|B χ[0,1) (t). (2.1) |fu (t)| ≤ 0
0
For t ≥ 1 we observe that |fu (t)| ≤
[t]−1 t−j t−j−1
j=0 [t]−1
≤
e
−ν(t−s)
e−νj
t−j
|u(s)| ds +
{t}
0
|u(s)| ds + e−ν[t]
t−j−1
j=0
0
e−ν(t−s) |u(s)| ds {t}
|u(s)| ds
(2.2)
[t]−1
≤
˜u (t − j − 1) + (c eν |u|B )e−νt . e−νj λ
j=0
˜ u (t − j − 1)χ[j+1,∞) (t). Since B is For j ∈ N, let λj : R+ → R+ , λj (t) = λ ˜ u |B , ˜ invariant to translations and λu ∈ B, we have that λj ∈ B and |λj |B = |λ for all j ∈ N. From (2.1) and (2.2), setting m = c |u|B , we deduce that |fu (t)| ≤
∞
e−νj λj (t) + eν m e−νt + m χ[0,1) (t),
∀t ≥ 0.
j=0
Using Lemma 2.7, from the above inequality we deduce that fu ∈ B and ˜ u |B /(1 − e−ν )) + eν m |eν |B + m |χ[0,1) |B . In addition we have |fu |B ≤ (|λ that ∞ t+j+1 ∞ ˜ u (t + j), ∀t ≥ 0. e−ν(s−t) |u(s)| ds ≤ e−νj λ |gu (t)| ≤ j=0
t+j
j=0
˜ u |B /(1 − e−ν ). Using Lemma 2.8, we obtain that gu ∈ B and |gu |B ≤ |λ
Notation. Let (X, || · ||) be a real or complex Banach space. For every B ∈ Q(R+ ) we denote by B(R+ , X), the linear space of all Bochner measurable functions u : R+ → X with the property that the mapping Nu : R+ → R+ , Nu (t) = ||u(t)|| lies in B. Endowed with the norm ||u||B(R+ ,X) := |Nu |B , B(R+ , X) is a Banach space.
120
A. L. Sasu and B. Sasu
IEOT
3. Admissibility and uniform dichotomy of evolution families Let X be a real or complex Banach space. The norm on X and on B(X)-the Banach algebra of all bounded linear operators on X, will be denoted by ||·||. Denote by Id the identity operator on X. Definition 3.1. A family U = {U (t, s)}t≥s≥0 ⊂ B(X) is called an evolution family if the following properties hold: (i) U (t, t) = Id , for t ≥ 0 and U (t, s)U (s, t0 ) = U (t, t0 ), for all t ≥ s ≥ t0 ≥ 0; (ii) there are M ≥ 1 and ω > 0 such that ||U (t, s)|| ≤ M eω(t−s) , for all t ≥ s ≥ 0; (iii) for every t0 ≥ 0 and every x ∈ X the mapping t → U (t, t0 )x is continuous on [t0 , ∞) and the mapping s → U (t0 , s)x is continuous on [0, t0 ]. Definition 3.2. We say that the evolution family U = {U (t, s)}t≥s≥0 is uniformly dichotomic if there exist a constant K ≥ 1 and a family of projections {P (t)}t≥0 such that the following conditions are satisfied: (i) U (t, s)P (s) = P (t)U (t, s), for all t ≥ s ≥ 0; (ii) ||U (t, s)x|| ≤ K ||x||, for all x ∈ Im P (s) and all t ≥ s ≥ 0; 1 ||y||, for all y ∈ Ker P (s) and all t ≥ s ≥ 0; (iii) ||U (t, s)y|| ≥ K (iv) U (t, s)| : Ker P (s) → Ker P (t) is an isomorphism, for all t ≥ s ≥ 0. Let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). Definition 3.3. The pair (O(R+ , X), I(R+ , X)) is admissible for the evolution family U = {U (t, s)}t≥s≥0 if for every v ∈ I(R+ , X) there exists a continuous function f ∈ O(R+ , X) such that the pair (f, v) satisfies the equation t f (t) = U (t, s)f (s) + U (t, τ )v(τ ) dτ, ∀t ≥ s ≥ 0. (EU ) s
In what follows we consider the initial stable subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)} and we will work in the hypothesis that Xs (0) is closed and complemented in X. Let H be a closed complement of Xs (0), i.e. Xs (0) ⊕ H = X. For every t0 > 0 and every x ∈ X we consider the function U (t, t0 )x, t ≥ t0 , ϕxt0 : R+ → X, ϕxt0 (t) = 0, t ∈ [0, t0 ). We define the stable subspace at t0 > 0, by Xs (t0 ) := {x ∈ X : ϕxt0 ∈ O(R+ , X)}. Moreover, for every t0 ≥ 0 we define the unstable subspace at t0 by Xu (t0 ) = U (t0 , 0)H. Lemma 3.4. If the pair (O(R+ , X), I(R+ , X)) is admissible for the evolution family U = {U (t, s)}t≥s≥0 , then for every v ∈ I(R+ , X) there exists a unique continuous function fv ∈ O(R+ , X) such that fv (0) ∈ Xu (0) and the pair (fv , v) satisfies the equation (EU ).
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
121
Proof. Let v ∈ I(R+ , X). According to our hypothesis, there is a continuous function f ∈ O(R+ , X) such that the pair (f, v) satisfies the equation (EU ). Let xs ∈ Xs (0), xu ∈ Xu (0) be such that f (0) = xs + xu . We define fv : R+ → X, fv (t) = f (t) − U (t, 0)xs . Then fv is a continuous function with fv ∈ O(R+ , X) and fv (0) = xu ∈ Xu (0). An easy computation shows that the pair (f, v) satisfies the equation (EU ). To prove the uniqueness, let f˜ be a continuous function with f˜ ∈ O(R+ , X), f˜(0) ∈ Xu (0) such that the pair (f˜, v) satisfies the equation (EU ). Setting ϕ = f − f˜ we have that ϕ is continuous, ϕ ∈ O(R+ , X), ϕ(0) ∈ Xu (0) and ϕ(t) = U (t, 0)ϕ(0), ∀t ≥ 0. (3.1) Since ϕ ∈ O(R+ , X), from (3.1) we deduce that ϕ(0) ∈ Xs (0). Then ϕ(0) ∈ Xs (0) ∩ Xu (0), so ϕ(0) = 0. Using (3.1), we obtain the uniqueness of f. Lemma 3.5. The linear space Θ(R+ , X) = {f ∈ O(R+ , X) : f|[0,1] is continuous and f (0) ∈ Xu (0)} endowed with the norm ||f ||Θ(R+ ,X) := ||f ||O(R+ ,X) + sup ||f (t)|| is a Banach t∈[0,1]
space.
Proof. Let (fn ) be a fundamental sequence in Θ(R+ , X). In particular, (fn ) is fundamental in O(R+ , X). Since O(R+ , X) is a Banach space, there is f ∈ O(R+ , X) such that fn → f in O(R+ , X). Setting ϕn = fn|[0,1] we have that (ϕn ) is fundamental in the sup-norm. This implies that there exists ϕ : [0, 1] → X such that ϕn → ϕ uniformly on [0, 1]. It follows that ϕ is continuous on [0, 1] and since ϕn (0) ∈ Xu (0), for every n ∈ N, and Xu (0) is closed, we deduce that ϕ(0) ∈ Xu (0). From fn → f in O(R+ , X) and Lemma 2.6 (ii) we deduce that there is a subsequence (fkn ) with fkn → f a.e. Then, we have that f|[0,1] = ϕ a.e. on [0, 1]. Since in O(R+ , X) one identifies the functions equal a.e., we may consider that f|[0,1] = ϕ. In conclusion, we obtain that f ∈ Θ(R+ , X) and fn → f in Θ(R+ , X), so Θ(R+ , X) is a Banach space. Remark 3.6. According to Lemma 3.4, if the pair (O(R+ , X), I(R+ , X)) is admissible for U, then it makes sense to consider the operator Γ : I(R+ , X) → Θ(R+ , X),
Γ(v) = fv
where fv is continuous, belongs to Θ(R+ , X) and the pair (fv , v) satisfies the equation (EU ). We have that Γ is correctly defined and it is easy to see that Γ is a linear operator. Theorem 3.7. Γ is a bounded linear operator. Proof. It is sufficient to prove that Γ is closed. Let (vn ) ⊂ I(R+ , X), v ∈ I(R+ , X) and f ∈ Θ(R+ X) be such that vn → v in I(R+ , X) and Γ(vn ) → f in Θ(R+ , X). We set fn = Γ(vn ), for all n ∈ N.
122
A. L. Sasu and B. Sasu
IEOT
Let M, ω > 0 be given by Definition 3.1. Let t > 0. Since I ∈ Q(R+ ) t there is c(t) > 0 such that 0 |u(τ )| dτ ≤ c(t) |u|I , for all u ∈ I. This implies that t t U (t, τ )vn (τ ) dτ − U (t, τ )v(τ ) dτ 0 0 t ≤ M eωt ||vn (τ ) − v(τ )|| dτ ≤ M eωt c(t) ||vn − v||I(R+ ,X) . (3.2) 0
From relation (3.2) we deduce that t t U (t, τ )vn (τ ) dτ → U (t, τ )v(τ ) dτ, 0
0
as n → ∞, ∀t > 0.
(3.3)
Since fn → f in Θ(R+ , X), in particular, it follows that fn (t) → f (t) as n → ∞, for every t ∈ [0, 1]. Then, using (3.3) we obtain that t f (t) = U (t, 0)f (0) + U (t, τ )v(τ ) dτ, ∀t ∈ [0, 1]. (3.4) 0
Moreover, from fn → f in O(R+ , X) and Lemma 2.6 (ii), it follows that there is a negligible set A ⊂ (1, ∞) and a subsequence (fkn ) ⊂ (fn ) such that fkn (t) → f (t), as n → ∞, for every t ∈ (1, ∞) \ A. Then using (3.3) we deduce that t f (t) = U (t, 0)f (0) + U (t, τ )v(τ ) dτ, ∀t ∈ (1, ∞) \ A. (3.5) 0
Since in O(R+ , X) we identify the functions equal a.e., according to relations (3.4) and (3.5), we may consider that t f (t) = U (t, 0)f (0) + U (t, τ )v(τ ) dτ, ∀t ≥ 0. 0
This implies that f is continuous and the pair (f, v) satisfies the equation (EU ), so f = Γ(v). It follows that Γ is closed and by the closed graph theorem we deduce that Γ is bounded. Theorem 3.8. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then there is K > 0 such that ||U (t, t0 )x|| ≤ K ||x||,
∀x ∈ Xs (t0 ), ∀t ≥ t0 ≥ 0.
Proof. Let α : R+ → [0, 2] be a continuous function with supp α ⊂ (0, 1) and 1 α(τ ) dτ = 1. Let M, ω > 0 be given by Definition 3.1. 0 Let t0 ≥ 0 and x ∈ Xs (t0 ). Since I is invariant to translations, the function αt0 : R+ → R, αt0 (t) = α(t − t0 )χ[t0 ,∞) (t) has the property that |αt0 |I = |α|I . We consider the functions v : R+ → X, v(t) = αt0 (t)U (t, t0 )x and t αt0 (τ ) dτ U (t, t0 )x. f : R+ → X, f (t) = 0
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
123
Since v ∈ C0c (R+ , X) we have that v ∈ I(R+ , X). The function f is continuous and f (0) = 0 ∈ Xu (0). Let λ = supt∈[0,t0 +1] ||f (t)||. Observing that f (t) = U (t, t0 )x, for all t ≥ t0 + 1, it follows that ||f (t)|| ≤ λ χ[0,t0 +1) (t) + ||U (t, t0 )x||χ[t0 +1,∞) (t), If t0 = 0, let
ϕx0
: R+ → X,
ϕxt0 : R+ → X,
∀t ≥ 0.
(3.6)
ϕx0 (t)
= U (t, 0)x and if t0 > 0 let U (t, t0 )x, t ≥ t0 , x ϕt0 (t) = 0, t ∈ [0, t0 ).
Since x ∈ Xs (t0 ) we have that ϕxt0 ∈ O(R+ , X), for all t0 ≥ 0. Using relation (3.6) and Lemma 2.6 (i), we deduce that f ∈ O(R+ , X), so f ∈ Θ(R+ , X). It is easy to see that the pair (f, v) satisfies the equation (EU ), so f = Γ(v). It follows that ||f ||O(R+ ,X) ≤ ||f ||Θ(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) .
(3.7)
From ||v(t)|| = αt0 (t) ||U (t, t0 )x|| ≤ M e ||x|| αt0 (t), for all t ≥ 0, we have that (3.8) ||v||I(R+ ,X) ≤ M eω ||x|| |α|I . Let t ≥ t0 + 2. Since f (τ ) = U (τ, t0 )x, for all τ ≥ t0 + 1, we deduce that ω
||U (t, t0 )x|| χ[t−1,t) (s) ≤ M eω ||U (s, t0 )x|| χ[t−1,t) (s) ≤ M eω ||f (s)||, ∀s ≥ 0. From the invariance to translations of the space O and from the above inequality, it follows that ||U (t, t0 )x|| FO (1) ≤ M eω ||f ||O(R+ ,X) .
(3.9)
From (3.7), (3.8) and (3.9) we obtain that ||U (t, t0 )x|| ≤
M 2 e2ω ||Γ|| |α|I ||x||, FO (1)
∀t ≥ t0 + 2.
Setting K = max{M e2ω , (M 2 e2ω ||Γ|| |α|I )/FO (1)} we have ||U (t, t0 )x|| ≤ K ||x||, for all t ≥ t0 . Taking into account that K does not depend on t0 or x, we conclude that ||U (t, t0 )x|| ≤ K ||x||, for all x ∈ Xs (t0 ) and t ≥ t0 ≥ 0. Theorem 3.9. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then there is K > 0 such that 1 ||x||, ∀x ∈ Xu (t0 ), ∀t ≥ t0 ≥ 0. ||U (t, t0 )x|| ≥ K Proof. Let α : R+ → [0, 2] be a continuous function with supp α ⊂ (0, 1) and 1 α(τ ) dτ = 1. Let M, ω > 0 be given by Definition 3.1. 0 Let t ≥ t0 ≥ 0 and x ∈ Xu (0). Since I is invariant to translations, the function αt : R+ → R, αt (s) = α(s−t)χ[t,∞) (s) belongs to I and |αt |I = |α|I . We consider the functions v : R+ → X, v(s) = −αt (s) U (s, 0)x and ∞ f : R+ → X, f (s) = αt (τ ) dτ U (s, 0)x. s
We have that f ∈ Cc (R+ , X) and v ∈ C0c (R+ , X), so in particular, f ∈ O(R+ , X) and v ∈ I(R+ , X). Moreover, f is continuous with
124
A. L. Sasu and B. Sasu
IEOT
f (0) = x ∈ Xu (0), so f ∈ Θ(R+ , X). An easy computation shows that the pair (f, v) satisfies the equation (EU ), so f = Γ(v). This implies that ||f ||Θ(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) .
(3.10)
We observe that f (s) = U (s, 0)x, for all s ∈ [0, t]. If t0 ∈ [0, 1], since ||f ||Θ(R+ ,X) = ||f ||O(R+ ,X) + supt∈[0,1] ||f (t)|| from (3.10) it follows that ||U (t0 , 0)x|| = ||f (t0 )|| ≤ ||Γ|| ||v||I(R+ ,X) .
(3.11)
If t0 > 1 then ||U (t0 , 0)x|| χ[t0 −1,t0 ) (s) ≤ M eω ||U (s, 0)x|| χ[t0 −1,t0 ) (s) ≤ M eω ||f (s)||, for all s ≥ 0. Using the invariance to translations of the space O, from the above inequality we deduce that ||U (t0 , 0)x|| FO (1) ≤ M eω ||f ||O(R+ ,X) .
(3.12)
Then, from (3.10) and (3.12) we obtain that ||U (t0 , 0)x|| ≤
M eω ||Γ|| ||v||I(R+ ,X) . FO (1)
(3.13)
Setting γ = max{||Γ||, M eω ||Γ||/FO (1)}, from relations (3.11) and (3.13) we have that (3.14) ||U (t0 , 0)x|| ≤ γ ||v||I(R+ ,X) . Moreover, from ||v(s)|| = αt (s) ||U (s, 0)x|| ≤ αt (s) M eω ||U (t, 0)x||, for all s ≥ 0, it follows that ||v||I(R+ ,X) ≤ M eω |α|I ||U (t, 0)x||.
(3.15)
Setting K = γM e |α|I , from relations (3.14) and (3.15) we obtain that ||U (t, 0)x|| ≥ (1/K) ||U (t0 , 0)x||. Since K does not depend on x, t0 or t, we deduce that ||U (t, 0)x|| ≥ (1/K) ||U (t0 , 0)x||, for all x ∈ Xu (0) and all t ≥ t0 ≥ 0. Since Xu (t0 ) = U (t0 , 0)Xu (0), for all t0 ≥ 0, the proof is complete. ω
Theorem 3.10. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then for every t0 ≥ 0, the subspaces Xs (t0 ) and Xu (t0 ) are closed. Proof. Step 1. We prove that Xs (t0 ) is closed, for all t0 ≥ 0. Let α : R+ → [0, 2] 1 be a continuous function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Let M, ω > 0 be given by Definition 3.1. Let t0 > 0 and αt0 : R+ → R, αt0 (t) = α(t − t0 )χ[t0 ,∞) (t). Then αt0 ∈ I and |αt0 |I = |α|I . Let (xn ) ⊂ Xs (t0 ) with xn → x. For every n ∈ N we consider the functions vn : R+ → X, vn (t) = αt0 (t)U (t, t0 )xn t fn : R+ → X, fn (t) = αt0 (τ ) dτ U (t, t0 )xn . 0
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
125
Using similar arguments as in the proof of Theorem 3.8, we deduce that vn ∈ I(R+ , X), fn ∈ O(R+ , X) and that fn = Γ(vn ), for all n ∈ N. Let v : R+ → X,
v(t) = αt0 (t)U (t, t0 )x
and let f = Γ(v). Then we have that ||fn −f ||O(R+ ,X) ≤ ||fn −f ||Θ(R+ ,X) ≤ ||Γ|| ||vn −v||I(R+ ,X) ,
∀n ∈ N. (3.16)
Observing that ||vn (t)−v(t)|| ≤ αt0 (t) M e ||xn −x||, for all n ∈ N, we deduce that (3.17) ||vn − v||I(R+ ,X) ≤ |α|I M eω ||xn − x||, ∀n ∈ N. From (3.16) and (3.17) it follows that fn → f in O(R+ , X). From Lemma 2.6 (ii) we obtain that there is a subsequence (fkn ) ⊂ (fn ) such that fkn → f a.e. on R+ . In particular, there is h > t0 + 1 such that fkn (h) → f (h). We observe that fn (t) = U (t, t0 )xn , for all t ≥ t0 + 1 and all n ∈ N. Then, we deduce that f (h) = lim fkn (h) = lim U (h, t0 )xkn = U (h, t0 )x. ω
n→∞
n→∞
Since f = Γ(v), we have that f (t) = U (t, h)f (h) = U (t, t0 )x, for all t ≥ h. Because f ∈ O(R+ , X) and O ∈ V(R+ ) it follows that the function U (t, t0 )x, t ≥ h, ϕ : R+ → X, ϕ(t) = 0, t ∈ [0, h) belongs to O(R+ , X). Considering the function U (t, t0 )x, x x ϕt0 : R+ → X, ϕt0 (t) = 0,
t ≥ t0 t ∈ [0, t0 ),
we have that ||ϕxt0 (t)|| ≤ M eω(h−t0 ) ||x|| χ[t0 ,h) (t) + ||ϕ(t)||, for all t ≥ 0. This implies that ϕxt0 ∈ O(R+ , X), so x ∈ Xs (t0 ). In conclusion, the subspace Xs (t0 ) is closed. Step 2. We prove that Xu (t0 ) is closed, for all t0 ≥ 0. Let K > 0 be given by Theorem 3.9. Let t0 > 0 and (yn ) ⊂ Xu (t0 ) with yn → y. Since yn ∈ Xu (t0 ) = U (t0 , 0)Xu (0), there is zn ∈ Xu (0) such that yn = U (t0 , 0)zn , for all n ∈ N. From 1 ||yn − ym || = ||U (t0 , 0)(zn − zm )|| ≥ ||zn − zm ||, ∀n, m ∈ N K we deduce that the sequence (zn ) is fundamental in Xu (0). Since Xu (0) is closed, it follows that there is z ∈ Xu (0) such that zn → z. Then y = lim yn = lim U (t0 , 0)zn = U (t0 , 0)z. n→∞
n→∞
It follows that y ∈ U (t0 , 0)Xu (0) = Xu (t0 ), so Xu (t0 ) is closed.
Lemma 3.11. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then Xs (t0 ) ∩ Xu (t0 ) = {0},
∀t0 ≥ 0.
Proof. Let t0 > 0 and x ∈ Xs (t0 ) ∩ Xu (t0 ). Since x ∈ Xu (t0 ), there is z ∈ Xu (0) such that x = U (t0 , 0)z. From x ∈ Xs (t0 ) we have that the function U (t, t0 )x, t ≥ t0 , x x ϕt0 : R+ → X, ϕt0 (t) = 0, t ∈ [0, t0 )
126
A. L. Sasu and B. Sasu
IEOT
belongs to O(R+ , X). Let ϕ : R+ → X, ϕ(t) = U (t, 0)z. If M, ω > 0 are given by Definition 3.1, we obtain that ||ϕ(t)|| ≤ M eωt0 ||z|| χ[0,t0 ) (t) + ||ϕxt0 (t)||, for all t ≥ 0. Since O ∈ V(R+ ) from the above inequality we deduce that ϕ ∈ O(R+ , X). This implies that z ∈ Xs (0), so z ∈ Xs (0) ∩ Xu (0). It follows that z = 0, so x = 0, and the proof is complete. Lemma 3.12. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then Xs (t0 ) + Xu (t0 ) = X,
∀t0 ≥ 0.
Proof. Let t0 > 0 and x ∈ X. Let α : R+ → [0, 2] be a continuous function t +1 with supp α ⊂ (t0 , t0 + 1) and t00 α(τ ) dτ = 1. We consider the function v : R+ → X,
v(t) = α(t)U (t, t0 )x.
Since v ∈ C0c (R+ , X) we have that v ∈ I(R+ , X). Let f = Γ(v). Then for every t ≥ t0 + 1 we have that t U (t, τ )v(τ ) dτ = U (t, t0 )(f (t0 ) + x). (3.18) f (t) = U (t, t0 )f (t0 ) + t0
Let
ϕ : R+ → X,
ϕ(t) =
U (t, t0 )(f (t0 ) + x), 0,
t ≥ t0 , t ∈ [0, t0 ).
If M, ω > 0 are given by Definition 3.1, then using (3.18) we deduce that ||ϕ(t)|| ≤ M eω ||f (t0 )+x|| χ[t0 ,t0 +1) (t)+||f (t)||, for all t ≥ 0. Since O ∈ V(R+ ) the above inequality implies that ϕ ∈ O(R+ , X), so f (t0 ) + x ∈ Xs (t0 ). In addition, we observe that f (t0 ) = U (t0 , 0)f (0). From f = Γ(v), we have that f (0) ∈ Xu (0), so f (t0 ) ∈ Xu (t0 ). In conclusion, we have that x = (x + f (t0 )) − f (t0 ) ∈ Xs (t0 ) + Xu (t0 ) and the proof is complete.
The main result of this section is: Theorem 3.13. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). If the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X, then the evolution family U is uniformly dichotomic. Proof. Let Xu (0) be a closed linear subspace of X such that X = Xs (0) ⊕ Xu (0). For every t0 > 0 we consider Xs (t0 ) - as the linear subspace of all x ∈ X with the property that the function U (t, t0 )x, t ≥ t0 , x x ϕt0 : R+ → X, ϕt0 (t) = 0, t ∈ [0, t0 )
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
127
belongs to O(R+ , X). We set Xu (t0 ) = U (t0 , 0)Xu (0), for all t0 > 0. Then, from Theorem 3.10, Lemma 3.11 and Lemma 3.12 we obtain that Xs (t0 ) ⊕ Xu (t0 ) = X, for all t0 ≥ 0. For every t0 ≥ 0 let P (t0 ) be the projection such that Im P (t0 ) = Xs (t0 ) and Ker P (t0 ) = Xu (t0 ). Then U (t, t0 )P (t0 ) = P (t)U (t, t0 ),
∀t ≥ t0 ≥ 0.
From the definition of the spaces Xu (t0 ) and from Theorem 3.9 we deduce that the restriction U (t, t0 )| : Ker P (t0 ) → Ker P (t) is an isomorphism, for all t ≥ t0 ≥ 0. Finally, from Theorem 3.8 and Theorem 3.9 we obtain that the evolution family U is uniformly dichotomic.
4. Uniform exponential dichotomy In this section we continue the investigation begun for the case of uniform dichotomy and we will give necessary and sufficient conditions for uniform exponential dichotomy of evolution families on the half-line in terms of the admissibility of pairs of function spaces. Let X be a real or complex Banach space. Definition 4.1. We say that an evolution family U = {U (t, s)}t≥s≥0 is uniformly exponentially dichotomic if there exist two constants K ≥ 1, ν > 0 and a family of projections {P (t)}t≥0 such that the following conditions are satisfied: (i) (ii) (iii) (iv)
U (t, s)P (s) = P (t)U (t, s), for all t ≥ s ≥ 0; ||U (t, s)x|| ≤ K e−ν(t−s) ||x||, for all x ∈ Im P (s) and all t ≥ s ≥ 0; 1 ||U (t, s)y|| ≥ K eν(t−s) ||y||, for all y ∈ Ker P (s) and all t ≥ s ≥ 0; U (t, s)| : Ker P (s) → Ker P (t) is an isomorphism, for all t ≥ s ≥ 0.
Let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). Let U = {U (t, s)}t≥s≥0 be an evolution family on X. In what follows we suppose that the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X. Let Xu (0) be a closed linear subspace such that X = Xs (0) ⊕ Xu (0). As in the previous section, for every t0 > 0 we consider Xs (t0 ) - the stable subspace and Xu (t0 ) - the unstable subspace. In this context, according to the main result in Section 3, if the pair (O(R+ , X), I(R+ , X)) is admissible for U, then U is uniformly dichotomic with respect to a family of projections {P (t)}t≥0 such that Im P (t0 ) = Xs (t0 )
and
Ker P (t0 ) = Xu (t0 ),
∀t0 ≥ 0.
In what follows, we prove that if O ∈ F(R+ )
or
I ∈ L(R+ )
then the admissibility of the pair (O(R+ , X), I(R+ , X)) for U implies the uniform exponential dichotomy of U.
128
A. L. Sasu and B. Sasu
IEOT
Lemma 4.2. Suppose that the following assertions hold: (i) there is δ > 0 such that ||U (t, t0 )x|| ≤ δ ||x||, for all x ∈ Im P (t0 ) and all t ≥ t0 ≥ 0; (ii) there is r > 0 such that ||U (t0 + r, t0 )x|| ≤ 1e ||x||, for all x ∈ Im P (t0 ) and all t0 ≥ 0. Then ||U (t, t0 )x|| ≤ Ke−ν(t−t0 ) ||x||, for all x ∈ Im P (t0 ) and all t ≥ t0 ≥ 0, where K = δe and ν = 1/r. Proof. Let t ≥ t0 ≥ 0 and x ∈ Im P (t0 ). If k ∈ N and s ∈ [0, r) are such that t = t0 + kr + s, then ||U (t, t0 )x|| ≤ δ ||U (t0 + kr, t0 )x|| ≤ δe−k ||x|| ≤ Ke−ν(t−t0 ) ||x||.
Theorem 4.3. If O ∈ F(R+ ) or I ∈ L(R+ ) and the pair (O(R+ , X), I(R+ , X)) is admissible for U, then there are K, ν > 0 such that ||U (t, t0 )x|| ≤ Ke−ν(t−t0 ) ||x||,
∀x ∈ Im P (t0 ), ∀t ≥ t0 ≥ 0.
Proof. Let δ > 0 be such that ||U (t, t0 )x|| ≤ δ ||x||,
∀t ≥ t0 ≥ 0, ∀x ∈ Im P (t0 ).
(4.1)
Case 1: O ∈ F(R+ ). Then supt>0 FO (t) = ∞. Let α : R+ → [0, 2] be a 1 continuous function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Let p > 0 be such that FO (p) ≥ e δ 2 ||Γ|| |α|I . (4.2) Let t0 ≥ 0 and let x ∈ Im P (t0 ). From I ∈ Q(R+ ) we have that the function αt0 : R+ → R, αt0 (t) = α(t − t0 )χ[t0 ,∞) (t) lies in I and |αt0 |I = |α|I . If U (t0 + 1, t0 )x = 0, we consider the functions v : R+ → X,
v(t) =
αt0 (t) U (t, t0 )x ||U (t, t0 )x||
t
αt0 (τ ) dτ U (t, t0 )x. ||U (τ, t0 )x|| 0 We have that v ∈ C0c (R+ , X), so v ∈ I(R+ , X). We observe that f is continuous with f (0) = 0. Moreover, setting t0 +1 αt0 (τ ) dτ λ := ||U (τ, t0 )x|| t0 f : R+ → X,
f (t) =
we have that f (t) = λ U (t, t0 )x, for all t ≥ t0 + 1. If t0 > 0 let U (t, t0 )x, t ≥ t0 , ϕxt0 : R+ → X, ϕxt0 (t) = 0, t ∈ [0, t0 ) and if t0 = 0 let ϕx0 : R+ → X, ϕx0 (t) = U (t, 0)x. Since x ∈ Im P (t0 ) = Xs (t0 ) we have that ϕxt0 ∈ O(R+ , X). Observing that ||f (t)|| ≤ λδ||x|| χ[t0 ,t0 +1) (t) + λ ||ϕxt0 (t)||, for all t ≥ 0 we deduce that
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
129
f ∈ O(R+ , X). Thus f ∈ Θ(R+ , X). An easy computation shows that the pair (f, v) satisfies the equation (EU ), so f = Γ(v). This implies that ||f ||O(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) .
(4.3)
From ||v(t)|| = αt0 (t), for all t ≥ 0, we have that ||v||I(R+ ,X) = |αt0 |I = |α|I . Then, from (4.3) we deduce that ||f ||O(R+ ,X) ≤ ||Γ|| |α|I .
(4.4)
It is easy to see that λ ≥ 1/(δ||x||). Then from ||U (t0 + p + 1, t0 )x|| χ[t0 +1,t0 +p+1) (s) ≤ δ ||U (s, t0 )x|| χ[t0 +1,t0 +p+1) (s) δ ||f (s)|| χ[t0 +1,t0 +p+1) (s) ≤ δ 2 ||x|| ||f (s)||, ∀s ≥ 0 λ using the invariance to translations of the space O and relation (4.4) we deduce that =
||U (t0 + p + 1, t0 )x|| FO (p) ≤ δ 2 ||Γ|| |α|I ||x||.
(4.5)
Setting r = p + 1 from relation (4.2) and (4.5) we obtain that ||U (t0 + r, t0 )x|| ≤ (1/e) ||x||. If U (t0 + 1, t0 )x = 0 then this inequality is obvious. Taking into account that r does not depend on t0 or x, it follows that 1 ||x||, ∀x ∈ Im P (t0 ), ∀t0 ≥ 0. e From relation (4.6), using Lemma 4.2 we obtain the conclusion. ||U (t0 + r, t0 )x|| ≤
Case 2: I ∈ L(R+ ). Let c > 0 be such that 1 |w(τ )| dτ ≤ c |w|O , 0
∀w ∈ O.
(4.6)
(4.7)
Since I ∈ L(R+ ) according to Remark 2.13, there is u : R+ → R+ with u ∈ I \ L1 (R+ , R). Let p > 0 be such that p u(s) ds ≥ e δ 2 c ||Γ|| |u|I . (4.8) 0
Let u0 = u and for t0 > 0, let ut0 : R+ → R, ut0 (t) = u(t − t0 )χ[t0 ,∞) (t). From the invariance to translations of I we have that |ut0 |I = |u|I , for all t0 ≥ 0. Let t0 ≥ 0 and x ∈ ImP (t0 ). Let ϕ : R+ → [0, 1] be a continuous function with compact support such that ϕ(t) = 1, for all t ∈ [0, t0 + p] and ϕ(t) = 0, for all t ≥ t0 + p + 1. We define the functions v : R+ → X, and
v(t) = ut0 (t) ϕ(t) U (t, t0 )x
ψ : R+ → R + ,
ψ(t) =
ϕ(t)||U (t, t0 )x||, ||x||,
t > t0 , t ∈ [0, t0 ].
130
A. L. Sasu and B. Sasu
IEOT
Then ||v(t)|| = ut0 (t)ψ(t), for all t ≥ 0. Since ψ ∈ Cc (R+ , R), ut0 ∈ I and I ∈ L(R+ ) we deduce that ut0 ψ ∈ I. It follows that v ∈ I(R+ , X). In addition ||v(t)|| ≤ ut0 (t)||U (t, t0 )x|| ≤ δ ||x|| ut0 (t), for all t ≥ 0. Hence we have that ||v||I(R+ ,X) ≤ δ ||x|| |ut0 |I = δ ||x|| |u|I . Let
f : R+ → X,
f (t) = 0
(4.9)
t
ut0 (τ ) ϕ(τ ) dτ U (t, t0 )x
t +p+1 ut0 (τ )ϕ(τ ) dτ. Then f is continuous, f (0) = 0 and f (t) = and λ := t00 λ U (t, t0 )x, for all t ≥ t0 + p + 1. Using similar arguments with those from Case 1, we deduce that f ∈ O(R+ , X), so f ∈ Θ(R+ , X). We observe that the pair (f, v) satisfies the equation (EU ), so f = Γ(v). Then from relation (4.9) we deduce that ||f ||O(R+ ,X) ≤ ||f ||Θ(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) ≤ δ ||Γ|| |u|I ||x||.
(4.10)
We consider the function g : R+ → X, g(t) = f (t + t0 + p + 1). From Lemma 2.8 we have that g ∈ O(R+ , X) and ||g||O(R+ ,X) ≤ ||f ||O(R+ ,X) . Using relation (4.7) it follows that 1 ||g(τ )|| dτ ≤ c ||g||O(R+ ,X) ≤ c ||f ||O(R+ ,X) . 0
Taking into account that ||U (t0 + p + 2, t0 )x|| ≤ δ ||U (s, t0 )x||, for all s ∈ [t0 + p + 1, t0 + p + 2], we deduce that t0 +p+2 δ t0 +p+2 ||U (s, t0 )x|| ds = ||f (s)|| ds ||U (t0 + p + 2, t0 )x|| ≤ δ λ t0 +p+1 t0 +p+1 δ 1 δc ||f ||O(R+ ,X) . ||g(τ )|| dτ ≤ (4.11) = λ 0 λ We note that t0 +p ut0 (τ )ϕ(τ ) dτ = λ≥ t0
t0 +p
t0
ut0 (τ ) dτ =
p
u(τ ) dτ
(4.12)
0
and then from relations (4.8), (4.10), (4.11) and (4.12) it follows that ||U (t0 + p + 2, t0 )x|| ≤ (1/e) ||x||. We set r = p + 2 and since r does not depend on t0 or x, we obtain that ||U (t0 + r, t0 )x|| ≤ (1/e) ||x||, for all x ∈ Im P (t0 ) and all t0 ≥ 0. Finally, by applying Lemma 4.2 we deduce the conclusion. Lemma 4.4. Suppose that the following assertions hold: (i) there is δ > 0 such that ||U (t, t0 )x|| ≥ 1δ ||x||, for all x ∈ Ker P (t0 ) and all t ≥ t0 ≥ 0; (ii) there is r > 0 such that ||U (t0 + r, t0 )x|| ≥ e ||x||, for all x ∈ Ker P (t0 ) and all t0 ≥ 0. 1 Then ||U (t, t0 )x|| ≥ K eν(t−t0 ) ||x||, for all x ∈ Ker P (t0 ) and all t ≥ t0 ≥ 0, where K = δe and ν = 1/r.
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
131
Proof. Let t ≥ t0 ≥ 0 and x ∈ Ker P (t0 ). Let k ∈ N and s ∈ [0, r) be such that t = t0 + kr + s. Then 1 1 1 ν(t−t0 ) e ||x||. ||U (t, t0 )x|| ≥ ||U (t0 + kr, t0 )x|| ≥ ek ||x|| ≥ δ δ K Theorem 4.5. If O ∈ F(R+ ) or I ∈ L(R+ ) and the pair (O(R+ , X), I(R+ , X)) is admissible for U, then there are K, ν > 0 such that ||U (t, t0 )x|| ≥
1 ν(t−t0 ) ||x||, e K
∀x ∈ Ker P (t0 ), ∀t ≥ t0 ≥ 0.
Proof. Let M, ω > 0 be given by Definition 3.1. Let δ > 0 be such that 1 ||x||, ∀x ∈ Ker P (t0 ), ∀t ≥ t0 ≥ 0. δ Case 1: O ∈ F(R+ ). Let α : R+ → [0, 2] be a continuous function with 1 supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Let p > 0 be such that ||U (t, t0 )x|| ≥
FO (p) ≥ e δ 2 ||Γ|| |α|I .
(4.13)
Let t0 ≥ 0 and x ∈ Ker P (t0 ) \ {0}. Since I ∈ Q(R+ ) the function αt0 +p : R+ → R, αt0 +p (t) = α(t − t0 − p)χ[t0 +p,∞) (t) has the property that |αt0 +p |I = |α|I . We consider the functions v : R+ → X,
f : R+ → X,
f (t) =
v(t) = −
⎧ ∞ ⎪ ⎨ t
αt0 +p (t) U (t, t0 )x, ||U (t, t0 )x||
αt0 +p (τ ) ||U(τ,t0 )x||
dτ U (t, t0 )x, λ U (t, t0 )x, λ U (t0 , t)−1 | x,
⎪ ⎩
t ≥ t0 + p, t ∈ (t0 , t0 + p), t ∈ [0, t0 ],
denotes the inverse of the operator where U (t0 , t)−1 | U (t0 , t)| : Ker P (t) → Ker P (t0 ), for all t ∈ [0, t0 ] and
t0 +p+1
λ := t0 +p
αt0 +p (τ ) dτ. ||U (τ, t0 )x||
Since v ∈ C0c (R+ , X) we have that v ∈ I(R+ , X). From ||v(t)|| = αt0 +p (t), for all t ≥ 0, we deduce that ||v||I(R+ ,X) = |α|I . We observe that f ∈ Cc (R+ , X), so f ∈ O(R+ , X). In addition, we note that f (0) = λ U (t0 , 0)−1 | x ∈ Ker P (0) = Xu (0) and hence f ∈ Θ(R+ , X). An easy computation shows that the pair (f, v) satisfies the equation (EU ). Thus f = Γ(v), so ||f ||O(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) = ||Γ|| |α|I . From ||U (t, t0 )x|| ≥ (1/δ)||x||, for all t ∈ [t0 , t0 + p) we have that ||x|| χ[t0 ,t0 +p) (t) ≤ δ ||U (t, t0 )x|| χ[t0 ,t0 +p) (t) ≤ (δ/λ) ||f (t)||,
(4.14)
132
A. L. Sasu and B. Sasu
IEOT
for all t ≥ 0. Now, using the invariance to translations of the space O we deduce that ||x|| FO (p) ≤ (δ/λ) ||f ||O(R+ ,X) . (4.15) From relations (4.13)-(4.15) it follows that e δ ||x|| ≤ 1/λ.
(4.16)
From ||U (t0 + p + 1, t0 )x|| ≥ (1/δ) ||U (τ, t0 )x||, for all τ ∈ [t0 + p, t0 + p + 1], we obtain that 1 . (4.17) λ≥ δ ||U (t0 + p + 1, t0 )x|| Then, from (4.16) and (4.17) it follows that ||U (t0 + p + 1, t0 )x|| ≥ e ||x||. We set r = p + 1 and taking into account that r does not depend on t0 or x we deduce that ||U (t0 + r, t0 )x|| ≥ e ||x||, for all x ∈ Ker P (t0 ) and all t0 ≥ 0. By applying Lemma 4.4 we obtain the conclusion. Case 2: I ∈ L(R+ ). Let c ≥ 1 be such that 1 |w(τ )| dτ ≤ c |w|O , 0
∀w ∈ O.
(4.18)
Since I ∈ L(R+ ) there is u : R+ → R+ with u ∈ I \ L1 (R+ , R). Let p > 0 be such that p u(s) ds ≥ e c M eω δ ||Γ|| |u|I . (4.19) 0
Let u0 = u and for t0 > 0 let ut0 : R+ → R, ut0 (t) = u(t − t0 )χ[t0 ,∞) (t). From the invariance to translations of the space I we have that |ut0 |I = |u|I , for all t0 ≥ 0. Let t0 ≥ 0 and x ∈ Ker P (t0 ). Let ϕ : R+ → [0, 1] be a continuous function with compact support such that ϕ(t) = 1, for t ∈ [0, t0 + p] and ϕ(t) = 0, for t ≥ t0 + p + 1. We consider the functions v : R+ → X, v(t) = −ut0 (t)ϕ(t)U (t, t0 )x ∞ t > t0 , t ut0 (τ )ϕ(τ ) dτ U (t, t0 )x, f : R+ → X, f (t) = λ U (t0 , t)−1 x, t ∈ [0, t0 ], | t0 +p+1 ut0 (τ )ϕ(τ ) dτ . Using similar arguments as in the proof where λ := t0 of Theorem 4.3, Case 2, we obtain that v ∈ I(R+ , X). From ||U (t0 + p + 1, t0 )x|| ≥ (1/δ) ||U (t, t0 )x||, for all t ∈ [t0 , t0 + p + 1], we deduce that ||v(t)|| = ut0 (t)ϕ(t)||U (t, t0 )x|| ≤ δ ut0 (t)||U (t0 + p + 1, t0 )x||, for all t ≥ 0. This implies that ||v||I(R+ ,X) ≤ δ |u|I ||U (t0 + p + 1, t0 )x||. Since f ∈ Cc (R+ , X) we have that f ∈ O(R+ , X). In addition, f (0) = λ U (t0 , 0)−1 | x ∈ Ker P (0) = Xu (0),
(4.20)
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
133
so f ∈ Θ(R+ , X). An easy computation shows that the pair (f, v) satisfies the equation (EU ), so f = Γ(v). Using (4.20), it follows that ||f ||Θ(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) ≤ δ ||Γ|| |u|I ||U (t0 + p + 1, t0 )x||.
(4.21)
If t0 ∈ [0, 1], then λ ||x|| = ||f (t0 )|| ≤ supt∈[0,1] ||f (t)|| ≤ ||f ||Θ(R+ ,X) . Hence using (4.21) we have that λ ||x|| ≤ δ ||Γ|| |u|I ||U (t0 + p + 1, t0 )x||.
(4.22)
If t0 > 1, then let g : R+ → X, g(t) = f (t + t0 − 1). Since f ∈ O(R+ , X), g ∈ O(R+ , X) and ||g||O(R+ ,X) ≤ ||f ||O(R+ ,X) (see Lemma 2.8). We observe that λx = U (t0 , t)f (t), for all t ∈ [t0 − 1, t0 ]. Thus, using relations (4.18) and (4.21) we successively deduce that t0 1 ω ω ||f (τ )|| dτ = M e ||g(τ )|| dτ ≤ c M eω ||g||O(R+ ,X) λ ||x|| ≤ M e 0
t0 −1
≤ c M eω ||f ||O(R+ ,X) ≤ c M eω δ ||Γ|| |u|I ||U (t0 + p + 1, t0 )x||. (4.23) p Observing that λ ≥ 0 u(τ ) dτ , from (4.22), (4.23) and (4.19) we obtain that ||U (t0 + p + 1, t0 )x|| ≥ e ||x||. We set r = p + 1 and since r does not depend on t0 or x we have that ||U (t0 + r, t0 )x|| ≥ e ||x||, for all t0 ≥ 0 and all x ∈ Ker P (t0 ). By applying Lemma 4.4 we obtain the conclusion. The main result of this section is: Theorem 4.6. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). If O ∈ F(R+ ) or I ∈ L(R+ ), then the following assertions hold: (i) if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X, then U is uniformly exponentially dichotomic; (ii) if I ⊂ O and one of the spaces I or O belongs to B(R+ ), then U is uniformly exponentially dichotomic if and only if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) is closed and complemented in X. Proof. (i) This follows from Theorem 3.13, Theorem 4.3 and Theorem 4.5. (ii) Necessity. Suppose that U is uniformly exponentially dichotomic with respect to the family of projections {P (t)}t≥0 and the constants K, ν > 0. Then, we have that L := supt≥0 ||P (t)|| < ∞ (see e.g.[15]). Let v ∈ I(R+ , X). We consider the function t ∞ U (t, τ )P (τ )v(τ ) dτ − U (τ, t)−1 f : R+ → X, f (t) = | (I − P (τ ))v(τ ) dτ. 0
t
We have that f is continuous and for all t ≥ 0 t −ν(t−τ ) ||f (t)|| ≤ LK e ||v(τ )|| dτ + (L + 1)K 0
t
∞
e−ν(τ −t) ||v(τ )|| dτ. (4.24)
134
A. L. Sasu and B. Sasu
IEOT
If I ∈ B(R+ ), since v ∈ I(R+ , X) it follows that the function t+1 ||v(s)|| ds λv : R+ → R+ , λv (t) = t
lies in I. Using Proposition 2.15, relation (4.24) and the fact that I ⊂ O we deduce that f ∈ O(R+ , X). If O ∈ B(R+ ), then from I ⊂ O and v ∈ I(R+ , X) we have that v ∈ O(R+ , X). Then, from Proposition 2.15 and relation (4.24) we obtain that f ∈ O(R+ , X). It is easy to see that the pair (f, v) satisfies the equation (EU ). It follows that the pair (O(R+ , X), I(R+ , X)) is admissible for U. We prove that Xs (0) = Im P (0). Let x ∈ Im P (0). Then ||U (t, 0)x|| ≤ Ke−νt ||x||, for all t ≥ 0. Since O ∈ V(R+ ), using Lemma 2.7 it follows that U (·, 0)x ∈ O(R+ , X), so x ∈ Xs (0). This shows that Im P (0) ⊂ Xs (0). Conversely, let x ∈ Xs (0). Then, the function ϕ : R+ → X, ϕ(t) = U (t, 0)x belongs to O(R+ , X). Setting xs = P (0)x and xu = (I − P (0))x, we have that ||ϕ(t)|| ≥ ||U (t, 0)xu || − ||U (t, 0)xs || ≥ (1/K) eνt ||xu || − K||xs ||, for all t ≥ 0. It follows that t+1 1 νt e ||xu || − K ||xs ||, ∀t ≥ 0. ||ϕ(τ )|| dτ ≥ (4.25) K t t+1 From Lemma 2.9 we have that supt≥0 t ||ϕ(τ )|| dτ < ∞. Then using (4.25) we obtain that xu = 0, so x = P (0)x. This shows that x ∈ Im P (0), so Xs (0) = Im P (0). Thus, we conclude that Xs (0) is closed and complemented in X. Sufficiency. This follows from (i).
5. Connections and applications In this section we present some applications of the main results of the paper and establish the connections with other dichotomy results published in the past few years. We will emphasize the generality of our conditions and provide some optimal methods in the study of dichotomy. The section is organized in two parts. In the first part, we prove that the result obtained in Theorem 4.6 is the most general in this topic and, in the second part, we obtain as consequences several criteria with wide applicability area. Let Cb (R+ , X) denote the space of all continuous functions v : R+ → X with the property that supt≥0 ||v(t)|| < ∞ and let C0 (R+ , X) be the space
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
135
of all continuous functions v : R+ → X with lim v(t) = 0. With respect to t→∞ the norm |||v||| := sup ||v(t)|| t≥0
the spaces Cb (R+ , X) and C0 (R+ , X) are Banach spaces. The first natural question arises whether the hypothesis “O ∈ F(R+ ) or I ∈ L(R+ )” can be removed. Specifically, the main question is: if O ∈ F(R+ ) and I ∈ L(R+ ), then is the admissibility of the pair (O(R+ , X), I(R+ , X)) a sufficient condition for uniform exponential dichotomy? Generally, the answer is negative. To argue this fact we need a technical lemma. Lemma 5.1. Let O be a Banach function space with O ∈ V(R+ ) \ F (R+ ). Then C0 (R+ , R) ⊂ O. Proof. Let γ := supt>0 FO (t). Let u ∈ C0 (R+ , R). Then there is an unbounded increasing sequence (tn ) ⊂ (0, ∞) such that |u(t)| ≤ 1/(n + 1), for all t ≥ tn and all n ∈ N. Setting un = u χ[0,tn ) we have that |un+p − un |O ≤ 1/(n + 1) |χ[tn ,tn+p ) |O ≤ γ/(n + 1), for all p ∈ N∗ and all n ∈ N. The above inequality shows that the sequence (un ) is fundamental in the Banach space O, so there is v ∈ O such that un → v in O. From Lemma 2.6 (ii) there exists a subsequence (ukn ) such that ukn → v a.e. This implies that u = v a.e., so u = v in O. Thus u ∈ O and the proof is complete. In what follows we prove that the assumptions on the structure of the underlying function spaces are indeed necessary. Example. Let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). We suppose that O ∈ F(R+ ) and I ∈ L(R+ ). This implies that I ⊂ L1 (R+ , R) and from Lemma 5.1 we have that C0 (R+ , R) ⊂ O. Let X = R2 with the norm ||(x1 , x2 )|| = |x1 | + |x2 |, for all (x1 , x2 ) ∈ X. For every t ≥ s ≥ 0 we consider the operator
s+1 t−s x1 , e x2 . U (t, s) : X → X, U (t, s)(x1 , x2 ) = t+1 Then U = {U (t, s)}t≥s≥0 is an evolution family on X. We prove that the pair (O(R+ , X), I(R+ , X)) is admissible for U. Let v = (v1 , v2 ) ∈ I(R+ , X). Then v1 , v2 ∈ L1 (R+ , R). We consider the function
t ∞ τ +1 v1 (τ ) dτ, − f : R+ → X, f (t) = e−(τ −t) v2 (τ ) dτ . 0 t+1 t We have that f is correctly defined, continuous and an easy computation shows that the pair (f, v) satisfies the equation (EU ). Setting t ∞ τ +1 v1 (τ ) dτ, f2 (t) = − e−(τ −t) v2 (τ ) dτ, ∀t ≥ 0, f1 (t) = t + 1 0 t it is obvious that f2 ∈ C0 (R+ , R).
136
A. L. Sasu and B. Sasu
IEOT
We prove that f1 ∈ C0 (R+ , R).Let ε > 0. From v1 ∈ L1 (R+ , R) it ∞ follows that there is δ > 0 such that δ |v1 (τ )| dτ < ε/2. Then, for t > δ, we have that δ 1 ε (5.1) (τ + 1)|v1 (τ )| dτ + . |f1 (t)| ≤ t+1 0 2 Let δ˜ > δ be such that δ ε 1 ˜ (5.2) (τ + 1) |v1 (τ )| dτ < , ∀t ≥ δ. t+1 0 2 ˜ so Then from (5.1) and (5.2) it follows that |f1 (t)| < ε, for all t ≥ δ, f1 ∈ C0 (R+ , R). This shows that f ∈ C0 (R+ , X), so f ∈ O(R+ , X). It follows that the pair (O(R+ , X), I(R+ , X)) is admissible for U. Let Xs (0) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)}. It is easy to see that R × {0} ⊂ Xs (0). Conversely, let x = (x1 , x2 ) ∈ Xs (0). Then 1 |x1 | + et |x2 | ≥ et |x2 |, ∀t ≥ 0. (5.3) ||U (t, 0)x|| = t+1 Since U (·, 0)x ∈ O(R+ , X) using similar arguments as in the proof of necessity of Theorem 4.6 (ii), from relation (5.3) it follows that x2 = 0, so x = (x1 , 0) ∈ R × {0}. Hence Xs (0) = R × {0}, so Xs (0) is closed and complemented in X. Suppose that U is uniformly exponentially dichotomic and let {P (t)}t≥0 be a family of projections such that U is uniformly exponentially dichotomic with respect to this family. Using similar arguments with those in the necessity of Theorem 4.6, we have that Im P (0) = Xs (0), so Im P (0) = R × {0}. Let K, ν > 0 be given by Definition 4.1. Then ||U (t, 0)x|| ≤ Ke−νt ||x||, for all t ≥ 0 and all x ∈ Im P (0). This implies that [1/(t + 1)] |x1 | ≤ Ke−νt |x1 |, for all t ≥ 0 and x1 ∈ R, which is absurd. It follows that U is not uniformly exponentially dichotomic. Remark 5.2. The above example shows that the hypotheses from Theorem 4.6 cannot be dropped and that the results given by Theorem 4.6 are the most general in this topic. It is important to point out that the input-output characterizations for exponential dichotomy relay on the verification of the solvability of an integral equation for diverse input functions. The inputs are test functions and therefore, in order to improve the admissibility concept, the input set should be ”small”, while the output space should be a ”large” space. In this context, we note that all the classes of Banach function spaces, considered throughout this paper, are closed to finite intersections. This fact allows various optimizations of the input space. Remark 5.3. Let M 1 (R+ , R) be the linear space of all measurable functions t+1 u : R+ → R with supt≥0 t |u(s)| ds < ∞. With respect to the norm t+1 ||u||M 1 := sup |u(s)| ds < ∞ t≥0
t
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
137
this is a Banach function space, which belongs to B(R+ ). Moreover, from Lemma 2.9 it follows that M 1 (R+ , R) is the largest space in V(R+ ). Theorem 5.4. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let I be a Banach function space in L(R+ ). The evolution family U is uniformly exponentially dichotomic if and only if the pair (M 1 (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ M 1 (R+ , X)} is closed and complemented in X. Remark 5.5. The above result allows the optimization of the input-output characterizations for uniform exponential dichotomy of evolution families on the half-line. Precisely, since the output space is the largest in V(R+ ), it suffices to consider smaller and smaller input spaces. Corollary 5.6. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X, let n ∈ N∗ , p1 , . . . , pn ∈ (1, ∞) and let I = Lp1 (R+ , R) ∩ . . . ∩ Lpn (R+ , R) ∩ C00 (R+ , R). Then U is uniformly exponentially dichotomic if and only if the pair (M 1 (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ M 1 (R+ , X)} is closed and complemented in X. Theorem 5.7. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X. Let n ∈ N∗ and let Oϕ , Oψ1 , . . . , Oψn be Orlicz spaces with ϕ(t) ∈ (0, ∞), for all t > 0 and ψk (1) < ∞, for all k ∈ {1, . . . , n}. Let I = Oϕ ∩ Oψ1 ∩ . . . Oψn ∩ C00 (R+ , R). The family U is uniformly exponentially dichotomic if and only if the pair (Oϕ (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ Oϕ (R+ , X)} is closed and complemented in X. Proof. From Remark 2.12 and Lemma 2.14 we have that Oϕ ∈ F(R+ )∩B(R+ ). By applying Theorem 4.6 the proof is complete. Corollary 5.8. Let p ∈ [1, ∞) and let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X. Let n ∈ N∗ , q1 , . . . , qn ∈ [1, ∞) and let I(R+ , X) = Lp (R+ , X) ∩ Lq1 (R+ , X) ∩ . . . ∩ Lqn (R+ , X) ∩ C00 (R+ , X). Then, the family U is uniformly exponentially dichotomic if and only if the pair (Lp (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ Lp (R+ , X)} is closed and complemented in X. Proof. This immediately follows from Theorem 5.7.
138
A. L. Sasu and B. Sasu
IEOT
Remark 5.9. Using different arguments, the above corollary was recently proved in [23] (see Theorem 3.3). Corollary 5.10. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X, let n ∈ N∗ , p1 , . . . , pn ∈ (1, ∞) and let I = Lp1 (R+ , R) ∩ . . . ∩ Lpn (R+ , R) ∩ C00 (R+ , R). Then U is uniformly exponentially dichotomic if and only if the pair (Cb (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ Cb (R+ , X)} is closed and complemented in X. Proof. We note that I ∈ L(R+ ). Hence, we apply Theorem 4.6 for O = L∞ (R+ , R), taking into account that a continuous function f lies in L∞ (R+ , R) if and only if f ∈ Cb (R+ , R). Remark 5.11. The above corollary was recently proved in [24], employing discrete-time arguments (see [24], Theorem 4.2). This also generalizes the main results of the papers [15] and [19] (see [15] Theorem 3.2 and [19] Theorem 4.3). Lemma 5.12. If V ∈ V(R+ ), then either V ∈ F(R+ ) or V ∈ L(R+ ). Proof. Indeed, suppose by contrary that there exists V ∈ V(R+ ) such V ∈ F(R+ ) and V ∈ L(R+ ). Then γ := supt>0 FV (t) < ∞ V ⊂ L1 (R+ , R). From V ⊂ L1 (R+ , R) it follows that there is c > 0 such ||v||1 ≤ c |v|V , for all v ∈ V . In particular, for v = χ[0,t) we deduce t ≤ c |χ[0,t) |V = c FV (t) ≤ cγ, for all t > 0, which is absurd.
that and that that
Theorem 5.13. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let E ∈ V(R+ ). The following assertions hold: (i) if the pair (E(R+ , X), E(R+ , X)) is admissible for U and the subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ E(R+ , X)} is closed and complemented in X, then U is uniformly exponentially dichotomic; (ii) if E ∈ B(R+ ), then U is uniformly exponentially dichotomic if and only if the pair (E(R+ , X), E(R+ , X)) is admissible for U and the subspace Xs (0) is closed and complemented in X. Proof. This follows from Theorem 4.6 and Lemma 5.12.
Lemma 5.14. Let B ∈ V(R+ ) and let ν > 0. Then, for every u : R+ → R+ in B, the functions fu , gu : R+ → R+ given by t ∞ −ν(t−τ ) fu (t) = e u(τ ) dτ and gu (t) = e−ν(τ −t) u(τ ) dτ 0
t
belong to Cb (R+ , R+ ). Proof. This follows using similar arguments as in Proposition 2.15 and using t+1 that, according to Lemma 2.9, we have that sup t u(τ ) dτ < ∞. t≥0
Vol. 66 (2010)
Integral Equations, Dichotomy and Applications
139
Corollary 5.15. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let E ∈ B(R+ ). Then U is uniformly exponentially dichotomic if and only if the pair (Cb (R+ , X) ∩ E(R+ , X), E(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ E(R+ , X)} is closed and complemented in X. Proof. Necessity follows from Theorem 4.6 (ii) and Lemma 5.14 and Sufficiency follows from Theorem 5.13. Remark 5.16. Using different arguments, the above corollary was proved by in [10] (see Theorem 4.2), under the hypothesis that the Banach function space E lies in a subclass of B(R+ ). We note that according to our proofs, conditions (1) and the Tτ− -invariance (from Definition 2.3 in [10]) may be removed. Acknowledgment The authors wish to express their special thanks to Professor Mihail Megan for very helpful discussions on this topic.
References [1] A. Ben-Artzi, I. Gohberg, Dichotomies of systems and invertibility of linear ordinary differential operators, Oper. Theory Adv. Appl. 56 (1992), 90-119. [2] A. Ben-Artzi, I. Gohberg, Dichotomies of perturbed time-varying systems and the power method, Indiana Univ. Math. J. 42 (1993), 699-720. [3] A. Ben-Artzi, I. Gohberg, M. A. Kaashoek, Invertibility and dichotomy of differential operators on the half-line, J. Dynam. Differential Equations 5 (1993), 1–36. [4] J. A. Ball, I. Gohberg, L. Rodman, Interpolation of rational matrix functions, Operator Theory: Advances and Applications 45 Birkh¨ auser Verlag, Basel, 1990. [5] C. Bennett, R. Sharpley, Interpolation of Operators. Pure Appl. Math. 129, 1988. [6] C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys and Monographs 70 Amer. Math. Soc. 1999. [7] W. A. Coppel, Dichotomies in Stability Theory, Springer Verlag, Berlin, Heidelberg, New-York, 1978. [8] J. L. Daleckii, M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, Trans. Math. Monographs, vol. 43, Amer. Math. Soc., Providence R.I., 1974. [9] J. K. Hale, S. M. Verduyn-Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences 99, New York, NY: Springer-Verlag, 1993. [10] N. Thieu Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal. 235 (2006), 330-354.
140
A. L. Sasu and B. Sasu
IEOT
[11] Y. Latushkin, T. Randolph, R. Schnaubelt, Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces, J. Dynam. Differential Equations 10 (1998), 489-509. [12] J. J. Massera, J. L. Sch¨ affer, Linear Differential Equations and Function Spaces. Academic Press, New-York, 1966. [13] M. Megan, B. Sasu, A. L. Sasu, On uniform exponential stability of evolution families, Riv. Mat. Univ. Parma 4 (2001), 27-43. [14] M. Megan, A. L. Sasu, B. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory 44 (2002), 71–78. [15] M. Megan, A. L. Sasu, B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dynam. Systems 9 (2003), 383–397. [16] M. Megan, A. L. Sasu, B. Sasu, The Asymptotic Behavior of Evolution Families, Mirton Publishing House, 2003. [17] M. Megan, C. Stoica, On uniform exponential trichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory 60 (2008), 499-506. [18] P. Meyer-Nieberg, Banach Lattices. Springer Verlag, Berlin, Heidelberg, New York, 1991. [19] N. Van Minh, F. R¨ abiger, R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the halfline, Integral Equations Operator Theory 32 (1998), 332–353. [20] K. J. Palmer, Exponential dichotomy and expansivity, Ann. Mat. Pura Appl. (4) 185 (2006), 171-185. [21] A. L. Sasu, B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces, Integral Equations Operator Theory 54 (2006), 113-130. [22] B. Sasu, A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z. 253 (2006), 515-536. [23] B. Sasu, A. L. Sasu, Exponential dichotomy and (p , q )-admissibility on the half-line, J. Math. Anal. Appl. 316 (2006), 397-408. [24] B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl. 323 (2006), 1465-1478. [25] A. L. Sasu, Integral equations on function spaces and dichotomy on the real line, Integral Equations Operator Theory 58 (2007), 133-152. Adina Luminit¸a Sasu and Bogdan Sasu Department of Mathematics Faculty of Mathematics and Computer Science West University of Timi¸soara Pˆ arvan Blvd. No. 4 300223 Timi¸soara Romania e-mail:
[email protected] [email protected] Submitted: April 2, 2009.
Integr. Equ. Oper. Theory 66 (2010), 141–152 DOI 10.1007/s00020-009-1732-8 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators Nikolai Vasilevski Abstract. We present here a quite unexpected result: Apart from already known commutative C ∗ -algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C ∗ -algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols. Mathematics Subject Classification (2010). Primary 47B35; Secondary 47L80, 32A36. Keywords. Toeplitz operator, weighted Bergman space, commutative Banach algebra, quasi-radial quasi-homogeneous symbol.
1. Introduction As it often happens a multidimensional extension of the one-dimensional results brings more surprises that one might expect. The paper is a next illustration of that. Recall first that the C ∗ -algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit disk were completely classified in [2]. Under some technical assumption on “richness” of a class of generating symbols the result was as follows. A C ∗ -algebra generated by Toeplitz operators is commutative on each weighted Bergman space if and only if the corresponding symbols of Toeplitz operators are constant on cycles of a pencil of hyperbolic geodesics on the unit disk, or if and only if the corresponding symbols of Toeplitz operators are invariant under the action a maximal commutative subgroup of the M¨ obius transformations of the unit This work was partially supported by CONACYT Project 80503, M´exico.
142
N. Vasilevski
IEOT
disk. We note that the commutativity on each weighted Bergman space was crucial in the part “only if” of the above result. Generalizing this result to Toeplitz operators on the unit ball, it was proved in [6, 7] that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C ∗ -algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each weighted Bergman space. The geometric description of corresponding symbols in terms of so-called Lagrangian foliations (which generalize the notion of a pencil of hyperbolic geodesics to multidimensional case) was also given. It turned out that for the unit ball of dimension n there are n + 2 essentially different “model” commutative C ∗ -algebras, all others are conjugated with one of them via biholomorphisms of the unit ball. It was firmly expected that the above algebras exhaust all possible algebras of Toeplitz operators on the unit ball which are commutative on each weighted Bergman space. Inspired by [8], we present here a quite unexpected result. There exist other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C ∗ -algebra, and for n = 1 all of them collapse to the C ∗ -algebra, which is generated by Toeplitz operators with radial symbols.
2. Preliminaries Let Bn be the unit ball in Cn , that is, Bn = {z = (z1 , . . . , zn ) ∈ Cn : |z|2 = |z1 |2 + . . . + |zn |2 < 1}, and let Sn be the corresponding unit sphere, the boundary of the unit ball Bn . In what follows we will use the notation τ (Bm ) for the base of the unit ball Bm , considered as a Reinhard domain, i.e., 2 ∈ [0, 1)}. τ (Bm ) = {(r1 , . . . , rm ) = (|z1 |, . . . , |zm |) : r2 = r12 + . . . + rm
Given a multi-index α = (α1 , α2 , . . . , αn ) ∈ Zn+ we will use the standard notation, |α| = α1 + α2 + . . . + αn , α! = α1 ! α2 ! · · · αn !, z α = z1α1 z2α2 · · · znαn . Two multi-indices α and β are called orthogonal, α ⊥ β, if α · β = α1 β1 + α2 β2 + . . . + αn βn = 0.
(2.1)
Vol. 66 (2010)
Quasi-Radial Quasi-Homogeneous Symbols
143
Denote by dV = dx1 dy1 . . . dxn dyn , where zl = xl + iyl , l = 1, 2, . . . , n, the standard Lebesgue measure in Cn ; and let dS be the corresponding surface measure on Sn . We introduce the one-parameter family of weighted measures, Γ(n + λ + 1) (1 − |z|2 )λ dV (z), λ > −1, dvλ (z) = n π Γ(λ + 1) which are probability ones in Bn ; and recall two known equalities (see, for example, [9, Section 1.3]) 2π n α! β , (2.2) ξ α ξ dS(ξ) = δα,β (n − 1 + |α|)! n S α! Γ(n + λ + 1) . (2.3) z α z β dvλ (z) = δα,β Γ(n + |α| + λ + 1) n B We introduce the weighted space L2 (Bn , dvλ ) and its subspace, the weighted Bergman space A2λ = A2λ (Bn ), which consists of all functions analytic in Bn . The (orthogonal) Bergman projection Bλ of L2 (Bn , dvλ ) onto A2λ (Bn ) is known to have the following integral form ϕ(ζ) dvλ (ζ) . (Bλ ϕ)(z) = n+λ+1 n B (1 − z · ζ) Finally, given a function a(z) ∈ L∞ (Bn ), the Toeplitz operator Ta with symbol a acts on A2λ (Bn ) as follows Ta : ϕ ∈ A2λ (Bn ) −→ Bλ (aϕ) ∈ A2λ (Bn ).
3. Quasi-radial symbols Let k = (k1 , . . . , km ) be a tuple of positive integers whose sum is equal to n: k1 + . . . + km = n. The length of such a tuple may obviously vary from 1, for k = (n), to n, for k = (1, . . . , 1). Given a tuple k = (k1 , . . . , km ), we rearrange the n coordinates of z ∈ Bn in m groups, each one of which has kj , j = 1, . . . , m, entries and introduce the notation z(1) = (z1,1 , . . . , z1,k1 ), z(2) = (z2,1 , . . . , z2,k2 ), . . . , z(m) = (zm,1 , . . . , zm,km ). We represent then each z(j) = (zj,1 , . . . , zj,kj ) ∈ Bkj in the form z(j) = rj ξ(j) , where rj = |zj,1 |2 + . . . + |zj,kj |2 and ξ(j) ∈ Skj . Given a tuple k = (k1 , . . . , km ), a bounded measurable function a = a(z), z ∈ Bn , will be called k-quasi-radial if it depends only on r1 , . . . , rm . Varying k we have a collection of the partially ordered by inclusion sets Rk of k-quasi-radial functions. The minimal among these sets is the set R(n) of radial functions and the maximal one is the set R(1,...,1) of separately radial functions. There is some ambiguity in the above definition. Indeed given a tuple k there are many corresponding sets Rk which differ by perturbation
144
N. Vasilevski
IEOT
of coordinates. At the same time each perturbation of coordinates of z is a biholomorphism, say κ, of the unit ball Bn , which generates the unitary equivalence of the Toeplitz operators Ta and Ta◦κ . Thus it is sufficient, in fact, to consider only one of these perturbation different sets. To avoid all possible repetitions and ambiguities in what follows we will always assume first, that k1 ≤ k2 ≤ . . . ≤ km , and second, that z1,1 = z1 , z1,2 = z2 , . . . , z1,k1 = zk1 , z2,1 = zk1 +1 , . . . , z2,k2 = zk1 +k2 , . . . , zm,km = zn .
(3.1)
Given k = (k1 , . . . , km ) and any n-tuple α = (α1 , . . . , αn ), we define α(1) = (α1 , . . . , αk1 ), α(2) = (αk1 +1 , . . . , αk1 +k2 ), . . . , α(m) = (αn−km +1 , . . . , αn ). As each set Rk is a subset of the set R(1,...,1) of separately radial functions, the Toeplitz operator Ta with symbol a ∈ Rk , by [6], is diagonal with respect to the standard monomial basis in A2λ (Bn ). The exact form of the corresponding spectral sequence gives the next lemma. Lemma 3.1. Given a k-quasi-radial function a = a(r1 , . . . , rm ), we have Ta z α = γa,k,λ (α) z α ,
α ∈ Zn+ ,
where γa,k,λ (α) = γa,k,λ (|α(1) |, . . . , |α(m) |) =
2m Γ(n + |α| + λ + 1) Γ(λ + 1) m j=1 (kj − 1 + |α(j) |)! ×
m
2|α(j) |+2kj −1
rj
τ (Bm )
a(r1 , . . . , rm )(1 − |r|2 )λ
drj .
j=1
Proof. We calculate Ta z α , z α = az α , z α =
Γ(n + λ + 1) π n Γ(λ + 1)
Bn
a(r1 , . . . , rm )z α z α (1 − |z|2 )λ dV (z).
Changing the variables z(j) = rj ξ(j) , where rj ∈ [0, 1] and ξ(j) ∈ Skj , j = 1, . . . , m, we have m Γ(n + λ + 1) 2|α |+2kj −1 az α , z α = n a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj π Γ(λ + 1) τ (Bm ) j=1 m α(j) α × ξ(j)(j) ξ (j) dS(ξ(j) ) j=1
Skj
Vol. 66 (2010)
=
Quasi-Radial Quasi-Homogeneous Symbols
145
2m Γ(n + λ + 1) α! m Γ(λ + 1) j=1 (kj − 1 + |α(j) |)! m 2|α |+2kj −1 a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj . × τ (Bm )
j=1
Then the result follows by (2.3).
Remark 3.2. For extreme cases of radial symbols (k = (n)) and separately radial symbols (k = (1, . . . , 1))) the above formula for γa,k,λ reduces to the corresponding formula from [1] and [6], respectively. Given k = (k1 , . . . , km ), we use the representations z(j) = rj ξ(j) , j = 1, . . . , m, to define the vector ξ = (ξ(1) , ξ(2) , . . . , ξ(m) ) ∈ Sk1 × Sk2 × . . . × Skm . We introduce now an extension of k-quasi-radial functions, which may be called following [3, 5, 8] the quasi-homogeneous functions. A function ϕ(z) is called quasi-homogeneous (or k-quasi-homogeneous) function if it has the form ϕ(z) = ϕ(z(1) , z(2) , . . . , z(m) ) s
s
s
(1) (2) (m) ξ(2) . . . ξ(m) , = a(r1 , r2 , . . . , rm ) ξ s = a(r1 , r2 , . . . , rm ) ξ(1)
where a(r1 , r2 , . . . , rm ) ∈ Rk and s ∈ Zn . After separating positive and negative entries in s, it admits the unique representation s = p − q, where p, q ∈ Zn+ and p ⊥ q. Then ξ s , for s ∈ Zn , is always understood as q
ξs = ξp ξ , where s = p − q, with p, q ∈ Zn+ and p ⊥ q. We will call the pair (p, q) the quasi-homogeneous degree of the k-quasi-homogeneous function q a(r1 , r2 , . . . , rm ) ξ p ξ . Lemma 3.3. The Toeplitz operator Ta ξp ξq with k-quasi-homogeneous symbol q
a ξ p ξ acts on monomials z α , α ∈ Zn+ as follows 0, if ∃ l such that αl < ql − pl α , Ta ξ p ξ q z = γ a,k,p,q,λ (α) z α+p−q , if ∀ l αl ≥ ql − pl where γ a,k,p,q,λ (α) =
2m Γ(n + |α + p − q| + λ + 1) (α + p)! m (3.2) Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m |2α +p −q |+2kj −1 × a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) (j) (j) drj . τ (Bm )
j=1
146
N. Vasilevski
IEOT
Proof. For each two multi-indices α, β ∈ Zn+ , we calculate q
Ta ξp ξq z α , z β = a ξ p ξ z α , z β Γ(n + λ + 1) q = n a(r1 , . . . , rm ) ξ p ξ z α z β (1 − |z|2 )λ dV (z). π Γ(λ + 1) Bn Changing the variables z(j) = rj ξ(j) , where rj ∈ [0, 1) and ξ(j) ∈ Skj , j = 1, . . . , m, we have Γ(n + λ + 1) p q α β a ξ ξ z , z = n a(r1 , . . . , rm )(1 − |r|2 )λ π Γ(λ + 1) τ (Bm ) × ×
m j=1 m j=1
|α(j) +β(j) |+2kj −1
rj
= δα+p,β+q ×
α
Skj
τ (Bm )
ξ(j)(j)
drj
+p(j) β(j) +q(j) ξ (j)
dS(ξ(j) )
2m Γ(n + λ + 1) (α + p)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! a(r1 , . . . , rm )(1 − |r|2 )λ
m
|λ(j) +β(j) |+2kj −1
rj
drj .
j=1
Now the right-hand side is non zero (for a generic a ∈ Rk ) if and only if α + p = β + q and αl + pl − ql ≥ 0, for each l = 1, 2, . . . , n. Then, for β = α + p − g with αl + pl − ql ≥ 0 for each l = 1, 2, . . . , n, we have by (2.3), z β , z β = z α+p−g , z α+p−g = and the result follows.
(α + p − q)! Γ(n + λ + 1) , Γ(n + |α + p − q| + λ + 1)
4. Commutativity results A particular case of the next theorem when k = (n) and λ = 0 was proved in [8]. Theorem 4.1. Let k = (k1 , k2 , . . . , km ) and p, q be a pair of orthogonal multiindices. Then for each pair of non identically zero k-quasi-radial functions a1 and a2 , the Toeplitz operators Ta1 and Ta2 ξp ξ q commute on each weighted Bergman space A2λ (Bn ) if and only if |p(j) | = |q(j) | for each j = 1, 2, . . . , m. Proof. For those multi-indices α with αl + pl − ql ≥ 0, for each l = 1, 2, . . . , n,
Vol. 66 (2010)
Quasi-Radial Quasi-Homogeneous Symbols
147
by Lemmas 3.1 and 3.3 we have Ta2 ξp ξ q Ta1 z α =
2m Γ(n + |α + p − q| + λ + 1) (α + p)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m |2α +p −q |+2kj −1 × a2 (r1 , . . . , rm )(1 − |r|2 )λ rj (j) (j) (j) drj τ (Bm )
j=1
2m Γ(n + |α| + λ + 1) × Γ(λ + 1) m j=1 (kj − 1 + |α(j) |)! m 2|α |+2kj −1 × a1 (r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj z α+p−q τ (Bm )
j=1
and Ta1 Ta2 ξp ξq z α =
2m Γ(n + |α + p − q| + λ + 1) m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) − q(j) |)! m 2|α +p −q |+2kj −1 a1 (r1 , . . . , rm )(1 − |r|2 )λ rj (j) (j) (j) drj × τ (Bm )
j=1
2m Γ(n + |α + p − q| + λ + 1) (α + p)! m × Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m |2α +p −q |+2kj −1 a2 (r1 , . . . , rm )(1 − |r|2 )λ rj (j) (j) (j) drj × τ (Bm )
×z
j=1
α+p−q
.
α
That is Ta2 ξp ξq Ta1 z = Ta1 Ta2 ξp ξq z α (for each pair of k-quasi-radial functions a1 and a2 ) if and only if |p(j) | = |q(j) | for each j = 1, 2, . . . , m Remark 4.2. For those j with kj = 1 both p(j) and q(j) are of length one, and the condition |p(j) | = |q(j) | is equivalent to p(j) = q(j) = 0. We note that under the condition |p(j) | = |q(j) |, for each j = 1, 2, . . . , m, formula (3.2) reads as γ a,k,p,q,λ (α) =
2m Γ(n + |α| + λ + 1) (α + p)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m 2|α |+2kj −1 2 λ × a(r1 , . . . , rm )(1 − |r| ) rj (j) drj τ (Bm )
m
= m
j=1 (kj
j=1 (kj
=
m j=1
(4.1)
j=1
− 1 + |α(j) |)! (α + p)!
− 1 + |α(j) + p(j) |)! (α + p − q)!
γa,k,λ (α)
(α(j) + p(j) )! (kj − 1 + |α(j) |)! γa,k,λ (α). (kj − 1 + |α(j) + p(j) |)! (α(j) + p(j) − q(j) )!
As surprising corollaries we have:
148
N. Vasilevski
IEOT
Corollary 4.3. Given k = (k1 , k2 , . . . , km ), for each pair of orthogonal multiindices p and q with |p(j) | = |q(j) |, for all j = 1, 2, . . . , m, and each a(r1 , r2 , . . . , rm ) ∈ Rk , we have Ta Tξp ξq = Tξp ξq Ta = Taξp ξq . Given k = (k1 , k2 , . . . , km ), and a pair of orthogonal multi-indices p and q with |p(j) | = |q(j) |, for all j = 1, 2, . . . , m, let p(j) = (0, . . . , 0, p(j) , 0, . . . , 0)
and
q(j) = (0, . . . , 0, q(j) , 0, . . . , 0).
Then, of course, p = p(1) + p(2) + . . . + p(m) and q = q(1) + q(2) + . . . + q(m) . For each j = 1, 2, . . . , m, we introduce the Toeplitz operator Tj = Tξp(j) ξ q(j) . Corollary 4.4. The operators Tj , j = 1, 2, . . . , m, mutually commute. Given an h-tuple of indices (j1 , j2 , . . . , jh ), where 2 ≤ h ≤ m, let ph = p(j1 ) + p(j2 ) + . . . + p(jh )
and
qh = q(j1 ) + q(j2 ) + . . . + q(jh ) .
Then h g=1
Tjg = Tξph ξqh .
In particular, m
Tj = Tξ p ξ q .
j=1
Given k = (k1 , k2 , . . . , km ), we consider two bounded measurable kq v quasi-homogeneous symbols a(r1 , r2 , . . . , rm )ξ p ξ and b(r1 , r2 , . . . , rm )ξ u ξ , which satisfy the conditions of Theorem 4.1, i.e., a(r1 , r2 , . . . , rm ) and b(r1 , r2 , . . . , rm ) are arbitrary k-quasi-radial functions, p ⊥ q, u ⊥ v, and |p(j) | = |q(j) |
and |u(j) | = |v(j) |, q
for all j = 1, 2, . . . , m. v
Theorem 4.5. Let a(r1 , r2 , . . . , rm )ξ p ξ and b(r1 , r2 , . . . , rm )ξ u ξ be as above. Then the Toeplitz operators Taξp ξq and Tbξu ξv commute on each weighted Bergman space A2λ (Bn ) if and only if for each l = 1, 2, . . . , n one of the next conditions is fulfilled 1. 2. 3. 4.
pl = ql = 0; ul = vl = 0; pl = ul = 0; ql = vl = 0.
Proof. We calculate and compare first Taξp ξq Tbξu ξv z α and Tbξu ξ v Taξp ξq z α for those multi-indices α when both these expressions are non zero.
Vol. 66 (2010)
Quasi-Radial Quasi-Homogeneous Symbols
149
By (4.1) we have Taξp ξq Tbξu ξv z α =
2m Γ(n + |α| + λ + 1) (α + u − v + p)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + u − v + p − q)! m 2|α |+2kj −1 a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj × τ (Bm )
j=1 m
2 Γ(n + |α| + λ + 1) (α + u)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + u(j) |)! (α + u − v)! m 2|α |+2kj −1 b(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj × ×
τ (Bm )
×z
α+u−v+p−q
j=1
,
and Tbξu ξ v Taξp ξq z α =
2m Γ(n + |α| + λ + 1) (α + p − q + u)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + u(j) |)! (α + p − q + u − v)! m 2|α |+2kj −1 b(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj × τ (Bm )
j=1
2m Γ(n + |α| + λ + 1) (α + p)! m × Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m 2|α |+2kj −1 a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj × τ (Bm )
×z
α+p−q+u−v
j=1
.
α
That is, Taξp ξq Tbξu ξv z = Tbξu ξ v Taξp ξq z α (for each pair of k-quasi-radial functions a and b) if and only if (α + p − q + u)! (α + p)! (α + u − v + p)! (α + u)! = . (α + u − v)! (α + p − q)! Varying α it is easy to see that the last equality holds if and only if for each l = 1, 2, . . . , n one of the next conditions is fulfilled 1. pl = ql = 0; 2. ul = vl = 0; 3. pl = ul = 0; 4. ql = vl = 0. To finish the proof we mention that under either of the above conditions both quantities Taξp ξq Tbξu ξv z α and Tbξu ξv Taξp ξq z α are zero or non zero simultaneously only. Example. Let n = 7 and k = (2, 5). Then by Theorem 4.1 the Toeplitz q operators with symbols a(r1 , r2 ) ∈ Rk and b ξ p ξ , where b(r1 , r2 ) ∈ Rk , p = (1, 0, 0, 3, 0, 1, 0), q = (0, 1, 1, 0, 1, 0, 2), commute. We mention that here p(1) = (1, 0), p(2) = (0, 3, 0, 1, 0) and q(1) = (0, 1), q(2) = (1, 0, 1, 0, 2).
150
N. Vasilevski
IEOT
As easy to see, all pairs (u, v) of orthogonal multi-indices such that (by Theorem 4.5) the Toeplitz operators with k-quasi-homogeneous symbols having that quasi-homogeneous degrees mutually commute, and commute with both Ta and Tbξp ξ q are of the form u = (u1 , 0, 0, u4 , 0, u6 , 0),
v = (0, v2 , v3 , 0, v5 , 0, v7 ),
(4.2)
where u1 , u4 , u6 ∈ Z+ , v2 , v3 , v5 , v7 ∈ Z+ , and u1 = v2 ,
u4 + u6 = v3 + v5 + v7 .
(4.3)
That is, the Banach algebra generated by all Toeplitz operators Ta ξu ξv , where a(r1 , r2 ) ∈ Rk , and the orthogonal multi-indices u and v of the form (4.2) satisfy the condition (4.3), is commutative. We formalize the above example as follows. First, to avoid the repetition of the unitary equivalent algebras and to simplify the classification of the (non unitary equivalent) algebras, in addition to (3.1), we can rearrange the variables zl and correspondingly the components of multi-indices in p and q so that (i) for each j with kj > 1, we have p(j) = (pj,1 , . . . , pj,hj , 0, . . . , 0) and q(j) = (0, . . . , 0, qj,hj +1 , . . . , qj,kj ); (4.4) (ii) if kj = kj with j < j , then hj ≤ hj . Now, given k = (k1 , . . . , km ), we start with m-tuple h = (h1 , . . . , hm ), where hj = 0 if kj = 1 and 1 ≤ hj ≤ kj − 1 if kj ≥ 1; in the last case, if kj = kj with j < j , then hj ≤ hj . We denote by Rk (h) the linear space generated by all k-quasi-homogeneous functions q a(r1 , r2 , . . . , rm ) ξ p ξ , where a(r1 , r2 , . . . , rm ) ∈ Rk , and the components p(j) and q(j) , j = 1, 2, . . ., m, of multi-indices p and q are of the form (4.4) with pj,1 + . . . + pj,hj = qj,hj +1 + . . . + qj,kj , pj,1 , . . . , pj,hj , qj,hj +1 , . . . , qj,kj ∈ Z+ . We note that Rk ⊂ Rk (h) and that the identity function e(z) ≡ 1 belongs to Rk (h). The main informatory result of the paper gives the next corollary. Corollary 4.6. The Banach algebra generated by Toeplitz operators with symbols from Rk (h) is commutative. We would like to emphasize the following features of such algebras: – for different k and h these algebras are not conjugated via biholomorphisms of the unit ball; – these algebras are just Banach and not C ∗ -algebras; extending them to C ∗ -algebras they become non commutative;
Vol. 66 (2010)
Quasi-Radial Quasi-Homogeneous Symbols
151
– given k = (1, 1, . . . , 1), there is a finite number of different m-tuples h and thus a finite number of different corresponding commutative algebras; – these algebras remain commutative for each weighted Bergman space A2λ (Bn ), with λ > −1, – for n = 1 all of them collapse to the single C ∗ -algebra generated by Toeplitz operators with radial symbols. We finish the paper presenting another application of Theorems 4.1 and 4.5. Studying commutativity properties of Toeplitz operators on the Bergman space on the unit disk I. Louhichi and N. V. Rao [4] conjectured that if two Topelitz operators commute with a third one, none of them being the identity, then they commute with each other. As next example shows, this conjecture is wrong when formulated for Toeplitz operators on the unit ball Bn , with n > 1. Example. Given n > 1, let k = (2, 1, . . . , 1). Consider the following three symbols a0 = a(r1 , r2 , . . . , rn−1 ), (1,0) (0,1)
a1 = b(r1 , r2 , . . . , rn−1 )ξ(1) ξ (1) , (0,1) (1,0)
a2 = c(r1 , r2 , . . . , rn−1 )ξ(1) ξ (1) , where a, b, c ∈ Rk . Then by Theorem 4.1 Ta0 commutes with both Ta1 and Ta2 , while by Theorem 4.5 the operators Ta1 and Ta2 do not commute.
References [1] S. Grudsky, A. Karapetyants, and N. Vasilevski. Toeplitz operators on the unit ball in Cn with radial symbols. J. Operator Theory, 49:325–346, 2003. [2] S. Grudsky, R. Quiroga-Barranco, and N. Vasilevski. Commutative C ∗ -algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal., 234(1):1–44, 2006. [3] I. Louhichi and N.V. Rao. On Toeplitz operators with quasihomogeneous symbols. Atch. Math., 851:248–257, 2005. [4] I. Louhichi and N.V. Rao. Bicommutants of Toeplitz operators. Atch. Math., 91:256–264, 2008. [5] Issam Louhichi, Elizabeth Strouse, and Lova Zakariasy. Products of Toeplitz operators on the Bergman space. Integral Equations Operator Theory, 54(4):525– 539, 2006. [6] Raul Quiroga-Barranco and Nikolai Vasilevski. Commutative C ∗ -algebras of Toeplitz operators on the unit ball, I. Bargmann-type transforms and spectral representations of Toeplitz operators. Integral Equations Operator Theory, 59(3):379–419, 2007.
152
N. Vasilevski
IEOT
[7] Raul Quiroga-Barranco and Nikolai Vasilevski. Commutative C ∗ -algebras of Toeplitz operators on the unit ball, II. Geometry of the level sets of symbols. Integral Equations Operator Theory, 59(1):89–132, 2008. [8] Ze-Hua Zhou and Xing-Tang Dong. Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball. Integral Equations and Operator Theory (to appear), 18 p. [9] Kehe Zhu. Spaces of Holomorphic Functions in the Unit Ball. Springer Verlag, 2005. Nikolai Vasilevski Departamento de Matem´ aticas CINVESTAV Apartado Postal 14-740 07000, M´exico, D.F. M´exico e-mail:
[email protected] Submitted: March 4, 2009. Revised: July 10, 2009.
Integr. Equ. Oper. Theory 66 (2010), 153–181 DOI 10.1007/s00020-010-1742-6 Published online January 29, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators A. A. Albanese, D. Jornet and A. Oliaro Abstract. In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, W F∗ , of classical distributions. In particular, we have the following inclusion W F∗ (u) ⊂ W F∗ (P u) ∪ Σ,
u ∈ D (Ω),
where Ω is an open subset of Rn , P is a linear partial differential operator with coefficients in a suitable ultradifferentiable class, and Σ is the characteristic set of P . Some applications are also investigated. Mathematics Subject Classification (2010). Primary 46F05; Secondary 35A18, 35A21. Keywords. Quasianalytic weight function, wave front set, propagation of singularities.
1. Introduction Classes of ultradifferentiable functions have been investigated intensively since the 20ies of the last century. According to the theorem of Denjoy– Carleman they split in two groups: the quasianalytic and the nonquasianalytic classes. Several authors have introduced the classes in different ways (e.g., Beurling [1]; Komatsu [16]; H¨ ormander [14]; Braun, Meise, and Taylor [9]). In general, the classes defined in one way cannot be defined in another way (see [8]). On the other hand, independently of the definition there are several known methods to construct functions with prescribed properties in nonquasianalytic classes, while this is not the case for quasianalytic classes, for which other techniques are needed, for example, complex variable methods. The research of the authors was partially supported by MEC and FEDER, Project MTM2007-62643, and MEC, Project MTM2007-30904-E, and Conselleria d’Educaci´ o de la GVA, Ajuda complementaria ACOMP/2009/253.
154
Albanese, Jornet and Oliaro
IEOT
The notion of wave front set was introduced by H¨ ormander in 1970 to simplify the study of the propagation of singularities of distributional solutions of linear partial differential operators. In 1971, H¨ ormander [14] established the following microlocal regularity result: W FL (u) ⊂ Σ ∪ W FL (P u),
u ∈ D (Ω),
(1)
where P is a linear partial differential operator with analytic coefficients, and Σ is the characteristic set of P . This type of result represents a fundamental tool in the study of propagation of singularities. This inclusion has been also applied, modified and adapted several times since the seventies of the last century to study the problem of iterates from a microlocal point of view (we refer to [2, 3, 5, 24], among others; see also [4] for a survey on this topic). In the inclusion (1) the wave front set W FL (u) is defined with respect to the ultradifferentiable class C L of Roumieu type and u is always a classical distribution on an open set Ω of Rn . The function spaces C L were introduced in [14] and cover the classical spaces of Gevrey functions and of analytic functions. In the case of the analytic wave front set, W FA , such an inclusion was proved by Kato [15] for hyperfunctions, but for a very different definition of W FA . We also mention that a version of inclusion (1) for wave front sets with respect to the C ∞ -class and an operator P with C ∞ -coefficients can be found in [13, Chapter VIII]. In this paper we present a version of inclusion (1) for wave front sets defined with respect to the ultradifferentiable classes as introduced by Braun, Meise, and Taylor [9] and classical distributions. By modifying the arguments in [14] in a suitable way, we cover both quasianalytic and non-quasianalytic cases at the same time. These classes have the advantage that can be studied using the decay properties of their Fourier transform and the decay behaviour of their derivatives. We give two versions of the result. First, we prove the Beurling case in Theorem 4.1. In this case, we require the assumption ω(t) = o(t), as t tends to infinity, on the weight function. The Roumieu version is obtained as a consequence of the Beurling case and the description of Roumieu wave front set of a classical distribution as the closure of the union of all the Beurling wave front sets contained in it. Such a description is proved for arbitrary weight functions, quasianalytic or not, in Proposition 4.5 (see [11] for a version for non-quasianalytic weight functions). To show our inclusion results, Theorem 4.1 and Theorem 4.8, we need to assume that the weight functions are equivalent to subadditive weight functions (see [18, 19]). In particular, our results apply to the most relevant cases considered by Komatsu [16] (see [8]). Finally, our results are applied to study the propagation of singularities of the operator P = ∂xn in Rn .
2. Notation and preliminaries In this preliminary section we fix the notation and provide a number of basic results that will be used in the subsequent sections. Throughout this article
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
155
| · | denotes the Euclidian norm on Rn or Cn and Br (a) denotes the open ball of radius r and center a. Definition 2.1. A weight function is a continuous increasing function ω : [0, ∞[→ [0, ∞[ with the following properties: (α) (β) (γ) (δ)
there exists L ≥ 0 such that ω(2t) ≤ L(ω(t) + 1) for all t ≥ 0, ω(t) = O(t), as t tends to ∞, log(t) = o(ω(t)) as t tends to ∞, ϕ : t → ω(et ) is convex. A weight function ω is called quasianalytic if ∞ ω(t) dt = ∞. t2 1
If this integral is finite, then ω is called nonquasianalytic. A weight function ω is equivalent to a sub-additive weight if, and only if, it has the property (α0 ) ∃D > 0 ∃t0 > 0 ∀λ ≥ 1 ∀t ≥ t0 : ω(λt) ≤ λDω(t). The condition above should be compared with [19, p.19] and [18, Lemma 1]. The Young conjugate ϕ∗ : [0, ∞[→ R of ϕ is given by ϕ∗ (s) := sup{st − ϕ(t), t ≥ 0}. There is no loss of generality to assume that ω vanishes on [0, 1]. Then ϕ∗ has only non-negative values, it is convex and ϕ∗ (t)/t is increasing and tends to ∞ as t → ∞ and ϕ∗∗ = ϕ. For more details on properties of ω and ϕ∗ we refer to [9, 12]. Example 2.2. The following are examples of weight functions (eventually after a change on the interval [0, δ] for a suitable δ > 0): (1) (2) (3) (4)
ω(t) = tα , 0 < α < 1; β ω(t) = (log(1 + t)) , β > 1; −β ω(t) = t (log(e + t)) , β > 0; ω(t) = t.
The weight function in (3) is quasianalytic for β ∈]0, 1] and nonquasianalytic for β > 1. The weight function in (4) is also quasianalytic. Moreover, all the weight functions above satisfy property (α0 ). For further examples of quasianalytic weight functions we refer to [8]. Definition 2.3. Let ω be a weight function. For an open set Ω ⊂ Rn we set E(ω) (Ω) :={f ∈ C ∞ (Ω) : f K,λ < ∞, for every K ⊂⊂ Ω and every λ > 0}, and E{ω} (Ω) := {f ∈ C ∞ (Ω) : for every K ⊂⊂ Ω, there exists λ > 0 such that f K,λ < ∞},
156
where
Albanese, Jornet and Oliaro
IEOT
|α|
f K,λ := sup sup |f (α) (x)| exp − λϕ∗ . λ x∈K α∈NN 0
The spaces E(ω) (Ω) and E{ω} (Ω) are endowed with their natural topologies. Then E(ω) (Ω) is a nuclear Fr´echet space, while E{ω} (Ω) is a countable projective limit of (DFN)-spaces, which is reflexive and complete. If ω is nonquasianalytic, the space E{ω} (Ω) is ultrabornological, see [9, Proposition 4.9]. If ω is quasianalytic, the space E{ω} (Ω) is surely ultrabornological under the assumption of convexity of Ω and, this follows from [21, Satz 3.25], together with [22, Theorem 3.4], and [23, Theorem 3.5]. The elements in the space E(ω) (Ω) (respectively, in the space E{ω} (Ω)) are called ω-ultradifferentiable functions of Beurling type (respectively, of (Ω) and E{ω} (Ω) we denote the duals of E(ω) (Ω) Roumieu type) in Ω. By E(ω) and E{ω} (Ω). When ω is quasianalytic the elements of E(ω) (Ω) (respectively, E{ω} (Ω)) are called quasianalytic functionals of Beurling (respectively, Roumieu) type. We observe that in the case ω(t) = tα , 0 < α ≤ 1, the corresponding Roumieu class is the Gevrey class with exponent s = 1/α. In particular, E{ω} (Ω) coincides with the space A(Ω) of all real analytic functions on Ω. We will write ∗ to denote (ω) or {ω} when it is not necessary to distinguish between both cases. If ω is quasianalytic, the elements with compact support in E{ω} (Ω) or in E(ω) (Ω) are trivial. While, if ω is nonquasianalytic, the space D∗ (K) := E∗ (Ω) ∩ D(K) = {0}, being K ⊂ Ω a compact set. Then D∗ (Ω) := indn D∗ (Kn ), where (Kn ) is any compact exhaustion of Ω. The elements of (Ω) (respectively, D{ω} (Ω)) are called ω-ultradistributions of Beurling D(ω) (respectively, Roumieu) type. The ∗-singular support of a classical distribution u ∈ D (Ω), denoted by sing∗ supp u, is the complementary in Ω of the biggest open set U ⊂ Ω satisfying u|U ∈ E∗ (U ). Remark 2.4. We also recall the following properties (see, for example, [12, Remark 2.8]): (a) If σ(t) = o(ω(t)) as t tends to infinity, then E{ω} (Ω) ⊂ E(σ) (Ω) with continuous inclusion. (b) If ω(t) = o(t) as t tends to infinity, then for each constant l ∈ N, there is a constant Cl > 0 such that y + Cl , y > 0. y log y ≤ y + lϕ∗ l
3. Quasianalytic wave front sets and properties In this section, we introduce and study Beurling and Roumieu quasianalytic and non quasianalytic wave front sets of a classical distribution. To provide
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
157
this, we begin with two lemmas which clarify the behaviour of the Fourier transform with respect to the weight function in Roumieu and Beurling settings. The first one treats the Roumieu case. Lemma 3.1. Let f be a continuous function defined in a cone Γ ⊂ Rn \ {0} taking values in [0, +∞[. The following statements are equivalent: (i) There exist constants C, ε > 0 such that f (ξ) ≤ C exp(−εω(ξ)),
ξ ∈ Γ,
(ii) There exists a constant C > 0 such that f (ξ) ≤ C
N +1
N!
1 ω(ξ)
N N = 0, 1, 2, . . . , ξ ∈ Γ,
,
(iii) There exists a constant C > 0 such that f (ξ) ≤ C
N
CN ω(ξ)
N = 0, 1, 2, . . . , ξ ∈ Γ,
,
(iv) There exist constants C > 0 and k ∈ N satisfying 1
|ξ|N f (ξ) ≤ Ce k ϕ
∗
(N k)
N = 0, 1, 2, . . . , ξ ∈ Γ.
,
(v) There exist constants C > 0 and k ∈ N such that 1
|ξ|N f (ξ) ≤ C N +1 e k ϕ
∗
(N k)
N = 0, 1, 2, . . . , ξ ∈ Γ.
,
Proof. (i) ⇐⇒ (ii) ⇐⇒ (iii) can be proved as in [20, Lemma 1.6.2]. We first show (i) ⇐⇒ (iv). To prove this, it is sufficient to check, for t > 1, that: 1
1
e− k ω(t) ≤ inf t−N e k ϕ
∗
(N k)
N ∈N0
1
≤ e− k ω(t)+log t .
(2)
Indeed, since log(t) = o(ω(t)) as t → ∞, for 0 < ε < k1 there is t0 > 0 so that − k1 ω(t) + log t ≤ −εω(t) for all t ≥ t0 . The first inequality in (2) follows by observing that (ϕ∗ )∗ = ϕ and hence, we have ω(t) = sup{s log t − ϕ∗ (s)} s>0
≥ sup {N k log t − ϕ∗ (N k)} N ∈N0
= k sup {N log t − N ∈N0
1 ∗ ϕ (N k)}. k
158
Albanese, Jornet and Oliaro
IEOT
The second inequality in (2) follows by observing that ϕ∗ is increasing and hence, we have ω(t) = sup{s log t − ϕ∗ (s)} s>0 sup
= sup N ∈N0
N k≤s≤(N +1)k
{s log t − ϕ∗ (s)}
≤ sup {(N + 1)k log t − ϕ∗ (N k)} N ∈N0
= k log t + sup {N k log t − ϕ∗ (N k)} N ∈N0 1 = k log t + sup {N log t − ϕ∗ (N k)} . k N ∈N0 It is obvious that (iv) implies (v). To finish the proof, it remains to show that (v) implies (iv). We proceed as in [10, p. 404]. We take s ∈ N to be the smallest natural number such that C ≤ es , being C the constant that appears in (iv). Let m the smallest natural number bigger than kLs , where k is the constant of (iv) and L > 1 the one that appears in property (α) of the weight function ω. We have s 1 1 j 1 ∗ ϕ (N k) + N s ≤ ϕ∗ (N m) + L , k m m j=1 and hence, 1
C N +1 e k ϕ
∗
(N k)
1
≤ CeN s+ k ϕ
∗
(N k)
1
≤ Ce m
s
j=1
Lj
1
emϕ
∗
(N m)
.
The following lemma treats the Beurling case. Lemma 3.2. Let f be a continuous function defined in a cone Γ ⊂ Rn \ {0} taking values in [0, +∞[. The following statements are equivalent: (i) For every k ∈ N there exists a constant Ck > 0 such that f (ξ) ≤ Ck exp(−kω(ξ)),
ξ ∈ Γ,
(ii) For every k ∈ N there exists a constant Ck > 0 such that N 1 f (ξ) ≤ Ck N ! , N = 0, 1, 2, . . . , ξ ∈ Γ, kω(ξ) (iii) For every k ∈ N there exists a constant Ck > 0 such that N N f (ξ) ≤ Ck , N = 0, 1, 2, . . . , ξ ∈ Γ, kω(ξ) (iv) For every k ∈ N there exists a constant Ck > 0 such that |ξ|N f (ξ) ≤ Ck ekϕ
∗
(N/k)
N = 0, 1, 2, . . . , ξ ∈ Γ.
,
(v) There exists a constant C > 0 such that for every k ∈ N there exists a constant Ck > 0 for which |ξ|N f (ξ) ≤ Ck C N ekϕ
∗
(N/k)
,
N = 0, 1, 2, . . . , ξ ∈ Γ.
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
159
Proof. It follows easily that (i) implies (ii) by using the Taylor expansion of the exponential function. It is clear that (ii) implies (iii). On the other hand, (iii) implies (ii) since N N ≤ eN N ! and therefore, if f satisfies (iii) for some sequence {Ck }, then for every k ∈ N, −N
f (ξ) ≤ C3k N ! [k ω(ξ)]
,
N = 0, 1, 2, . . . , ξ ∈ Γ.
(ii) implies that, for each k ∈ N, f (ξ)k N ω(ξ)N /N ! ≤ C2k (1/2)N for all N = 0, 1, 2, . . . . So, using the expansion of the exponential function, we obtain f (ξ) exp(k ω(ξ)) ≤ 2C2k , and hence, (i) is satisfied. (i) ⇐⇒ (iv) is proved in a similar way to Lemma 3.1. It remains to prove that (v) implies (iv). As in the Roumieu setting, we take s ∈ N such that C ≤ es , being C the constant that appears in (v). By (v) for each k ∈ N there is Ck > 0 such that, for Ak = k(Ls + · · · + L), f (ξ) ≤ Ck C N ekL
s
ϕ∗ (N/(kLs ))
≤ Ck eAk ekϕ
∗
(N/k)
.
The following proposition describes the ∗-singular support of a classical distribution. The proof follows the lines of [14] (see also [12, Lemma 4.7]). Proposition 3.3. Let Ω ⊂ Rn be an open set, u ∈ D (Ω) and x0 ∈ Ω. (a) Then u is a E{ω} -function in some neighborhood of x0 if and only if for some neighborhood U of x0 there exists a bounded sequence uN ∈ E (Ω) which is equal to u in U and satisfies, for some C > 0 and k ∈ N, the estimates 1
uN (ξ)| ≤ Ce k ϕ |ξ|N |
∗
(N k)
,
N = 1, 2, . . . ,
ξ ∈ Rn .
(3)
(b) Then u is a E(ω) -function in some neighborhood of x0 if and only if for some neighborhood U of x0 there exists a bounded sequence uN ∈ E (Ω) which is equal to u in U and such that for every k ∈ N there exists a constant Ck > 0 satisfying uN (ξ)| ≤ Ck ekϕ |ξ|N |
∗
(N/k)
,
N = 1, 2, . . . ,
ξ ∈ Rn .
(4)
Proof. (a) Necessity. Let u ∈ E{ω} (U ) with U = B3r (x0 ) and choose χN ∈ D(Ω) so that χN = 1 in Br (x0 ) and χN = 0 on (B2r (x0 ))c in such a way that |Dα χN | ≤ (C1 N )|α| ,
|α| ≤ N,
(5)
where C1 does not depend on N (we refer to [14] for a proof of the existence of sequences {χN } satisfying (5)). Now, we put uN = χN u. Then, for |α| ≤ N ,
α ∗ 1 (6) (C1 N )|β| Dr e m ϕ (m|α−β|) , |Dα (χN u)| ≤ β β≤α
being Dr the value of the seminorm u B2r (x0 ),1/m . Since ϕ∗ is a convex 1 ∗ 1 ∗ 1 ∗ function, we have m ϕ (m|α − β|) + m ϕ (m|β|) ≤ m ϕ (m|α|) and therefore,
160
Albanese, Jornet and Oliaro
IEOT
the right hand side of (6) is less than or equal to
α ∗ 1 1 ϕ∗ (m|α|) m e|β| log(C1 N )− m ϕ (m|β|) Dr e β β≤α
N
≤ 2 Dr e
∗ 1 m ϕ (mN )
1
sup e|β| log(C1 N )− m ϕ
∗
(m|β|)
|β|≤N 1
∗
(mN )
e m sup|β|≤N {m|β| log(C1 N )−ϕ
1
∗
(mN )
emϕ
≤ 2N Dr e m ϕ ≤ 2N Dr e m ϕ
1
1
1
∗∗
1
∗
≤ Dr eN + m ω(C1 N )+ m ϕ
∗
(m|β|)}
(log(C1 N ))
(mN )
,
where we have used the fact that ϕ∗∗ = ϕ, the definition of ϕ and |α| ≤ N . Then, we have, for |α| = N ,
∗ 1 1 α −ix,ξ α
N (ξ)| = e D uN (x)dx
≤ CDr eN + m ω(C1 N )+ m ϕ (mN ) , |ξ u where the constant C depends on the Lebesgue measure of B2r (x0 ). Now, we select i = 1, . . . , n such that |ξi | = max1≤j≤n |ξj | and set α = N ei , where ei is the i−th vector of the canonical basis of Rn . Then, |ξ|N ≤ nN/2 max |ξj |N = nN/2 |ξi |N = nN/2 |ξ α |. 1≤j≤n
Consequently, uN (ξ)| ≤ nN/2 |ξ α u N (ξ)| |ξ|N | 1
1
≤ CDr nN/2 eN + m ω(C1 N )+ m ϕ N
= CDr e 2
∗
(N m)
1 1 log n+N + m ω(C1 N )+ m ϕ∗ (N m)
.
Since ω(t) = O(t) as t tends to infinity, we can find a positive constant C2 such that ω(t) ≤ C2 t + C2 for all t > 0. Now, proceeding as in the proof Lemma 3.2, (v) ⇒ (iv), we take s ∈ N greater than 1 + log2 n + C1mC2 . Then, if k is the smallest natural number bigger than mLs , where L > 1 is the constant that appears in property (α) of the weight function ω, we have s 1 1 j 1 ∗ ϕ (N m) + N s ≤ ϕ∗ (N k) + L . m k k j=1
Hence, we obtain |ξ|N | uN (ξ)| ≤ CDr eC2 +N (
log n ∗ 1 1 2 +1+ m C2 C1 )+ m ϕ (N m) 1
≤ CDr eC2 eN s+ m ϕ 1
s
∗
(N m) j
1
≤ Cm e k ϕ
∗
(N k)
,
where the constant Cm = CDr eC2 + k j=1 L depends on m, on the weight ω, on r, and the selection of χN . Sufficiency. For x ∈ U , we have Dα u(x) = (2π)−n ξ α u N (ξ)eix,ξ dξ,
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
161
when N ≥ |α| + n + 1, for ξ α u N (ξ) is then integrable since (3) is satisfied by hypothesis. Now, as {uN } is a bounded sequence in E (Ω), an application of the Banach-Steinhaus Theorem gives, for all N ∈ N, | uN (ξ)| ≤ C1 (1 + |ξ|)M , for some constants C1 > 0 and M ∈ N that do not depend on N . Hence, for x ∈ U , |Dα u(x)| ≤ (2π)−n |ξ α || uN (ξ)|dξ = I1 + I2 , 1 where I1 denotes the integral when |ξ| ≤ exp( kN ϕ∗ (kN )), and I2 denotes 1 ϕ∗ (kN )). Since |ξ α | ≤ |ξ||α| , the integral when |ξ| ≥ exp( kN M |α| ∗ ∗ 1 , I1 ≤ mn C1 e kN ϕ (N k) 1 + e kN ϕ (kN ) 1 being mn the Lebesgue measure of the set {ξ : |ξ| ≤ exp( kN ϕ∗ (kN ))} n in R , that is less than or equal to the Lebesgue measure of the hypercube 1 ϕ∗ (kN ))},
· ∞ denotes the maximum norm in {ξ : ξ ∞ ≤ exp( kN n where n R , which is equal to 2n exp kN ϕ∗ (kN ) . Summing up, we have 1 ∗ n+|α|+M 1 ∗ ≤ 2n+M C1 e k ϕ (k(n+|α|+M )) , I1 ≤ 2n+M C1 e kN ϕ (kN )
when N = n + |α| + M . On the other hand, by (3), I2 ≤ |ξ|N −n−1 | uN (ξ)|dξ 1 ∗ |ξ|≥e N k ϕ (N k) 1 ∗ 1 ≤ Ce k ϕ (N k) dξ 1 ϕ∗ (kN ) n+1 |ξ| kN |ξ|≥e 1 ∗ 1 ≤ Ce k ϕ (N k) dξ. n+1 |ξ| |ξ|≥1 Therefore, we deduce that there is a constant C > 0, that depends only on n and M , such that 1
|Dα u| ≤ C e k ϕ
∗
(k(|α|+n+M ))
,
on U . Now, from the convexity of ϕ∗ we obtain 1 ∗ 1 ∗ 1 ∗ ϕ (k(|α| + n + M )) ≤ ϕ (2k|α|) + ϕ (2k(n + M )), k 2k 2k 1
∗
1
∗
and hence, C e k ϕ (k(|α|+n+M )) ≤ Ck e m ϕ (m|α|) , with m = 2k, which concludes the proof of (a). (b) The proof is similar to the one of (a), and we only indicate the main changes. Necessity. As in (a), we put uN = χN u. Then, for |α| ≤ N and k ∈ N there is Ck > 0 such that
α ∗ α |D (χN u)| ≤ Ck (7) (C1 N )|β| ekϕ (|α−β|/k) . β β≤α
162
Albanese, Jornet and Oliaro
IEOT
Proceeding as in the Roumieu case, the right hand side of (7) is less than or equal to Ck eN +kω(C1 N )+kϕ
∗
(N/k)
,
and therefore, N
|ξ|N | uN (ξ)| ≤ DCk e 2
log n+N +kω(C1 N )+kϕ∗ (N/k)
,
where the constant D depends on the Lebesgue measure of B2r (x0 ). Now, since ω(t) = o(t) as t tends to infinity, we can find, for each k ∈ N, a positive constant Ak such that kω(t) ≤ t + Ak for t > 0. Hence, if we put log n C k = DCk eAk and B = e 2 +1+C1 , we obtain |ξ|N | uN (ξ)| ≤ C k B N ekϕ
∗
(N/k)
.
Proceeding as in Lemma 3.2, (v) ⇒ (iv), we obtain an estimate like (4). Sufficiency. As in (a), for x ∈ U , and a fixed k ∈ N, |Dα u(x)| ≤ (2π)−n |ξ α || uN (ξ)|dξ = I1 + I2 , ∗ N , and I2 denotes where I1 denotes the last integral when |ξ| ≤ exp 2k Nϕ 2k 2k ∗ N the integral when |ξ| ≥ exp N ϕ 2k . Since |ξ α | ≤ |ξ||α| , M 2k|α| ∗ N 2k ∗ N , I1 ≤ mn C1 e N ϕ ( 2k ) 1 + e N ϕ ( 2k ) ∗ N } in Rn . being mn the Lebesgue measure of the set {ξ : |ξ| ≤ exp 2k Nϕ 2k We obtain, as in (a), 2k ∗ N n+|α|+M ∗ n+|α|+M ≤ 2n+M C1 e2kϕ ( 2k ) , I1 ≤ 2n+M C1 e N ϕ ( 2k ) when N = n + |α| + M , and by (4), I2 ≤ |ξ|N −n−1 | uN (ξ)|dξ 2k ϕ∗ N ( 2k ) |ξ|≥e N ∗ N 1 dξ. ≤ Ce2kϕ ( 2k ) n+1 |ξ|≥1 |ξ| Therefore, we deduce that there is a constant C > 0, that depends only on n and M , such that ∗ |α|+n+M |Dα u| ≤ C e2kϕ ( 2k ) , on U . Now, from the convexity of ϕ∗ we obtain |α| + n + M ∗ 2kϕ ≤ kϕ∗ (|α|/k) + kϕ∗ ((n + M )/k). 2k
We can now give the definition of quasianalitic (and non quasianlytic) wave front set of a classical distribution in Roumieu and Beurling settings.
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
163
Definition 3.4. Let Ω ⊂ Rn be an open set and u ∈ D (Ω). Let ω be a weight function. The {ω}−wave (respectively (ω)−wave) front set W F{ω} (u) (respectively W F(ω) (u)) of u is the complement in Ω × (Rn \ 0) of the set of points (x0 , ξ0 ) such that there exist an open neighborhood U of x0 in Ω, a conic neighborhood Γ of ξ0 and a bounded sequence uN ∈ E (Ω) equal to u in U satisfying (3) (respectively satisfying (4)) in Γ. Lemma 3.5. Let u ∈ D (Ω) and let K be a compact subset of Ω with nonempty interior, F a closed cone in Rn . Let ω be a weight function. Suppose that {χN } ⊂ D(K) and, for every α ∈ Nn0 and N ∈ N, that |β|
|Dα+β χN | ≤ Cα (Cα N ) Then {χN u} is a bounded sequence in E hood of K. Moreover:
M
,
|β| ≤ N.
(8)
if u is of order M in a neighbor-
(a) If W F{ω} (u) ∩ (K × F ) = ∅, there exist a constant C > 0 and k ∈ N such that 1
N +1 k ϕ e |ξ|N |χ N u(ξ)| ≤ C
∗
(kN )
,
N = 1, 2, . . . ,
ξ ∈ F.
(9)
(b) If W F(ω) (u) ∩ (K × F ) = ∅ and ω(t) = o(t) as t tends to infinity, there is a constant C > 0 such that for every k ∈ N there is Ck > 0 for which N kϕ |ξ|N |χ N u(ξ)| ≤ Ck C e
∗ N (k
)
,
N = 1, 2, . . . ,
ξ ∈ F.
(10)
Proof. The condition (8) with β = 0 implies that the sequence χN is bounded M in D(K) and hence χN u is bounded in E if u is of order M in a neighborhood of K. Let x0 ∈ K, ξ0 ∈ F \ {0} and choose U , Γ and uN according to Definition 3.4. If the support of χN is in U , we have χN u = χN uN . We first prove (a). By hypothesis uN satisfies in Γ 1
|ξ|N | uN (ξ)| ≤ D1 e k ϕ
∗
(N k)
,
N = 1, 2, . . . ,
(11)
for some constant D1 > 0 that does not depend on N . On the other hand, | uN (ξ)| ≤ D2 (1 + |ξ|)M for some constants D2 , M > 0 and all ξ ∈ Rn , as the sequence uN is bounded in E M . Since the function ϕ∗ (x)/x is increasing and k ≥ 1, it follows from (8) that |β| ∗ 1 |Dα+β χN | ≤ Cα Cα e kN ϕ (N k) , |β| ≤ N, where Cα denotes again a suitable positive constant depending only on α. Therefore, we obtain 1 ∗ N exp kN ϕ (N k) N +1 −n−1−M , η ∈ Rn . (12) | χN (η)| ≤ C 1 N (1 + |η|) |η| + exp kN ϕ∗ (N k) The properties of the Fourier transform give −n uN (ξ − η)dη. χ N (η) χ N u(ξ) = (2π)
164
Albanese, Jornet and Oliaro
IEOT
Let 0 < c < 1. We split the integral into two parts: χ N u(ξ) −n
= (2π)
χ N (η) uN (ξ − η)dη +
χ N (η) uN (ξ − η)dη .
|η|≤c|ξ|
(13)
|η|≥c|ξ| −1
In the second integral we have |ξ−η| ≤ (1+c )|η|, and from the boundedness of {uN }, it can be estimated by −1 M | χN (η)|(1 + |η|)M dη. (14) D2 (1 + c ) |η|≥c|ξ|
Now, we estimate the first integral in (13) by uN (ξ − η)|.
χN L1 sup |
(15)
|η|≤c|ξ|
On the other hand, for a conic closed neighborhood Γ1 of ξ0 contained in Γ \ {0} we can choose the constant c so that η ∈ Γ if ξ ∈ Γ1 and |ξ − η| ≤ c|ξ|. We observe that |η| ≥ (1−c)|ξ| and that the last supremum can be written as uN (η)|. Therefore, we conclude, from formulas (11), (12), (14) sup|ξ−η|≤c|ξ| | and (15), and for N ≥ 0, −N sup |ξ|N |χ
χN L1 sup | uN (η)| · |η|N N u(ξ)| ≤ (1 − c) Γ1 Γ −1 N +M M N + D2 (1 + c ) χN (η)|dη (1 + |η|) |η| | 1
≤ (1 − c)−N C1 C N D1 e k ϕ
∗
(N k)
1
+ D2 (1 + c−1 )N +M C2 e k ϕ
∗
(N k)
,
where the constants C1 , C2 , D1 , D2 do not depend on N . Now, F can be covered by a finite number of neighborhoods like Γ1 so that (9) is valid if supp χN ⊂ U for a sufficiently small neighborhood of x0 . We can cover K by such neighborhoods Uj , j = 1, . . . , J, and choose χN,j ∈ D(Uj ) so that χN,j = 1 in K and χN,j satisfies (8) for j = 1, . . . , J. But, if χN ∈ D(K) satisfies (8), the same is valid also for the product χN,j χN with some other constants. Hence (9) is valid with χN replaced by χN,j χN . The proof of (a) is complete since χN,j χN = χN . We now prove (b). By hypothesis, uN satisfies in Γ |ξ|N | uN (ξ)| ≤ Ck ekϕ
∗
(N/k)
,
N = 1, 2, . . . ,
for some sequence {Ck } of positive constants that does not depend on N . On the other hand, | uN (ξ)| ≤ D(1 + |ξ|)M for some constants D, M > 0 and all n ξ ∈ R , as the sequence uN is bounded in E M . It follows from (8) that | χN (η)| ≤ C N +1
NN N
(|η| + N )
(1 + |η|)−n−1−M ,
η ∈ Rn .
Now we can proceed as in the Roumieu case (a) taking into account that N N ≤ eN N ! and the following property: as ω(t) = o(t) as t tends to infinity,
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
165
from Remark 2.4(b) and Lemma 3.2, it follows that for every B > 0 and λ > 0 there is a constant CB,λ > 0 such that ∗ n B n n! ≤ CB,λ eλϕ ( λ ) ,
n ∈ N.
At this point, we can give some properties and consequences. Theorem 3.6. Let Ω ⊂ Rn be an open set and ω be a weight function. Then the projection of W F∗ (u) in Ω is equal to sing∗ supp u if u ∈ D (Ω). Proof. We give the proof only for the Roumieu case. The Beurling case is similar. If u is a E{ω} -function in a neighborhood of x0 it follows from Proposition 3.3 that (x0 , ξ0 ) ∈ / W F{ω} (u) for each ξ0 ∈ Rn \ {0}. Assume that (x0 , ξ0 ) ∈ / W F{ω} (u) for every ξ0 ∈ Rn \ {0}. Then we can choose a compact neighborhood K of x0 so that W F{ω} (u)∩(K ×Rn ) = ∅. By Lemma 3.5 there is a sequence χN ∈ D(K) which is equal to 1 in a neighborhood of x0 such / sing{ω} supp u that χN u is bounded in E and satisfies (3). Therefore, x0 ∈ by Proposition 3.3. The condition (8) is satisfied by any fixed function in E∗ (Ω) with support in K, where ∗ = {ω} or (ω). Therefore, if ω is a non-quasianalytic weight function, an equivalent definition of wave front set is given by the following proposition. Proposition 3.7. Let Ω ⊂ Rn be an open set and u ∈ D (Ω). Let ω be a nonquasianalytic weight function. Then (x0 , ξ0 ) ∈ Ω × (Rn \ 0) is not in the wave front set W F{ω} (u) (resp. W F(ω) (u)) of u if and only if there is a neighborhood U ⊂ Ω of x0 , a conic neighborhood Γ of ξ0 and v ∈ E{ω} (Ω) (resp. v ∈ E(ω) (Ω)) which is equal to u in U and has a Fourier transform satisfying (3) (resp. (4)) in Γ. Combining Proposition 3.7 with Proposition 3.3 and Lemmas 3.1 and 3.2, we recover the definition of wave front set in the Gevrey setting, [20, p. 36], and in the nonquasianalytic Beurling setting, [11]. The properties of the Young conjugate ϕ∗ω of ϕω for a given weight function ω lead to the following property: Proposition 3.8. Let Ω ⊂ Rn be an open set. If ω and σ are two weight functions such that ω = O(σ), then W F{ω} (u) ⊂ W F{σ} (u), and W F(ω) (u) ⊂ W F(σ) (u), for each u ∈ D (Ω). Since log t = o(ω), and ω = O(ω1 ), where ω1 (t) = max(t−1, 0) for every weight function ω, we observe that W F (u) ⊂ W F{ω} (u) ⊂ W FA (u),
u ∈ D (Ω),
being W F (u) the classical wave front set, and W FA (u) the Roumieu wave front set with respect to ω1 . Moreover, if ω(t) = o(t) as t tends to infinity, we have W F (u) ⊂ W F{ω} (u) ⊂ W F(ω) (u) ⊂ W FA (u),
u ∈ D (Ω).
166
Albanese, Jornet and Oliaro
IEOT
Proposition 3.9. If Ω ⊂ Rn is an open set and S is a closed cone in Ω×(Rn \0), then there is u ∈ D (Ω) with W F (u) = W F{ω} (u) = S
(respectively, W F (u) = W F(ω) (u) = S)
for every weight function ω (respectively, for every weight function ω satisfying ω(t) = o(t) as t tends to infinity). Proof. By [13, Theorem 8.4.14], there is u ∈ D (Ω) such that W F (u) = W FA (u) = S. Since W F (u) ⊂ W F{ω} (u) ⊂ W FA (u) (respectively, W F (u) ⊂ W F(ω) (u) ⊂ W FA (u)) for every weight function ω (respectively, for every weight function ω satisfying ω(t) = o(t) as t tends to infinity), the result follows. The conditions in (8) remain valid if we multiply all χN by the same function in E∗ (Ω). Therefore, we obtain Theorem 3.10. W F∗ (au) ⊂ W F∗ (u) if a ∈ E∗ (Ω) and u ∈ D (Ω). On the other hand, it is clear that W F{ω} (∂u/∂xj ) ⊂ W F{ω} (u). In fact,
∂u
N +1 ∗ 1
N +1 (ξ) = |ξj u N +1 (ξ)| ≤ C|ξ| e (N +1)k ϕ ((N +1)k) /|ξ|
∂xj
1
= Ce k ϕ
∗
((N +1)k)
1
/|ξ|N ≤ C|ξ|−N e 2k ϕ
∗
(2N k)
1
e 2k ϕ
∗
(2k)
.
If we combine this property with Theorem 3.10, we obtain Theorem 3.11. Let Ω ⊂ Rn be an open set and P (x, D) = |α|≤m aα (x)Dα be a linear partial differential operator with coefficients in E∗ (Ω). Then W F{ω} (P (x, D)u) ⊂ W F{ω} (u),
u ∈ D (Ω),
W F(ω) (P (x, D)u) ⊂ W F(ω) (u),
u ∈ D (Ω))
(respectively,
for every weight function ω (respectively, for every weight function ω satisfying ω(t) = o(t) as t tends to infinity).
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
167
4. Propagation of singularities In this section we prove a converse of Theorem 3.11 related to the propagation of singularities of solutions of linear partial differential operators. We first study the Beurling case. The corresponding Roumieu version is then obtained as a consequence of a description of Roumieu wave front sets via a suitable union of Beurling wave front sets. Theorem 4.1. Let Ω ⊂ Rn be an open set. Let σ be a weight function satisfying property (α0 ) and ω be a weight function such that ω(t) = o(σ(t)) as t tends to infinity. If P := P (x, D) = |α|≤m aα (x)Dα is a linear partial differential operator with coefficients in the Roumieu class E{σ} (Ω), then (16) W F(ω) (u) ⊂ W F(ω) (P u) ∪ Σ, u ∈ D (Ω), α where Pm (x, ξ) = |α|=m aα (x)ξ is the principal symbol of P and Σ the characteristic set of P which is defined by Σ = {(x, ξ) ∈ Ω × (Rn \ 0) : Pm (x, ξ) = 0} . Proof. We must prove that if (x0 , ξ0 ) does not belong to the right hand side of / W F(ω) (u). If we assume this hypothesis, we (16) and ξ0 = 0, then (x0 , ξ0 ) ∈ can choose a compact neighborhood K of x0 and a closed conic neighborhood Γ of ξ0 in Rn \ 0 such that Pm (x, ξ) = 0 in K × Γ, (K × Γ) ∩ W F(ω) (P u) = ∅.
(17) (18)
Now, we take a sequence χN ∈ D(K) equal to 1 in a fixed neighborhood U of x0 satisfying property (8) for every α. Then, the sequence uN = χ2N u is bounded in E and equal to u in U . The theorem will be proved if we show that (4) is valid for the weight ω when ξ ∈ Γ and |ξ| ≥ N , since (4) is true for |ξ| ≤ N. In fact, when |ξ| ≤ N we can argue in the following way. As M ≤ C1 (1 + N )M ≤ C N for some in Proposition 3.3, |u N (ξ)| ≤ C1 (1 + |ξ|) positive constants C1 , C and a natural number M ∈ N. On the other hand, since ω(t) = o(t) as t tends to infinity, we have that, for each k ∈ N, there ∗ exists Ck > 0 satisfying N N ≤ Ck ekϕω (N/k) . Then, we obtain ∗
N N |ξ|N |u ≤ Ck C N ekϕω (N/k) . N (ξ)| ≤ C N
To estimate u N (ξ) in Γ we will solve in an approximate way the equation P t v(x) = χ2N (x)e−ix,ξ .
(19)
−ix,ξ
w/Pm (x, ξ) and observe that the principal Following [14], we put v = e symbol of P t is Pm (x, −ξ). Hence, we obtain instead of (19) an equation of the form (20) w − Rw = χ2N , R = R1 + R2 + · · · + Rm where Rj |ξ|j is a differential operator of order less than or equal to j with E{σ} -coefficients which is homogeneous of degree 0 with respect to ξ for ξ ∈ Γ and x ∈ K.
168
Albanese, Jornet and Oliaro
We now write
wN =
IEOT
Rj1 · · · Rjk χ2N .
(21)
j1 +···+jk ≤N −m
From this formula, we obtain
wN − RwN = χ2N −
Rj1 · · · Rjk χ2N
(22)
j1 +···+jk >N −m≥j2 +···+jk
= χ2N − eN , which means that P t (x, D)(e−ix,ξ wN (x, ξ)/Pm (x, ξ)) = e−ix,ξ (χ2N (x) − eN (x, ξ)). With integrals denoting action of distributions we obtain, with f = P (x, D)u, u(x)χ2N (x)e−ix,ξ dx = u(x)eN (x, ξ)e−ix,ξ dx (23) + f (x)e−ix,ξ wN (x, ξ)/Pm (x, ξ)dx. To estimate the right-hand side of (23) we need the following lemma: Lemma 4.2. There exists a constant C > 0 such that for j = j1 + · · · + jk and j + |β| ≤ 2N , 1 ∗ j+|β| |Dβ Rj1 · · · Rjk χ2N | ≤ C N +1 e 2hN ϕσ (2hN ) |ξ|−j , ξ ∈ Γ. (24) Proof of Lemma 4.2. By homogeneity it suffices to prove the lemma when |ξ| = 1 (x ∈ K). But then, this lemma is a consequence of the next one. Lemma 4.3. Let K ⊂ Ω be a compact set and χN ∈ D(K) be a sequence satisfying property (8). If a1 , . . . , aj−1 are functions in E{σ} (Ω) such that, for some constant C > 0 and some h ∈ N, that 1
∗
sup |Dα as | ≤ Ce h ϕσ (h|α|) K
(25)
for every α ∈ Nn0 and s = 1, . . . , j − 1, we have, for some constant C1 > 0 that depends only on C and the sequence {χN }, and j ≤ N , 1 ∗ j (26) sup |Di1 a1 Di2 · · · aj−1 Dij χN | ≤ C1j+1 e hN ϕσ (hN ) K
Proof of Lemma 4.3. Since σ(t) = O(t) as t tends to infinity, N N ≤ ∗ C N eϕσ (N ) for a sufficiently large constant C > 0. Therefore, from (8), for α = 0, we obtain |β| ∗ 1 , |β| ≤ N. (27) |Dβ χN | ≤ C0 C0 e N ϕσ (N ) It is clear that Di1 a1 Di2 · · · aj−1 Dij χN is a sum of terms of the form (Dα1 a1 ) · · · (Dαj−1 aj−1 )Dαj χj with |α1 | + · · · + |αj | = j.
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
169
If there are Ck1 ,...,kj terms with |α1 | = k1 ,. . . ,|αj | = kj , we have |Di1 a1 Di2 · · · aj−1 Dij χN | 1 ∗ kj
1 ∗ 1 ∗ k ≤ Ck1 ,...,kj C j−1 e h ϕσ (hk1 ) · · · e h ϕσ (hkj−1 ) C0 C0 j e N ϕσ (N )
e ≤ Ck1 ,...,kj k1 ! · · · kj−1 !C j−1
1 ∗ h ϕσ (hk1 )
k1 !
···
e
1 ∗ h ϕσ (hkj−1 )
kj−1 !
(28)
1 ∗ kj k C0 C0 j e N ϕσ (N ) .
As in [10], since σ satisfies property (α0 ), we can suppose that σ is equivalent to a sub-additive weight and then, we have 1
∗
1
∗
1
∗
e h ϕσ (hkj−1 ) e h ϕσ (h(j−kj )) e h ϕσ (hk1 ) ··· ≤ . k1 ! kj−1 ! (j − kj )! We also observe that k1 ! · · · kj−1 ! k1 ! · · · kj !j! k1 ! · · · kj ! = ≤ 2j (j − kj )! (j − kj )!kj !j! j! and that Ck1 ,...,kj k1 ! · · · kj ! = (2j − 1)!!. We can assume that C, C0 > 1 k and hence, if we put C1 = CC0 , we have C j−1 C0 C0 j ≤ C1j+1 . Since ϕ∗σ (x)/x is increasing, from (28), we obtain |Di1 a1 Di2 · · · aj−1 Dij χN | 1 ∗ kj
1 ∗ ≤C1j+1 2j /j! Ck1 ,...,kj k1 ! · · · kj !e h ϕσ (h(j−kj )) e hN ϕσ (hN ) 1 ∗ kj
1 ϕ∗ (h(j−kj )) j−kj ≤(2C1 )j+1 /j! Ck1 ,...,kj k1 ! · · · kj ! e h(j−kj ) σ e hN ϕσ (hN ) 1 ∗ j (2j − 1)!! ≤(4C1 )j+1 e hN ϕσ (hN ) , j!2j which concludes the proof of the lemma since
(2j−1)!! j!2j
≤ 1.
We now finish the proof of Theorem 4.1. If M is the order of u in a neighborhood of K, we can estimate the first term on the right-hand side of (23) for large N and |ξ| > N, with ξ ∈ Γ, by
(1 + |ξ|)M −|α| sup |Dα eN (x, ξ)|. C |α|≤M
x
The number of terms in eN cannot exceed 2N , and each term can be estimated by means of (24), where j1 + . . . + jk = j > N − m by (22), which gives the bound 1 ∗ N +|α| |Dxα eN (x, ξ)| ≤ C N +1 2N e 2hN ϕσ (2hN ) |ξ|−N +m . Therefore, the first term on the right-hand side of (23) can be estimated by 1 ∗ N +M |ξ|M −N +m . (29) C N +1 2N +M e 2hN ϕσ (2hN )
170
Albanese, Jornet and Oliaro
IEOT
Now, as ϕ∗σ is convex and ϕ∗σ (x)/x is increasing, we have N +M N +M 1 ∗ ∗ 1 e 2hN ϕσ (2hN ) ≤ e 2h(N +M ) ϕσ (2h(N +M )) ≤e
∗ 1 2h ϕσ (2h(N +M ))
= Ch,M e
∗ 1 4h ϕσ (4hN )
≤e
∗ 1 4h ϕσ (4hM )
(30) e
∗ 1 4h ϕσ (4hN )
.
Since ω(t) = o(σ(t)) as t tends to ∞, if N is replaced by N + m + M , the bound (29) and (30) imply an estimate of the form (11) for the first integral on the right of (23). Indeed, from ω(t) = o(σ(t)) as t tends to ∞ we deduce that for every k ∈ N there exists dk > 0 such that N 1 ∗ ∗ ϕσ (4hN ) ≤ dk + kϕω , N ∈ N. (31) 4h k Now, combining (29) togheter (31) and (30) with N replaced by N + m + M , we easily obtain an estimate of the form (11) for the first integral on the right of (23). To estimate the last term in (23) we observe that (24) gives 1 ∗ |β| , |β| ≤ N, ξ ∈ Γ, |ξ| > N. (32) |Dβ wN | ≤ C1N +1 e 2hN ϕσ (2hN ) We have a similar bound for wN |ξ|m /Pm (x, ξ). The proof is completed by the following lemma: Lemma 4.4. Let f ∈ D (Ω). Let K ⊂ Ω be a compact set and Γ a closed cone ⊂ Rn \ 0 such that W F(ω) (f ) ∩ (K × Γ) = ∅. If wN ∈ D(K) and (32) is fulfilled, then there exists C2 > 0 such that for every k ∈ N there exists Ck > 0 such that N −M −n ∗ N −M −n k N |w (33) e N −M −n ϕω ( k ) /|ξ| N f (ξ)| ≤ Ck C2 if ξ ∈ Γ, |ξ| > N , and N > M +n. Here M is the order of f in a neighborhood of K. Proof. By Lemma 3.5 we can find a sequence fN which is bounded in E M and equal to f in a neighborhood of K so that there exists D > 0 such that for every k ∈ N there exists Dk > 0 for which k ∗ N |f N (η)| ≤ Dk DN e N ϕω (N/k) /|η| , N = 0, 1, 2, . . . , η ∈ Γ , where Γ is a conic neighborhood of Γ. Then wN f = wN fN , N = N −M −n. On the other hand, using (32) and proceeding as in Lemma 3.5 to show (12), we obtain that there exists E > 0 such that 1
∗
1
∗
|w N (η)| ≤ E N +1 (e 2hN ϕσ (2hN ) /(e 2hN ϕσ (2hN ) + |η|))N ,
η ∈ Rn .
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
171
Proceeding again as in the proof of Lemma 3.5, it follows (the integrals that appear there are convergent from the selection of N ), that there exists D > 0 such that for every k ∈ N there exists Dk > 0 for which ∗ ∗ 1 N sup |ξ|N |w ekϕω (N /k) + e 2h ϕσ (2hN ) , ξ ∈ F, |ξ| > N. N f (ξ)| ≤ Dk D ξ∈Γ
We now observe that the convexity of ϕ∗σ implies that 1 ∗ 1 ϕσ (2hN ) ≤ [ϕ∗ (4h(N − M − n)) + ϕ∗σ (4h(M + n))] , 2h 4h σ
(34)
N > M + n.
(35) On the other hand, ω(t) = o(σ(t)) as t tends to ∞ and hence, for every k ∈ N there exists Dk > 0 such that N 1 ∗ ∗ ϕσ (4hN ) ≤ Dk + kϕω , N ∈ N. (36) 4h k Combining (34) with (35) and (36), (33) follows. Thus the lemma is proved. The last lemma gives the key to estimate the second integral on the right-hand side of (23), and finishes the proof of Theorem 4.1. An examination of the proofs above shows that we can repeat the same arguments to prove the analogous result of Theorem 4.1 in the Roumieu setting, even assuming the coefficients of P (x, D) in E{ω} (Ω) with ω a weight function satisfying property (α0 ). We point out that it is necessary only to state and prove Lemma 4.4 appropriately. But, we also present another proof of the Roumieu version based on an application of Theorem 4.1 and of the following proposition, which is an extension to the quasianalytic case of [11, Proposition 2] and could be of independent interest. Given two weight functions σ0 and ω such that σ0 (t) = o(ω(t)) as t tends to infinity, we set S := {σ weight function : σ0 ≤ σ = o(ω)}. Proposition 4.5. Let ω be weight function. If σ0 and σ are as above, we have W F(σ) (u), u ∈ D (Ω). W F{ω} (u) = σ∈S
Proof. The inclusion ∪σ∈S W F(σ) (u) ⊂ W F{ω} (u) follows easily from the definition of S, and the facts that E{ω} (Ω) ⊂ E(σ) (Ω) if σ = o(ω) (see Remark 2.4) and that the wave front set is closed. Conversely, suppose that (x0 , ξ0 ) ∈ Γ := ∪σ∈S W F(σ) (u). We can then find a compact neighborhood K of x0 and a closed conic neighborhood F of ξ0 in Rn \ 0 such that (K × F ) ∩ ∪σ∈S W F(σ) (u) = ∅.
172
Albanese, Jornet and Oliaro
IEOT
For each N ∈ N let χN ∈ D(K) be equal to 1 in a fixed neighborhood U of x0 such that, for every α ∈ Nn0 , |Dα+β χN | ≤ Cα (Cα N )|β| ,
|β| ≤ N.
Then, by Lemma 3.5. (b), χN u is a bounded sequence in E M if M is the order of u in a neighborhood of K and, for every σ ∈ S and k ∈ N, there is Ckσ > 0 so that ∗
σ kϕσ (N/k) , |ξ|N |χ N u(ξ)| ≤ Ck e
ξ ∈ F, N = 0, 1, 2, . . . .
(37)
We will deduce from this fact that (x0 , ξ0 ) ∈ W F{ω} (u), after showing that there exist C > 0 and h ∈ N for which 1
∗
N +1 h ϕω (hN ) e , |ξ|N |χ N u(ξ)| ≤ C
ξ ∈ F, N = 0, 1, 2, . . . .
(38)
In order to prove such an inequality, we will proceed as follows. By (37) we obtain that, for every σ ∈ S, k ∈ N, and r > 0, gN (r) := rN and hence
sup |ξ|=r,ξ∈F
∗
σ kϕσ (N/k) |χ , N u(ξ)| ≤ Ck e
∗
sup gN (r) ≤ Ckσ ekϕσ (N/k) , r>0
N = 0, 1, 2, . . .
N = 0, 1, 2, . . . .
This implies that, for every σ ∈ S and k ∈ N, N aN := log sup gN (r) ≤ kϕ∗σ + log Ckσ , k r>0
N = 0, 1, 2, . . . .
(39)
We claim that (39) implies that there exist h ∈ N and C > 0 so that 1 (40) aN ≤ ϕ∗ω (N h) + C, N = 0, 1, 2, . . . . h Proceeding by contradiction we can construct an increasing sequence (N (h))h of positive integers (N (1) := 0) such that 1 aN (h) > ϕ∗ω (N (h)h) + Ch , (41) h for every h ∈ N (Ch := h). We will show that inequalities (41) are in contradiction with inequalities (39) by constructing a weight function σ ∈ S for which the inequality 1 ∗ ϕ (N (h)h) + Ch < ϕ∗σ (N (h)) + log C1σ (42) h ω does not hold for infinitely many indices h ∈ N. Without loss of generality, we can suppose that ω|[R,+∞[ is a C 1 function for some R ≥ 0 (see [9, Lemma 1.7]). Then the function ϕ := ϕω is a C 1 function too on [R, +∞[ and ϕ is a nondecreasing continuous function on [R, +∞[. In particular, limx→+∞ ϕ (x) = +∞ because log(1 + t) = o(ω(t)) as t tends to infinity. Then, ϕ ([0, +∞[) ⊃ [ϕ (R), +∞[. Consequently, we can find an increasing sequence (xh )h ⊂ [0, +∞[ (x1 := 0) satisfying xh → ∞ and ϕ (xh ) = N (h)h for all h ∈ N. Set x1 = y1 = z1 = 0. As xh → ∞, we can inductively define an increasing
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
173
sequence (h(n))n of positive integers (h(0) = h(1) = 1) and the sequences (yn )n and (zn )n with xh(2) > R, y2 = z2 = xh(2) and for all n ≥ 3 N (h(n)) >
N (h(n − 1))h(n − 1) , h(n − 3)
xh(n) > yn−1 + n, σ0 (ex ) ≤
(44)
ϕ(x) for all x ≥ xh(n) , h2 (n − 1)
ϕ(xh(n) ) ≥ h(n − 1)
n−1
(43)
(45)
ϕ(zi ),
(46)
h(n − 1) ϕ (xh(n) ), h(n − 2)
(47)
i=1
ϕ (yn ) = and [h(n − 1) − h(n − 2)]ϕ(zn )
= h(n − 1)ϕ(xh(n) ) − h(n − 2)ϕ(yn ) + h(n − 1)(yn − xh(n) )ϕ (xh(n) ).
(48)
We have that xh(n) ≤ zn ≤ yn
(49)
for all n ∈ N. From (47) we get that yn ≥ xh(n) as ϕ is a nondecreasing function. Hence, by (48) we get ϕ(zn ) − ϕ(xh(n) ) yn − xh(n) 1 = h(n − 1) − h(n − 2) h(n − 1)ϕ(xh(n−1) ) − h(n − 2)ϕ(yn ) − (h(n − 1) − h(n − 2))ϕ(xh(n) ) × yn − xh(n) h(n − 1) ϕ (xh(n) ) h(n − 1) − h(n − 2) ϕ(xh(n) ) − ϕ(yn ) h(n − 1) h(n − 2) ϕ (xh(n) ) + = h(n − 1) − h(n − 2) yn − xh(n) h(n − 1) − h(n − 2) ϕ(yn ) − ϕ(xh(n) ) h(n − 2) ≥ ≥ 0, ϕ (xh(n) ) − h(n − 1) − h(n − 2) yn − xh(n) +
because ϕ is a nondecreasing function. On the other hand, (48) also implies that [h(n − 1) − h(n − 2)]ϕ(zn ) = h(n − 1)[ϕ(xh(n) ) + (yn − xh(n) )ϕ (xh(n) )] − h(n − 2)ϕ(yn ) ≤ [h(n − 1) − h(n − 2)]ϕ(yn ) because ϕ is a convex function.
174
Albanese, Jornet and Oliaro
IEOT
We define a function ψ on [0, +∞[ by setting ⎧ n−2 1 ⎪ ϕ(xh(n) ) + i=1 h(i)−h(i−1) ⎪ h(i−1)h(i) ϕ(zi+1 ) ⎪ ⎨h(n−2) if xh(n) ≤ x < yn , x−xh(n) ψ(x) = + h(n−2) ϕ (xh(n) ) (50) ⎪ ⎪ ⎪ ⎩ 1 ϕ(x) + n−1 h(i)−h(i−1) ϕ(z ) if yn ≤ x ≤ xh(n+1) . i+1 i=1 h(i−1)h(i) h(n−1) From (47) and (48) it follows that ψ is a C 1 function. Moreover, it is convex because it consists of linear parts and of dilated and shifted parts of ϕ. We define σ by σ(t) = ψ(max(log t, 0)), which is again a C 1 function. Proceeding as in the proof of [9, Lemma 1.7], it is easily seen that 1 ϕ(x) for all x ∈ [xh(n) , xh(n+1) ], n ≥ 3; ψ(x) ≥ h(n − 1) hence by (45) we deduce 1 1 ϕ(x) ≤ ψ(x) σ0 (t) ≤ 2 h (n − 1) h(n − 1)
for all x ∈ [xh(n) , xh(n+1) ], n ≥ 3.
Therefore σ0 (t) = o(σ(t)) as t → ∞. We also have ψ(x) 2 ≤ ϕ(x) h(n − 2)
for all x ∈ [xh(n) , xh(n+1) ], n ≥ 2,
and hence, σ(t) = o(ω(t)) as t → ∞. Now, we have, for every n ∈ N, that ϕ∗ (N (h(n))h(n)) = N (h(n))h(n)xh(n) − ϕ(xh(n) ).
(51)
Indeed,
ϕ (xh(n) ) = N (h(n))h(n), and, since ϕ is nondecreasing, N (h(n))h(n) − ϕ (s) ≥ 0 for all 0 ≤ s ≤ xh(n) . Moreover, N (h(n))h(n)s − ϕ(s) → −∞ as s → +∞ and (N (h(n))h(n)s − ϕ(s))(0) = 0. On the other hand, by (50) and proceeding as above we deduce, for every n ∈ N, that ψ ∗ (N (h(n))) = N (h(n))ζn − ψ(ζn ),
(52)
where ψ (ζn ) = N (h(n)) (indeed, the function δn (s) = N (h(n))s − ψ(s) is C 1 with derivative δn (s) = N (h(n)) − ψ (s) and hence δn (s) ≥ 0 if and only if ψ (s) ≤ N (h(n))). In particular, ζn ∈ [yn−1 , xh(n) ]. In fact, if x ∈ [yn−1 , xh(n) ] then, by (50), ϕ (xh(n) ) ϕ (yn−1 ) ≤ ψ (x) ≤ . h(n − 2) h(n − 2) Moreover, by (43) and (47), ϕ (yn−1 ) 1 N (h(n − 1))h(n − 1) = ϕ (xh(n−1) ) = < N (h(n)), h(n − 2) h(n − 3) h(n − 3) while
ϕ (xh(n) ) N (h(n))h(n) = > N (h(n)). h(n − 2) h(n − 2)
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
175
Consequently, we have N (h(n)) ∈ ψ ([yn−1 , xh(n) ]) and so we can conclude from the fact that ψ is continuous. Therefore, we have constructed a C 1 –function σ : [0, +∞[→ [0, +∞[ such that σ0 (t) = o(σ(t)) and σ(t) = o(ω(t)) as t → +∞. Then, by [9, Lemma 1.7] we can find a weight function v such that σ(t) = o(v(t)) and v(t) = o(ω(t)) as t → +∞ (hence, v ∈ S), and ψ(t) ≤ ϕv (t)
for all t ≥ 0.
ϕ∗v (t) ≤ ψ ∗ (t)
for all t ≥ 0.
This implies that (53)
As v ∈ S, by (39) and (41) we then obtain 1 ϕ∗ (N (h(n))h(n)) + h(n) < ϕ∗v (N (h(n))) + log C1v , h(n) for all n ∈ N. Thus, by (53) we get 1 ϕ∗ (N (h(n))h(n)) + h(n) < ψ ∗ (N (h(n)) + log C1v , h(n) for all n ∈ N. By (51) and (52) we also have N (h(n))xh(n) −
ϕ(xh(n) ) + h(n) < N (h(n))ζn − ψ(ζn ) + log C1v . h(n)
Then, by (50), since ϕ is a nondecreasing function and since ϕ (xh(n) ) = N (h(n))h(n), we obtain N (h(n))(xh(n) − ζn ) + h(n) n−2
<
h(i) − h(i − 1) ϕ(xh(n) ) ϕ(ζn ) − − ϕ(zi+1 ) + log C1v h(n) h(n − 2) i=1 h(i − 1)h(i)
ϕ(xh(n) ) ϕ(ζn ) ϕ(ζn ) ϕ(ζn ) − + − + log C1v h(n) h(n) h(n) h(n − 2) xh(n) ϕ (s) ds + log C1v ≤ h(n) ζn xh(n) ϕ (xh(n) ) ds + log C1v ≤ h(n) ζn
<
= N (h(n))(xh(n) − ζn ) + log C1v ,
which is a contradiction. Clearly, (41) implies (38) and the proof is complete.
An immediate consequence of Proposition 4.5 is the following result. Corollary 4.6. Let ω be a weight function and Ω be an open set in Rn . Then E(σ) (Ω). E{ω} (Ω) = σ∈S
176
Albanese, Jornet and Oliaro
IEOT
Proof. Let g ∈ D (Ω). By Theorem 3.6, we have that g ∈ E{ω} (Ω) if, and only if, W F{ω} (g) is empty. Then, W F(σ) (g) is empty for all σ ∈ S, and the conclusion follows. There exists an unpublished version of Corollary 4.6, with a different proof, for open convex sets, due to J. Bonet and R. Meise, and is an extension for quasianalytic classes of [6, Proposition 3.5]. Another consequence of Proposition 4.5 is the following: Corollary 4.7. Let ω be a weight function and Ω be an open set in Rn . Then sing(ω) supp (u), u ∈ D (Ω). sing{ω} supp (u) = σ∈S
Proof. Fix u ∈ D (Ω). Then, combining Proposition 4.5 and Theorem 3.6, we easily obtain sing(ω) supp (u). sing{ω} supp (u) ⊂ σ∈S
The opposite inclusion follows from Remark 2.4(a) and the fact that the singular support is always a closed set. We now state and prove the Roumieu version of Theorem 4.1. We do not need any change of weight function, but the weight function must satisfy property (α0 ), as in the Beurling version (Theorem 4.1). Theorem 4.8. Let ω be a weight function satisfying property (α0 ) and Ω ⊂ Rn be an open set. If P (x, D) is a linear partial differential operator whose coefficients belong to E{ω} (Ω), then W F{ω} (u) ⊂ W F{ω} (P u) ∪ Σ,
u ∈ D (Ω),
where Σ is the characteristic set of P . Proof. Let S be the set of weight functions σ such that σ = o(ω). Then, for each σ ∈ S and u ∈ D (Ω), by Theorem 4.1 we have W F(σ) u ⊂ W F(σ) (P u) ∪ Σ,
u ∈ D (Ω),
as the coefficients of P (x, D) belong to E{ω} (Ω). Now, we apply Proposition 4.5 to conclude. Finally, we point out that from Theorems 4.1 and 4.8 we immediately get (see, for example, [20, p.105]): Corollary 4.9. Let P = P (x, D) be an elliptic linear partial differential operator defined on an open set Ω of Rn (elliptic means that Σ = ∅). Then: (a) If ω is a weight function satisfying property (α0 ) and the coefficients of P belong to E{ω} (Ω), we have W F{ω} (u) = W F{ω} (P u),
u ∈ D (Ω),
and hence, sing{ω} supp (u) = sing{ω} supp (P u),
u ∈ D (Ω).
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
177
(b) If ω and σ are two weight functions such that ω satisfies property (α0 ) and ω = o(σ), and the coefficients of P belong to E{σ} (Ω), then W F(ω) (u) = W F(ω) (P u),
u ∈ D (Ω),
and hence, sing(ω) supp (u) = sing(ω) supp (P u),
u ∈ D (Ω).
We conclude the paper by studying the wave front set of the solutions of the following (non hypoelliptic) partial differential operator of principal type in Rn : ∂ P = . ∂xn Observe that the characteristic set of P is Σ = {(x, ξ) ∈ R2n : ξn = 0, ξ = 0}. Moreover, we point out that u ∈ D (Rn ) is a solution of P u = 0 if, and only if, u = v ⊗ 1 for some v ∈ D (Rn−1 ), being 1 the function identically 1 in the xn -variable. Indeed, if u is of the form v ⊗ 1, then ∂x∂n (v ⊗ 1) = 0. On the other hand, if P u = 0, then u satisfies τh u = u for every h = (0, . . . , 0, hn ), where τh u denotes the h-translation of the distribution u (see, for example, [20]). From this fact, by an approximation procedure, it is easy to conclude that u must be of the form v ⊗ 1 for some distribution v ∈ D (Rn−1 ). We can now state the following result: Proposition 4.10. Let ω be a weight function satisfying property (α0 ), and write, as usual, ∗ for {ω} or (ω). Let u ∈ D (Rn ) be a solution of the equation P u = 0. If (x0 , ξ0 ) ∈ W F∗ (u), then (x0 , ξ0 ) ∈ Σ, and splitting Rn x = (x , xn ) = (x1 , . . . , xn−1 , xn ), we have that the straight line L = {(x0 , xn , ξ0 ), xn ∈ R} is contained in W F∗ (u). Moreover, if ω is non-quasianalytic, for every (x0 , ξ0 ) ∈ Σ there exists a solution u ∈ D∗ (Rn ) of P u = 0 whose ∗-wave front set is given by the set {(x0 , xn , λξ0 ), xn ∈ R, λ > 0}. Proof. Since u ∈ D (Rn ) is a solution of P u = 0, by Theorems 4.1 and 4.8 we have W F∗ (u) ⊂ Σ and u = v ⊗ 1 for some suitable v ∈ D (Rn−1 ). We claim that W F∗ (u) = {(x, ξ) ∈ Σ : (x , ξ ) ∈ W F∗ v}.
(54)
First, we show property (54) in Beurling case proceeding as follows. Let / W F(ω) (v) then, by Definition 3.4, there exist an open (x, ξ) ∈ Σ. If (x , ξ ) ∈ neighborhood U of x , a conic neighborhood Γ of ξ and a bounded sequence vN ∈ E (Rn−1 ), vN = v in U , such that for every k ∈ N there exists a positive constant Ck satisfying |ξ |N | vN (ξ )| ≤ Ck ekϕ
∗
(N/k)
(55)
178
Albanese, Jornet and Oliaro
IEOT
in Γ , for N = 1, 2, . . . . Let χ ∈ D(R), be a function equal to 1 in a neighborhood I of xn . Then, we have that uN := vN ⊗ χ is a bounded sequence in E (Rn ), uN = u in U := U × I. Let Γ be a conic neighborhood of (ξ , 0) (since (x, ξ) ∈ Σ we have ξ n = 0) satisfying Γ ∩ {ξn = 0} ⊂ Γ . Then there exists a positive constant c1 such that |ξn | ≤ c1 |ξ | for ξ = (ξ , ξn ) ∈ Γ.
(56)
We also observe that u ˆN (ξ) = v N (ξ ) χ(ξn ). N ⊗ χ(ξ) = v Now, from (55) and (56), it follows that for every k ∈ N, N uN (ξ)| ≤ |ξ | + |ξn | | vN (ξ )| | χ(ξn )| |ξ|N | vN (ξ )| | χ(ξn )| ≤ (1 + c1 )N |ξ |N | ≤ c2 Ck (1 + c1 )N ekϕ
∗
(N/k)
,
for each N ∈ N and ξ ∈ Γ. In view of Definition 3.4 and Lemma 3.2, this inequality implies that (x, ξ) ∈ / W F(ω) (u). Then, we deduce that W F(ω) (u) ⊂ {(x, ξ) ∈ Σ : (x , ξ ) ∈ W F(ω) (v)}. / W F(ω) (u). By Definition 3.4 Conversely, let (x, ξ) ∈ Σ such that (x, ξ) ∈ there exist an open neighborhood U of x, a conic neighborhood Γ of ξ and a bounded sequence uN ∈ E (Rn ), uN = u in U , such that (4) is satisfied in Γ. Since u = v ⊗1, without loss of generality we can assume that uN = vN ⊗χN , eventually multiplying uN by a tensor product test function equal to 1 in a neighborhood V of x with V ⊂ U (see Lemmas 3.5 and 3.2 and remarks before Proposition 3.7). Therefore, there exists a sequence Ck of positive constants such that |ξ|N | uN (ξ)| = |ξ|N | vN (ξ )| | χN (ξn )| ≤ Ck ekϕ
∗
(N/k)
(57)
for N ∈ N and ξ ∈ Γ. We may also suppose that χN =: χ is independent of N and satisfies χ (0) = 0. So, from (57) it follows that |ξ |N | vN (ξ )| ≤
Ck kϕ∗ (N/k) e | χ(0)|
for N = 1, 2, . . . and ξ ∈ Γ = Γ ∩ {ξn = 0}. Then (x , ξ ) ∈ / W F(ω) (v), and so we have shown that {(x, ξ) ∈ Σ : (x , ξ ) ∈ W F(ω) (v)} ⊂ W F(ω) (u). This concludes the proof of (54). In the Roumieu case the proof is similar with minor changes. Now, it follows immediately that if (x, ξ) ∈ W F∗ (u), being u a solution of P u = 0, then every point of the kind (x0 , xn , ξ0 ), with xn ∈ R, belongs to W F∗ (u). Thereby, the straight line L is contained in W F∗ (u).
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
179
In the following, let ω be a non-quasianalytic weight function. Now, we show that there exists an ∗-ultradistribution u ∈ D∗ (Rn ) solution of P u = 0 with the prescribed wave front set W F∗ (u) = {(x, ξ) ∈ Rn × Rn \ {0} : x = (x0 , xn ), ξ = λξ0 , xn ∈ R, λ > 0}. (58) Fix (x0 , ξ0 ) ∈ Σ. Proceeding in a similar way as in Example 1 in [11], we can construct v˜ ∈ E∗ (Rn−1 ) satisfying v) W F∗ (˜ = {(x , ξ ) ∈ Rn−1 × Rn−1 \ {0} : x = 0, ξ = (0, . . . , 0, ξn−1 ), ξn−1 > 0}. By a linear change of variable and a translation, we then find v satisfying W F∗ (v) = {(x , ξ ) ∈ Rn−1 × Rn−1 \ {0} : x = x0 , ξ = λξ0 , λ > 0}. Set u = v ⊗ 1 ∈ D∗ (Rn ). Then, we have that P u = 0 and that, by (54), equality (58) is satisfied. Observe that an analogous result holds for the equation P u = f , with f ∈ C ∞ (Rn ). Indeed, every solution u ∈ D (Rn ) of P u = f can be written as xn u(x) = u0 (x) + f (x , t) dt, x = (x , xn ) ∈ Rn , 0
/ W F∗ (f ), then where u0 is a solution of P u = 0. If (x0 , ξ0 ) ∈ Σ and (x0 , ξ0 ) ∈ (x0 , ξ0 ) ∈ W F∗ (u) implies that (x0 , xn , ξ0 ) ∈ W F∗ (u) for xn in a suitable / W F∗ (f ) then there exists a interval I containing xn,0 . In fact, if (x0 , ξ0 ) ∈ neighborhood U of (x0 , ξ0 ) with empty intersection with W F∗ (f ). So, in a neighborhood of x0 the wave front set of u coincides with the wave front set of u0 . Acknowledgement The authors wish to thank professor Luigi Rodino for proposing the problem. The second author is indebted to professor Rodino for his hospitality and his helpful suggestions on mathematics during his different visits in Torino. He thanks also the members of the Department of Mathematics of the University of Lecce for their hospitality, where part of the present paper was written.
References [1] A. Beurling, Quasi-analiticity and general distributions, Lecture 4 and 5, AMS Summer Institute, Stanford, 1961. [2] P. Bolley, J. Camus, R´egularit´e Gevrey et it´er´es pour une classe d’op´erateurs hypoelliptiques, Comm. Partial Differential Equations 6 (1981), no. 10, 1057– 1110. [3] P. Bolley, J. Camus, C. Mattera, Analyticit´e microlocale et it´er´es d’operateurs, S´eminaire Goulaouic-Schwartz, 1978–79. [4] P. Bolley, J. Camus, L. Rodino, Hypoellipticit´e analytique-Gevrey et it´er´es d’op´erateurs, Rend. Sem. Mat. Univ. Politec. Torino 45 (1987), no. 3, 1–61 (1989).
180
Albanese, Jornet and Oliaro
IEOT
[5] C. Bouzar, L. Chaili, A Gevrey microlocal analysis of multi-anisotropic differential operators, Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), no. 3, 305–317. [6] J. Bonet, C. Fern´ andez, R. Meise, Characterization of the ω-hypoelliptic convolution operators on ultradistributions, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 2, 261–284. [7] J. Bonet, A. Galbis, and S. Momm, Nonradial H¨ ormander algebras of several variables and convolution operators, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2275–2291. [8] J. Bonet, R. Meise, S.N. Melikhov, A comparison of two different ways to define classes of ultradifferentiable functions, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 3, 425–444. [9] R.W. Braun, R. Meise, B.A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990) 206–237. [10] C. Fern´ andez, A. Galbis, Superposition in classes of ultradifferentiable functions, Publ. RIMS, Kyoto Univ. 42 (2006), 399–419. [11] C. Fern´ andez, A. Galbis, D. Jornet, Pseudodifferential operators of Beurling type and the wave front set, J. Math. Anal. Appl. 340 (2008) 1153–1170. [12] T. Heinrich, R. Meise, A support theorem for quasianalytic functionals, Math. Nachr. 280, No. 4, 364–387 (2007). [13] L. H¨ ormander, The Analysis of Linear Partial Differential Operators I, Springer, Berlin 1983. [14] L. H¨ ormander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971), 671–704. [15] M. Kato, Some results on potential scattering, Proc. Internat. Conf. on Functional Analysis and Related topics (Tokyo, 1969), 91–94, Univ. of Tokyo Press, Tokyo, 1970. [16] H. Komatsu, Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Tokyo Sec. IA 20 (1973), 25–105. [17] R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford Univ. Press, Oxford, 1997. [18] J. Peetre, Concave majorants of positive functions, Acta Math. Acad. Sci. Hungaricae 21 (1970), 327–333. [19] H.J. Petzsche, D. Vogt, Almost analytic extension of ultradifferentiable functions and the boundary values of holomorphic functions, Math. Ann. 267 (1984), 17–35. [20] L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Pub. (1993). [21] T. R¨ osner, Surjectivit¨ at partieller Differentialoperatoren auf quasianalytischen Roumieu-Klassen, Dissertation, D¨ usseldorf 1997. [22] D. Vogt, Topics in projective spectra of (LB)-spaces, in “Advances in the theory of Fr´echet spaces” (ed. T. Terzioglu), NATO Advanced Sciencie Institute, Series C, 287, 11–27. [23] J. Wengenroth, Acyclic inductive spectra of Fr´echet spaces, Studia Math., 120 (1996), 247–258.
Vol. 66 (2010)
Quasianalytic Wave Fronts Sets
181
[24] L. Zanghirati, Iterati di operatori e regolarit` a Gevrey microlocale anisotropa, Rend. Sem. Mat. Univ. Padova, 67 (1982). A. A. Albanese Dipartimento di Matematica “E. De Giorgi” Universit` a del Salento-Lecce Via Per Arnesano, P.O. Box 193 I-73100 Lecce Italy e-mail:
[email protected] D. Jornet Instituto Universitario de Matem´ atica Pura y Aplicada IUMPA-UPV Universidad Polit´ecnica de Valencia C/Camino de Vera, s/n E-46022 Valencia Spain e-mail:
[email protected] A. Oliaro Dipartimento di Matematica Universit` a di Torino Via Carlo Alberto, 10 I-10123 Torino Italy e-mail:
[email protected] Submitted: April 8, 2009. Revised: October 1, 2009.
Integr. Equ. Oper. Theory 66 (2010), 183–195 DOI 10.1007/s00020-010-1740-8 Published online January 26, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
A Levi-Civit´a Equation on Compact Groups and Nonabelian Fourier Analysis Jinpeng An and Dilian Yang Abstract. In this paper, we study the following Levi-Civit´ a equation m w(xy) + w(yx) = fi (x)gi (y) (LC) i=1
on a compact group G, where w, fi ’s, and gi ’s are continuous complex-valued functions to determine. Our main ingredient is (nonabelian) Fourier analysis on compact groups. We apply the Fourier transform to Eq. (LC) on the product group G × G so that we obtain its several equivalent operator equations. Using those equivalent equations, we derive some crucial properties of solutions to Eq. (LC). Consequently, Eq. (LC) with m ≤ 2 is completely solved. In particular, a Wilson type equation arising from and playing a central role in [4] is solved on compact groups. Mathematics Subject Classification (2010). Primary 39B52; Secondary 22C05. Keywords. Functional equation, nonabelian Fourier analysis.
1. Introduction Let G be a group, and let w and f be complex-valued functions on G. The equation w(xy) + w(yx) = 2f (x)w(y) + 2w(x)f (y) (1.1) came up recently in [4], when Davison studied a pre-d’Alembert equation, a generalization of the d’Alembert functional equation. (For the d’Alembert and related equations, refer to [1, 2, 7, 10].) It turns out that Eq. (1.1) plays a significant role in [4]. In particular, Davison [4] used Eq. (1.1) to study the Wilson functional equation g(xy) + g(xy −1 ) = 2g(x)f (y)
(1.2)
J. An is partially supported by the NSFC grant 10901005 and the FANEDD grant 200915. D. Yang is partially supported by an NSERC Discovery grant and a Start-up grant.
184
An and Yang
IEOT
on groups. Surprisingly, he proved that, if f satisfies the pre-d’Alembert equation with some other properties, the dimension of the space consisting of the solutions g of Eq. (1.2) is either 2 or 4. We should mention that [3, Corollary 6.3] implies that this holds true on compact groups without any restriction on f except that f = 0. According to our best knowledge, the above result has not been well-understood yet in general. We remark that Eq. (1.1) has not been solved although some nice properties of its solutions were given in [4]. In this paper, we study a much more general functional equation, which is called a (nonclassical ) Levi-Civit´ a equation, i.e., w(xy) + w(yx) =
m
fi (x)gi (y)
(LC)
i=1
on a compact group G, where w, f1 , ..., fm , g1 , ..., gm are the unknown complex-valued functions on G, and m ≥ 1 is an integer. When G is abelian, Eq. (LC) is nothing but the classical Levi-Civit´ a equation, which includes many important and well-known equations, such as the Cauchy equation, the Pexider equation, and trigonometric equations, as special cases. Refer to [1, 12] for more details. In [1, 12], the classical Levi-Civit´ a equation was studied on abelian groups. The classical one was solved on any locally compact group in [8], where only the role of w in the solution was emphasized. The argument there is slick, but unfortunately heavily depends on the form of the left hand side on the classical Levi-Civit´ a equation being w(xy), rather than w(xy) + w(yx). It seems that the argument there can not be adapted to the above nonclassical case. For more information about the classical Levi-Civit´a equation, refer to the recent work [11] and the references therein. Our main tool to solve Eq. (LC) is (nonabelian) Fourier analysis on compact groups. In our recent paper [3], Fourier analysis on a compact group G was used to solve a class of functional equations on G. For Eq. (LC), we find that it is more convenient (especially for the proof of Theorem 5.1(ii) below) to use Fourier analysis on the product group G × G. By regarding the functions on both sides of Eq. (LC) as functions on G × G and taking the Fourier transform on G × G, we obtain several equivalent operator equations which have very nice properties and are easy to analyze. After exploring these operator equations, we are able to obtain some crucial properties of solutions to Eq. (LC). Making use of these properties, we completely solve the LeviCivit´ a equation (LC) in the case of m ≤ 2. In particular, Eq. (1.1) is solved. Furthermore, we shall see the assumption that f satisfies the pre-d’Alembert equation given in [4] is redundant on compact groups. This paper is organized as follows. In Section 2, we construct two types of solutions of Eq. (LC). It will be shown later that those solutions are generic in certain cases. With the aid of the Fourier transform on G × G, in Section 3 Eq. (LC) is converted to its several equivalent operator equations. In Section 4 some useful lemmas are given, which are crucial for obtaining the main results in Section 5. The last section gives our main theorems, from which,
Vol. 66 (2010)
A Levi-Civitá Equation and Fourier Analysis
185
particularly, Eq. (1.1) in [4] is easily solved. Furthermore, it turns out that the solutions constructed in Section 2 indeed exhaust all of the cases of m ≤ 2. We should mention that Eq. (LC) is also meaningful even when G is a monoid, which is actually the context used in [4]. However, since, as mentioned above, we will use the Fourier transform, G needs to be a group, not just a monoid. We end the introduction with some notation. Throughout the paper, we suppose that G is a compact group, unless otherwise specified. Let dx be the normalized Haar measure on G, and L2 (G) the Hilbert space of all square integrable functions on G with respect to dx. We always assume that representations of G and solutions of an equation on G are continuous. Let Mm,n (C) denote the space of all m × n complex matrices. If m = n, then we simply write it as Mn (C). By diag(a1 , ..., an ), we mean the diagonal matrix with (i, i)-th entry ai . The notation Rn stands for the (row) space Cn whose vectors are regarded as row vectors.
2. Constructing solutions Consider the Levi-Civit´ a equation (LC) on a compact group G. We denote its solution by a row vector (w, f1 , . . . , fm , g1 , . . . , gm ) of continuous complexvalued functions. It is natural to focus on the case where both {fi }m i=1 and are linearly independent (over C). Such solutions are said to be non{gi }m i=1 degenerate. The aim of this section is to construct two types of solutions of Eq. (LC). As we shall see later, these solutions actually exhaust all solutions in certain cases. 2.1. Construction 1. Let π1 , . . . , πm be m 1-dimensional representations of G. We want to find solutions of Eq. (LC) which are linear combinations of π1 , . . . , πm . That is, we wish to find a matrix Q ∈ Mm,2m+1 (C) such that (w, f1 , . . . , fm , g1 , . . . , gm ) = (π1 , . . . , πm )Q
(2.1) t
is a solution of Eq. (LC). We write Q = (ε A B), where ε = (ε1 , . . . , εm ) ∈ Rm , and A, B ∈ Mm (C). Then Eq. (LC) becomes 2(π1 (x), . . . , πm (x)) diag(ε1 , . . . , εm )(π1 (y), . . . , πm (y))t = (π1 (x), . . . , πm (x))AB t (π1 (y), . . . , πm (y))t . Thus if AB t = 2 diag(ε1 , . . . , εm ), then the row vector of functions determined by (2.1) is a solution of Eq. (LC). Moreover, such a solution is nondegenerate iff π1 , . . . , πm are distinct and all components εi ’s of ε are nonzero. Indeed, it suffices to notice that π1 , ..., πm distinct is the same as π1 , ..., πm linearly independent. This is a direct consequence of Schur’s orthogonality relations ([5]).
186
An and Yang
IEOT
2.2. Construction 2. This construction applies only to the case of m = 2. We first prove a simple lemma. We denote sl(2, C) = {A ∈ M2 (C) : Tr A = 0}. Lemma 2.1. Let W ∈ sl(2, C). Then for any X, Y ∈ M2 (C) we have Tr(W XY + W Y X) = Tr(W X) Tr Y + Tr(W Y ) Tr X. Proof. It is easy to check that AB + BA = Tr(AB)I
for all A, B ∈ sl(2, C).
Substituting A and B with W and X − Tr(X)I/2, respectively, we get W X + XW = Tr(W X)I + Tr(X)W. In order to obtain the desired formula, it suffices to multiply Y from the right at both sides and then take the trace. Now let π : G → U (2) be a representation, and χπ the character of π: for all x ∈ G. 0 Let W ∈ sl(2, C), and C, D ∈ M2 (C) with CDt = 1 the functions on G defined by χπ (x) = Tr π(x)
w(x) = Tr(W π(x)),
1 . We claim that 0
(f1 , f2 , g1 , g2 ) = (w, χπ )(C D)
give a solution of Eq. (LC) with m = 2. Indeed, by Lemma 2.1, we have f1 (x)g1 (y) + f2 (x)g2 (y) = (f1 (x), f2 (x))(g1 (y), g2 (y))t = (w(x), χπ (x))CDt (w(y), χπ (y))t = Tr(W π(x)) Tr π(y) + Tr(W π(y)) Tr π(x) = Tr(W π(xy) + W π(yx)) = w(xy) + w(yx). Furthermore, one can check that if π is irreducible, the solution (w, f1 , f2 , g1 , g2 ) constructed above is nondegenerate iff W = 0. In fact, this is a straight-forward consequence of Schur’s orthogonality relations, and the fact that both {f1 , f2 } and {g1 , g2 } are linearly independent iff so is {w, χπ }.
3. Operator Equations via the Fourier transform 3.1. The Fourier transform In this subsection, we briefly review some background in Fourier analysis on compact groups, which will be used later. For more information, refer to [5, Chapter 5] and [6, Section 27]. ˆ a complete set of mutually Let G be a compact group. We denote by G ˆ let inequivalent unitary irreducible matrix representations of G. For π ∈ G,
Vol. 66 (2010)
A Levi-Civitá Equation and Fourier Analysis
187
dπ stand for the dimension of the representation space of π. According to ˆ [6, Theorem 27.43], we may choose G × G = {π ⊗ ρ : π, ρ ∈ G}. For f ∈ L2 (G), the Fourier transform of f is defined by ˆ f (x)π(x)−1 dx for all π ∈ G. fˆ(π) = dπ G
So fˆ(π) ∈ Mdπ (C). Note that, for convenience, our definition is different from the one in [5] by a factor dπ . Then the Fourier inversion formula is given by Tr fˆ(π)π(x) for all x ∈ G. f (x) = ˆ π∈G
Note that if f is continuous, the above series converges pointwise. We denote ˆ : fˆ(π) = 0} and supp{fˆ1 , ..., fˆm } = m supp(fˆi ). supp(fˆ) = {π ∈ G i=1 3.2. Operator equations equivalent to Eq. (LC) In this subsection, with the aid of the Fourier transform, we convert Eq. (LC) to some equivalent forms, which will be used later to explore some crucial properties of solutions to Eq. (LC). Let n Eij ⊗ Eji ∈ Mn (C) ⊗ Mn (C), Pn = i,j=1
where Eij , as usual, is the matrix unit whose (i, j)-entry is 1 and others 0. It is easy to verify that Pn has the following properties: Pn2 = In ⊗ In ,
Pn (A ⊗ B)Pn = B ⊗ A,
where A, B ∈ Mn (C). The importance of Pn for us lies in the following lemma. ˆ we have Lemma 3.1. For π, ρ ∈ G, 0, −1 dπ π(x) ⊗ ρ(x) dx = Pdπ , G
π= ρ; π = ρ.
Proof. By Schur’s orthogonality relations, we have dρ dπ −1 dπ π(x) ⊗ ρ(x) dx = dπ πij (x)ρlk (x)Eij ⊗ Ekl dx G
i,j=1 k,l=1
G
⎧ ⎪ π = ρ; ⎪ ⎨0, dπ dπ = ⎪ δil δjk Eij ⊗ Ekl = Pdπ , π = ρ. ⎪ ⎩ i,j=1 k,l=1
This proves the lemma. Proposition 3.2. Eq. (LC) is equivalent to m Pdπ (w(π) ˆ ⊗ Idπ ) + (w(π) ˆ ⊗ Idπ ) Pdπ , ˆ fi (π) ⊗ gˆi (ρ) = 0, i=1
π = ρ; π = ρ.
(3.1)
188
An and Yang
IEOT
Proof. The idea of the proof is simple: Regard both sides of Eq. (LC) as functions on G × G, and then take their Fourier transforms on G × G. The value of the Fourier transform of the function (x, y) → w(xy) + w(yx) at π ⊗ ρ is given by (w(xy) + w(yx))π(x)−1 ⊗ ρ(y)−1 dxdy dπ dρ G×G = d π dρ w(x)(π(y)π(x)−1 ) ⊗ ρ(y)−1 dxdy G×G + dπ dρ w(x)(π(x)−1 π(y)) ⊗ ρ(y)−1 dxdy G×G = dπ dρ π(y) ⊗ ρ(y)−1 dy w(x)π(x)−1 ⊗ Idρ dx G G −1 w(x)π(x) ⊗ Idρ dx π(y) ⊗ ρ(y)−1 dy + dπ dρ G G 0, π = ρ, = Pdπ (w(π) ˆ ⊗ Idπ ) + (w(π) ˆ ⊗ Idπ )Pdπ , π = ρ, where Lemma 3.1 was used to get the last equality. The value of the Fourier transform of the function fi ⊗ gi : (x, y) → fi (x)gi (y) at π ⊗ ρ is fi (x)gi (y)π(x)−1 ⊗ ρ(y)−1 dxdy fi ⊗ gi (π ⊗ ρ) =dπ dρ G×G = d π dρ fi (x)π(x)−1 dx ⊗ gi (y)ρ(y)−1 dy G
G
= fˆi (π) ⊗ gˆi (ρ). The proposition now immediately follows from the above calculations and the Fourier inversion formula. In order to obtain our main results, we need to convert the operator equation (3.1) of Eq. (LC) to another more convenient one. To this end, we need to introduce more notation first. For any n ∈ N, define a linear isomorphism R : Mn (C) → Rn2 by R(A) = (v1 · · · vn ), where vi (i = 1, ..., n) is the i-th row vector of A ∈ Mn (C). Then for any m, n, the linear mapping Mat : Mm (C) ⊗ Mn (C) → Mm2 ,n2 (C) determined by Mat(A ⊗ B) = R(A)t R(B), is also a linear isomorphism, where A ∈ Mm (C), B ∈ Mn (C). Finally, we define for any n a linear mapping Φ : Mn (C) → Mn2 (C) by Φ(W ) = Mat (Pn (W ⊗ In ) + (W ⊗ In )Pn ) . Keeping the notation as above and by Proposition 3.2, we have
Vol. 66 (2010)
A Levi-Civitá Equation and Fourier Analysis
189
Corollary 3.3. Eq. (LC) is equivalent to
Φ(w(π)), ˆ R(fˆi (π)) R(ˆ gi (ρ)) = 0, i=1
m
t
π = ρ; π = ρ.
(3.2)
ˆ = {π1 , π2 , . . .}, then (3.2) can be rewritten as If G m
(R(fˆi (π1 )) · · · )t (R(ˆ gi (π1 )) · · · ) = diag(Φ(w(π ˆ 1 )), . . .),
(3.3)
i=1
which is an equation on matrices of infinite size.
4. Some auxiliary lemmas As we shall see later, the mapping Φ defined in Section 3 plays an important role in analyzing properties of solutions to Eq. (LC). We take a closer look at it in this section. Lemma 4.1. For any 0 = W ∈ Mn (C), we have rank Φ(W ) ≥ n. In particular, Φ is injective. Proof. Set W = (Wij ). Direct computations give Pn (W ⊗ In ) + (W ⊗ In )Pn = =
=
n
n
r,s=1 q=1 n
n
(Esr W + W Esr ) ⊗ Ers
r,s=1
Wrq Esq +
n
Wps Epr
⊗ Ers
p=1
(δps Wrq + δrq Wps )Epq ⊗ Ers .
p,q,r,s=1
But Mat(Epq ⊗ Ers ) = R(Epq )t R(Ers ) = E(p−1)n+q,(r−1)n+s . So the ((p − 1)n + q, (r − 1)n + s)-th entry of Φ(W ) is Φ(W )(p−1)n+q,(r−1)n+s = δps Wrq + δrq Wps .
(4.1)
If W is not diagonal, then there are p, s ∈ Nn = {1, ..., n} with p = s s, from (4.1) we have such that Wps = 0. We fix such a pair p, s. As p = Φ(W )(p−1)n+q,(r−1)n+s = δrq Wps . Let I = {(p − 1)n + q : q ∈ Nn }
and J = {(r − 1)n + s : r ∈ Nn }.
Then the submatrix {(Φ(W ))i,j : i ∈ I, j ∈ J } of Φ(W ) has exactly one nonzero entry in each row and each column, and so it has rank n. This implies that rank Φ(W ) ≥ n. If W is diagonal, then it follows from (4.1) that Φ(W )(p−1)n+q,(r−1)n+s = δps δqr (Wpp + Wqq ).
190
An and Yang
IEOT
This implies that rank Φ(W ) = |{(p, q) : Wpp + Wqq = 0}|. Let J = {(p, p) : Wpp = 0} ∪ {(p, q) : Wpp = 0, Wqq = 0}. Then (p, q) ∈ J implies that Wpp + Wqq = 0. If k(≥ 1) is the number of nonzero diagonal entries in W , then |J| = k(n − k + 1) ≥ n. Hence we also have rank Φ(W ) ≥ |J| ≥ n. The proof of the following lemma is an easy exercise of linear algebra, and left to the reader. m Lemma 4.2. Let α1 , . . . , αm , β1 , . . . , βm ∈ RN . Then rank ( i=1 αit βi ) ≤ m. m The equality holds iff both {αi }m i=1 and {βi }i=1 are linearly independent. In this case, for u1 , . . . , um , v1 , . . . , vm ∈ RN , m i=1
uti vi
=
m
αit βi
⇐⇒
ui =
i=1
m
cji αj , vi =
j=1
m
dji βj
j=1
for some matrices C = (cij ), D = (dij ) ∈ Mm (C) with CDt = Im . For n = 2, the following lemma gives some very nice characterizations of rank Φ(W ) = 2, which is the key to completely solving Eq. (LC) in the case of m = 2. Lemma 4.3. Let 0 = W ∈ M2 (C). Then rank Φ(W ) = 2 ⇐⇒ W ∈ sl(2, C) ⇐⇒ Φ(W ) = R(W )t R(I) + R(I)t R(W ). Proof. From (4.1) we see that ⎛ 2W11 ⎜ W12 Φ(W ) = ⎜ ⎝ W21 0
W12 0 Tr W W12
W21 Tr W 0 W21
⎞ 0 W12 ⎟ ⎟. W21 ⎠ 2W22
Suppose that rank Φ(W ) = 2. Let vi be the i-th row of Φ(W ). If W ∈ sl(2, C), then v2 and v3 are linearly independent. Hence v1 − v4 = (2W11 , 0, 0, −2W22 ) must be a linear combination of v2 and v3 . This implies W11 = W22 = 0, a contradiction. So we have W ∈ sl(2, C). Conversely, if W ∈ sl(2, C), it is straightforward to verify that Φ(W ) = R(W )t R(I) + R(I)t R(W ), which, by Lemmas 4.1 and 4.2, is a matrix of rank 2.
Vol. 66 (2010)
A Levi-Civitá Equation and Fourier Analysis
191
5. Main theorems 5.1. The general case The first main theorem in this section provides some properties of solutions to the Levi-Civit´ a equation (LC). Theorem 5.1. Suppose that (w, f1 , . . . , fm , g1 , . . . , gm ) is a nondegenerate solution of Eq. (LC) on a compact group G. Then we have (i) supp w ˆ = supp{fˆ1 , . . . , fˆm } = supp{ˆ g1 , . . . , gˆm }; d ≤ m. In particular, supp w, ˆ supp fˆi and supp gˆi (i = (ii) π π∈supp w ˆ 1, ..., m) all 2are finite; (iii) π∈supp w ˆ dπ ≥ m. ˆ If π ∈ supp w, Proof. (i) Let π ∈ G. ˆ then from (3.2) and the injectivity of Φ, ˆ ˆ ˆ ⊆ supp{fˆ1 , . . . , fˆm }. we see that π ∈ supp{f1 , . . . , fm }, so supp w ˆ we have If π ∈ / supp w, ˆ then, by (3.2), for all 1 ≤ k ≤ d2π and ρ ∈ G, m
R(fˆi (π))k gˆi (ρ) = 0,
i=1
where R(fˆi (π))k is the k-th component of R(fˆi (π)). By the Fourier inversion m m ˆ formula, we get i=1 R(fi (π))k gi = 0. Since {gi }i=1 is linearly indepenˆ ˆ dent, we have R(fi (π))k = 0. Thus fi (π) = 0 for all 1 ≤ i ≤ m, and so ˆ = supp{fˆ1 , . . . , fˆm }. Similarly, π ∈ / supp{fˆ1 , . . . , fˆm }. This proves supp w we have supp w ˆ = supp{ˆ g1 , . . . , gˆm }. This proves (i). ˆ from (3.3) we have (ii) For any distinct π1 , . . . , πk ∈ supp w, m
(R(fˆi (π1 )) · · · R(fˆi (πk )))t (R(ˆ gi (π1 )) · · · R(ˆ gi (πk )))
i=1
ˆ k ))). = diag(Φ(w(π ˆ 1 )), . . . , Φ(w(π By Lemmas 4.1 and 4.2, the matrix on the right hand side of the above k equation has rank at least j=1 dπj , while the one on the left hand side has k rank at most m. Hence j=1 dπj ≤ m. Now (ii) follows from the arbitrariness of πj . The rest of (ii) is obvious. (iii) Let ˆ 1 ≤ i, j ≤ dπ }. H = span{πij : π ∈ supp w, By (i) and the Fourier inversion formula, we have {f1 , ..., fm } ⊆ H.
2 Then by Schur’s orthogonality relations, we have dim H = π∈supp w ˆ dπ m (which is finite from (ii)). Now the linear independence of {fi }i=1 yields m ≤ π∈supp wˆ d2π . It immediately follows from Theorem 5.1 that one can strengthen the above results in the two extremal cases, as shown below.
192
An and Yang
IEOT
Corollary 5.2. Suppose that (w, f1 , . . . , fm , g1 , . . . , gm ) is a nondegenerate solution of Eq. (LC) on a compact group G. Then we have (i) dπ = m for some π ∈ supp w ˆ =⇒ supp w ˆ = {π}; (ii) dπ = 1 for all π ∈ supp w ˆ ⇐⇒ | supp w| ˆ = m. Corollary 5.3. If | supp w| ˆ = m, then any nondegenerate solution of Eq. (LC) is given by Construction 1 in Section 2. More precisely, there are distinct representations π1 , ..., πm : G → U (1) and Q ∈ Mm,2m+1 (C) such that (w, f1 , . . . , fm , g1 , . . . , gm ) = (π1 , . . . , πm )Q,
(5.1)
where Q = (ε A B) for some ε = (ε1 , . . . , εm ) ∈ Rm , A, B ∈ Mm (C) with AB t = 2 diag(ε1 , . . . , εm ). t
Proof. By Theorem 5.1 (i) and Corollary 5.2 (ii), we have that dπ = 1 for all π ∈ supp w ˆ and that supp w ˆ = supp{fˆ1 , ..., fˆm } = supp{gˆ1 , ..., gˆm } = {π1 , ..., πm }. Let ˆ i ), A = (fˆj (πi )), B = (ˆ gj (πi )). εi = w(π Then the Fourier inversion formula implies that (5.1) holds true. From (3.3) we see that ε, A, B satisfy all required conditions. Remark 5.4. Although, by Theorem 5.1, we have nice estimates on supp w ˆ and the dimensions of representation spaces in supp w, ˆ it is far from completely solving Eq. (LC). It is still unsolved even for the extremal case (i) of Corollary 5.2, because here we are not able to obtain a result analogous to Lemma 4.3. However, we can completely solve Eq. (LC) for the cases of m ≤ 2. We are devoted to considering these cases in the rest of the paper. 5.2. The case m = 1 For this case, the general solution is of the form obtained from Construction 1 in Section 2. Corollary 5.5. Let (w, f, g) be a nondegenerate solution of the equation w(xy) + w(yx) = f (x)g(y)
(5.2)
on a compact group G. Then there are a representation π : G → U (1) and nonzero constants a, b ∈ C such that ab π, aπ, bπ . (5.3) (w, f, g) = 2 Proof. By Theorem 5.1 (i), (ii), and Corollary 5.2 (i), we see that supp w ˆ= ˆ ˆ supp f = supp gˆ = {π} for some π ∈ G with dπ = 1. Then the theorem follows from the Fourier inversion formula. Actually, the general solution (without any regularity) of Eq. (5.2) on any (not necessarily compact) group G is in fact of the form in Corollary 5.5. It is probably known in the literature. But we are lack of a reference; we give a short proof here for completeness.
Vol. 66 (2010)
A Levi-Civitá Equation and Fourier Analysis
193
Lemma 5.6. Any nondegenerate solution (without any regularity) of Eq. (5.2) on an arbitrary group G is of the form (5.3) for some homomorphism π : G → C∗ and nonzero complex numbers a, b. Proof. Letting x = e (resp. y = e) in Eq. (5.2) gives w(y) = f (e) 2 g(y) (resp. w(x) = g(e) f (x)). Since w ≡ 0, we have f (e) = 0 and g(e) = 0. Thus 2 g(e) f (x) g(x) = f (e) f (x). Put π(x) = f (e) . It is now easy to check that π(xy) + π(yx) = 2π(x)π(y). From [9], π is indeed a homomorphism from G to C∗ .
5.3. The case m = 2 We are now in a position to obtain the general solution to Eq. (LC) with m = 2. It turns out that every solution has to be one of the forms constructed in Section 2. Theorem 5.7. Suppose that (w, f1 , f2 , g1 , g2 ) is a nondegenerate solution of the equation w(xy) + w(yx) = f1 (x)g1 (y) + f2 (x)g2 (y)
(5.4)
on a compact group G. Then either (i) there are two distinct homomorphisms π1 , π2 : G → U (1) and Q = (ε A B) ∈ M2,5 (C) with AB t = 2 diag(ε1 , ε2 ) such that (w, f1 , f2 , g1 , g2 ) = (π1 , π2 )Q, where ε = (ε1 , ε2 ) , and A, B ∈ M2 (C); or (ii) there exist an irreducible representation π : G → U (2), 0 = W 0 1 ∈ sl(2, C), C, D ∈ M2 (C) with CDt = such that 1 0 t
w(x) = Tr(W π(x)),
(f1 , f2 , g1 , g2 ) = (w χπ )(C D).
Proof. Applying Theorem 5.1 to the case of m = 2, we see that either supp w ˆ = supp{fˆ1 , fˆ2 } = supp{gˆ1 , gˆ2 } = {π1 , π2 } ˆ with dπ = dπ = 1; or for some distinct π1 , π2 ∈ G 1 2 supp w ˆ = supp{fˆ1 , fˆ2 } = supp{gˆ1 , gˆ2 } = {π} ˆ with dπ = 2. for some π ∈ G For the former case, the conclusion follows directly from Corollary 5.3. For the latter case, if we denote W = w(π), ˆ it follows from Corollary 3.3 that 2 R(fˆi (π))t R(ˆ gi (π)) = Φ(W ). i=1
By Lemmas 4.1 and 4.2, both sides of the above equation have rank 2. Then Lemma 4.3 implies 0 = W ∈ sl(2, C)
and
Φ(W ) = R(W )t R(I) + R(I)t R(W ).
194
An and Yang
IEOT
It now follows from the last part of Lemma 4.2 that there are C = (cij ), ˜ = (dij ) ∈ M2 (C) with C D ˜ t = I such that D R(fˆi (π)) = c1i R(W ) + c2i R(I),
R(gˆi (π)) = d1i R(I) + d2i R(W ).
Hence we have fˆi (π) = c1i W + c2i I, gˆi (π) = d1i I + d2i W 0 1 ˜ D. Then from the Fourier inversion foras R is injective. Let D = 1 0 mula, we see that the solution is as in case (ii) of the theorem. As a consequence of Theorem 5.7, we can easily solve Eq. (1.1), which plays an important role in [4]. However, in [4], Eq. (1.1) is only solved in the context where f is given to be a solution of the pre-d’Alembert equation. In Theorem 5.8 below, we derive the solution formulas for Eq. (1.1) without assuming that f satisfies the pre-d’Alembert equation, but in contrast to [4] we require that the group G is compact. In the following theorem, we discard the trivial solutions: w = 0, and f arbitrary. Theorem 5.8. Any nontrivial solution (w, f ) of the equation w(xy) + w(yx) = 2f (x)w(y) + 2w(x)f (y), on a compact group G has one of the following forms: (i) There are a representation π : G → U (1) and 0 = a ∈ C such that 1 w(x) = aπ(x) and f (x) = π(x). 2 (ii) There are two distinct representations π, π : G → U (1) and 0 = a ∈ C such that 1 w(x) = a(π(x) − π (x)) and f (x) = (π(x) + π (x)). 2 (iii) There are an irreducible representation π : G → U (2) and 0 = W ∈ sl(2, C) such that 1 w(x) = Tr(W π(x)) and f (x) = Tr(π(x)). 2 Proof. If w and f are linearly dependent, a straightforward calculation using Corollary 5.5 shows that (w, f ) is of the form in (i). Otherwise, (w,2f,w,w,2f ) is a nondegenerate solution of Eq. (5.4). Applying Theorem 5.7 to the case where f1 = g2 = 2f and f2 = g1 = w, we obtain (ii) and (iii). By Theorem 5.8, one can easily see that f satisfies the pre-d’Alembert equation defined in [4] automatically. It should be mentioned that f in [4] is normalized, i.e., f (e) = 1. So only f ’s given in (ii) or (iii) of Theorem 5.8 have such a property. Acknowledgment We are very grateful to the anonymous referee for his/her careful reading and many valuable comments.
Vol. 66 (2010)
A Levi-Civitá Equation and Fourier Analysis
195
References [1] J. Acz´el, Lectures on functional equations and their applications, Academic Press, New York-London, 1966. [2] J. Acz´el and J. Dhombres, Functional equations in several variables, Cambridge University Press, Cambridge, 1989. [3] J. An and D. Yang, Nonabelian harmonic analysis and functional equations on compact groups, preprint, 2008. [4] T. M. K. Davison, D’Alembert’s functional equation on topological monoids, preprint, 2008. [5] G. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton, FL, 1995. [6] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. II, Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Berlin, 1970. [7] Pl. Kannappan, Functional equations and inequalities with applications, Springer Monographs in Mathematics. Springer, New York, 2009. [8] E. V. Shulman, Group representations and stability of functional equations, J. London Math. Soc. 54 (1996), 111–120. [9] H. Stetkær, On multiplicative maps, Semigroup Forum 63 (2001), 466–468. [10] H. Stetkær, Functional equations on groups—recent results, presented in an invited talk at the 42nd International Symposium on Functional Equations, Opava, Czech Republic, 2004. [11] H. Stetkær, Functional equations and matrix-valued spherical functions, Aequationes Math. 69 (2005), 271–292. [12] L. Sz´ekelyhidi, Convolution type functional equations on topological abelian groups, World Scientific Publishing Co., Inc., Teaneck, NJ, 1991. Jinpeng An LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China e-mail:
[email protected] Dilian Yang Department of Mathematics & Statistics, University of Windsor, Windsor, ON, N9B 3P4, Canada e-mail:
[email protected] Submitted: October 1, 2009. Revised: November 6, 2009.
Integr. Equ. Oper. Theory 66 (2010), 197–214 DOI 10.1007/s00020-010-1741-7 Published online February 5, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Summability Properties for Multiplication Operators on Banach Function Spaces O. Delgado and E. A. S´anchez P´erez Abstract. Consider a couple of Banach function spaces X and Y over the same measure space and the space X Y of multiplication operators from X into Y . In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X Y . At this end, using the “generalized K¨ othe duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with x ∈ X and y ∈ Y . Mathematics Subject Classification (2010). Primary 46E30; Secondary 47B38. Keywords. Banach function spaces, K¨ othe dual and generalized dual spaces, product spaces, multiplication operators, summability properties.
1. Introduction Let (Ω, Σ, µ) be a fixed σ-finite measure space and consider a couple of Banach function spaces X and Y related to µ. In this paper we introduce a technique based on topological products of Banach function spaces for analyzing the summability properties of the multiplication operators from X into Y . For the definition of such topologies we use the so called generalized duality for Banach function spaces, which was originally studied by Maligranda and Persson in [9]. The Y -dual space of X, denoted by X Y , is the space of measurable functions g defining a multiplication operator (also denoted by g) from X into Y , that is, g, x = gx ∈ Y for all x ∈ X. This notion includes 1 the classical K¨othe dual (or associate) space X = X L . The first author thanks the support by UPV (PAID-06-08 Ref. 3093), MEC (TSGD-08 and D.G.I. #MTM2006-13000-C03-01) (Spain) and FEDER. The second author thanks the support by UPV (PAID-06-08 Ref. 3093), MEC (D.G.I. #MTM2006-11690-C02-01) (Spain) and FEDER.
198
Delgado and Sánchez Pérez
IEOT
Given 1 ≤ p < ∞ and Z another Banach function space related to µ, our goal is to characterize when a multiplication operator g ∈ X Y is what we call (p, Z)-summing, that is, when there exists K > 0 such that for every x1 , ..., xn ∈ X, n n 1/p 1/p gxi pY ≤ K sup f xi pZ . (1.1) i=1
f ∈BX Z
i=1
An operator satisfying this kind of inequality is interesting as it transforms sequences which are summable in a certain weak sense into strongly summable sequences. Some relevant well known geometric and topological properties involving vector norm inequalities for operators can be written as particular examples of this general class of inequalities when Z is chosen adequately. For instance, in the case when X is order continuous, the positive p-summing multiplication operators coincide with the (p, L1 )-summing ones, or in the case when X is order semi-continuous and p-convex with constant 1, the pconcave multiplication operators coincide with the (p, Lp )-summing ones (see Section 4). Inspired in part by the representation theory of operator ideals as dual spaces of topological tensor products (see for instance [5]), we show that the subspace of X Y of all (p, Z)-summing multiplication operators can be described as the K¨ othe dual of a product space with a particular normed topology given by a certain dp,Z -norm. Actually, there is an abuse of the notation as the “dp,Z -product space” of X and Y consists of infinite sums of products of the type xy with x ∈ X and y ∈ Y . As a consequence of the above description, some factorization theorems for multiplication operators which play a central role in the theory of the Banach function spaces (Reisner and Maurey-Rosenthal’s theorems) provide sufficient conditions for (p, Z)summability type properties to hold. The paper is organized as follows. Section 2 contains the definitions and some results concerning product spaces which will be necessary for our work. In Section 3 we introduce the dp,Z -product spaces which will allow us to characterize in Section 4 the (p, Z)-summing multiplication operators. Moreover, in Section 4 we show conditions on X, Y and Z guaranteeing that every multiplication operator from X into Y is (p, Z)-summing. Examples in which these conditions hold are provided in Section 5 by using the already quoted factorization theorems.
2. Preliminaries and first results Let (Ω, Σ, µ) be a fixed σ-finite measure space and denote by L0 the space of all (a.e. classes of) real measurable functions defined on Ω. A Banach function space is a Banach space X ⊂ L0 with norm · X , satisfying that if f ∈ L0 , g ∈ X and |f | ≤ |g| a.e. then f ∈ X and f X ≤ gX . Note that in this case, X is a Banach lattice for the pointwise a.e. order. A Banach function
Vol. 66 (2010)
Summability Properties for Multiplication Operators
199
space is order continuous if every order bounded increasing sequence is norm convergent. A Banach function space X has the Fatou property if for every sequence (fn ) ⊂ X such that 0 ≤ fn ↑ f a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X . A Banach function space X is order semicontinuous if f, fn ∈ X with 0 ≤ fn ↑ f a.e. implies fn X ↑ f X . Of course, if a Banach function space X is order continuous or has the Fatou property, then X is order semi-continuous. For issues related to Banach function spaces, see [12, Ch. 15] considering the function norm ρ defined as ρ(f ) = f X if f ∈ X and ρ(f ) = ∞ in other case. Given two Banach function spaces X and Y , the Y -dual space of X is defined by X Y = {h ∈ L0 : hf ∈ Y for all f ∈ X}, that is, the space of functions in L0 defining a continuous linear operator from X into Y . The continuity follows from the fact that every positive linear operator between Banach lattices is continuous, see [6, p. 2]. The space X Y is a Banach function space with norm hX Y = sup hf Y , f ∈BX
for h ∈ X Y ,
if and only if X is saturated, that is, there is no A ∈ Σ with µ(A) > 0 such that f χA = 0 a.e. for all f ∈ X, see [9, Proposition 2] and [2, p. 3]. The saturation property is equivalent to the following one: for all A ∈ Σ with µ(A) > 0 there exists B ∈ Σ such that B ⊂ A, µ(B) > 0 and χB ∈ X. This is also equivalent to X having a weak unit, i.e. a function g ∈ X such that g > 0 a.e. As we have already noted, the classical K¨othe dual space X coincide with 1 X L , the L1 -dual space of X. In this case, X is saturated whenever X is so. However, the generalized dual X Y of X may be non saturated even if X is saturated. For these and other comments about saturation involving the spaces X Y see [2]. Let us introduce now the product spaces which will be the basic setting for defining the dp,Z -product spaces in Section 3. 0 Definition 2.1. The π–product space XπY is the space of functions z ∈ L such that |z| ≤ i≥1 |xi yi | a.e. for some sequences (xi ) ⊂ X and (yi ) ⊂ Y satisfying i≥1 xi X yi Y < ∞. For z ∈ XπY , consider π(z) = inf xi X yi Y , i≥1
where the infimum is taken over all sequences (xi ) ⊂ X and (yi ) ⊂ Y such that |z| ≤ i≥1 |xi yi | a.e. and i≥1 xi X yi Y < ∞. The space XπY is clearly an ideal of L0 and π(v) ≤ π(z) whenever |v| ≤ |z| a.e. It can be routinely checked that π is a seminorm. However, there are cases in which π is not a norm. Example. Let [0, 1], B([0, 1]), m be the fixed measure space, where B([0, 1]) is the σ–algebra of all Borel sets of [0, 1] and m is the Lebesgue measure on
200
Delgado and Sánchez Pérez
IEOT
i 1 [0, 1], and consider the product space L πL2 . For the intervals Ani = [ i−1 n , n] n with i = 1, ..., n, we have that χ[0,1] ≤ i=1 χAni and
π(χ[0,1] ) ≤
n
χAni L1 χAni L2 =
i=1
n 1 1 12 1 =√ . n n n i=1
Then, taking limit as n → ∞ we have that π(χ[0,1] ) = 0, while χ[0,1] > 0. So, π is not a norm. Saturation conditions will be crucial for π to be a norm under which XπY is a Banach function space. We write “X →c Y ” (“X →i Y ”) if X is continuously contained in Y with xY ≤ c xX ( xY = xX ) for all x ∈ X. If X = Y with equal norms, we write X ≡ Y . Proposition 2.2. The following conditions are equivalent: (a) XπY is a saturated Banach function space. (b) X, Y and X Y are saturated. Moreover, if (a)-(b) holds, we have that
(i) XπY →1 (X Y ) , (ii) (XπY ) ≡ X Y ≡ Y X . Proof. (a) ⇒ (b) Let us see that X is saturated. If this is not the case, there exists A ∈ Σ with µ(A) > 0 such that xχA = 0 a.e. for all x ∈ X. Since XπY is saturated we can take B ∈ Σ such that B ⊂ A, µ(B) > 0 and χB ∈ XπY . Let (xi ) ⊂ X and (yi ) ⊂ Y be such that i≥1 xi X yi Y < ∞ and χB ≤ i≥1 |xi yi | a.e. Then, χB = χB · χA ≤ i≥1 |xi χA yi | = 0 a.e. and so µ(B) = 0, which is a contradiction. Similarly, Y is saturated. Then, the spaces Y and X Y are Banach function spaces. Let us prove that X Y ≡ (XπY ) and so we will have that X Y is saturated as (XπY ) is so. Let h ∈ X Y . Given z ∈ XπY and (xi ) ⊂ X, (yi ) ⊂ Y with i≥1 xi X yi Y < ∞ such that |z| ≤ i≥1 |xi yi | a.e., using the monotone convergence theorem, we have hxi Y yi Y ≤ hX Y xi X yi Y . |hxi yi |dµ ≤ |hz|dµ ≤
i≥1
i≥1
i≥1
Then, |hz|dµ ≤ hX Y · π(z). So, h ∈ (XπY ) and h(XπY ) ≤ hX Y . Consider now h ∈ (XπY ) . For every x ∈ X and y ∈ Y , we have that xy ∈ XπY and so hxy ∈ L1 (µ). Then, hx ∈ Y for every x ∈ X, that is, h ∈ X Y . Moreover, since π(xy) ≤ xX yY for x ∈ X and y ∈ Y , it follows hX Y = sup sup |hxy|dµ ≤ sup |hz|dµ = h(XπY ) . x∈BX y∈BY
z∈BXπY
(b) ⇒ (a). Note that from the hypothesis (X Y ) is a Banach function space. Let us see that XπY →1 (X Y ) . Given z ∈ XπY , (xi ) ⊂ X, (yi ) ⊂ Y
Vol. 66 (2010)
Summability Properties for Multiplication Operators
201
such that i≥1 xi X yi Y < ∞ and |z| ≤ i≥1 |xi yi | a.e., for every h ∈ XY , hxi Y yi Y ≤ hX Y xi X yi Y |zh|dµ ≤ |hxi yi |dµ ≤
i≥1
i≥1
i≥1
and so |zh|dµ ≤ hX Y · π(z). Then, z ∈ (X Y ) and z(X Y ) ≤ π(z). Hence, π(z) = 0 implies z = 0 a.e. That is, π is a norm on XπY . Let (zn )n≥1 ⊂ XπY be such that zn ≥ 0 and n≥1 π(zn ) < ∞. Let us prove that n≥1 zn ∈ XπY (i.e. XπY has the Riesz-Fischer property) and so XπY will be complete, see [12, Ch.15, §64, Theorem 2]. Given ε > 0 there exist (xnj )j ⊂ X and (yjn )j ⊂ Y such that zn ≤ j≥1 |xnj yjn | a.e. and ε n n n n j≥1 xj X yj Y ≤ π(zn ) + 2n . So, n≥1 zn ≤ n≥1 j≥1 |xj yj | a.e. and xnj X yjn Y ≤ π(zn ) + ε < ∞. n≥1 j≥1
Note that z := we have that
n≥1 zn
n≥1
< ∞ a.e., since taking h ∈ X Y such that h > 0 a.e.,
|hz|dµ ≤ hX Y
xnj X yjn Y < ∞
n≥1 j≥1
which implies that |hz| < ∞ a.e. Then, z ∈ XπY and π(z) ≤ n≥1 π(zn ). Therefore, it follows that XπY is a Banach function space. Moreover, given A ∈ Σ with µ(A) > 0, since X is saturated, there exists B ∈ Σ such that B ⊂ A, µ(B) > 0 and χB ∈ X. Since Y is also saturated, there exists C ∈ Σ such that C ⊂ B, µ(C) > 0 and χC ∈ Y . Then, χC = χB · χC ∈ XπY and so XπY is saturated. Suppose (a)-(b) holds. Claim (i) has been proved in (b) ⇒ (a). The first equivalence in (ii) has been obtained in (a) ⇒ (b). For the second equivalence, just note that XπY ≡ Y πX. The proof of the completeness of XπY in the previous proposition can also be obtained as a consequence of the fact that XπY is saturated. This can be found in [8], where these notions are developed in the general frame work of the function norms. The space defined in a similar way as XπY by taking finite sums has been independently studied in [11] obtaining similar results, although for the completeness the pointwise product BX · BY of the unit balls of X and Y is required to be convex. Remark 2.3. Suppose (a)-(b) in Proposition 2.2 holds.Then, if (xi ) ⊂ X and (yi ) ⊂ Y with i≥1 xi X yi Y < ∞, we have that i≥1 |xi yi | ∈ XπY . In deed, taking h ∈ X Y such that h > 0 a.e., it follows that h i≥1 |xi yi |dµ < ∞ and so i≥1 |xi yi | < ∞ a.e. As a consequence, every z ∈ XπY is actually an infinite sum of products of the type xy with x ∈ X and y ∈ Y . Indeed, if (xi ) ⊂ X and (yi ) ⊂ Y with i≥1 xi X yi Y < ∞ and |z| ≤ ˜i = vz χsupp (v) |xi | ∈ X (as i≥1 |xi yi | a.e., taking v = i≥1 |xi yi | and x
202
Delgado and Sánchez Pérez
IEOT
∈ BL∞ ), we have that z = i≥1 x ˜i |yi | a.e. This fact may fail if X is not saturated. The series can be even divergent a.e. For instance, taking xni = χAni ∈ L1 and yin = n1 χAni ∈ L2 for i = 1, ..., n in Example 2, we have that n n 1 1 1 1 12 · xni L1 · yin L2 = = 3 < ∞, n n n n2 n≥1 i=1 n≥1 i=1 n≥1 n while n≥1 i=1 |xni yin | = n≥1 n1 = ∞ a.e. z v χsupp (v) Y
Example. Let us show some particular cases of π-product spaces. (i) XπL∞ ≡ X even if X is not saturated. This is direct from the definition of π-product space. (ii) If X is saturated, from a classical Lozanovskii’s result ([7, Theorem 6]) it follows that XπX ≡ L1 . (iii) Let 1 ≤ p < ∞. The p-power of a saturated Banach function space X is the Banach function space given by X p = {x ∈ L0 : |x|p ∈ X} 1/p with norm xX p = |x|p X for x ∈ X p , see [9, Proposition 1]. If 1 ≤ r, q < ∞ satisfy 1/r = 1/p + 1/q, from [9, Lemma 1], it follows that X p πX q ≡ X r . Moreover, if Y is another Banach function space and 0 < θ < 1 we obtain the Calder´ on-Lozanovskii interpolation space X θ Y 1−θ 1/θ as the π-product space X πY 1/(1−θ) (see [3] and [11, Section 2]). We end this section with a result which will be useful along the paper. Lemma 2.4. Assume that X, Y and Z are saturated Banach function spaces such that X Z and Z Y are saturated. Then, X Z πZ Y is a saturated Banach function space and satisfies X Z πZ Y →1 X Y . Y
Proof. Let us see that (X Z )(Z ) is saturated and so, by Proposition 2.2, we will have that X Z πZ Y is a saturated Banach function space. Take x ∈ X such that x > 0 a.e. and y ∈ Y such that y > 0 a.e. Then xy > 0 a.e. and for every f ∈ X Z and g ∈ Z Y , as Z Y →1 Z Y ≡ (Y )Z (see for instance [2, §2(3) and Lemma 3.1(a)] and Proposition 2.2(ii)), it follows |xy f g| dµ ≤ xf Z y gZ < ∞ Y
and so xy ∈ (X Z )(Z ) . Given z ∈X Z πZ Y , consider sequences (fi )⊂ X Z and (gi ) ⊂ Z Y satisfying that i≥1 fi X Z gi Z Y < ∞ and |z| ≤ i≥1 |fi gi | a.e. For every x ∈ X, we have that |zx| ≤ i≥1 |fi gi x| a.e. with fi gi xY ≤ fi xZ gi Z Y ≤ xX fi X Z gi Z Y i≥1
i≥1
i≥1
and so zx ∈ Y with zxY ≤ xX · π(z). Hence, z ∈ X Y and satisfies zX Y ≤ π(z).
Vol. 66 (2010)
Summability Properties for Multiplication Operators
203
Note that the hypothesis of Lemma 2.4 are satisfied for instance if X is saturated and X ⊂ Z ⊂ Y since in this case L∞ is contained in both X Z and Z Y .
3. The dp,Z -product spaces Throughout this section, X, Y and Z will be saturated Banach function spaces such that X Z and Z Y are saturated. Then, by Lemma 2.4, we can consider the saturated Banach function space X Z πZ Y which is contained in Y Y X . In particular, X is saturated and so, by Proposition 2.2, we also can consider the saturated Banach function space XπY . Let 1 ≤ p ≤ ∞. For any Banach space E and (ei ) ⊂ E, we will denote (ei )E,p =
ei pE
1/p
i≥1
if p < ∞ and for the case p = ∞, (ei )E,∞ = sup ei E . i≥1
Definition 3.1. Thedp,Z -product space Xdp,Z Y is the space of functions h ∈ L0 such that |h| ≤ i≥1 |xi yi | a.e. for some (xi ) ⊂ X and (yi ) ⊂ Y satisfying (yi )Y,p · sup (f xi )Z,p < ∞,
(3.1)
f ∈BX Z
where 1 ≤ p ≤ ∞ is such that 1/p + 1/p = 1. For h ∈ Xdp,Z Y , we denote dp,Z (h) = inf (yi )Y,p · sup (f xi )Z,p , f ∈BX Z
where the infimum is taken over all (xi ) ⊂ X, (yi ) ⊂ Y satisfying (3.1) such that |h| ≤ i≥1 |xi yi | a.e. Proposition 3.2. The space Xdp,Z Y is a Banach function space with norm dp,Z . Moreover,
Xdp,Z Y →1 (X Z πZ Y ) . Proof. Let h ∈ Xdp,Z Y and take (xi ) ⊂ X, (yi ) ⊂ Y satisfying (3.1) such that |h| ≤ i≥1 |xi yi | a.e. Consider a function ξ ∈ X Z πZ Y and (fj ) ⊂ X Z , (gj ) ⊂ Z Y ≡ Y Z with |ξ| ≤ j≥1 |fj gj | a.e. and j≥1 fj X Z gj Y Z < ∞.
204
Delgado and Sánchez Pérez
Then |hξ| dµ ≤
|xi yi fj gj | dµ ≤
j≥1 i≥1
≤
gj Y Z
j≥1
≤
xi fj Z yi gj Z
j≥1 i≥1
xi fj Z yi Y
i≥1
gj Y Z (fj xi )i Z,p (yi )Y,p
j≥1
≤
IEOT
fj X Z gj Y Z · (yi )Y,p · sup (f xi )Z,p . f ∈BX Z
j≥1
So, h ∈ (X Z πZ Y ) and h(X Z πZ Y ) ≤ dp,Z (h). In particular, dp,Z (h) = 0 implies h = 0 a.e. Note that if h ∈ Xdp,Z Y and (xi ) ⊂ X, (yi ) ⊂ Y are such that |h| ≤ (f xi )Z,p < ∞, then i≥1 |xi yi | a.e. and satisfy 0 < (yi )Y,p · supf ∈BX Z there exists (˜ xi ) ⊂ X and (˜ yi ) ⊂ Y such that |h| ≤ i≥1 |˜ xi y˜i | a.e., 1/p , (˜ yi )Y,p = (yi )Y,p · sup (f xi )Z,p f ∈BX Z
and 1/p sup (f x ˜i )Z,p = (yi )Y,p · sup (f xi )Z,p .
f ∈BX Z
f ∈BX Z
Indeed, the vectors 1/p
x ˜i = (yi )Y,p ·
−1/p
y˜i = (yi )Y,p ·
sup (f xi )Z,p
f ∈BX Z
−1/p
· xi ,
1/p sup (f xi )Z,p ) · yi
f ∈BX Z
work. Let (hn ) ∈ Xdp,Z Y such that n≥1 dp,Z (hn ) < ∞. Let us prove that h = n≥1 hn ∈ Xdp,Z Y with dp,Z (h) ≤ n≥1 dp,Z (hn ) and so we will have that dp,Z satisfies the triangular inequality and Xdp,Z Y has the RieszFischer property. Given ε > 0, we can take (xni )i ⊂ X, (yin )i ⊂ Y satisfying that hn ≤ i≥1 |xni yin | a.e. and (yin )i Y,p · sup (f xni )i Z,p ≤ dp,Z (hn ) + f ∈BX Z
ε . 2n
Note that h ∈ L0 , since taking ξ ∈ X Z πZ Y such that ξ > 0 a.e. and (fj )j ∈ X Z , (gj )j ∈ Z Y ≡ Y Z with j≥1 fj X Z gj Y Z < ∞ and ξ ≤ j≥1 |fj gj |
Vol. 66 (2010)
Summability Properties for Multiplication Operators
205
˜= a.e., denoting h n≥1 |hn | we have that ˜ |xni yin fj gj | dµ ≤ xni fj Z yin gj Z h ξ dµ ≤ j≥1 n≥1 i≥1
≤
gj Y Z
j≥1
≤
gj Y Z
j≥1
≤
j≥1 n≥1 i≥1
xni fj Z yin Y
n≥1 i≥1
j≥1
≤
(yin )i Y,p · (fj xni )i Z,p
n≥1
(yin )i Y,p · sup (f xni )i Z,p gj Y Z fj X Z · f ∈BX Z
n≥1
gj Y Z fj X Z · ε + dp,Z (hn ) < ∞
j≥1
n≥1
n ˜ < ∞ a.e. We can assume that (y n )i Y,p ·sup and so h f ∈BX Z (f xi )i Z,p > 0 i as in other case dp,Z (hn ) = xni )i and (˜ yin )i as 0 and nsonhn = 0 a.e. Consider (˜ xi y˜i | a.e. and it can be checked that above. Then, |h| ≤ n≥1 i≥1 |˜ (yin )i Y,p · sup (f xni )i Z,p . (˜ yin )n,i Y,p · sup (f x ˜ni )n,i Z,p ≤ f ∈BX Z
n≥1
f ∈BX Z
Thus, h ∈ Xdp,Z Y and dp,Z (h) ≤ n≥1 dp,Z (hn ). The remaining conditions for Xdp,Z Y to be a Banach function space are clear and we have already shown in the beginning of the proof that Xdp,Z Y →1 (X Z πZ Y ) . The norm π of XπY can be described as follows. For every z ∈ XπY , π(z) = inf (xi )X,p · (yi )Y,p (3.2) where the infimum is taken over all sequences (xi ) ⊂ X and (yi ) ⊂ Y such that |z| ≤ i≥1 |xi yi | a.e. and (xi )X,p · (yi )Y,p < ∞. The proof of this fact is a routine computation after noting that if (xi ) ⊂ X and (yi ) ⊂ Y are such that |z| ≤ i≥1 |xi yi | a.e., then for x ˜i = (xi X yi Y )1/p xxi i X ∈ X
and y˜i = (xi X yi Y )1/p yyi iY ∈ Y we have that x ˜i y˜i = xi yi and (˜ xi )X,p · X ∞ (˜ yi )Y,p = i≥1 xi X yi Y . Since X ≡ L (see [9, Theorem 1]) and sup (f xi )X,p = (xi )X,p ,
f ∈BL∞
from (3.2), it follows that XπY ≡ Xdp,X Y . Hence, the π-product spaces are particular cases of the dp,Z -product spaces. Proposition 3.3. The space Xdp,Z Y is saturated and satisfies XπY →1 Xdp,Z Y.
206
Delgado and Sánchez Pérez
IEOT
Proof.Let h ∈ XπY . By (3.2) there exist (xi ) ⊂ X, (yi ) ⊂ Y such that |h| ≤ i≥1 |xi yi | a.e. and (xi )X,p · (yi )Y,p < ∞. Since sup (f xi )Z,p ≤ (xi )X,p ,
f ∈BX Z
we have that h ∈ Xdp,z Y and dp,Z (h) ≤ π(z). Then, XπY →1 Xdp,Z Y and in particular, Xdp,Z Y is saturated.
4. (p,Z)-summing multiplication operators Let us recall the definition given in (1.1). Given X, Y , Z saturated Banach function spaces, a multiplication operator g : X → Y is (p, Z)-summing if there exists a constant K > 0 such that for every x1 , ..., xn ∈ X, n
gxi pY
1/p
f ∈BX Z
i=1
if 1 ≤ p < ∞ and
n
≤ K sup
sup gxi Y ≤ K sup
1/p
i=1
f ∈BX Z
i=1,...,n
f xi pZ
sup f xi Z
i=1,...,n
if p = ∞.
Some relevant classes of multiplication operators between Banach function spaces can be obtained as particular (p, Z)-summing operators. Let us show some examples. (I) p-concave multiplication operators. Let 1 ≤ p < ∞. Recall that a linear operator T : E → F , from a Banach lattice E into a Banach space F , is p-concave if there exists C > 0 such that for every x1 , ..., xn ∈ E, n
T (xi )pF
1/p
n 1/p ≤ C |xi |p .
i=1
E
i=1
Every (p, Lp )-summing multiplication operator g : X → Y is p-concave. p p Indeed, noting that X →1 X L L (see for instance [2, §2(3)]), we have that there exists K > 0 such that for every x1 , ..., xn ∈ X, n
gxi pY
1/p
≤K
i=1
sup f ∈BX Lp
=K
sup f ∈BX Lp
n
f xi pLp
i=1
n 1/p |xi |p f
Lp
i=1
n 1/p =K |xi |p i=1 n
≤K
i=1
1/p
p Lp
XL
|xi |p
1/p . X
(4.1)
Vol. 66 (2010)
Summability Properties for Multiplication Operators
207
Recall that a Banach lattice E is p-convex with constant K if for all x1 , ..., xn ∈ E, n n 1/p 1/p |xi |p xi pE . ≤K i=1
E
i=1
If the contrary inequality holds, then E is called p-concave. In the case when X is p-convex with constant 1 and order semi-continuous, the inequality (4.1) is an equality (see [2, Proposition 5.3(ii)]) and so the class of the p-concave multiplication operators from X into Y coincides with the class of the (p, Lp )-summing ones. (II) Positive p-summing multiplication operators. Let 1 ≤ p < ∞. An operator T : E → F , with E a Banach lattice and F a Banach space, is positive p-summing if there exists K > 0 such that for every x1 , ..., xn ∈ E, n n 1/p 1/p T (|xi |)p ≤ K sup |x∗ , |xi ||p , i=1
x∗ ∈BE ∗
i=1
∗
where E is the topological dual of E, see [1]. Every (p, L1 )-summing multiplication operator g : X → Y is positive p-summing. Indeed, noting that each f ∈ X can be identified with an element of the dual space X ∗ via f, x = f x dµ for all x ∈ X with f X = f X ∗ , we have that there exists K > 0 such that for every x1 , ..., xn ∈ X, n n 1/p 1/p p g|xi | Y = gxi pY i=1
i=1
≤ K
n
sup f ∈B
1 XL
= K sup ≤K
sup
x∗ ∈BX ∗
1/p
i=1
n
f ∈BX
f xi pL1
|f |, |xi | p
1/p
i=1
n
|x∗ , |xi ||p
1/p
.
(4.2)
i=1
In the case when X is order continuous, the K¨ othe dual X can be iden∗ tified with the whole space X (see for instance [6, p. 29]) and then the inequality (4.2) is just an equality, so the class of the positive p-summing multiplication operators from X into Y coincides with the class of the (p, L1 )-summing ones. The analogous result holds for p = ∞. From now and on X, Y and Z will be saturated Banach function spaces such that X Z and Z Y are saturated. In this case, by Lemma 2.4, we have that X Z πZ Y is a saturated Banach function space. Moreover, since Z Y is Y also saturated (as it contains Z ), we can consider the space Xdp,Z Y . Let us show now our main result which gives a characterization of the space of all (p, Z)-summing multiplication operators from X into Y , whenever Y is order semi-continuous. Note that, adopting the notation given in Section 3,
208
Delgado and Sánchez Pérez
IEOT
a function g ∈ X Y is (p, Z)-summing if there exists a constant K > 0 such that for every x1 , ..., xn ∈ X, (gxi )Y,p ≤ K sup (f xi )Z,p . f ∈BX Z
In this case, the inequality also holds for infinite sequences. Theorem 4.1. Assume that Y is order semi-continuous and let g : X → Y be a multiplication operator. Then, g is (p, Z)-summing if and only if g ∈ Xdp,Z Y . Proof. Suppose that g is (p, Z)-summing. Given h ∈ Xdp,Z Y and(xi ) ⊂ X, (yi ) ⊂ Y satisfying (3.1) for Y instead of Y and such that |h| ≤ i≥1 |xi yi | a.e., we have that |gh| dµ ≤
|gxi yi | dµ
i≥1
≤
gxi Y yi Y
i≥1
≤ (gxi )Y,p · (yi )Y ,p ≤ K (yi )Y ,p · sup (f xi )Z,p < ∞ f ∈BX Z
and so g ∈ (Xdp,Z Y ) . Let us prove the converse. Suppose that g ∈ (Xdp,Z Y ) and let x1 , ..., xn ∈ X. Suppose first that 1 ≤ p < ∞. Given ε > 0, since yY = yY for all y ∈ Y as Y is order semi-continuous (see
for instance [2, p. 4,5]), there exists yi ∈ BY such that gxi Y ≤ εc + |gxi yi | dµ, where n p 1/p n p−1 c= . Then, denoting y˜i = gxi p−1 / Y yi , i=1 gxi Y i=1 gxi Y we have that n
gxi pY =
i=1
n i=1
≤
n
gxi p−1 Y gxi Y gxi Yp−1 εc + |gxi yi | dµ
i=1
=ε
n
gxi pY
i=1
≤ε
n i=1
gxi pY
1/p 1/p
+
|g|
n
|xi y˜i | dµ
i=1
+ g(Xdp,Z Y ) dp,Z
n i=1
|xi y˜i | .
Vol. 66 (2010)
Summability Properties for Multiplication Operators
209
Noting that n n n 1/p 1/p dp,Z |xi y˜i | ≤ ˜ yi pY · sup f xi pZ i=1
≤
i=1 n
f ∈BX Z
(p−1)p gxi Y
1/p
i=1
· sup
n
f ∈BX Z
i=1
f xi pZ
1/p
,
i=1
we obtain
n n n 1/p 1/p p p p gxi Y ≤ gxi Y · ε + g(Xdp,Z Y ) sup f xi Z i=1
f ∈BX Z
i=1
i=1
and so n
gxi pY
1/p
n
≤ ε + g(Xdp,Z Y ) sup
f ∈BX Z
i=1
f xi pZ
1/p
.
i=1
Since ε is arbitrary, g is (p, Z)-summing. For the case p = ∞, given ε > 0, there exists yi ∈ BY such that gxi Y ≤ ε + |gxi yi | dµ, and then sup gxi Y ≤ ε + sup |gxi yi | dµ i=1,...,n
i=1,...,n
≤ ε + g(Xd∞,Z Y ) sup d∞,Z (xi yi ) i=1,...n
≤ ε + g(Xd∞,Z Y ) sup
i=1,...n
≤ ε + g(Xd∞,Z Y ) sup
f ∈BX Z
yi Y · sup f xi Z
f ∈BX Z
sup f xi Z ,
i=1,...n
so g is (∞, Z)-summing.
Note that under conditions of Theorem 4.1, from the proof it follows that g(Xdp,Z Y ) is the smallest constant K satisfying the inequality of the definition of (p, Z)-summing. By Proposition 3.2 and Proposition 3.3, XπY →1 Xdp,Z Y →1 X Z πZ Y . Then, X Z πZ Y →1 Xdp,Z Y →1 XπY (see for instance [2, and Lemma 3.1(b)]) and since X Z πZ Y →1 X Z πZ Y →1 X Z πZ Y (XπY ) ≡ X Y (see Proposition 2.2(ii)), we have that X Z πZ Y →1 Xdp,Z Y →1 X Y . (4.3) On the other hand, from Lemma 2.4, it follows that
X Z πZ Y →1 X Y →1 X Y . Rewriting Theorem 4.1, whenever Y is order semi-continuous, we get g ∈ X Y : g is (p, Z)-summing = X Y ∩ Xdp,Z Y .
(4.4)
210
Delgado and Sánchez Pérez
IEOT
In the case when Y has the Fatou property (i.e. Y ≡ Y ), from (4.3), it follows that the space of (p, Z)-summing multiplication operators from X into Y is just Xdp,Z Y , in particular, it has the Fatou property as it coincides with the K¨ othe dual of a Banach function space. From (4.3), (4.4) and Theorem 4.1, we obtain that under the assumption of order semi-continuity for Y , if g ∈ X Z πZ Y then g is a (p, Z)-summing multiplication operator from X into Y . A direct computation proves that this holds also without any assumption on Y . In particular, if a multiplication operator g : X → Y factorizes through Z via two multiplication operators (i.e. g = f h for some f ∈ X Z and h ∈ Z Y ), then g is (p, Z)-summing. Let us show a useful consequence of (4.3), (4.4) and Theorem 4.1. Corollary 4.2. Let Y a Banach function space with the Fatou property and suppose that X Z πZ Y = X Y . Then g ∈ X Y : g is (p, Z)-summing = (Xdp,Z Y ) = X Y . The Fatou property for Y is necessary in the result above. An easy counterexample can be given if it is not satisfied. Take X = ∞ , Y = c0 and Z = ∞ . Then X Y = c0 and X Z πZ Y = ∞ πc0 = c0 = X Y . However, 1 = ∞ π 1 = ∞ dp,∞ 1 (see the comments before Proposition 3.3), and then (Xdp,Z Y ) = ( ∞ dp,∞ 1 ) = ∞ . Corollary 4.2 provides conditions guaranteeing that the space consisting of all (p, Z)-summing multiplication operators from X into Y coincides with the whole space X Y . This is not a general fact, as the following example shows. Example. Consider 1 < p < q < r < ∞ and the spaces X = p , Y = q and Z = r . Note that, since X Z ≡ X Y ≡ ∞ and Z Y ≡ s for 1/s = 1/q−1/r (see [9, Theorem 2 and Proposition 3]), in this case X Z πZ Y ≡ ∞ π s ≡ s X Y . The space of (t, r )-summing multiplication operators from p into q is just the space of sequences g ∈ ∞ satisfying (gxi )q ,t ≤ K sup (f xi )r ,t = K (xi )r ,t f ∈B∞
for some constant K > 0 and for every x1 , ..., xn ∈ p . Clearly, there exist elements of ∞ which do not satisfy the above inequality (e.g. g = (1, 1, 1, ...)). Note that, since X Z πZ Y ≡ s , every g ∈ s is (t, r )-summing for all 1 ≤ t ≤ ∞. Other conditions different from those in Corollary 4.2 under which every multiplication operators from X into Y is (p, Z)-summing are presented in the following result. Proposition 4.3. Assume Y is p-concave, Z is p-convex and X →i X ZZ . Then, every g ∈ X Y is (p, Z)-summing.
Vol. 66 (2010)
Summability Properties for Multiplication Operators
211
Proof. Let g ∈ X Y . Given x1 , ..., xn ∈ X, we have that n n 1/p 1/p gxi pY ≤ K1 |gxi |p i=1
Y
i=1
n 1/p ≤ K1 gX Y |xi |p i=1
= K1 gX Y
X
n
sup
f ∈BX Z
≤ K1 K2 gX Y
|f xi |p
1/p
i=1
sup f ∈BX Z
n
f xi pZ
Z
1/p
,
i=1
where K1 is the p-concavity constant of Y , K2 is the p-convexity constant of Z and for the equality we have used that xX = xX ZZ for all x ∈ X. Actually in Proposition 4.3 X and X Z being saturated are enough instead of the saturation conditions required before Theorem 4.1. Conditions under which X →i X ZZ are studied in [2]. Finally, note that every multiplication operator from X into Y is (p, X)-summing and (p, Y )-summing.
5. Applications Let us finish the paper by applying several important factorization theorems for multiplication operators to the results obtained for the dp,Z -product spaces. Summability properties of these operators are obtained in a straightforward way from Corollary 4.2. 5.1. Reisner’s theorem 1 ∞ Let Z be a Banach function space satisfying that L∞ F ⊂ Z ⊂ Lloc , where LF ∞ denotes the space of functions in L with support having finite measure and L1loc denotes the space of locally integrable functions. Given 1 ≤ p < q ≤ ∞, consider r defined by 1/r = 1/p − 1/q. If Z is p-convex with constant K1 and p q-concave with constant K2 , then for every ε > 0 and g ∈ (Lq )L ≡ Lr (see [9, q p Proposition 3]), the multiplication operator g : L → L has a factorization as Lq HH j H f H
g
Z
- Lp * h
p
(i.e. g = f h) where f ∈ (Lq )Z and h ∈ Z L are such that f (Lq )Z · hZ Lp ≤ (1 + ε)K1 K2 g(Lq )Lp p
(see [10, Theorem 1]) and so g ∈ (Lq )Z πZ L with π(g) ≤ K1 K2 g(Lq )Lp . p p p Hence, (Lq )L →K1 K2 (Lq )Z πZ L . Note that Z L and (Lq )Z are saturated, see the comments in [2] after Proposition 5.3 and Theorem 5.4. Then, by
212
Delgado and Sánchez Pérez
IEOT
p
p
p
p
Lemma 2.4, we have (Lq )Z πZ L →1 (Lq )L . Hence, (Lq )Z πZ L = (Lq )L (with equal norms if K1 K2 ≤ 1). Therefore, from Corollary 4.2, we obtain the following result. Proposition 5.1. Let 1 ≤ p < q ≤ ∞ and r such that 1/r = 1/p − 1/q. If Z is 1 a p-convex and q-concave Banach function space such that L∞ F ⊂ Z ⊂ Lloc , q then the space of (s, Z)-summing multiplication operators from L into Lp (for any 1 ≤ s ≤ ∞) coincides with the whole space Lr , which also coincides with (Lq ds,Z Lp ) . p
Note that under conditions of Proposition 5.1, the norm π of (Lq )Y πY L p is equivalent to π ˜ , defined on g ∈ (Lq )Y πY L as p π ˜ (g) = inf f (Lq )Y hY Lp : g = f h with f ∈ (Lq )Y and h ∈ Y L . (5.1) The norms π and π ˜ coincide whenever K1 K2 ≤ 1. 5.2. Maurey-Rosenthal’s theorem Let 1 ≤ p < ∞. Recall that a linear operator T : E → F , from a Banach space E into a Banach lattice F , is p-convex if there exists C > 0 such that for every x1 , ..., xn ∈ E, n n 1/p 1/p |T (xi )|p xi pE . ≤C F
i=1
i=1
Consider a saturated Banach function space Y being order semi-continuous and p-concave with constant K1 . If T : E → Y is p-convex with constant K2 , then there exists a function 0 ≤ h ∈ (Lp )Y and an operator R : E → Lp such that T factorizes as E
T
H R
H H j H
Lp
- Y * h
and h(Lp )Y RE→Lp ≤ K1 K2 (see [4, Corollary 2]). If a saturated Banach function space X is p-convex with constant K, it is direct to check that every multiplication operator g : X → Y is p-convex with constant KgX Y . Then, there exists 0 ≤ h ∈ (Lp )Y and an operator R : X → Lp such that gx = hR(x) for all x ∈ X and h(Lp )Y RX→Lp ≤ K1 KgX Y . Taking f := hg χ[h>0] , p we have that f ∈ X L (as |f x| ≤ |R(x)| for all x ∈ X) and g = f h (as gx = hR(x) with 0 < x ∈ X implies g(ω) = 0 whenever h(ω) = 0). Then, p p g ∈ X L π(Lp )Y with π(g) ≤ K1 KgX Y . Hence, X Y →K1 K X L π(Lp )Y . Suppose that (Lp )Y is saturated. For instance, this is the case when L∞ F ⊂ p Y ⊂ L1loc , as Y is p-concave. Since X L is saturated (as X is p-convex), p p by Lemma 2.4 we have that X L π(Lp )Y →1 X Y . So, X L π(Lp )Y = X Y (with equal norms if K1 K ≤ 1). Therefore, from Corollary 4.2, we obtain the following result.
Vol. 66 (2010)
Summability Properties for Multiplication Operators
213
Proposition 5.2. Let 1 ≤ p < ∞. Given two saturated Banach function spaces X and Y such that X is p-convex and Y is p-concave, has the Fatou property and satisfies that (Lp )Y is saturated, then the space of (s, Lp )-summing multiplication operators from X into Y (for any 1 ≤ s ≤ ∞) coincides with the whole space X Y , which also coincides with (Xds,Lp Y ) . p
Note that under conditions of Proposition 5.2, the norm π of X L π(Lp )Y p is equivalent to π ˜ defined on X L π(Lp )Y in a similar way as (5.1). If K1 K ≤ 1, the norms π and π ˜ coincide. Acknowledgment The authors thank the referee for providing the reference to the paper by Luxemburg and Zaanen quoted after the proof of Proposition 2.2.
References [1] O. Blasco, Positive p-summing operators on Lp -spaces, Proc. Amer. Math. Soc. 100 (1987), 275–280. [2] J. M. Calabuig, O. Delgado and E. A. S´ anchez P´erez, Generalized perfect spaces, Indag. Math. 19 (2008), 359–378. [3] A. P. Calder´ on, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. [4] A. Defant, Variants of the Maurey-Rosenthal Theorem for quasi-K¨ othe function spaces, Positivity 5 (2001), 153–175. [5] A. Defant and K. Floret, Tensor norms and operator ideals, North Holland Math. Studies, Amsterdam, 1993. [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer-Verlag, Berlin, 1979. [7] G. Ya. Lozanovskii, On some Banach lattices, Siberian Math. J. 10 (1969), 419–430. [8] W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach function spaces, Note III, Nederl. Akad. Wet., Proc., Ser. A 66 (1963), 239-250. [9] L. Maligranda and L. E. Persson, Generalized duality of some Banach function spaces, Indag. Math. 51 (1989), 323–338. [10] S. Reisner, A factorization theorem in Banach lattices and its application to Lorentz spaces, Annales de l´institut Fourier 31 (1981), 239–255. [11] A. R. Schep, Products and factors of Banach function spaces, preprint. [12] A. C. Zaanen, Integration, 2nd rev. ed. North Holland, Amsterdam; Interscience, New York, 1967. O. Delgado and E. A. S´ anchez P´erez Instituto Universitario de Matem´ atica Pura y Aplicada Universidad Polit´ecnica de Valencia 46071 Valencia Spain e-mail:
[email protected] [email protected]
214
Delgado and Sánchez Pérez
Submitted: February 24, 2009. Revised: October 14, 2009.
IEOT
Integr. Equ. Oper. Theory 66 (2010), 215–229 DOI 10.1007/s00020-010-1747-1 Published online February 5, 2010 © The Author(s). This article is published with open access at Springerlink.com 2010
Integral Equations and Operator Theory
The Non-symmetric Discrete Algebraic Riccati Equation and Canonical Factorization of Rational Matrix Functions on the Unit Circle A.E. Frazho, M.A. Kaashoek and A.C.M. Ran Abstract. Canonical factorization of a rational matrix function on the unit circle is described explicitly in terms of a stabilizing solution of a discrete algebraic Riccati equation using a special state space representation of the symbol. The corresponding Riccati difference equation is also discussed. Mathematics Subject Classification (2010). Primary 47A68, 15A24; Secondary 47B35, 42A58, 39A99. Keywords. Toeplitz operators, rational matrix functions, canonical factorization, discrete algebraic Riccati equation, Riccati difference equation, Schur complements, finite section method.
1. Introduction Throughout R is a rational m × m matrix function with no poles on the unit ∞ j circle T, and we write R(z) = j=−∞ z Rj for the Laurent expansion of R on T. The corresponding (block) Toeplitz operator on 2+ (Cm ) is denoted by T , that is, ⎡ ⎤ R0 R−1 R−2 · · · ⎢R1 R0 R−1 · · ·⎥ ⎢ ⎥ (1.1) T = ⎢R2 R1 on 2+ (Cm ). R0 · · ·⎥ ⎣ ⎦ .. .. .. .. . . . . Recall (see, e.g., [9], Section XXIV.3) that R is said to admit a right canonical factorization (with respect to T) if R can be factored as R(z) = Ψ(z)Θ(z),
z ∈ T,
(1.2)
216
Frazho, Kaashoek and Ran
IEOT
where Θ and Ψ are regular m × m matrix rational functions such that Θ(z) and Θ(z)−1 have no poles in {z ∈ C : |z| ≤ 1}, while Ψ(z) and Ψ(z)−1 have no poles in {z ∈ C : |z| ≥ 1} infinity included. It is well-known that R admits such a factorization if and only if the Toeplitz operator T is invertible. Moreover, in that case, T −1 = TΘ−1 TΨ−1 , where TΘ−1 and TΨ−1 are the Toeplitz operators defined by Θ(z)−1 and Ψ(z)−1 , respectively. In this paper we analyze canonical factorization of R using a special state space representation, namely R(z) = R0 + zC(I − zA)−1 B + γ (zI − α)
−1
β.
(1.3)
Here A and α are square matrices of sizes n × n and ν × ν, say, which are assumed to be stable, that is, the eigenvalues of A and α are contained in the open unit disc. The I’s in (1.3) stand for identity matrices of appropriate sizes, and B, C, β, γ in (1.3) are matrices, again of appropriate sizes. We shall refer to (1.3) as a stable representation of R. With the representation (1.3) we associate the algebraic Riccati equation Q = αQA + (β − αQB)(R0 − γQB)−1 (C − γQA).
(1.4)
We say that Q is a stabilizing solution to this Riccati equation if the matrix R0 − γQB is invertible, Q is a solution to (1.4), and the matrices A◦ = A − B(R0 − γQB)−1 (C − γQA) −1
α◦ = α − (β − αQB)(R0 − γQB)
γ
(1.5) (1.6)
are both stable. We are now ready to state the main result of this note. Theorem 1.1. Let R be an m × m rational matrix function with no poles on the circle, and let (1.3) be a stable representation for R. Then R admits a right canonical factorization with respect to the unit circle if and only if the algebraic Riccati equation (1.4) has a stabilizing solution Q, and in that case a canonical factorization R(z) = Ψ(z)Θ(z) is obtained by taking Θ(z) = D + zC◦ (I − zA)−1 B,
Ψ(z) = δ + γ(zI − α)−1 β◦ .
(1.7)
Here
(1.8) C◦ = δ −1 (C − γQA), β◦ = (β − αQB)D−1 , and δ and D are any invertible matrices satisfying δD = R0 −γQB. Moreover, the inverses of the factors are given by Θ(z)−1 = D−1 − zD−1 C◦ (I − zA◦ )−1 BD−1 Ψ(z)−1 = δ −1 − δ −1 γ(zI − α◦ )−1 β◦ δ −1 ,
(1.9) (1.10)
where A◦ and α◦ are defined by (1.5) and (1.6), respectively. Finally, if (1.4) has a stabilizing solution, then this solution is unique and given by ⎡ ⎤ C ⎢ CA ⎥ ⎢ ⎥ Q = β αβ α2 β · · · T −1 ⎢CA2 ⎥ , (1.11) ⎣ ⎦ .. .
Vol. 66 (2010)
Discrete Algebraic Riccati Equation and Canonical Factorization
217
where T is the block Toeplitz operator (1.1) defined by R. In Section 2 below we prove Theorem 1.1 and specify this theorem for the case when the Toeplitz operator T is tri-diagonal and for the case when R has no poles on the closed unit disc. Theorem 1.1 is of particular interest for the case when the values of R on the unit circle are hermitian matrices. In that case, one takes α = A∗ , β = C ∗ , and γ = B ∗ . The corresponding Riccati equation is then given by Q = A∗ QA + (C ∗ − A∗ QB)(R0 − B ∗ QB)−1 (C − B ∗ QA),
(1.12)
and the rational matrix function R(z) plays the role of the so-called Popov function or spectral density. This symmetric algebraic Riccati equation originates from stochastic realization theory (see the books [3], [5], [6], [12]), and for this symmetric case Theorem 1.1 is basically known. See, for instance, [15] where R(z) is positive definite on the unit circle and the resulting canonical factorization is a spectral factorization (cf., Appendix A1 in [7]). When R(z) is just hermitian on the unit circle symmetric canonical factorization is usually referred to as a J-spectral factorization, and for this type of factorization the relation with the stabilizing solution of (1.12) is covered by Lemma 12.4.1 (iv) in [11]. The existence of a (unique) stabilizing solution of the symmetric algebraic Riccati equation and its connection with the invertibility of the Toeplitz operator can be found in Section 4.7 of [13]. For further references on the symmetric algebraic Riccati equation and a more detailed description of the history of the subject, see the books [3], [11], [13], and [14]. For the non-symmetric case Theorem 1.1 seems to be new. Our proof of Theorem 1.1, which is self-contained, is based on a Schur complement argument as in [6]. This proof and formula (1.11) for the stabilizing solution also may be of interest for the symmetric case. We see Theorem 1.1 as an addition to [10], where canonical factorization of R and invertibility of T are described explicitly in terms of a different state space representation, namely R(z) = I + C(zG − A)−1 B, where zG − A is a square matrix pencil which is invertible for z on the unit circle. In [10] canonical factorization is obtained by matching of spectral subspaces of the pencils zG − A and zG − (A − BC) (see also Chapter 2 of [1] for the special case when G = I). In Section 3 below we reconsider the example discussed in Section 10 of [10] (cf., Section XXIV.10 in [9]), and we use this example to illustrate Theorem 1.1. Solving (1.4) by iteration leads to the Riccati difference equation QN +1 = αQN A + (β − αQN B)(R0 − γQN B)−1 (C − γQN A). As one may expect from formula (1.11) for the stabilizing solution, solving this equation is closely related to inverting Toeplitz operators by the finite section method. This connection is the main topic of Section 4, the final section of the paper. Other solutions of (1.4), not just the stabilizing ones, are also of interest. For instance, if Q is an arbitrary solution of (1.4), then the rational matrix functions Θ and Ψ defined by (1.7) and (1.8) are analytic on the closed unit
218
Frazho, Kaashoek and Ran
IEOT
disc and R(z) = Ψ(z)Θ(z). Moreover, this factorization is a so-called pseudocanonical factorization (see [16] or Section 9.2 in [2]) of R if, in addition, the matrices A◦ and α◦ defined by (1.5) and (1.6) have eigenvalues only in the closed unit disc. Note that in this case (1.11) does not hold, and to prove that conversely any pseudo-canonical factorization of R is obtained in this way one has to require additional minimality conditions on the representation (1.3). We will come back to this in a future publication. Finally, we mention that canonical factorization of rational matrix functions with respect to the real line or the imaginary axis and its connection with continuous time algebraic Riccati equations is well understood; see, e.g., Chapter 5 in [2], and Chapter 19 in [14], and the references in these books.
2. Proof of Theorem 1.1 Let T be the (block) Toeplitz operator on 2+ (Cm ) defined in (1.1). The Toeplitz structure of T allows us to partition T as a 2 × 2 operator matrix
R0 Γ Cm T = (2.1) on 2 m . + (C ) Ξ T Here Γ is the row operator and Ξ is the column operator defined by ⎡ ⎤ R1 2 m ⎢R2 ⎥ m Γ = R−1 R−2 · · · : + (C ) → C , Ξ = ⎣ ⎦ : Cm → 2+ (Cm ). .. . If T is an invertible operator on 2+ (Cm ), then (see, e.g., pages 28, 29 in [2]) the Schur complement ∆ = R0 − ΓT −1 Ξ is a well-defined invertible operator on Cm . Moreover, the inverse of T admits the block matrix representation
∆−1 −∆−1 ΓT −1 Cm −1 on 2 m . (2.2) T = + (C ) −T −1Ξ∆−1 T −1 + T −1 Ξ∆−1 ΓT −1 This yields the following useful result. Lemma 2.1. Assume that T is an invertible Toeplitz operator on 2+ (Cm ), and let ∆ = R0 − ΓT −1 Ξ be the corresponding Schur complement. Then the following identities hold: T −1 = ST −1 S ∗ + (E − ST −1 Ξ)(R0 − ΓT −1 Ξ)−1 (E ∗ − ΓT −1 S ∗ ), (2.3) S ∗ T −1 = T −1 S ∗ − T −1 Ξ∆−1 (E ∗ − ΓT −1 S ∗ ), T
−1
S = ST
−1
− (E − ST
−1
−1
Ξ)∆
ΓT
−1
.
(2.4) (2.5)
Here E denotes the canonical embedding of Cm onto the first coordinate space of 2+ (Cν ), and S is the block forward shift on 2+ (Cm ). Note that the identity in (2.3) can be viewed as an algebraic Riccati equation with T −1 as the solution. We shall see below (in Part 1 of the proof of Theorem 1.1) that equation (1.4) follows from (2.3) in a straightforward way.
Vol. 66 (2010)
Discrete Algebraic Riccati Equation and Canonical Factorization
219
Proof. A simple computation shows that
0 0 −1 ∗ ST S = . 0 T −1 This identity, together with the identity (2.2), yields T
−1
− ST
−1
∆−1 −∆−1 ΓT −1 S = −1 −1 −1 −T Ξ∆ T Ξ∆−1 ΓT −1
I = ∆−1 I −ΓT −1 . −T −1Ξ ∗
Next observe that
I 0 I = = E − ST −1 Ξ, − 0 T −1 Ξ −T −1 Ξ I −ΓT −1 = I 0 − 0 ΓT −1 = E ∗ − ΓT −1 S ∗ .
(2.6)
(2.7) (2.8)
Using the latter identities in (2.6), we obtain T −1 = ST −1 S ∗ + (E − ST −1 Ξ)∆−1 (E ∗ − ΓT −1 S ∗ ).
(2.9)
This is precisely equation (2.5). Multiplying (2.3) by S ∗ on the left yields (2.4). Likewise multiplying (2.3) by S on the right gives (2.5). Proof of Theorem 1.1. The proof is broken up into four parts. In the first two parts we assume that R admits a right canonical factorization, or equivalently, that T is invertible, and we show that the matrix Q defined by (1.11) is a stabilizing solution to the algebraic Riccati equation (1.4). In the third part we start from a stabilizing solution to (1.4) and derive the desired canonical factorization. The final part deals with uniqueness statement. Part 1. Assume that T is invertible, and let Q be the matrix defined by (1.11). In this part we show that Q is a solution to (1.4). Notice that Q can be written Q = ωT −1W , where W is the observability and ω is the controllability operator defined by ⎡ ⎤ C ⎢ CA ⎥ ⎢ ⎥ W = ⎢CA2 ⎥ : Cn → 2+ (Cm ), ⎣ ⎦ .. . ω = β αβ α2 β · · · 2+ (Cm ) → Cν . By comparing (1.1) with (2.1), and using the representation (1.3), we obtain Ξ = W B, Γ = γ,
C = E∗W β = ωE
and S ∗ W = W A,
and ωS = α.
(2.10) (2.11)
220
Frazho, Kaashoek and Ran
IEOT
Using the first identities in (2.10) and (2.11) together with Q = ωT −1W , we see that ∆ = R0 − ΓT −1 Ξ = R0 − γωT −1W B = R0 − γQB.
(2.12)
Furthermore, the identities in (2.10) and (2.11) yield (E ∗ − ΓT −1 S ∗ )W = C − γωT −1 W A = C − γQA, ω(E − ST
−1
Ξ) = β − αωT
−1
W B = β − αQB.
(2.13) (2.14)
Now multiplying equation (2.3) on the left by ω and on the right by W , we obtain that Q = ωT −1 W is a solution to the algebraic Riccati equation (1.4); here we used (2.12), (2.13) and (2.14). Part 2. As in the previous part Q = ωT −1W . In this part we show that for this choice of Q the matrices A◦ and α◦ defined by (1.5) and (1.6) are stable matrices. In fact, we shall only prove that A◦ is stable. The proof of the stability of α◦ can be obtained in the same way using a duality argument. First we note that S ∗ T −1 W = T −1 W A◦ . This identity follows from (2.4) together with (2.10) and (2.13). Indeed, we have S ∗ T −1 W = T −1 S ∗ − Ξ∆−1 (E ∗ − ΓT −1 S ∗ ) W = T −1 W A − W B∆−1 (C − γQA = T −1 W A◦ . Next, we decompose Cn as Cn = X1 ⊕ X2 , where X2 = Ker W and X1 = C Ker W . Notice that X2 is an invariant subspace for A, and C|X2 = 0. We claim that QA|X2 = 0. This follows from the fact that n
QAX2 ⊆ QX2 = ωT −1 W X2 = {0}. By using C|X2 = 0 and QA|X2 = 0 in (1.5), we see that A◦ |X2 = A|X2 and X2 is also an invariant subspace for A◦ . In other words, A◦ admits a matrix representation of the form
A11 0 X1 A◦ = on (2.15) A21 A22 X2 where A22 = A|X2 on X2 . Since X2 is an invariant subspace for A and A is stable, A22 is also stable. Let E1 be the natural embedding of X1 into Cn = X1 ⊕ X2 . Let W1 be the one to one operator defined by W1 = W E1 mapping X1 into 2+ (Cm ). Using S ∗ T −1 W = T −1 W A◦ with A11 = E1∗ A◦ E1 , we obtain S ∗ T −1 W1 = S ∗ T −1 W E1 = T −1 W A◦ E1 = T −1 W1 E1 A◦ E1 = T −1 W1 A11 . In other words, S ∗ T −1 W1 = T −1 W1 A11 . Because W1 is one to one, T −1 W1 is also one to one. Notice that S ∗n T −1 W1 = T −1 W1 An11 for all integers n ≥ 0. Since S ∗n converges to zero in the strong operator topology and A11 acts on a finite dimensional space, An11 converges to zero. Therefore A11 is stable. Recall that A22 = A|X2 is stable. By consulting the matrix representation for A◦ in (2.15), we see that A◦ is stable.
Vol. 66 (2010)
Discrete Algebraic Riccati Equation and Canonical Factorization
221
Part 3. In this part Q is a stabilizing solution of (1.4), and we derive the desired canonical factorization. Let Θ(z) and Ψ(z) be the rational m × m matrix functions defined by (1.7) and (1.8). First we prove that R(z) = Ψ(z)Θ(z). Note that C − γQA = δC◦ ,
β − αQB = β◦ D,
R0 − γQB = δD.
(2.16)
Using these identities we see that the Riccati equation (1.4) can be rewritten as a Stein equation (a discrete Lyapunov equation), namely Q = αQA + β◦ C◦ .
(2.17)
From the identity (2.17) we see that zβ◦ C◦ = z(Q − αQA) = (zI − α)Q − (zI − α)Q(I − zA) + zQ(I − zA). It follows that γ(zI − α)−1 (zβ◦ C◦ )(I − zA)−1 B = γQ(I − zA)−1 B − γQB + zγ(zI − α)−1 QB = zγQA(I − zA)−1 B + γQB + γ(zI − α)−1 αQB, and hence
Ψ(z)Θ(z) = δ + γ(zI − α)−1 β◦ D + zC◦ (I − zA)−1 B = δD + γ(zI − α)−1 β◦ D + zδC◦ (I − zA)−1 B + γ(zI − α)−1 (zβ◦ C◦ )(I − zA)−1 B = (δD + γQB) + γ(zI − α)−1 (β◦ D + αQB)+ + z(γQA + δC◦ )(I − zA)−1 B.
From the third identity in (2.16) we see that δD + γQB = R0 , the second identity in (2.16) yields β◦ D + αQB = β, and the first identity in (2.16) shows that γQA + δC◦ = C. But then we see that Ψ(z)Θ(z) is equal to the right hand side of (1.3), that is, R(z) = Ψ(z)Θ(z). It remains to show that this factorization is a right canonical one, i.e., we have to show (see, e.g., [9], Section XXIV.3) that Θ(z) and Θ(z)−1 have no poles in {z ∈ C : |z| ≤ 1}, while Ψ(z) and Ψ(z)−1 have no poles in {z ∈ C : |z| ≥ 1} infinity included. But these properties follow directly from the stability of the matrices A, α, A◦ , and α◦ . Thus R(z) = Ψ(z)Θ(z) is a right canonical factorization of R relative to the unit circle. Part 4. In this part we prove the uniqueness of the stabilizing solution. In fact, we show that the stabilizing solution to (1.4) is given by (1.11). So let Q be a stabilizing solution to (1.4). By the result of the previous part, R admits a right canonical factorization R(z) = Ψ(z)Θ(z), where Θ and Ψ are given by (1.7). It follows that the block Toeplitz operator T defined by R is invertible. Moreover, its inverse can be expressed explicitly in terms of the × × −1 at zero and the Taylor coeffiTaylor coefficients Θ× 0 , Θ1 , Θ2 , . . . of Θ(z) × × × −1 cients Ψ0 , Ψ1 , Ψ2 , . . . of Ψ(z) at infinity. In fact (see Theorem XXIV.4.1
222
Frazho, Kaashoek and Ran
IEOT
in [9]), we have T −1 = TΘ−1 TΨ−1 , where ⎤ ⎡ × ⎡ × 0 0 ··· Θ0 Ψ0 × ⎥ ⎢Θ× ⎢ Θ 0 · · · 0 ⎥ ⎢ 1 ⎢ 0 TΘ−1 = ⎢Θ× Θ× Θ× ⎥ , TΨ−1 = ⎢ 0 1 0 ⎦ ⎣ 2 ⎣ .. .. .. .. . . . .
Ψ× 1 Ψ× 0 0 .. .
Ψ× 2 Ψ× 1 Ψ× 0
⎤ ··· · · ·⎥ ⎥ ⎥ . (2.18) ⎦ .. .
As Θ(z)−1 and Ψ(z)−1 are given by (1.9) and (1.10), we see that −1 , Θ× 0 = D
Ψ× 0
=δ
−1
,
−1 −1 Θ× C◦ Aj−1 ◦ BD j = −D
Ψ× j
= −δ
−1
−1 γαj−1 ◦ β◦ δ
(j = 1, 2, . . .),
(j = 1, 2, . . .).
Using these identities and (2.18) we have ωTΘ−1 = β αβ α2 β · · · , where ⎞ ⎛ ∞ β = βD−1 − α ⎝ αj βD−1 C◦ Aj◦ ⎠ BD−1 ,
(2.19)
j=0
and
⎡ ⎤ C ⎢ CA ⎥ ⎢ ⎥ TΨ−1 W = ⎢CA 2⎥ , ⎣ ⎦ .. .
where
⎞ ⎛ ∞ = δ −1 C − δ −1 γ ⎝ C αj◦ β◦ δ −1 CAj ⎠ A.
(2.20)
j=0
= C◦ . To do this set ∆ = R0 − γQB. We shall prove that β = β◦ and C Because Q is a solution to the algebraic Riccati equation (1.4) and δD = ∆, we have Q = αQA + (β − αQB)(R0 − γQB)−1 (C − γQA) = αQ(A − B∆−1 (C − γQA)) + β∆−1 (C − γQA) = αQA◦ + βD−1 C◦ . In other Q = αQA◦ + βD−1 C◦ . Since α and A◦ are both stable, ∞ words, j −1 Q = j=0 α βD C◦ An◦ . So according to (2.19) and the second identity in (1.8), we see that β = βD−1 −αQBD−1 = (β−αQB)D−1 = β◦ . Analogously, Q = αQA + (β − αQB)∆−1 (C − γQA) = α − (β − αQB)∆−1 γ QA + (β − αQB)∆−1 C = α◦ QA + β◦ δ −1 C. In other Q = α◦ QA + β◦ δ −1 C. Since α◦ and A are both stable, ∞words, j −1 Q = CAj . So according to (2.20) and the first identity in j=0 α◦ β◦ δ (1.8), we see that C◦ = C.
Vol. 66 (2010)
Discrete Algebraic Riccati Equation and Canonical Factorization
223
To complete the proof, recall that Q also satisfies the Lyapunov equation ∞
Q = αQA + β◦ C◦ .
Hence Q = j=0 αj β◦ C◦ Aj . By consulting (2.19) and (2.20) with β = β◦ = C◦ , we obtain and C ωT −1 W = ωTΘ−1 TΨ−1 W =
∞
αj β◦ C◦ Aj = Q.
j=0
Therefore Q = ωT −1W and the stabilizing solution Q is unique.
Next we specify Theorem 1.1 for the simple case when R has no poles on the closed unit disc D. In this case the block Toeplitz operator T on 2+ (Cm ) defined by R is block lower triangular, and R admits a representation of the form R(z) = R0 + zC(I − zA)−1 B
with A a stable n × n matrix.
(2.21)
2+ (Cm )
Notice that T defines an invertible operator on if and only if det R(z) had no zeros in D, or equivalently, R0 is invertible and A − BR0−1 C is stable. By choosing α = 0 on the zero space {0}, and β = 0 from Cm into {0}, and γ = 0 from {0} into Cm , we see that (2.21) is of the form (1.3). In this case the corresponding Riccati equation (1.4) is just the equation Q = 0, where Q maps Cn into {0}. Hence Q = 0 is the only solution to (1.4). Moreover, α◦ = 0 and A◦ = A − BR0−1 C. So Q is a stabilizing solution if and only if A − BR0−1 C is stable, or equivalently, T is invertible. By choosing D = R0 and δ = I, we see that R(z) = Ψ(z)Θ(z) is a right canonical factorization with Θ = R and Ψ = I. From the simple case considered in the previous paragraphs it already follows that the algebraic Riccati equation (1.4) may not have any stabilizing solution. For example, take R as in (2.21) above, with A = 0, B = −2, C = 1 and R0 = 1. Then Q = 0 is the only solution to the corresponding Riccati equation (1.4) and A◦ = 2 is unstable. The next proposition is a corollary of Theorem 1.1 for the case when the Toeplitz operator is tri-diagonal. Proposition 2.2. Assume R(z) = R0 + zR1 + z −1 R−1 . Then R admits a right canonical factorization if and only if the equation Q = R−1 (R0 − Q)−1 R1
(2.22)
has a solution Q with R0 − Q invertible and with A◦ = −(R0 − Q)−1 R1
and
α◦ = −R−1 (R0 − Q)−1
(2.23)
being stable matrices. In this case, take any δ and D invertible so that δD = R0 − Q. Then the corresponding canonical factorization is given by R(z) = Ψ(z)Θ(z), where the factors Ψ and Θ and their inverses are determined by
224
Frazho, Kaashoek and Ran
Θ(z) = D + zδ −1 R1 ,
IEOT
Ψ(z) = δ + z −1 R−1 D−1 ,
Θ(z)−1 = D−1 + zA◦ (I + zA◦ )−1 D−1 , −1
Ψ(z)
=δ
−1
+δ
−1
−1
(z − α◦ )
(2.24)
α◦ .
Proof. To see this we simply set A = α = 0 on Cm , and B = γ = I on Cm . Moreover, take C = R1 and β = R−1 . Then this proposition follows from Theorem 1.1. Put Y = R0 − Q. Then (2.22) can be rewritten as R0 = Y + R−1 Y −1 R1 .
(2.25)
So we can also reformulate the above proposition in terms of Y rather than in terms of Q. The stabilizing solution in this reformulation is the one for which Y is invertible and A◦ = −Y −1 R1 and α◦ = −R−1 Y −1 are stable matrices. With R0 positive definite, and R−1 = R1∗ , equation (2.25) is studied in [4]. The case when R is a trigonometric polynomial, R(z) = pj=−t Rj , is also of special interest. In this case we obtain a representation (1.3) by taking ⎤ ⎡ ⎤ ⎡ R1 0 I ⎥ ⎢R2 ⎥ ⎢ . . . . ⎥ ⎢ ⎥ ⎢ . . A=⎢ ⎥ , B = ⎢ .. ⎥ , C = I 0 · · · 0 , ⎣ . ⎦ ⎣ 0 I⎦ 0
⎡
0 ⎢I ⎢ α=⎢ ⎣
0 .. .
.. I
.
⎤
⎥ ⎥ ⎥, ⎦ 0
Rp ⎡ ⎤ I ⎢0 ⎥ ⎢ ⎥ β = ⎢.⎥ , ⎣ .. ⎦ 0
γ = R−1
R−2 · · ·
R−t .
Using (1.11) one computes that for this special representation of R the solution Q of the correponding algebraic Riccati equation is just equal to the p × t block matrix in the top left corner of the operator T −1 .
3. An example To illustrate how one can use Theorem 1.1, we reconsider the example analyzed in Section 10 of [10] (cf., Section XXIV.10 in [9]). Consider the rational matrix function
1 − z −1 12 z −1 . R(z) = −3z 1+z As in [10] we seek a canonical factorization of R with respect to the unit circle. For this R we have R0 = I and a stable representation is obtained by taking
0 1 A = 0, B = −3 1 , C = , α = 0, β = −1 12 , γ = . 1 0
Vol. 66 (2010)
Discrete Algebraic Riccati Equation and Canonical Factorization
In this setting, Q = q is a scalar,
1 + 3q −q (1 + 3q)−1 R0 − γqB = , (R0 − γqB)−1 = 0 1 0
225
q(1 + 3q)−1 . 1
The corresponding Riccati equation is now determined by the following scalar equation:
(1 + 3q)−1 q(1 + 3q)−1 0 −1 1 q = β(R0 − γqB) C = −1 2 0 1 1 1 q 1+q = − = . 2 1 + 3q 2 + 6q Rewriting this leads to the quadratic equation 6q 2 + q − 1 = 0. The zeros of this equation are −1/2 and 1/3. Since A and α are zero, equations (1.5) and (1.6) yield A◦ = −B(R0 − γqB)−1 C
and α◦ = −β(R0 − γqB)−1 γ.
A simple calculation shows that A◦ and α◦ are both equal to −1/(1 + 3q). So the stabilizing solution is obtained with q = 1/3, and A◦ = α◦ = −1/2. We now take δ = I and D = R0 − γqB, that is
1 1 2 − 13 −1 D= and D = 2 6 . 0 1 0 1 Then we compute the factors Θ(z) and Ψ(z) in (1.7):
0 0 2 − 13 2 − 13 +z = , Θ(z) = 0 1 −3 1 −3z 1 + z and Ψ(z) = I +
1 1 −1 z 0
1 2
D−1 = I +
1 1 1 −2 z 0
1 3
=
1 1 − 2z 0
1 3z
1
.
This is exactly the canonical factorization of R derived in Section 10 of [10].
4. Riccati iteration and finite sections Throughout this section R is a rational m × m matrix function given by the stable representation (1.3), and (1.4) is the corresponding Riccati equation. Solving (1.4) by iteration leads to a Riccati difference equation (cf., Appendix A2 in [7]): QN +1 = αQN A + (β − αQN B)(R0 − γQN B)−1 (C − γQN A).
(4.1)
Let us assume that starting from an initial condition at N = k, at each step of the iteration the matrix R0 − γQN B is invertible. In this way we obtain from (4.1) a sequence QN , QN +1 , QN +2 , . . . of ν × n matrices. Moreover, assume that this sequence converges with limit Q and the matrix R0 − γQB is invertible. Then Q is a solution to the Riccati equation (1.4) and this solution will be the stabilizing solution of (1.4) provided the matrices A◦ and α◦ defined by (1.5) and (1.6) are both stable. In that case the Toeplitz operator
226
Frazho, Kaashoek and Ran
IEOT
defined by R is invertible and a right canonical factorization for R is given by R(z) = Ψ(z)Θ(z) where Ψ and Θ are defined by (1.7). Formula (1.11) for the stabilizing solution suggests to define the iterates QN in terms of the finite sections of T . By definition the N -th section of T is the N × N block matrix TN given by ⎤ ⎡ R−1 · · · R1−N R0 ⎢ R1 R0 · · · R2−N ⎥ ⎥ ⎢ TN = ⎢ . .. ⎥ . . .. .. ⎣ .. . . ⎦ RN −1 RN −2 · · · R0 In what follows we assume that the Toeplitz operator T defined by R is invertible and that the same holds true for its block transpose T # = [Tk−j ]∞ j,k=0 . # −1 Note that the latter is equivalent to requiring that R (z) = R(z ) admits a right canonical factorization relative to the circle. Thus in the scalar case the invertibility of T # follows from the invertibility of T and conversely. This is also true in the symmetric case (when R(z) is Hermitian for each z on the unit circle). Since R is a continuous matrix symbol, we know (see Section VIII.5 in [8]) that invertibility of T and T # implies (in fact, is equivalent to the statement) that the finite section method for T converges. In particular, in this case, there exists a positive integer k such that TN is invertible for N ≥ k and lim ωn TN−1 WN = ωT −1 W. n→∞
Here
⎡
ωj = β
αβ
α2 β
···
αj−1 β ,
C CA CA2 .. .
⎤
⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Wj = ⎢ ⎥, ⎥ ⎢ ⎦ ⎣ CAj−1
j ≥ 1.
In checking the invertibility of the finite section the following lemma is useful. Lemma 4.1. Assume the N -the section TN of T is invertible, and put QN = ωN TN−1 WN . Then TN +1 is invertible if and only if R0 − γQN B is invertible, and in that case the matrix QN +1 = ωN +1 TN−1+1 WN +1 is given by QN +1 = αQN A + (β − αQN B)(R0 − γQN B)−1 (C − γQN A). Note that the matrix R0 − γQB is a square matrix of order m while TN +1 is of order m(N + 1). Hence, in general, checking the invertibility of R0 − γQB will be a much easier task than checking the invertibility TN +1 . Proof. Note that TN +1 admits the 2 × 2 block matrix representation:
R0 γωN TN +1 = . (4.2) WN B TN
Vol. 66 (2010)
Discrete Algebraic Riccati Equation and Canonical Factorization
227
Moreover, the Schur complement ∆N of TN corresponding 2 × 2 block matrix in (4.2) is given by ∆N = R0 − γωN Tn−1 WN B = R0 − γQN B.
(4.3)
Hence TN +1 will be invertible if and only if R0 − γQN B is invertible. The fact that QN +1 = ωN +1 TN−1+1 WN +1 is then given by the right hand side of (4.3) follows by proving the analogue of Lemma 2.1 with T being replaced by TN +1 and using the same type of arguments as in Part 1 of the proof of Theorem 1.1. We summarize the preceding discussion with the following proposition which is a partial converse to the result stated in the second paragraph of this section. Proposition 4.2. Let R be given by the stable representation (1.3), and consider the Riccati difference equation in (4.1). Assume the Toeplitz operator T defined by R and its block transpose T # are invertible. Then there exists a positive integer k such that the following holds (i) Tk is invertible; (ii) R0 − γQN B is invertible for all N ≥ k where QN is the solution to (4.1) subject to the initial condition Qk = ωk Tk−1 Wk ; (iii) QN converges to Q and R0 − γQB is invertible; (iv) the matrices α◦ and A◦ are stable. In this case, Q is the unique stabilizing solution to the Riccati equation (1.4). It can happen that Qn converges to Q and R0 − γQB is invertible and α◦ or A◦ may not be stable. In fact, this follows from the example considered in the final paragraphs of the previous section. Indeed, take R(z) = 1 − 2z, and represent R(z) as in (1.3) with A, B, C matrices of size 1 × 1, A = 0, B = −2, C = 1, and with α = 0 on {0}, β = 0 from C into {0}, and γ = 0 from {0} into C. Then (4.1) and (1.4) reduce to QN = 0 and Q = 0. Thus limN →∞ QN = Q, but Q is not a stabilizing solution. In this case, R does not admit a right canonical factorization. In conclusion we note that for the example considered in Section 3 the Riccati difference equation is given by
(1 + 3qN )−1 qN (1 + 3qN )−1 0 1 + qN 1 = . qN +1 = −1 2 1 2 + 6qN 0 1 Starting with the initial condition q0 = 0, we see that the sequence qN converges to 1/3. In fact, qN = 1/3 for all N ≥ 11, and the first eleven values for qN are given by 1 3 13 51 205 819 3277 13107 52429 209715 1 , , , , , , , , , , . 2 10 38 154 614 2458 9830 39322 157286 629146 3
228
Frazho, Kaashoek and Ran
IEOT
References [1] H. Bart, I. Gohberg, and M.A. Kaashoek, Wiener-Hopf integral equations, Toeplitz matrices and linear systems, in: Toeplitz Centennial (ed. I. Gohberg), OT 4, Birkh¨ auser Verlag, 1982; pp. 85-135. [2] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorization of matrix and operator functions: the state space method, OT 178, Birkh¨ auser Verlag, Basel, 2008. [3] P.E. Caines, Linear Stochastic Systems, Wiley, New York, 1988. [4] J.C. Engwerda, A.C.M. Ran and A. Rijkeboer, Necessary and sufficient conditions for the existence of a positive solution of the matrix equation X + A∗ X −1 A = I, Lin. Alg. Appl. 184 (1993), 255-275. [5] P. Faurre, M. Clerget and F. Germain, Op´erateurs rationels positifs, Dunod, Paris, 1979. [6] P.L. Faurre, Stochastic realization algorithms, in: System Identification: Advances and case studies, (R.K. Mehra and D.F. Lainiotis, Eds), Academic Press, New York, 1976. [7] C. Foias, A.E. Frazho, I. Gohberg, M. A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, OT 100, Birkh¨ auser-Verlag, Basel, 1998. [8] I.C. Gohberg and I.A. Fel’dman, Convolution equations and projection methods for their solution, Transl. Math. Monographs, vol 41, Amer. Math. Soc., Providence, R.I., 1974. [9] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Volume I, OT 63, Birkh¨ auser Verlag, Basel, 1993. [10] I. Gohberg and M.A. Kaashoek, Block Toeplitz operators with rational symbols, in: Contributions to Operator Theory and its Applications (Eds. I. Gohberg, J.W. Helton and L. Rodman), OT 35, Birkh¨ auser Verlag, Basel, 1988; pp. 385-440. [11] B. Hassibi, A.H. Sayed and T. Kailath. Indefinite-Quadratic Estimation and Control, A Unified Approach to H 2 and H ∞ Theories., SIAM, Philadelphia, 1999. [12] C. Heij, A.C.M. Ran and F. van Schagen, Introduction to Mathematical Systems Theory, Birkh¨ auser Verlag, Basel, 2007. [13] V. Ionescu, C. Oara and M. Weiss. Generalized Riccati Theory and robust control. A Popov function approach, John Wiley, Chichester, 1999. [14] P. Lancaster and L. Rodman, Algebraic Riccati equations, Clarendon Press, Oxford, 1995. [15] B.P. Molinari, The stabilizing solution of the discrete algebraic Riccati equation, IEEE Trans. Automat. Control 20 (1975), 396-399. [16] L. Roozemond, Canonical pseudo-spectral factorization and Wiener-Hopf integral equations, in: Constructive methods of Wiener-Hopf factorization, OT 21, Birkh¨ auser Verlag, Basel, 1986, pp. 127–156.
Vol. 66 (2010)
Discrete Algebraic Riccati Equation and Canonical Factorization
229
A.E. Frazho Department of Aeronautics and Astronautics Purdue University West Lafayette, IN 47907 USA e-mail:
[email protected] M.A. Kaashoek and A.C.M. Ran Afdeling Wiskunde Faculteit der Exacte Wetenschappen Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail:
[email protected] [email protected] Submitted: July 11, 2009. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Integr. Equ. Oper. Theory 66 (2010), 231–251 DOI 10.1007/s00020-010-1739-1 Published online January 29, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System B. Fritzsche, B. Kirstein and A. L. Sakhnovich Abstract. Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form φ(λ) exp{−2iλD}, where φ is a strictly proper rational matrix function and D = D∗ ≥ 0 is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case. Mathematics Subject Classification (2010). Primary 34A55; Secondary 34B20; 47G10; 34A05. Keywords. Skew-self-adjoint Dirac system, Weyl function, inverse problem, semiseparable operator, operator identity, explicit solution.
1. Introduction A skew-self-adjoint Dirac-type system has the form d u(x, λ) = iλj + jV (x) u(x, λ), x ≥ 0, (1.1) dx where Ip 0 v 0 j= , (1.2) , V = v∗ 0 0 −Ip Ip is the p × p identity matrix, and v is a p × p matrix function. Such systems have been actively studied in analysis and soliton theory (see, for instance, [1, 12] and the references therein). The system (1.1) is an auxiliary linear The work of A. L. Sakhnovich was supported by the Austrian Science Fund (FWF) under Grant no. Y330.
232
Fritzsche, Kirstein and Sakhnovich
IEOT
system for the matrix nonlinear Schr¨ odinger equation, Sine-Gordon and other important integrable equations. In this paper we consider an inverse problem for the system (1.1), namely, the problem to recover the system (1.1) from its Weyl function. We provide an explicit procedure to recover the potential v when the Weyl function is of the form ϕ(λ) = φ(λ) exp{−2iλD}R,
D ≥ 0,
(1.3)
where φ is a strictly proper rational p×p matrix function, D is a p×p diagonal matrix, and R is a p×p unitary matrix. First, we derive the new Theorem 2.3, which allows us to apply to a wider class of functions, including functions of the form (1.3), the general (non-explicit) scheme for solving inverse problems presented in [29, 33]. Next, we modify the expression for the solution of the inverse problem so that this solution is presented in a more traditional and convenient form. Finally, we use the result mentioned above and the results on the inversion of semiseparable operators from [18] to construct the explicit solution of the inverse problem. The solution of the inverse problem given in this paper has a simple form for the case D = 0, see Corollary 4.10. To present the solution for this case, we first recall that all strictly proper rational matrix functions admit representations (realizations in the terminology of system theory) of the form ϕ(λ) = iθ1∗ (λIn − β)−1 θ2 ,
(1.4)
where β is an n × n (n > 0) matrix and θm (m = 1, 2) are n × p matrices. Using this realization, it is shown in Section 4 that the potential v is given by −1 θ2 , v(l) = 2θ1∗ ρ− 11 (l) − where ρ− mj (l) (m, j = 1, 2) are n × n blocks of ρ (l), iβ θ2 θ2∗ , ρ− (l) := e−2lζ , ζ := θ1 θ1∗ iβ ∗
and det ρ− 11 (l) = 0. In another way the inverse problem for this case was solved in [21]. Let us consider connections with the self-adjoint case in some detail and give some references. The inverse problem to recover a self-adjoint Dirac-type system from its Weyl or spectral function is closely related to the inversion of integral operators with difference kernels, see [9, 26, 32, 36, 37] and various references therein. For the discrete analogues of Dirac systems, Toeplitz matrices appear instead of the operators with difference kernels [7, 10, 15, 38]. (Various results on Toeplitz matrices and related j-theory one can find, for instance, in [5, 8, 13, 14].) When the Weyl functions of the self-adjoint Dirac-type system are rational, one can solve the inverse problem explicitly. One of the approaches to solve the inverse problem explicitly is connected with a version of the B¨acklund-Darboux transformation and some notions from system theory [20, 22]. (See also [15, 16, 24, 27] for this approach, and see [39] and the
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
233
references therein for explicit formulas for the radial Dirac equation.) Another method is to apply the general theory. It proves [2] that for the case of rational Weyl functions the corresponding operators with difference kernels can be inverted explicitly by formulas from [4]. The case of the skew-self-adjoint Dirac-type system with the rational Weyl function was treated in [21] . It was shown that any strictly proper rational p × p matrix function is the Weyl function of a skew-self-adjoint Dirac type system on semi-axis and the solution of the inverse problem was constructed explicitly similar to the self-adjoint case treated in [20]. The analogues of the operators with difference kernel for the skew-selfadjoint system (1.1) are bounded operators Sl in L2p (0, l) (0 < l < ∞), which have the form [30, 33] 1 l x+t z + x − t z + t − x ∗ k dzf (t)dt, (1.5) Sl f = Sf = f (x) + k 2 0 |x−t| 2 2 where sup0<x
0. The class of bounded linear operators acting from H1 into H2 is denoted by {H1 , H2 }, identity operators are denoted by I, and the spectrum of an operator A is denoted by σ(A).
2. Inverse problem. Preliminaries First, normalize the fundamental solution u(x, λ) of system (1.1) by the initial condition (2.1) u(0, λ) = I2p ,
234
Fritzsche, Kirstein and Sakhnovich
IEOT
and assume that the real and imaginary parts vij and vij of the entries vij of the potential v are measurable functions. If sup v(x) ≤ M,
(2.2)
0<x<∞
the unique p×p Weyl matrix function ϕ(λ) of the skew-self-adjoint Dirac type system (1.1) on the semi-axis [0, ∞) can be defined [29] (see also [6, 21, 33]) by the inequality ∞ ϕ(λ) ϕ(λ)∗ Ip u(x, λ)∗ u(x, λ) dx < ∞, (2.3) Ip 0 which holds for all λ in the halfplane λ < −M < 0. Under condition (2.2) such a Weyl function always exists. An example of bounded potentials is given by the class of so called pseudo-exponential potentials [21], which are denoted by the acronym PE. A potential v ∈ PE is determined by three parameter matrices, that is, by the n × n matrix α (n > 0) and two p × n matrices θ1 and θ2 , which satisfy the identity (2.4) α − α∗ = i(θ1 θ1∗ + θ2 θ2∗ ). The pseudo-exponential potential has the form ∗
v(x) = 2θ1∗ eixα Σ(x)−1 eixα θ2 , where
Σ(x) = In +
x
Λ(t)jΛ(t)∗ dt,
Λ(x) =
0
e−ixα θ1
(2.5)
eixα θ2
.
(2.6)
By Proposition 1.4 in [21] the pseudoexponential potential v, i.e., the potential given by (2.5) is bounded on the semi-axis. The Weyl function of the system (1.1) with v ∈ PE is a rational matrix function, which is also expressed in terms of the parameter matrices [21]: ϕ(λ) = iθ1∗ (λIn − β)−1 θ2 ,
β = α − iθ2 θ2∗ .
(2.7)
In spite of the requirement β − β ∗ = i(θ1 θ1∗ − θ2 θ2∗ ),
(2.8)
α − iθ2 θ2∗ ,
which is implied by the equalities (2.4) and β = any strictly proper rational matrix function can be presented in the form (2.7). The inverse problem to recover v from the strictly proper rational matrix function ϕ is solved explicitly in [21], using a minimal realization of ϕ and formula (2.5). When (2.2) is true, inequality (2.3) implies other inequalities: ixλ ϕ(λ) < ∞ for all 0 < l < ∞, (2.9) sup e u(x, λ) Ip x≤l, λ<−M which can be treated as a more general definition of the Weyl function. Definition 2.1. Let the system (1.1) be given on the semi-axis [0, ∞). Then a p × p matrix function ϕ(λ) analytic in some halfplane λ < −M < 0 is called a Weyl function of this system, if inequalities (2.9) hold.
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
235
If sup v(x) < ∞
0<x
for all 0 < l < ∞,
(2.10)
then there is at most one Weyl function. Definition 2.2. The inverse spectral problem (ISP) for system (1.1) on the semi-axis is the problem to recover v(x) satisfying (2.9) and (2.10) from the Weyl function ϕ. For an analytic matrix function ϕ satisfying the condition
sup λ2 ϕ(λ) − α/λ < ∞,
(2.11)
λ<−M
where α is some p × p matrix, the solution of the inverse problem always exists (see Lemma 1 [30] for the scalar version of this result and the matrix case can be proved in a quite similar way). The general (non-explicit) procedure to solve ISP is described in [28, 29, 30, 33]. Fix a positive value l (0 < l < ∞). The first step to solve ISP is to recover a p × p matrix function s(x) with the entries from L2 (0, l) (l < ∞), i.e., s(x) ∈ L2p×p (0, l) via the Fourier transform. That is, we put a i −ηx e l.i.m.a→∞ eiξx λ−1 ϕ(λ/2)dξ (λ = ξ + iη, η <−2M ), (2.12) s(x) = 2π −a the limit l.i.m. being the limit in L2p×p (0, l). As (2.12) has sense for any l < ∞ the matrix function s(x) is defined on the non-negative real semi-axis x ≥ 0. Moreover, it is easily checked that s is absolutely continuous, it does not depend on the choice of η < −2M , s is bounded on any finite interval, and s(0) = 0. To define the operator Sl we substitute k(x) = s (x) into (1.5). Next, denote the p × 2p block rows of u by ω1 and ω2 : ω1 (x) = [Ip
0]u(x, 0),
ω2 (x) = [0 Ip ]u(x, 0).
(2.13)
It follows from (1.1) that u(x, 0)∗ u(x, 0) = I2p . Hence, by (1.1) and (2.13) we have (2.14) v(x) = ω1 (x)ω2 (x)∗ , and ω1 , ω2 satisfy the equalities ω1 (0) = [Ip
0],
ω1 ω1∗ ≡ Ip ,
ω1 ω1∗ ≡ 0,
ω1 ω2∗ ≡ 0.
It is immediate that ω1 is uniquely recovered from ω2 using (2.15). Finally, we obtain ω2 via the formula l ∗ ω2 (l) = [0 Ip ] − Sl−1 s (x) [Ip s(x)]dx (0 < l < ∞),
(2.15)
(2.16)
0
where Sl−1 is applied to s columnwise. From the considerations in [29, 30] (see also similar constructions in [31], where the Weyl theory for the linear system auxiliary to the nonlinear optics equation is treated) it follows that one can solve ISP under requirements weaker than (2.11). Namely, we assume sup ϕ(λ) < ∞,
λ<−M
(2.17)
236
Fritzsche, Kirstein and Sakhnovich
ϕ(λ) ∈ L2p×p (−∞, ∞), and s(0) = 0,
λ = ξ + iη (−∞ < ξ < ∞) for all η < −M, (2.18)
sup k(x) < ∞ for all 0 < l < ∞,
0<x
IEOT
∞
0
k(x) := s (x),
e−cx k(x)dx < ∞
(2.19) (2.20)
for s given by (2.12) and for some c > 0. Theorem 2.3. Let the matrix function ϕ be analytic in the halfplane λ < −M and satisfy the relations (2.17) and (2.18). Assume also that the function s(x), defined via ϕ by formula (2.12), is absolutely continuous and satisfies (2.19) and (2.20). Then ISP has a unique solution, which is given by formulas (2.14)– (2.16), where Sl ≥ I has the form (1.5) with k = s .
3. Factorization of S and operator identity Consider again the operator S = Sl . It is easy to see that functions, which are bounded on the interval, can be approximated in the L1 -norm by the continuous functions. As k = s is bounded on the finite intervals, one can see that the kernel K of S, which is given by (1.6), is continuous with respect to x and t. Hence, the kernel of Sl−1 is continuous with respect to x, t, and l
([23], p. 185). Therefore, Sl−1 k has the form Sl−1 k (x) = k(x)+k1 (x), where k1 is continuous, and the matrix function Sl−1 k (l) is well-defined:
Sl−1 k (l) = k(l)+k1 (l) = k(l)+ lim Sl−1 k (x)−k(x) (0 < l < ∞). (3.1) x→l−0 It is useful to express v in terms of Sl−1 k (l). Theorem 3.1. Under the conditions of Theorem 2.3 we have
v(l) = Sl−1 s (l).
(3.2)
Proof. STEP 1. According to [34] there are triangular operators Vl ∈ {L2p (0, l), L2p (0, l)}, such that x x −1 V− (x, t)f (t)dt, Vl AVl = iω1 (x) ω1 (t)∗ · dt, (3.3) (Vl f )(x) = f (x)+ 0
0
where A = Al = i
0
x
· dt,
A ∈ {L2p (0, l), L2p (0, l)},
V− (x, t) does not depend on l, and the operators Vl and Vl−1 map functions with bounded derivatives into functions with bounded derivatives. Moreover, as bounded functions on an interval can be approximated in the L1 -norm by the continuous functions, it follows from the construction in [34] that V− (x, t) (x ≥ t) is continuous with respect to x and t.
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
237
Next, introduce the operator x V − (x − t)f (t)dt, (V l f )(x) = f (x) + where ω11
d −1 Vl ω11 (x), (3.4) V − (x) := dx 0 is the first p × p block of ω1 , and put x V− (x, t) · dt. (3.5) Vl := Vl V l = I + 0
It is easy to see that V l A = AV l , and so the second equality in (3.3) yields x −1 ω1 (t)∗ · dt. (3.6) Vl AVl = iω1 (x) 0
By (2.13) we see that ω11 (0) = Ip . Hence, using definition (3.4) one gets x
d −1 Vl ω11 (x − t)dt Vl Ip = Ip + (3.7) dx 0 x
d −1 Vl ω11 (t)dt = Vl−1 ω11 (x). = Ip + dx 0 Formula (3.7) implies Vl−1 ω11 = Ip . Moreover, from [29, 33] it follows that under the conditions of Theorem 2.3 the equalities −1
Vl ω1 (x) = [Ip s(x)] (3.8) and
Sl−1 = Vl∗ Vl
(3.9) are also true. STEP 2. Using (3.9) and changing variables l and x into x and t, correspondingly, we rewrite (2.16) in the form x (3.10) ω2 (x) = [0 Ip ] − Vx s (t)∗ Vx [Ip s(t)]dt. 0
As V− does not depend on l we have Vx s (t) = Vl s (t) for t ≤ x ≤ l. Thus, according to (3.8) and (3.10), we get
(3.11) ω2 (x) = − Vl s (x)∗ ω1 (x). Multiplying both sides of (3.11) by ω1∗ from the right and taking into account (2.14) and (2.15), one derives −v(x)∗ = − Vl s (x)∗ , i.e., the equality
v(x) = Vl s (x) (3.12) is true. is continuous, taking into account (3.4) and (3.5) we see
V− (x, t) As that Vl s (x) − s (x) is continuous. It is also immediate from (3.5) that l ∗ V− (t, x)∗ f (t)dt. (3.13) (Vl f )(x) = f (x) + x
Hence, according to (3.1), (3.9), and (3.13) we have Sl−1 s (l) = Vl s (l). Finally, formula (3.2) follows from (3.12) and (3.14).
(3.14)
238
Fritzsche, Kirstein and Sakhnovich
IEOT
By (3.9) the equality
∗ −1 ∗ Vl AS − SA∗ = Vl−1 Vl AVl−1 − Vl AVl−1 is valid for S = Sl . Therefore, taking into account (3.6) and (3.8) one can see that S satisfies the operator identity AS − SA∗ = iΠΠ∗ ,
Π = [Φ1
Φ2 ],
Φ1 g ≡ g,
Φ2 g = s(x)g.
(3.15)
Here Φm ∈ {Cp , L2p (0, l)} (m = 1, 2) and C denotes the complex plane. This identity differs from the identity AS − SA∗ = i(Φ1 Φ∗2 + Φ2 Φ∗1 ) [35, 36] for an operator with difference kernel. Matrices satisfying a discrete analogue of (3.15) were treated in [17]. The operator identity (3.15) for the case, when k in (1.5) is a vector, was studied in [25]. It could be useful also to prove (3.15) directly. In fact, we prove below a somewhat more general identity. Proposition 3.2. Let the operator S in L2p (0, l) (0 < l < ∞) be defined by 1 l x+t z + x − t z + t − x Sf = f (x) + k k dzf (t)dt, (3.16) 2 0 |x−t| 2 2
k(x) < ∞. Then S satisfies the operator identity where sup0<x
x 0
k(t)dt, ψ(x) =
l
0
x 0
Ip + ψ(x)ψ(t) · dt,
(3.17)
k(t)dt.
Proof. Taking into account (3.16) and changing the order of integration we have l γ1 (x, t)f (t)dt, (3.18) ASf = Af + i 0 1 x y+t z + y − t z + t − y k γ1 (x, t) := k dzdy. (3.19) 2 0 |y−t| 2 2 Denote the kernel of S by K(x, t). Using the equality (A∗ f, g)l =(f, Ag)l for the scalar product (·, ·)l in L2p (0, l), we see that 0
l
K(x, t)(A∗ f )(t)dt =
0
l
AK(x, ·)∗ (t)∗ f (t)dt = −i
l 0
0
t
K(x, y)dyf (t)dt.
Hence, we rewrite SA∗ in the form l SA∗ f = A∗ f − i γ2 (x, t)f (t)dt, (3.20) 0 1 t x+y z + x − y z + y − x γ2 (x, t) := k k dzdy. (3.21) 2 0 |x−y| 2 2
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
239
First, consider the case t ≥ x. From (3.19), after changes of variables ξ = (z + y − t)/2 and η = t − y + ξ, we get 1 x y+t z + y − t z + t − y k dzdy (3.22) γ1 (x, t) = k 2 0 t−y 2 2 x y x x = k(ξ) k(t − y + ξ)dξdy = k(ξ) k(t − y + ξ)dydξ 0
0
x
0
t
k(ξ) k(η)dηdξ.
= 0
ξ
t−x+ξ
Next, calculate γ2 (x, t) (t ≥ x). From (3.21) it follows that γ2 (x, t) = γ21 (x, t) + γ22 (x, t), where
(3.23)
z + x − y z + y − x k dzdy, (3.24) 2 2 0 x−y 1 t x+y z + x − y z + y − x k k dzdy. (3.25) γ22 (x, t) := 2 x y−x 2 2
γ21 (x, t) :=
1 2
x
x+y
k
Replace the variable z by η = (z + y − x)/2 in (3.24) , then change the order of integration, and after that put ξ = x − y + η and change the order of integration again to obtain x ξ k(ξ) k(η)dηdξ. (3.26) γ21 (x, t) = 0
0
In (3.25), replace z by ξ = (z + x − y)/2, change the order of integration and put η = y − x + ξ. We get x t−x+ξ γ22 (x, t) = k(ξ) k(η)dηdξ. (3.27) 0
ξ
By (3.22), (3.23), (3.26), and (3.27) the equality x t k(ξ) k(η)dηdξ = ψ(x)ψ(t) γ1 (x, t) + γ2 (x, t) = 0
(3.28)
0
is true for t ≥ x. Using similar calculations one can show that (3.28) holds also for x ≥ t, i.e., (3.28) is true for all 0 ≤ x, t ≤ l. Finally, formulas (3.18), (3.20), and (3.28) yield (3.17).
4. ISP and semiseparable operators Sl In this section we consider matrix functions of the form ϕ(λ) = iθ1∗ (λIn − β)−1 θ2 e−2iλD R, D = diag{d1 , . . . , dp }, dk1 ≥ dk2 ≥ 0
(4.1) for k1 > k2 ,
(4.2)
240
Fritzsche, Kirstein and Sakhnovich
IEOT
where θj (j = 1, 2) is an n × p matrix with the m-th column denoted by θj,m , β is an n × n matrix, R is a p × p matrix, and D is a p × p diagonal matrix. We do not suppose here that θj and β satisfy the identity (2.8). Proposition 4.1. Let the matrix function ϕ be given by (4.1). Then, the matrix function s, which is defined via ϕ by (2.12), has the form s = CR, where C = c1 c2 . . . cp , the columns cm (p ≥ m ≥ 1) being given by the formulas for 0 ≤ x ≤ dm , x−dm cm (x) = 2θ1∗ exp{2itβ}dtθ2,m
cm (x) = 0
0
(4.3) for
x ≥ dm ,
(4.4)
and the function ϕ is the Weyl function of system (1.1) with potential v satisfying (2.10). Proof. First, choose M > 0 such that σ(β + iM In ) ⊂ C+ , where σ means spectrum and C+ is the open upper halfplane. According to (4.1) ϕ(λ) is analytic and the function λϕ(λ) is bounded in the halfplane λ < −M . So, the conditions (2.17) and (2.18) on ϕ are fulfilled. The fact that s is absolutely continuous and satisfies conditions (2.19) and (2.20) is immediate from (4.3) and (4.4). Therefore, after we have proved (4.3) and (4.4) , it will follow from Theorem 2.3 that ϕ is the Weyl function of system (1.1) with potential v satisfying (2.10). Now, let us prove (4.3) and (4.4). As λϕ(λ) is bounded, one can rewrite (2.12) as a pointwise limit: ∞ 1 eiλ(x−dm ) (4.5) s = c1 c2 . . . cp R, cm (x) = − θ1∗ π −∞ ×λ−1 (λIn − 2β)−1 dξθ2,m
(λ = ξ + iη,
η < −2M ).
Introduce the counterclockwise oriented contours, where ξ may take complex values: {ξ : |ξ| = a, ξ > 0}, Γ− {ξ : |ξ| = a, ξ < 0}. Γ+ a = [−a, a] a = [−a, a] For λ = ξ + iη and for the fixed values of η < −2M , it follows from (4.5) that 1 eiλ(x−dm ) λ−1 (λIn − 2β)−1 dξθ2,m (4.6) cm (x) = − θ1∗ lim π a→∞ Γ+ a in the case x ≥ dm , and 1 eiλ(x−dm ) λ−1 (λIn − 2β)−1 dξθ2,m cm (x) = θ1∗ lim π a→∞ Γ− a
(4.7)
in the case x ≤ dm . As eiλ(x−dm ) λ−1 (λIn − 2β)−1 is analytic with respect to ξ inside Γ− a and on the contour itself, equality (4.3) is immediate from (4.7). Next, consider the case x ≥ dm . For sufficiently large a all the poles of (λIn − 2β)−1 (and the pole ξ = −iη of λ−1 ) are contained inside Γ+ a and taking into account (4.6) we have 1 ∗ eiλ(x−dm ) λ−1 (λIn − 2β)−1 dξθ2,m . (4.8) cm (x) = − θ1 π Γ+ a
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
241
Let us approximate β by matrices βε such that β − βε < ε and det βε = 0 (if det β = 0 we put β = βε ). It is easy to see that
λ−1 (λIn − 2βε )−1 = (2βε )−1 (λIn − 2βε )−1 − λ−1 In . (4.9) For sufficiently small ε all the poles of (λIn − 2βε )−1 are contained inside Γ+ a and we have 1 eiξx (λIn − 2βε )−1 dξ = eηx exp(2ixβε ) (x ≥ 0). (4.10) 2πi Γ+ a Finally, using (4.8)-(4.10) we get cm (x) = lim
ε→0
2θ1∗
x−dm
0
exp{2itβε }dtθ2,m .
Hence, formula (4.4) is immediate.
Remark 4.2. Note that the matrix functions ϕ of the form (4.1) in general position do not satisfy (2.11) and so they do not satisfy in a scalar case conditions of Lemma 1 [30], but the conditions of Theorem 2.3 are fulfilled. By Proposition 4.1 the matrix function k in the expression (1.6) for the kernel of the operator Sl , generated by the Weyl function ϕ of the form (4.1), is given by the formula k(x) = s (x) = 2θ1∗ e2ixβ νχ(x)R,
ν := {exp(−2idm β)θ2,m }pm=1 , (4.11) 0, 0 ≤ x < dm , χ(x) = diag{χ1 (x), χ2 (x), . . . , χp (x)}, χm (x) := 1, x > dm .
According to (1.6) and (4.11) we have K(x, t) ∗ = 2θ1
(4.12) exp i(z + x − t)β Q(z, x, t) exp − i(z + t − x)β ∗ dzθ1 ,
x+t
|x−t|
where
z + t − x z + x − t (4.13) Q(z, x, t) = νχ RR∗ χ ν∗. 2 2 The matrix function Q(z, x, t) is piecewise constant with respect to z and without loss of generality we assume Q(0, x, t) = 0. It is easy to see that Q(z, x, t) has only a finite number of jumps {Qj }. Moreover, if σ(β)∩σ(β ∗ ) = ∅, the matrix identity i(βXj − Xj β ∗ ) = Qj always has the solution Xj . Therefore we have ∗ ∗ d izβ (4.14) e Xj e−izβ . eizβ Qj e−izβ = dz Hence, according to (4.12) and (4.14) we can express the kernel K(x, t) of S explicitly in terms of matrix exponents and {Xj }. It follows also from (1.6) that K(x, t) = K(t, x)∗ , (4.15) and so we need to simplify (4.12) only for x > t.
242
Fritzsche, Kirstein and Sakhnovich
IEOT
Another approach to the presentation of K in terms of matrix exponents is given in the following lemma. Lemma 4.3. Put gj (z) := [0
izβ
e
zEj
]e
Then we have
Ip 0
Ej :=
,
−iβ ∗ Qj
0 −iβ
.
(4.16)
d ∗ gj (z) = eizβ Qj e−izβ . dz
Proof. By (4.16) we have d gj = iβgj + [0 eizβ ]Ej ezEj dz
Ip 0
(4.17)
∗
= iβgj + eizβ Qj e−izβ − iβgj ,
and (4.17) is immediate.
Recall [18] that the operator S is called semiseparable, when K admits representation K(x, t) = F1 (x)G1 (t)
for x > t,
K(x, t) = F2 (x)G2 (t)
for x < t, (4.18)
where F1 and F2 are p × p matrix functions and G1 and G2 are p × p matrix functions. For the operator S to be semiseparable, assume RR∗ = Ip .
(4.19)
Then the matrix function Q has the form z + t − x Q(z, x, t) = νχ ν∗ 2 Rewrite (4.2) as D = diag{d 1 Ip1 , . . . , d r Ipr },
for x > t.
p1 + . . . + pr = p,
(4.20)
d k1 > d k2 ≥ 0 for k1 > k2 ,
(4.21)
Qj = νPj ν ∗ ,
(4.22)
and put Pj = diag{0, . . . , 0, Ipj , 0, . . . , 0}.
Remark 4.4. Some notations. Further we consider K(x, t) (x > t) on the intervals d m < t < min(x, d m+1 ), where we choose such m that the inequalities d m < x hold. If d 1 > 0 we put d 0 = 0 and include the interval d 0 < t < min(x, d 1 ) into consideration. If x > d r , we include the interval d r < t < d r+1 (d r+1 = x). Some matrix functions, like B(t) and C(t), will be considered on the intervals as above, but with x = l. In the following, in mall such cases we simply write dm < t < dm+1 . We also assume that j=1 . . . = 0, when m = 0. Proposition 4.5. Let the matrix function ϕ be given by (4.1), where D satisfies (4.21) and R is unitary. Assume also that the matrix identities i(βXj − Xj β ∗ ) = Qj ,
(4.23)
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
243
where Qj are given by (4.22), have solutions Xj . Then the operator S, which is defined via ϕ by formulas (1.5), k = s and (2.12), is semiseparable, and its kernel K(x, t) (0 < x, t < l) is given by relation (4.15) and by the equalities ∗ m θ1 (d m < t < d m+1 ) (4.24) K(x, t) = 2θ1∗ e2ixβ Zm e−2itβ − e2i(x−t)β Z for t < x < l. Here m Xj , Zm = j=1
m = Z
m
exp 2id j β Xj exp − 2id j β ∗ .
(4.25)
j=1
Moreover, there are self-adjoint solutions of (4.23) and we suppose Xj = Xj∗ in (4.23) and (4.25). Proof. First, note that we can choose Xj = Xj∗ because the adjoint of each solution of (4.23) also satisfies (4.23). Next, using (4.12), (4.20), and (4.22) we get equalities m x+t
exp i(z + x − t)β Qj K(x, t) = 2θ1∗ j=1
x−t+2d j
× exp − i(z + t − x)β ∗ dz θ1
(4.26)
for t < x and d m < t < d m+1 . From (4.14) and (4.26) it follows that (4.24) holds. Formula (4.15) was derived earlier. Remark 4.6. By (2.14)–(2.16) and (4.11) the equality v(x) = 0 is valid for 0 < x < d1 in the case d1 > 0. Therefore inequality (2.9) for l = d1 takes the form ixλ ixλj φ(λ)e−2iλD e e < ∞, sup (4.27) Ip x≤d1 , λ<−M and can be easily checked directly.
l When the operator S = I + 0 K(x, t) · dt is semiseparable and its kernel K is given by (4.18), the kernel of the operator T = S −1 is expressed in terms of the 2 p × 2 p solution U of the differential equation d U (x) = H(x)U (x), x ≥ 0, U (0) = I2 p , (4.28) dx where H(x) := B(x)C(x), (4.29) −G1 (x) , C(x) = F1 (x) F2 (x) . B(x) = G2 (x) Namely, we have (see, for instance, [18]) l T (x, t) · dt, T = S −1 = I + T (x, t) =
(4.30)
0
C(x)U (x) I2 p − P × U (t)−1 B(t), x > t, −C(x)U (x)P × U (t)−1 B(t), x < t.
(4.31)
244
Fritzsche, Kirstein and Sakhnovich
IEOT
Here P × is given in terms of the p × p blocks U21 (l) and U22 (l) of U (l): 0 0 × P = , (4.32) U22 (l)−1 U21 (l) Ip and the invertibility of U22 (l) is a necessary and sufficient condition for the invertibility of S. If K admits the representation
CexA I2 p − P e−tA B, x > t, (4.33) K(x, t) = −CexA P e−tA B, x < t, where A, B, and C are constant matrices, then U is calculated explicitly [19]. In our case a representation
Cm exA I2 p − Pm e−tA Bm , t < x < l, d m < t < d m+1 , (4.34) K(x, t) = −Cm exA Pm e−tA Bm , x < t < l, d m < x < d m+1 , where p = n and
A = 2i
β 0
0
β∗
,
(4.35)
easily follows from (4.15) and (4.24). However, (4.34) is insufficient for the explicit construction of U and we shall construct U and T explicitly, using more general formulas (4.28)-(4.32). For this purpose we introduce B(x) and C(x) (0 < x < l) by the equalities −2ixβ √ m − Zm e−2ixβ ∗ e Z (4.36) θ1 (d m < x < d m+1 ), B(x) = 2 ∗ e−2ixβ √ m e2ixβ ∗ C(x) = 2θ1∗ e2ixβ e2ixβ Zm − Z (d m < x < d m+1 ), (4.37) ∗ ∗ m = Z m where Zm = Zm and Z are defined in (4.25).
Proposition 4.7. Let the conditions of Proposition 4.5 be fulfilled and let S be defined via ϕ by formulas (1.5), k = s and (2.12). Then the operator T = S −1 is given by formulas (4.30)-(4.32), (4.36), (4.37), and U (x) = Ωm e−xA exAm Ξ−1 m U (dm )
(d m ≤ x ≤ d m+1 ),
×
where A is defined by (4.35) and A× m := A + 2Ym , Ωm :=
In 0
Ym := −Zm In
,
m Z In
θ1 θ1∗
m −Z
In
U (x) JU (x) = J,
−1
U (x)
,
(4.39)
×
Ξm := Ωm e−dm A edm Am .
Moreover, we have ∗
U (0) = I2n , (4.38)
∗
∗
= JU (x) J ,
J :=
0 In
−In 0
(4.40) .
(4.41)
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
245
Proof. Recall that B and C are recovered from K by formulas (4.18) and (4.29). In view of (4.15) and (4.24) we have √ F2 (x) = G1 (x)∗ , (4.42) F1 (x) = 2θ1∗ e2ixβ , √
∗ m θ1 (d m < x < d m+1 ), G1 (x) = 2 Zm e−2ixβ − e−2ixβ Z (4.43) G2 (x) = F1 (x)∗ .
(4.44)
Therefore, formulas (4.29) and (4.42)-(4.44) imply that B and C corresponding to S are given by (4.36) and (4.37). It follows from (4.29) and (4.35)-(4.37) that H(x) = 2Ωm e−xA Ym exA Ω−1 (d m < x < d m+1 ), (4.45) m where Ym is given in (4.39), Ωm is given in (4.40), and In Zm −1 Ωm = . (4.46) 0 In According to (4.38), (4.39), and (4.45) we get d xA× m ) = H(x)U (x) m Ξ−1 U (d U (x) = Ωm e−xA A× m−A e m dx for d m < x < d m+1 , and so U of the form (4.38) satisfies (4.28). In other words, formulas (4.36)-(4.38) define explicitly B, C and U , which are used in the expressions (4.31) and (4.32) to construct the kernel of T = S −1 . It remains to prove (4.41). Note that JA∗ J ∗ = −A, ∗
JΩ∗m J ∗ = Ω−1 m ,
JYm∗ J ∗ = −Ym .
(4.47)
∗
Hence, we have JH J = −H, i.e.,
d U (x)∗ JU (x) ≡ 0. dx Formula (4.41) follows from (4.48) and from U (0) = I2n . Taking into account (4.36)-(4.38) we get √ m ), m ]exA× m Ξ−1 U (d F (x) := C(x)U (x) = 2θ1∗ [In − Z m √ m Z := U (t)−1 B(t) = 2U (d m )−1 Ξm e−tA× m G(t) θ1 , In for d m < x < d m+1 , d m < t < d m+1 ,
(4.48)
(4.49) (4.50)
and F (d m+1 ) is given by the substitution of x = d m+1 into the right-hand side of (4.49). In particular, we define in this way F (l) for l = d r+1 . Corollary 4.8. Let the conditions of Proposition 4.5 be fulfilled. Then the kernel T (x, t) of the operator T = Sl−1 has the form
x > t, F (x) I2n − P × G(t), (4.51) T (x, t) = × −F (x)P G(t), x < t, are given by (4.49) and (4.50). where F and G
246
Fritzsche, Kirstein and Sakhnovich
IEOT
By (4.11), (4.39), and (4.50) for d m < t < d m+1 we get m √ × In Pj R G(t)k(t) = 2U (d m )−1 Ξm e−tAm (2Ym )etA ν 0 j=1 m √ d −tA× In m etA = − 2U (d m )−1 Ξm ν Pj R. e 0 dt
(4.52)
j=1
From Theorem 3.1 and formulas (4.30), (4.51), and (4.52) the explicit solution of the inverse problem is immediate. Theorem 4.9. Let the Weyl matrix function ϕ be given by (4.1), where D satisfies (4.21) and R is unitary. Assume also that the matrix identities (4.23), where Qj are given by (4.22), have solutions Xj = Xj∗ . Then the ISP solution v is given by the formula N √
v(l) = k(l) + F (l) I2n − P × 2U (dm )−1 Ξm
m=1 −dm A× m dm A
× e
e
−dm+1 A× m dm+1 A
−e
e
(4.53)
m I n Pj R, ν 0 j=1
×
where k is given by (4.11), U is given by (4.38), P is given by (4.32), and Ξm is given by (4.40). The number N in the sum is chosen in the following way: if l < d 1 then N = 0; if d j < l < d j+1 then N = j; if l > d r then N = r. We put dm = d m for m ≤ N and dN +1 = l. Corollary 4.10. Let the Weyl matrix function ϕ be given by (4.1), where D = 0 and R = Ip . Then we have s (x) = k(x) = 2θ1∗ e2ixβ θ2 , the kernel K of the operator Sl is given by the formula t ∗ e2iyβ θ2 θ2∗ e−2iyβ dy θ1 (x > t), K(x, t) = 4θ1∗ e2i(x−t)β
(4.54)
(4.55)
0
and −1 v(l) = 2θ1∗ ρ− θ2 , 11 (l)
where
ρ− mj (l)
(4.56)
−
(m, j = 1, 2) are n × n blocks of ρ (l), iβ θ2 θ2∗ − −2lζ ρ (l) := e , ζ := , θ1 θ1∗ iβ ∗
(4.57)
and det ρ− 11 (l) = 0. Proof. By (4.11) the equality (4.54) is true. As D = 0 we get (see Remark 4.4): r = 1, d 1 = 0, d 2 = l. (4.58)
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
247
In view of (4.58) it follows from (4.26) after substitution 2y = z + t − x that K(x, t) = F1 (x)G1 (t) for x > t, where t ∗ ∗ 2ixβ −2itβ , G1 (t) = 2e e2iyβ θ2 θ2∗ e−2iyβ dy θ1 , (4.59) F1 (x) = 2θ1 e 0
and (4.55) is immediate. Next assume that there is X1 such that i(βX1 − X1 β ∗ ) = Q1 = θ2 θ2∗ ,
X1 = X1∗ .
(4.60)
According to (4.25) the equalities 1 = X1 Z1 = Z
(4.61)
hold. Taking into account (4.58) and (4.61) we derive from (4.40) that In −X1 In X1 = Ω1 = Ξ1 = , Ω−1 . (4.62) 1 0 In 0 In Moreover, in view of (4.49), (4.58), (4.61), and (4.62) we have √
F (l) = 2θ1∗ [In 0] exp lΩ1 A× Ω−1 . 1 Formulas (4.35), (4.60), and (4.62) imply iβ θ2 θ2∗ Ω1 AΩ−1 = 2 . 1 0 iβ ∗ Hence, using (4.39), (4.57), (4.62), and (4.64) we get X1 ∗ −X I Ω1 A× Ω−1 = Ω θ A + 2 θ Ω−1 n 1 1 1 1 1 1 = 2ζ. In Finally, by (4.63) and (4.65) we have √ F (l) = 2θ1∗ [In
0]e2lζ .
(4.63)
(4.64)
(4.65)
(4.66)
From (4.32), (4.53), (4.58), and (4.66) it follows that (4.67) v(l) = k(l) + 2θ1∗ [In 0]e2lζ × 0 In In Ip Ξ1 − e−lA e2ilβ θ2 . × 0 0 −U22 (l)−1 U21 (l) 0 According to (4.62) and (4.65) one can rewrite (4.67) as (4.68) v(l) = k(l) + 2θ1∗ [In 0]e2lζ In 0 In Ip − e−2lζ e2ilβ θ2 . × 0 0 −U22 (l)−1 U21 (l) 0 By (4.38), (4.62), and (4.65) we get ∗
U2,m (l) = e−2ilβ ρ+ 2m (l)
(m = 1, 2),
ρ+ (l) := e2lζ .
(4.69)
Taking into account (4.54) and (4.69) rewrite (4.68) in the form v(l) = 2θ1∗ e2ilβ θ2
+ + −1 + 2ilβ θ2 . + 2θ1∗ ρ+ ρ21 (l) In − ρ− 11 (l) − ρ12 (l)ρ22 (l) 11 (l)e
(4.70)
248
Fritzsche, Kirstein and Sakhnovich
IEOT
As ρ− (l) = ρ+ (l)−1 , one can see that + + −1 + −1 ρ21 (l) = ρ− . ρ+ 11 (l) − ρ12 (l)ρ22 (l) 11 (l)
Hence, formula (4.70) implies (4.56). Now consider an arbitrary strictly proper rational matrix function ϕ(λ) = iθ1∗ (λIn − β)−1 θ2 without requirement (4.60). Notice that for each ε > 0 there is a matrix βε , which satisfies condition (4.60) and inequality β − βε < ε. For matrices and functions corresponding to ϕε (λ) = iθ1∗ (λIn − βε )−1 θ2 we shall use the index ε. It is easily checked that lim vε (l) = v(l),
ε→0
− lim ρ− ε,11 (l) = ρ11 (l).
ε→0
(4.71)
+ Moreover, we have limε→0 Uε,22 (l) = U22 (l), limε→0 ρ+ ε,22 (l) = ρ22 (l), and −2ilβε∗ + −2ilβ ∗ + so Uε,22 (l) = e ρε,22 (l) implies U22 (l) = e ρ22 (l). As Sl is invertible and semiseparable, so det U22 (l) = 0, that is, det ρ+ 22 (l) = 0. The last inequality yields (4.72) det ρ− 11 (l) = 0. −1 (l) θ . So, by (4.71) and (4.72) It was shown above that vε (l) = 2θ1∗ ρ− 2 ε,11 formula (4.56) is valid without requirement (4.60).
Another way to prove Corollary 4.10 would be to use the procedure from [19] for explicitly inverting semiseparable operators. Acknowledgment The authors are grateful to the referee for very useful remarks.
References [1] M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and continuous nonlinear Schr¨ odinger systems, London Math. Soc. Lect. Note Ser. 302, Cambridge University Press, 2004 [2] D. Alpay, I. Gohberg, L. Lerer, M. A. Kaashoek, and A. L. Sakhnovich, Krein systems, OT: Adv. Appl. 191 (2009), Birkh¨ auser, 19–36. [3] D. Alpay, I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Direct and inverse scattering problem for canonical systems with a strictly pseudoexponential potential, Math. Nachr. 215 (2000), 5–31. [4] H. Bart, I. Gohberg, and M. A. Kaashoek, Convolution equations and linear systems, IEOT 5 (1982), 283–340. [5] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz operators, Springer Monographs in Mathematics, Berlin: Springer, 2006. [6] S. Clark and F. Gesztesy, On Self-adjoint and J-self-adjoint Dirac-type Operators: A Case Study, Contemp. Math. 412 (2006), 103–140. [7] Ph. Delsarte, Y. Genin, and Y. Kamp, Schur parametrization of positive definite block-Toeplitz systems, SIAM J. Appl. Math. 36:1 (1979), February, 34–46. [8] V. K. Dubovoj, B. Fritzsche, and B. Kirstein, Matricial version of the classical Schur problem, in: Teubner-Texte zur Mathematik [Teubner Texts in Mathematics] 129, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992.
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
249
[9] H. Dym and A. Iacob, Positive definite extensions, canonical equations and inverse problems, OT: Adv. Appl. 12 (1984), Birkh¨ auser, 141–240. [10] H. Dym, Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, OT: Adv. Appl. 34 (1988), Birkh¨ auser, 79–135. [11] Y. Eidelman and I. Gohberg, Algorithms for inversion of diagonal plus semiseparable operator matrices, IEOT 44 (2002), 172–211. [12] L. D. Faddeev and L. A. Takhtajan, Hamiltonian methods in the theory of solitons, Berlin: Springer, 1986. [13] B. Fritzsche, B. Kirstein, An extension problem for non-negative Hermitian block Toeplitz matrices, Math. Nachr., Part I: 130 (1987), 121–135; Part II: 131 (1987), 287–297; Part III: 135 (1988), 319–341; Part IV: 143 (1989), 329– 354; Part V: 144 (1989), 283–308. [14] B. Fritzsche, B. Kirstein, On the Weyl matrix balls associated with nondegenerate matrix-valued Caratheodory functions, Z. Anal. Anwendungen 12 (1993), 239–261. [15] B. Fritzsche, B. Kirstein, I. Roitberg, and A. L. Sakhnovich, Weyl matrix functions and inverse problems for discrete Dirac-type self-adjoint systems: explicit and general solutions, Operators and Matrices 2 (2008), 201-231. [16] B. Fritzsche, B. Kirstein, and A. L. Sakhnovich, Completion problems and scattering problems for Dirac type differential equations with singularities, J. Math. Anal. Appl. 317 (2006), 510–525. [17] B. Fritzsche, B. Kirstein, and A. L. Sakhnovich, On a new class of structured matrices related to the discrete skew-self-adjoint Dirac systems, ELA 17 (2008), 473-486. [18] I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators vol. 1, OT: Adv. Appl. 49, Birkh¨ auser, 1990. [19] I. Gohberg and M. A. Kaashoek, Time varying linear systems with boundary conditions and integral operators I. The transfer operator and its properties, IEOT 7 (1984), 325–391. [20] I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Canonical systems with rational spectral densities: explicit formulas and applications, Math. Nachr. 194 (1998), 93–125. [21] I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Pseudocanonical systems with rational Weyl functions: explicit formulas and applications, J. Diff. Eqs. 146 (1998), 375–398. [22] I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Scattering problems for a canonical system with a pseudo-exponential potential, Asymptotic Analysis 29 (2002), 1–38. [23] I. Gohberg and M.G. Krein, Theory and applications of Volterra operators in Hilbert space, Transl. of math. monographs 24, Providence, RI, 1970. [24] M. A. Kaashoek and A. L. Sakhnovich, Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model, J. Functional Anal. 228 (2005), 207–233. [25] I. Koltracht, B. Kon, and L. Lerer, Inversion of structured operators, IEOT 20 (1994), 410–480.
250
Fritzsche, Kirstein and Sakhnovich
IEOT
[26] M. G. Krein, Continuous analogues of propositions on polynomials orthogonal on the unit circle (Russian), Dokl. Akad. Nauk SSSR 105 (1955), 637–640. [27] R. Mennicken, A. L. Sakhnovich, and C. Tretter, Direct and inverse spectral problem for a system of differential equations depending rationally on the spectral parameter, Duke Math. J. 109:3 (2001), 413–449. [28] A. Sakhnovich, The mixed problem for nonlinear Shr¨ odinger equation and the inverse spectral problem, Preprint, Branch of Hydroacoustics, Inst. Hydromech. Acad. Sci. Ukrainian SSR, Odessa, 1989, Manuscript No. 3255-B89 deposited at VINITI AN SSSR, 1989. [29] A. L. Sakhnovich, Nonlinear Schr¨ odinger equation on a semi-axis and an inverse problem associated with it, Ukr. Math. J. 42:3 (1990), 316–323. [30] A. L. Sakhnovich, The Goursat problem for the sine-Gordon equation and the inverse spectral problem, Russ. Math. Iz. VUZ 36:11 (1992), 42–52. [31] A. L. Sakhnovich, Inverse spectral problem related to the N -wave equation, OT: Adv. Appl. 117 (2000), Birkh¨ auser, M.G. Krein volume, 323–338. [32] A. L. Sakhnovich, Dirac type and canonical systems: spectral and WeylTitchmarsh functions, direct and inverse problems. Inverse Problems 18 (2002), 331–348. [33] A. L. Sakhnovich, Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg-Marchenko theorems, Inverse Problems 22 (2006), 2083–2101. [34] L. A. Sakhnovich, Spectral analysis of Volterra’s operators defined in the space of vector-functions L2m (0, l), Ukr. Mat. Zh. 16 (1964), 259-268. [35] L. A. Sakhnovich, An integral equation with a kernel dependent on the difference of the arguments, Mat. Issled. 8 (1973), 138–146. [36] L. A. Sakhnovich, Factorisation problems and operator identities, Uspekhi Mat. Nauk 41 :1 (1986), 3–55; English transl. in Russian Math. Surveys 41 (1986), 1–64. [37] L. A. Sakhnovich, Spectral theory of canonical differential systems, method of operator identities, OT: Adv. Appl. 107, Birkh¨ auser, 1999. [38] B. Simon, Orthogonal polynomials on the unit circle, Parts 1,2, Colloquium Publications, American Mathematical Society 51, 54, Providence, RI, 2005. [39] G. Teschl, Deforming the point spectra of one-dimensional Dirac operators, Proc. Amer. Math. Soc. 126:10 (1998), 2873–2881. [40] R. Vandebril, M. Van Barel, G. Golub, and N. Mastronardi, A bibliography on semiseparable matrices, Calcolo 42 (2005), 249–270. B. Fritzsche and B. Kirstein Fakult¨ at f¨ ur Mathematik und Informatik Mathematisches Institut Universit¨ at Leipzig Johannisgasse 26 D-04103 Leipzig Germany e-mail: [email protected] [email protected]
Vol. 66 (2010)
Semiseparable Operators and an Inverse Problem
A. L. Sakhnovich Fakult¨ at f¨ ur Mathematik Universit¨ at Wien Nordbergstrasse 15 A-1090 Wien Austria e-mail: al [email protected] Submitted: April 20, 2009. Revised: September 18, 2009.
251
Integr. Equ. Oper. Theory 66 (2010), 253–264 DOI 10.1007/s00020-010-1746-2 Published online January 26, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Translation-Invariant Bilinear Operators with Positive Kernels Loukas Grafakos and Javier Soria Abstract. We study Lr (or Lr,∞ ) boundedness for bilinear translationinvariant operators with nonnegative kernels acting on functions on Rn . We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on R2n , while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L1 to L1 and from L1 to L1,∞ are equivalent properties, boundedness from L1 × L1 to L1/2 and from L1 × L1 to L1/2,∞ may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels. Mathematics Subject Classification (2010). Primary 42A85; Secondary 47A07. Keywords. Bilinear operators, convolution, positive kernels.
1. Introduction For a nonnegative regular Borel measure µ on Rn × Rn , we define the bilinear convolution operator: Tµ (f, g)(x) = f (x − y)g(x − z) dµ(y, z), (1) Rn ×Rn
n
where x ∈ R and f, g are nonnegative functions on Rn . If dµ(y, z) = K(y, z) dydz, for some nonnegative function K, then we denote TK (f, g)(x) = f (x − y)g(x − z) K(y, z) dydz, Rn
Rn
The first author was supported by the NSF under grant DMS 0900946. The second author was partially supported by grants MTM2007-60500 and 2005SGR00556.
254
Grafakos and Soria
IEOT
assuming no confusion occurs in the notation. We are interested in studying boundedness properties of these operators on different products of Lp (Rn ) spaces and on more general rearrangement-invariant quasi-Banach function spaces. We discuss necessary conditions for boundedness in terms of the range of the Lebesgue indices and of the kernels of such operators. A sufficient condition for boundedness is obtained in a particular case, see Theorem 3.2. Theorem 4.3 provides a characterization, in terms of the Lorentz space L1/2n,1/2 (R+ ), of the boundedness of TK from L1 × L1 to L1/2 , if K(y, z) = ϕ(|y| + |z|) and ϕ is decreasing. The study of bilinear operators within the context of harmonic analysis was initiated by Coifman and Meyer [2, 3] in the late seventies but recent attention in the subject was rekindled by the breakthrough work of Lacey and Thiele [9, 10] on the bilinear Hilbert transform. The behavior of this operator is still not understood on spaces near L1 × L1 . Although the results obtained in this paper are not applicable to the bilinear Hilbert transform, they suggest that bilinear translation-invariant operators may exhibit behavior at the endpoint L1 × L1 different from that of their linear counterparts on L1 (see Theorems 3.4 and 4.1). An interesting example of an operator of type (1) is given by the measure µ = δ0 (y + z)χ|y|≤1 on Rn × Rn , where δ0 denotes the Dirac delta mass on the diagonal in Rn . This operator (which appeared in the study of bilinear fractional integrals) can be written as f (x − y)g(x + y)χ|y|≤1 dy , B(f, g)(x) = Rn
and maps L1 (Rn ) × L1 (Rn ) to L1/2 (Rn ), as proved independently by Kenig and Stein [8] and Grafakos and Kalton [5]. The bilinear fractional integrals are also operators of the form (1) associated with the singular measures µα = δ0 (y + z)|y|−n+α on Rn × Rn , where 0 < α < n, and they map Lp × Lq → Lr , when 1/p + 1/q = α/n + 1/r.
2. Necessary conditions We begin by exhibiting a general restriction on a set of indices p, q, r for which an operator Tµ of the form (1) is bounded. The next result is analogous to H¨ormander’s [7] in the linear case; see also [6]. Proposition 2.1. Let µ be a nonnegative regular Borel measure on Rn × Rn . Suppose that the bilinear operator Tµ maps Lp (Rn ) × Lq (Rn ) to Lr (Rn ) for some 0 < p, q, r ≤ ∞. Then one has 1/p + 1/q ≥ 1/r. In particular, if Tµ : L1 (Rn ) × L1 (Rn ) → Lp (Rn ), then p ≥ 1/2. Proof. Fix 0 < p, q, r ≤ ∞. By translating µ if necessary, we may assume that there exists a compact set E ⊂ [1, M ]n × [1, M ]n (for some M > 1), such that 0 < µ(E) < ∞.
Vol. 66 (2010)
Bilinear Operators with Positive Kernels
255
n Let x = (x1 , . . . , xn ) in Rn . Taking f (x) = j=1 |xj |−α χ(1,∞)n (x), with n α > 1/p, and g(x) = j=1 |xj |−β χ(1,∞)n (x), with β > 1/q we have, for xj > M + 1, j = 1, . . . , n: n Tµ (f, g)(x) ≥ f (x − y)g(x − z) dµ(y, z) ≥ µ(E) (xj − 1)−(α+β) . E
r
j=1 n
Since Tµ (f, g) ∈ L (R ), this implies that α + β > 1/r, for all α > 1/p and all β > 1/q; i.e., 1/p + 1/q ≥ 1/r. In the endpoint case 1/p + 1/q = 1/r, we prove that the boundedness of the bilinear operator Tµ necessarily implies that the measure µ must be finite. In fact, this result is valid even under the weaker assumption that Tµ is of weak-type (p, q, r). We study this condition in detail in Section 3 where we give an example showing that, in general and contrary to what happens in the linear case, the finiteness of the measure (or the integrability of the kernel) is not a sufficient condition for the boundedness of the associated operator. Proposition 2.2. If µ is a nonnegative regular Borel measure and the operator Tµ : Lp (Rn ) × Lq (Rn ) → Lr,∞ (Rn ) for some 0 < p, q ≤ ∞ satisfying 1/p + 1/q = 1/r, then µ is a finite measure. In particular, if K ≥ 0 and TK : Lp (Rn ) × Lq (Rn ) → Lr,∞ (Rn ), for some 0 < p, q ≤ ∞ with 1/p + 1/q = 1/r, then K ∈ L1 (Rn × Rn ). Proof. We consider first the case 0 < r < ∞. Fix R > 0 such that µ(BR × BR ) > 0, where BR is the ball B(0, R) ⊂ Rn . Then, for every x ∈ BR we have: Tµ (χB2R , χB2R )(x) = µ(B(x, 2R) × B(x, 2R)) ≥ µ(BR × BR ) = λ > 0. Therefore BR Tµ (χB2R , χB2R ) > λ/2 , and |BR | ≤ Tµ (χB2R , χB2R ) > λ/2 2r C r ≤ |B2R |r/p |B2R |r/q λr C r 2r+n = |BR |. λr Hence, for every R > 0, we have that µ(BR × BR ) ≤ 21+n/r C, which proves the result when r < ∞ letting R → ∞. When r = ∞ we have 2 µ(BR × BR ) ≤ Tµ (χB2R , χB2R )L∞ ≤ C χB2R L∞ = C, and the conclusion follows letting R → ∞ as well.
3. Sufficient conditions We now study certain sufficient conditions for boundedness of operators of the form (1). We start with a couple of observations:
256
Grafakos and Soria
IEOT
If K ∈ L1 (Rn × Rn ), then TK : Lp (Rn ) × Lq (Rn ) → Lr (Rn ), where 1 ≤ p, q ≤ ∞ and 1/p + 1/q = 1/r ≤ 1. In fact, this statement can be strengthened as follows: Proposition 3.1. If µ is a nonnegative regular Borel measure and 1/p + 1/q = 1/r ≤ 1, then the following statements are equivalent: (a) Tµ : Lp (Rn ) × Lq (Rn ) → Lr (Rn ). (b) Tµ : Lp (Rn ) × Lq (Rn ) → Lr,∞ (Rn ). (c) µ is a finite measure. Proof. Obviously (a) implies (b) while the fact that (b) implies (c) is proved in Proposition 2.2. Using Minkowski’s integral inequality, we have: f (· − y)g(· − z)r dµ(y, z) Tµ (f, g)r ≤ Rn ×Rn ≤ f (· − y)p g(· − z)q dµ(y, z) Rn ×Rn
= µ(Rn × Rn )f p gq .
It is interesting that this result is false, in general, when 0 < r < 1. We show that there exists K ≥ 0, K ∈ L1 (in fact K ∈ L1 ∩ L∞ ) such that TK does not map L1 × L1 to L1/2,∞ ; see Theorem 3.4. A second observation is that if a kernel K satisfies |K(y, z)| ≤ K1 (y)K2 (z), 1
n
p
n
q
(2) n
r
n
where 0 ≤ Kj ∈ L (R ), then TK : L (R ) × L (R ) → L (R ), whenever 1 ≤ p, q ≤ ∞ and 1/p + 1/q = 1/r. In this case r ≥ 1/2 and K lies in L1 (Rn × Rn ), which is a necessary condition by Proposition 2.2. We now provide a weaker sufficient condition than (2), that yields the boundedness of TK in the nontrivial case 0 < r < 1: Theorem 3.2. Suppose that 1/p + 1/q = 1/r ≥ 1 and ϕ is a nonnegative function on R+ × R+ , decreasing in each variable separately and obeying the estimate: (ϕ(2j1 , 2j2 )2j1 n 2j2 n )r < ∞. j1 ∈Z j2 ∈Z
Let K be a function on Rn × Rn that satisfies |K(y1 , y2 )| ≤ ϕ(|y1 |, |y2 |). Then TK maps Lp (Rn ) × Lq (Rn ) to Lr (Rn ). Proof. For each j1 , j2 integers we set Kj1 ,j2 (y1 , y2 ) = K(y1 , y2 )χIj1 (y1 )χIj2 (y2 ), where Ijl = {2jl < |yl | ≤ 2jl +1 }. Then we have T (f1 , f2 )(x) ≤
j1 ∈Z j2 ∈Z
j1
j2
ϕ(2 , 2 )
2 l=1
I jl
|fl (x − yl )| dyl .
Vol. 66 (2010)
Bilinear Operators with Positive Kernels
257
We raise this expression to the power r and integrate over Rn . As we can pass the power r inside the sum we obtain that |T (f1 , f2 )(x)|r dx ≤ ϕ(2j1 , 2j2 )r Rn
j1 ∈Z j2 ∈Z
×
R
2 n l=1
Ijl
r |fl (x − yl )| dyl dx
and we apply H¨ older’s inequality to control the previous quantity by
p r/p j1 j2 r ϕ(2 , 2 ) |f1 (x − y1 )| dy1 dx C j1 ∈Z j2 ∈Z
×
Rn
Rn
≤ C
|f2 (x − y2 )| dy2 j1
j2 r
×
Rn
= C
Rn
I j2
dx
ϕ(2 , 2 )
j1 ∈Z j2 ∈Z
r/q
q
I j2
I j1
q
I j1
j1 n(p−1)
|f1 (x − y1 )| dy1 2
j2 n(q−1)
|f2 (x − y2 )| dy2 2
p
r/p dx
r/q dx
(ϕ(2j1 , 2j2 )2j1 n 2j2 n )r (f1 Lp f2 Lq )r
j1 ∈Z j2 ∈Z
≤ C (f1 Lp f2 Lq )r .
Remark 3.3. It is easy to see that the hypothesis on K can be equivalently written as |K(y1 , y2 )|r dy1 dy2 < ∞, n n 1−r Rn Rn (|y1 | |y2 | ) and in this case the monotonicity condition on ϕ is replaced by the condition: whenever |y1 | ≤ |y1 | we have |K(y1 , y2 )| ≥ |K(y1 , y2 )| and whenever |y2 | ≤ |y2 | we have |K(y1 , y2 )| ≥ |K(y1 , y2 )|. Under no extra conditions on K, and for the case 0 < r < 1, no positive results can be obtained, as the following result indicates: Theorem 3.4. There exists a nonnegative function K on Rn × Rn such that, if X is a rearrangement invariant (r.i.) quasi Banach space, then TK : L1 × L1 → X if and only if L∞ is a subspace of X. Proof. We work the details in the case n = 1, although the construction can be easily extended to Rn for n ≥ 2. For a < 0 and r > 0 set fa,r (x) =
1 χ(a−r,a+r) (x) . 2r
Also let a = {(x − a, x) : x ∈ R}
258
Grafakos and Soria
IEOT
be the line of slope 1 passing through the point (0, a). Then for almost all (x − a, x) ∈ R2 we have 1 K(y, z) dzdy TK (fa,r , f0,r )(x) = 2 4r (x−a−r,x−a+r) (x−r,x+r) and from this we deduce that TK (fa,r , f0,r )(x) → K(x − a, x)
(3)
as r → 0. In other words, (3) holds for almost every a < 0 and almost every point on the line a with respect to one-dimensional Lebesgue measure. For each k ∈ N, we construct a sequence of disjoint rectangles Rk as in Figure 1 with base length equal to 1/k 3 , height equal to 2k, and longest side parallel to the line a . We arrange that all these rectangles touch each other and are contained in the right angle −|x| ≤ y ≤ |x| on the (x, y) plane. We let P (Rk ) be the intersection of the smallest strip containing the longest side of Rk and the negative y-axis. Set ∞
R=
Rk
k=1
and K = χR . Then K1 = |R| =
∞
2/k 2 < ∞.
k=1
a
R3 R2 R1
P (R1 )
...
P (R2 ) (0, a) u P (R3 )
Figure 1.
Vol. 66 (2010)
Bilinear Operators with Positive Kernels
259
Suppose that for this kernel K the following estimate holds: TK (f, g)X ≤ Cf 1 g1 for all f, g nonnegative in L1 (Rn ). Then for any k ≥ 1, (3) holds √ ∞ functions −3 for almost all − 2 k=1 k < a < 0, with (0, a) in P (Rk ) (in particular for one such a), and for almost all points (x − a, x) in a . Since χ(0,k) (x) ≤ χRk (x − a, x) ≤ K(x − a, x), for all real x, using Fatou’s lemma and (3) we deduce that χ(0,k) X ≤ lim inf TK (fa,r , f0,r )X ≤ Cfa,r 1 f0,r 1 = C, r→0
for every k ∈ N. Thus, the fundamental function ϕX of X (see [1]) is bounded, which is equivalent to saying that L∞ is a subspace of X. Conversely, if L∞ is a subspace of X and K ∈ L∞ , then it is clear that TK : L1 × L1 → L∞ and thus TK maps L1 × L1 to X. Remark 3.5. If K ∈ L∞ , then we have just observed that, trivially, TK : L1 ×L1 → L∞ . Therefore, if K ∈ L1 ∩L∞ , 1/p+1/q ≤ 1 and 0 ≤ θ ≤ 1, using bilinear interpolation [1] for this estimate and Proposition 3.1, we obtain:
TK : Lp /(p −θ) × Lq /(q −θ) → Lpq/(θ(p+q)) . In particular, TK : Lp × Lp → Lp/2 whenever for 2 ≤ p ≤ ∞, and TK : Lp × Lp → Lp /2 whenever 1 ≤ p ≤ 2. Consequently, for K ∈ L1 ∩ L∞ such that TK : L1 × L1 → Lp for some p ≥ 1/2 (cf. Proposition 2.1), the boundedness TK : L1 × L1 → Lq holds for every q in [p, ∞]. It is then an interesting question to determine the least possible value of p in the interval [1/2, ∞], for which such an operator is bounded from L1 × L1 to Lp . We have indicated that there are examples showing that we can have the best possible situation (boundedness on L1/2 when K is a tensor product of two kernels in L1 ) and also the worst case (only bounded on L∞ , as in Theorem 3.4). See also Proposition 4.2. Modifications of bilinear fractional integrals also provide examples in the intermediate cases.
4. Other examples and estimates Well-known examples of bilinear singular integral operators, such as the bilinear Riesz transforms [6], indicate that boundedness from L1 × L1 to L1/2 may not hold although boundedness from L1 × L1 to L1/2,∞ is valid. These operators have kernels that change sign but the next result shows that there exist positive measures that provide examples of kernels with the same property. This situation should be contrasted with its linear version that fails: if a convolution operator with a positive Borel measure on Rn maps L1 (Rn ) to L1,∞ (Rn ), then the measure is finite and therefore the operator maps L1 (Rn ) to itself!
260
Grafakos and Soria
IEOT
Theorem 4.1. There exists a nonnegative regular finite Borel measure µ on R × R with the property that Tµ maps L1 × L1 to L1/2,∞ but does not map L1 × L1 to L1/2 . Proof. We first observe that if we want Tµ : L1 × L1 → L1/2,∞ , then necessarily µ must be a finite measure (Proposition 2.2). We choose a positive sequence {λj }j∈Z ∈ 1/2,∞ \ 1/2 and define µ = j λj δaj , where aj = (j, j) and δaj is the Dirac mass at aj . Clearly λj < ∞ . µ(R × R) =
j
Then, Tµ (f, g)(x) = j λj f (x − j)g(x − j). Let also Dµ (h)(x) = 1/2,∞ λ h(x − j). Using that {λ and [4, Lemma 3.5] we have j }j∈Z ∈ j j 1/2 1/2,∞ that Dµ : L →L , and hence,
Tµ (f, g)1/2,∞ = Dµ (f g)1/2,∞ ≤ Cf g1/2 ≤ Cf 1 g1 . / 1/2 , by [4, Theorem 3.1] (see also [11]), we have that Now, since{λj }j∈Z ∈ Dµ is not of strong-type L1/2 and, as before, Tµ L1 ×L1 →L1/2 ≥ sup f
Tµ (f, f )1/2 Dµ (h)1/2 = sup = ∞. 2 f 1 h1/2 h
We now consider some particular cases of kernels, defined in terms of a special function ϕ. The first example is K(y, z) = ϕ(y + z), where ϕ : R n → R+ . Proposition 4.2. Let 1 ≤ α ≤ ∞ and ϕ ∈ Lα (Rn ) be a positive function. Set K(y, z) = ϕ(y + z). Then, TK : Lp (Rn ) × Lq (Rn ) → Lr (Rn ),
(4)
where
1 1 1 1 = + + − 2 ≤ 1. r p q α Moreover, if ϕ ∈ Lα ∩ L∞ , then 0≤
TK : L1 (Rn ) × L1 (Rn ) → Lr (Rn ),
(5)
for every α ≤ r ≤ ∞ and the result is false, in general, if r < α. Proof. The main observation is that TK (f, g)(x) = (f ∗ g ∗ ϕ)(2x), and hence the result is a reformulation of Young’s inequality: TK (f, g)r ≤ f p g ∗ ϕβ , if 1 ≤ p ≤ β and 1/r = 1/p + 1/β − 1. Similarly, g ∗ ϕβ ≤ gq ϕα ,
if 1 ≤ q ≤ α and 1/β = 1/q + 1/α − 1, which proves (4). If p = q = 1 and ϕ ∈ L∞ , then TK : L1 (Rn ) × L1 (Rn ) → Lα (Rn ) that, together with the estimate TK : L1 (Rn ) × L1 (Rn ) → L∞ (Rn ), gives (5).
Vol. 66 (2010)
Bilinear Operators with Positive Kernels
261
To finish, take r < α, and define ϕ(t) = t(−1−ε)/α χ(1,∞) (t) ∈ Lα ∩ L∞ , where 0 < ε < α/r − 1. Set f = g = χ(0,1) . Then, if x > 3/2:
x ∞ χ(x−1,x) (y)(z + y)(−1−ε)/α dy dz TK (f, g)(x) = x−1 1−z
x ∞ ≥ χ(x−1,x) (y) dy (z + x)(−1−ε)/α dz x−1 1−z x ≥ (z + x)(−1−ε)/α dz ≥ (2x)(−1−ε)/α . x−1
Therefore TK (f, g)r = ∞. This proves the result if n = 1. The n-dimensional case follows by adapting this idea. Another example of interest comes when the kernel is defined as K(y, z) = ϕ(|y| + |z|), where ϕ : R+ → R+ is a decreasing function. We will study the behaviour of TK at the endpoints p = q = 1 and r = 1/2, for which we give a complete characterization in terms of the Lorentz space L1/2n,1/2 (R+ ): Theorem 4.3. Let ϕ : R+ → R+ be a decreasing function and define K(y, z) = ϕ(|y| + |z|). Then, TK : L1 (Rn ) × L1 (Rn ) → L1/2 (Rn ) if and only if ϕ ∈ L1/2n,1/2 (R+ ). Proof. Assume that TK : L1 (Rn ) × L1 (Rn ) → L1/2 (Rn ). Set Rk = {(y, z) ∈ Rn × Rn : 2k < |y| + |z| ≤ 2k+1 }, so that |Rk | ≈ 22kn . Fix 2j−1 < |x| ≤ 2j and δ ≤ 2l , with l ≤ j − 2. Consider also the functions f (x) = g(x) = χ{|x|<δ} (x). Then (TK (f, g)(x))1/2 dx ≤ Cδ n . (6) Rn
Observe that {y ∈ Rn : |x − y| < δ} × {z ∈ Rn : |x − z| < δ} ⊂ Rj−1 ∪ Rj ∪ Rj+1 , j+2
j−1
and |y| + |z| ≥ 2|x| − 2δ ≥ 2 . since |y| + |z| ≤ 2δ + 2|x| ≤ 2 Discretizing the operator, and using (7), we obtain: f (x − y)g(x − z)ϕ(|y| + |z|) dydz TK (f, g)(x) = k∈Z
Rk
≥ ϕ(2j+2 )
j+1 k=j−1
= Cn ϕ(2j+2 )δ 2n ≥ Cn ϕ(8|x|)δ 2n .
Rk
f (x − y)g(x − z) dydz
(7)
262
Grafakos and Soria
IEOT
Thus, by (6) and the previous estimate: (TK (f, g)(x))1/2 dx ≥ C δ n Cδ n ≥ C {|x|>2δ}
ϕ(8|x|) dx,
{|x|>2δ}
and hence,
∞
ϕ(t) tn
16δ
Letting δ → 0 we finally obtain:
1/2
ϕ1/2n,1/2 =
∞
dt ≤ C . t
ϕ(t) tn
0
dt < ∞. t
Conversely, since ϕ(|y| + |z|) ≤ ϕ(|y|) and ϕ(|y| + |z|) ≤ ϕ(|z|), we have
ϕ(|y| + |z|) ≤ ϕ(|y|) ϕ(|z|) , and therefore K is bounded from above by the tensor product of two functions in L1 (Rn ), since 1/2 ϕ(|y|) dy = Cϕ1/2n,1/2 < ∞, Rn
which implies the result (see (2)).
Remark 4.4. By Proposition 2.2 we know that the boundedness of TK in the previous theorem would imply that K ∈ L1 (Rn × Rn ). This condition is, in fact, equivalent to ϕ ∈ L1/2n,1 (R+ ): K1 =
ϕ(|y| + |z|) dydz
Rn
Rn ∞ ∞
ϕ(s + t)t
0
= = ≈ =
dt sn−1 ds
∞ ∞ n−1 ϕ(u)(u − s) du sn−1 ds C 0 s u
∞ n−1 n−1 ϕ(u) (u − s) s ds du C 0 ∞0 du ϕ(u)u2n u 0 ϕ1/2n,1 .
= C
n−1
0
Since L1/2n,1/2 (R+ ) L1/2n,1 (R+ ), we observe that Theorem 4.3 gives a stronger condition. We end by giving an analogous version of Proposition 3.1 in the case of linear convolution operators that, surprisingly enough, seems to be missing from the literature. For K ≥ 0, we define the averaging operator: 1 K(y) dy. A(K)(x, r) = |B(x, r)| B(x,r)
Vol. 66 (2010)
Bilinear Operators with Positive Kernels
263
We observe that A(K)(x, ·)L∞ = M (K)(x), where M is the Hardyr Littlewood maximal function. We use the following notation for the mixed norm space X[Y ]: F X[Y ] denotes the quasinorm in X of the function F (x, ·)Y . We consider first the case p = 1: Proposition 4.5. Let K ≥ 0, and TK (f )(x) =
Rn
f (x − y)K(y) dy.
Then, the following statements are equivalent: ∞ (a) A(K) ∈ L1,∞ x [Lr ]. ∞ 1,∞ (b) A(K) ∈ Lr [Lx ]. (c) K ∈ L1 . (d) TK : L1 → L1,∞ . (e) TK : L1 → L1 . Moreover, A(K)L∞ [Lx1,∞ ] ≈ A(K)Lx1,∞ [L∞ ] ≈ K1 . r
r
Proof. It is well known that A(K)L∞ [Lx1,∞ ] ≤ A(K)Lx1,∞ [L∞ ] ≈ K1 , r r i.e., (a) ⇔ (c) ⇒ (b). Taking r > 0 such that B(0,r/2) K(y) dy > 0 implies C n K ≤ A(K)(·, r)1,∞ rn , r ≤ C TK (χB(0,r) ) ≥ K B(0,r/2) B(0,r/2) hence (b) ⇒ (c). Clearly (c) ⇒ (e) ⇒ (d). Finally, if (d) holds, taking f = χB(0,r) , we obtain that TK (f )(x) = Crn A(K)(x, r), thus TK (f )1,∞ = Crn A(K)(·, r)1,∞ ≤ Crn . Therefore (d) ⇒ (b).
Remark 4.6. (i) It is easy to see that if 1 < p < ∞, then we also have that: K ∈ L1 ⇔ TK : Lp → Lp,∞ ⇔ TK : Lp → Lp . This should be compared to the bilinear case (cf. Theorem 4.1), where weaktype estimates do not imply, in general, the strong-type boundedness of the operator. (ii) For an r.i. Banach function space X for which the maximal operator M maps X to itself (e.g., X = Lp , 1 < p ≤ ∞), the equivalences: A(K)Xx [L∞ ≈ A(K)L∞ ≈ KX , r ] r [Xx ] are easy consequences of Fatou’s Lemma. Remark 4.7. The results of this article concerning positive bilinear operators easily adapt to the setting of m-linear positive convolution operators when m ≥ 3. The precise formulation of these statements and their proofs are analogous to the case m = 2 and are omitted.
264
Grafakos and Soria
IEOT
References [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc. 1988. [2] R. R. Coifman and Y. Meyer, Commutateurs d’int´ egrales singuli`eres et op´erateurs multilin´eaires, Ann. Inst. Fourier (Grenoble) 28 (1978), 177–202. [3] R. R. Coifman and Y. Meyer, Au d´el` a des op´erateurs pseudo-diff´ erentiels, Ast´erisque No. 57, Societ´e Mathematique de France, 1979. [4] L. Colzani, Translation invariant operators on Lorentz spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 257–276. [5] L. Grafakos and N. Kalton, Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001), 151–180. [6] L. Grafakos and R. H. Torres, Multilinear Calder´ on–Zygmund theory, Adv. in Math. 165 (2002), 124–164. [7] L. H¨ ormander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93–140. [8] C. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), 1–15. [9] M. Lacey and C. Thiele, Lp bounds for the bilinear Hilbert transform, p > 2, Ann. of Math. 146 (1997), 693–724. [10] M. Lacey and C. Thiele, On Calder´ on’s conjecture, Ann. of Math. 149 (1999), 475–496. [11] D. M. Oberlin, Translation-invariant operators on Lp (G), 0 < p < 1, Michigan Math. J. 23 (1976), 119–122. Loukas Grafakos Dept. of Mathematics University of Missouri Columbia MO 65211 USA e-mail: [email protected] Javier Soria Dept. Appl. Math. and Analysis University of Barcelona E-08007 Barcelona Spain e-mail: [email protected] Submitted: May 9, 2009.
Integr. Equ. Oper. Theory 66 (2010), 265–282 DOI 10.1007/s00020-010-1748-0 Published online January 28, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
Convergence of Collocation Method with Delta Functions for Integral Equations of First Kind Urve Kangro Abstract. Integral equations of first kind with periodic kernels arising in solving partial differential equations by interior source methods are considered. Existence and uniqueness of solution in appropriate spaces of linear analytic functionals is proved. Rate of convergence of collocation method with Dirac’s delta-functions as the trial functions is obtained in case of uniform meshes. In case of an analytic kernel the convergence rate is exponential. Mathematics Subject Classification (2010). Primary 45B05; Secondary 65R20. Keywords. Interior source method, integral equations, scattering.
1. Introduction This paper provides a theory of existence and approximation for a class of first kind Fredholm equations arising in two-dimensional electromagnetic and acoustic scattering problems and elsewhere. Usually, the kernels of such equations are weakly singular, and often the integral equations are of second kind, so that solvability is not hard to establish. However in some recent applications it has become necessary to investigate the case of first kind equations with analytic kernels and that is the subject of this paper. Integral equations with analytic kernels arise when solving elliptic boundary value problems using interior source methods, where the solution is represented as an integral over some curve different from the boundary of the domain. Such methods are proposed especially for electromagnetic scattering problems [1], [2] and many references therein, but also for exterior problems in acoustics [8]. These This work was supported by Estonian Science Foundation under grant No. 5221 and by Estonian Targeted Financing Project SF0180039s08.
266
Kangro
IEOT
methods have also been called discrete sources methods or fictitious current methods. In [11] Vainikko provided a theoretical framework for analysing a wide class of operator equations which, in fact, includes the weakly singular integral equations mentioned above as a special case. However, Vainikko’s theory, which is set in Sobolev spaces, is not directly applicable to analytic kernels. A main goal of this work is to extend the theory to cover the analytic situation. Our theory also covers integral equations which arise, for example, when using interior source methods for domains with nonsmooth boundaries. In this case the kernel is not analytic and one has to look for solutions in spaces of periodic analytic functionals. In the theory of [11], the kernel is represented as a sum of functions in which the dominant part generally contains the singular behavior. In the theory presented here the dominant part can be analytic. The smoothness of the kernel is often expressed by decay rates of its Fourier coefficients. Here we consider the case of exponential decay of the coefficients, instead of the algebraic decay of [11]. To solve the integral equation we propose to use the collocation method with Dirac’s δ-functions as the trial functions. This method has the advantage of being the simplest method to use (to set up the matrix of the discrete system, one just has to evaluate the kernel at a given set of points; no integration is needed). Also, when the integral equation is obtained by an interior source method from some differential equation, the resulting approximate solution of the integral equation generates a very simple approximate solution for the differential equation. For smooth data the convergence is very fast (exponential in the number of variables), and the method converges even for boundaries with corners, if one chooses the interior boundary, the supports of the δ-functions and the collocation points carefully. We will work out the details for the application to scattering problems in [4]. The convergence of the collocation method has been considered e.g. in [6] which considers the dependence of the convergence on the node distribution, but no convergence rates have been obtained. It is pointed out in [9] that interior source methods work best when the solution of the differential equation can be analytically extended as a solution of the same equation up to the interior boundary. Extendability of solutions of two-dimensional scattering problems is considered in [3]. From the results of current paper it follows that in certain cases the method can also be used without the extendability condition; in fact, it is also possible to use the interior source method in case of nonanalytic boundary, but in this case one looses the exponential convergence. For Laplace equation in interior domains exponential convergence of similar method has been proved in [5]. Dirac’s δ-functions have been used as trial functions in conventional boundary element methods as well (see e.g. [10]). Generally in this case the collocation (on the same mesh) cannot be used. Therefore in [10] splines
Vol. 66 (2010)
Collocation Method with Delta Functions
267
are used for test functions, which makes this method more difficult to use than ours. Also, when using δ-functions in boundary element methods, the approximation of the solution of the corresponding differential equation is poor near the boundary, because the solution blows up at the mesh points on the boundary. Interior source methods are free of this shortcoming. The discrete equations for the collocation method with δ-functions are actually the same as the equations for the quadrature method with equal weights, which in case of integral equations of second kind or weakly singular integral equations has been well analysed [11],[7]. In contrast, we discuss the case of integral equation of first kind with analytic kernel. Also the interpretation of the approximate solution is different in our case, and the norms we use to estimate the error are also different. The subsequent contents of the paper are as follows: in the next section we present an example to show how equations with analytic kernels arise in potential theory. Then we construct the spaces and operators for which the existence and other results will be obtained. Next we discuss an approximation procedure which is convenient for implementation and obtain an error estimate. The error estimate shows that, in suitable norms, convergence is exponentially fast in terms of the number of discretization nodes.
2. Interior source methods for solving scattering problems In this section we will present a simple example of an interior source method, where a first kind integral equation with an analytic kernel appears. Consider a two-dimensional acoustic scattering problem U + k 2 U = 0 in Ω,
(2.1)
U (x) = F (x), x ∈ Γ, x 1 · ∇U + ikU = o as |x| → ∞, |x| |x|
(2.2) (2.3)
where Ω ⊂ R2 is a complement of a compact set (the scattering body) with a smooth boundary Γ, and F (x) is a given incident field. An integral representation of a solution of the Helmholtz equation in Ω satisfying the outgoing wave condition (2.3) is given by (1) u(y)H0 (k|x − y|)dsy , x ∈ Ω, (2.4) U (x) = γ
(1) H0
is the zeroth order Hankel function of the first kind and γ is a where closed curve with γ Ω = ∅. If u satisfies the integral equation (1) H0 (k|x − y|)u(y)dsy = F (x), x ∈ Γ, (2.5) γ
then (2.4) gives the solution of (2.1)–(2.3).
268
Kangro
IEOT
In conventional potential theory γ = Γ and the equation (2.5) is weakly singular. In interior source methods, however, γ is chosen strictly inside the scattering body. Therefore smoothness of the kernel is determined by smoothness of the boundary and the auxiliary curve γ, and if they are analytic, the kernel is also analytic. Note that the kernel is a function of |x − y| and therefore may be expected to be close to a convolution kernel. In fact, if Γ and γ are concentric circles, then the integral equation is exactly of convolution type (using the polar angles as variables), and in general, it is possible to choose γ and parametrizations of the boundaries so that the dominant part (the most singular part) of the kernel is of convolution type. It may be nontrivial, since one needs to map the boundary into a real line so that the mapping is conformal in some (possibly one-sided) neighborhood of the boundary. This is simple if the boundary has an analytic parametrization. If the boundary is not analytic, but is simply connected, then such a mapping exists, but it may be nontrivial to find. For polygons, the Schwarz-Christoffel mapping can be used. In general, it can be shown that equation (2.5), under certain assumptions and after a suitable change of variables, can be rewritten as
2π
0
(κ0 (t − s) + κ1 (t − s)a1 (t, s) + a2 (t, s))v(s)ds = f (t),
t ∈ [0, 2π], (2.6)
where κ0 (t − s) = ln[2(cosh η − cos(t − s) )] and κ1 (t − s) = (cosh η − cos(t − s) ) ln[2(cosh η − cos(t − s) )], where η is a parameter characterizing the distance between Γ and γ, and a1 and a2 are 2π-biperiodic (2π-periodic with respect to both arguments), and smoother (in some sense) than κ0 and κ1 . For details see [4]. We note here that the existence and uniqueness of solution for (2.5) can easily be obtained in ordinary function spaces (e.g. in L2 (γ) or in C n (γ)), if the solution of the exterior problem (2.1)-(2.3) can be extended as a solution of the Helmholtz equation (2.1) up to the interior curve γ (under the condition that −k 2 is not an eigenvalue for with Dirichlet boundary conditions in the domain bounded by γ). If the solution of the exterior problem can be extended to a neighborhood of γ, and γ itself is analytic, then the solution of the integral equation is also analytic. For the extension results for the Helmholtz equation see [3], where an algorithm is given to determine the region, where the solution of the Helmholtz equation can be extended.
Vol. 66 (2010)
Collocation Method with Delta Functions
269
3. The integral equation In the following we will consider the integral equation (using the notation of [11]) k Au := Aj u = f, (3.1) j=0
where f is 2π-periodic, and the operators Aj , j = 0, . . . , k are of the form 2π (Aj u)(t) = κj (t − s)aj (t, s)u(s)ds, j = 0, . . . , k. (3.2) 0
Assume that κj , j = 0, . . . , k are 2π-periodic, a0 ≡ 1, and aj , j = 1, . . . , k are 2π-biperiodic. The integrals here may actually stand for linear functionals on certain periodic function spaces, e.g. κ0 may actually be a δ-function. In the case of interior source methods κj and aj are usually smooth functions, but the solution u may only exist in some space of linear functionals on analytic periodic functions. In both these cases the integrals have to be understood as the dual products in appropriate spaces. We will present the smoothness assumptions in terms of Fourier coefficients. We define the Fourier coefficients of an integrable 2π-periodic function v by 2π 1 v(t)e−int dt, n ∈ Z. vˆ(n) = 2π 0 Then v(t) = vˆ(n)eint . If v is a periodic distribution or a continuous linear n∈Z
functional on a space of periodic analytic functions, then the integral above should be replaced by the corresponding duality pairing. In this case the Fourier coefficients may be exponentially growing, and the convergence of the Fourier series will take place in appropriate spaces of linear functionals. For n ∈ Z, λ, α ∈ R we denote cn (λ, α) = e|n|λ max{1, |n|}α . We assume that there exist η, θ, α1 , α2 , nonnegative λ1 , λ2 and positive constants ε, C, C1 , C2 such that for all m, n ∈ Z, j = 1, . . . , k κ0 (n)| ≤ C2 cn (−η, −θ), C1 cn (−η, −θ) ≤ |ˆ
(3.3)
|ˆ κj (n)| ≤ Ccn (−η, −θ − ε),
(3.4)
|ˆ aj (m, n)| ≤ Ccm (−λ1 , −α1 )cn (−λ2 , −α2 ).
(3.5)
These assumptions are motivated by the example considered in Section 2. Assumption (3.5) guarantees that one can multiply by aj in the function spaces we will be working with; usually aj are smoother than needed. However, there are cases (e.g. using the interior source method for domains with nonanalytic boundaries), where the exact smoothness of aj is needed to determine the smoothness of the solution and the convergence rate of the method. Similar assumptions hold for integral equations obtained by using other integral representations of the solution of the scattering problem (2.1)-(2.3),
270
Kangro
IEOT
and also for integral equations obtained from other elliptic partial differential equations in (analytic) domains by representing the solution in an integral form, where the integral is taken over a suitable curve, which does not intersect the domain or the boundary of the domain. Conventional boundary integral equations also fall under these assumptions: they correspond to the case η = 0. We will consider the integral operators Aj on spaces Aλ,α , λ, α ∈ R defined by ∞ int 2 v (n)cn (λ, α))n=−∞ ∈ l Aλ,α = v(t) = vˆ(n)e (ˆ n∈Z
with the norm
v λ,α =
1/2 2
|ˆ v (n)cn (λ, α)|
.
n∈Z
For λ = 0 this is the space of 2π-periodic functions in the Sobolev space H α (0, 2π). For λ > 0 this is the space of 2π-periodic (real) analytic functions which can be extended to be complex analytic in the strip Dλ = {t + iτ | t, τ ∈ R, |τ | < λ} with the traces on the edges of the strip v(· ± iλ) ∈ H α (0, 2π). For λ < 0 the space is the dual space of A−λ,−α . We use lexicographic ordering ≺ for the indices of these spaces, i.e. (λ, α) ≺ (λ , α ) means that either λ < λ , or λ = λ and α < α . Obviously (λ, α) ≺ (λ , α ) if and only if Aλ,α ⊃ Aλ ,α . Recall that a continous linear operator between two Banach spaces is called a Fredholm operator of index 0, if its range is closed and the dimension of its null space and the codimension of its range are finite and equal. A continuous linear operator is a Fredholm operator of index 0 if and only if it can be represented as a sum of an invertible operator and a compact operator in the same spaces. If A : X → Y is a Fredholm operator of index 0, then the uniqueness of the zero solution of either Au = 0 in X or A∗ u = 0 in Y implies the unique solvability of Au = f for any f ∈ Y . The next theorem gives sufficient conditions for the operator A in the integral equation (3.1) to be a Fredholm operator of index 0 in appropriate function spaces. Theorem 3.1. Let the assumptions (3.3)-(3.5) be satisfied. Let λ, α satisfy the following conditions: i) (|λ|, max{ 12 , |α|}) ≺ (λ2 , α2 − 12 ), ii) (|λ + η|, max{ 12 , |α + θ|}) ≺ (λ1 , α1 − 12 ). Then the operator A : Aλ,α → Aλ+η,α+θ is a Fredholm operator of index 0.
ˆ 0 (n)ˆ v (n), we have A0 ∈ L(Aλ,α , Aλ+η,α+θ ) and Proof. Since A 0 v(n) = κ −1 A0 ∈ L(Aλ+η,α+θ , Aλ,α ) for any λ, α ∈ R by assumption (3.3). We will show in Section 5 that assumptions (3.4)-(3.5) together with i) and ii) imply that Aj , j = 1, . . . , k are compact operators from Aλ,α to Aλ+η,α+θ . The claim of the theorem then follows.
Vol. 66 (2010)
Collocation Method with Delta Functions
271
Corollary 3.2. Let the assumptions (3.3)-(3.5) be satisfied. Let (λ0 , α0 ) satisfy the assumptions i) and ii) of Theorem 3.1 and additionally, either the equation Au = 0 has only the zero solution in Aλ0 ,α0 , or the equation A∗ u = 0 has only the zero solution in A−λ0 −η,−α0 −θ . Then for any (λ, α) satisfying the assumptions i) and ii) of Theorem 3.1, and for any f ∈ Aλ+η,α+θ , the integral equation Au = f has a unique solution in Aλ,α . Proof. First note that the uniqueness of the zero solution of Au = 0 in Aλ0 ,α0 and A∗ u = 0 in A−λ0 −η,−α0 −θ are equivalent. For (λ, α) (λ0 , α0 ) the uniqueness of the zero solution of the homogeneous equation in Aλ,α follows from the uniqueness of the zero solution of the equation in Aλ0 ,α0 . For (λ, α) (λ0 , α0 ) the uniqueness of the zero solution of the adjoint homogeneous equation in A−λ−η,−α−θ follows from the uniqueness of the zero solution of the adjoint equation in A−λ0 −η,−α0 −θ . For any (λ, α) satisfying the assumptions i) and ii) of Theorem 3.1, and for any f ∈ Aλ+η,α+θ , the unique solvability of the integral equation Au = f in Aλ,α now follows. In practice, for checking the uniqueness, it may be preferable to choose (λ0 , α0 ) = (0, 0) or (λ0 , α0 ) = (−η, −θ). Then one has to check the uniqueness of the zero solution of either Au = 0 or A∗ u = 0 in L2 (0, 2π). In case of the interior source methods, often the uniqueness of the solution of the integral equation can be obtained by using the uniqueness results for the solution of the boundary value problem for the differential equation.
4. The collocation method We look for solutions of the integral equation (3.1) in the form u(s) =
N
cl δ(s − sl ),
sl ∈ [0, 2π]
l=1
, where δ(s − sl ) are Dirac’s δ-functions with supports at sl , and require that the equation (3.1) is exactly satisfied at points tm ∈ [0, 2π], m = 1, . . . , N . Then one has to solve the following linear system of equations N k
κj (tm − sl )aj (tm , sl )cl = f (tm ),
m = 1, . . . , N.
(4.1)
j=0 l=1
Note that the collocation method with delta-functions can be used without explicitly knowing aj and κj , j = 0, . . . , k. In fact, let the original equation be γ
K(x, y)u(y)dsy = F (x),
x∈Γ
, with γ and Γ some closed curves. Then the collocation method is N l=1
K(xm , yl )cl = F (xm ),
m = 1, . . . , N
272
Kangro
IEOT
, with xm , m = 1, . . . , N and yl , l = 1, . . . , N some points on Γ and γ respectively. On the other hand, to choose the points on the curves well, it is necessary to know parametrizations of Γ and γ, which transform the integral equation into a form (3.1). The collocation points xm , m = 1, . . . , N and the supports of the delta-functions yl , l = 1, . . . , N should then be chosen as the images of uniform meshes in the parameter space. Generally, the collocation method is very sensitive on the choice of the collocation points and the location of the supports of the δ-functions. In the following we assume that tl = sl = 2πl N , l = 1, . . . , N . Denote N cl δ(s − sl ) cl ∈ C . XN = l=1
1 −insl 1 −i(n+N p)sl e = 2π e Since the n-th Fourier coefficient of δ(s − sl ) is 2π for any p ∈ N, the Fourier coefficients of elements of XN are periodic with period N . Linear independence of the δ-functions now implies ∞ ins ˆ(n + N p) = u ˆ(n), p ∈ Z . XN = u = u ˆ(n)e u n=−∞
We will also use notations N N ZN = n ∈ Z − < n ≤ , 2 2
YN =
n∈ZN
dn eint dn ∈ C .
Then the approximate equation (4.1) can be written as QN AuN = QN f,
uN ∈ XN .
(4.2)
where QN is the trigonometric interpolation operator from Aλ+η,α+θ onto YN . The interpolation operator is well defined if (λ + η, α + θ) (0, 12 ). The convergence of the collocation method is given in the following theorem. Theorem 4.1. Let the assumptions of Corollary 3.2 be satisfied. Additionally, let κ0 satisfy κ ˆ 0 (n) = Ccn (−η, −θ), (4.3) for some C = 0, and let λ, α satisfy the following conditions: i) (λ, α) ≺ (0, − 12 ); ii) (λ + η, α + θ) (0, 12 ). Let f ∈ Aλ∗ +η,α∗ +θ be given, where λ∗ , α∗ satisfy the assumptions i) and ii) of Theorem 3.1, and let (λ, α) ≺ (λ∗ , α∗ ). Then for N large enough the discrete equation (4.2) is uniquely solvable and its solution uN satisfies
uN − u λ,α ≤ C inf u − v λ,α v∈XN
≤ C u λ∗ ,α∗ max eλN N α , e(λ−λ∗ )N/2 N α−α∗ ,
(4.4)
Vol. 66 (2010)
Collocation Method with Delta Functions
273
and
AuN − f λ+η,α+θ ≤ C f λ∗ +η,α∗ +θ max eλN N α , e(λ−λ∗ )N/2 N α−α∗ ,
(4.5) where u ∈ Aλ∗ ,α∗ is the unique solution of Au = f , and C, C and C do not depend on N . Remark 4.2. Conditions i) and ii) imply that (η, θ) (0, 1). It means that κ0 must be smooth enough for A0 to transform δ-functions into at least continuous functions, otherwise the method cannot be applied. Condition i) guarantees that XN ⊂ Aλ,α and condition ii) implies Aλ+η,α+θ ⊂ C[0, 2π]. Remark 4.3. The assumption (4.3) seems very restrictive. Nevertheless, it can be satisfied (redefining κ0 and the other parts, if necessary), if the most singular part of the original integral equation behaves as |x − y|β if β is not a nonnegative even number, or |x − y|β ln |x − y| if β is a nonnegative even number, and x and y are points on (possibly different) smooth curves, which can be parametrized “in the same way” (by a conformal mapping of the region involved, see [4]). This covers integral equations arising from solving two-dimensional elliptic differential equations. But (4.3) can not be satisfied κ0 (n)). In this case serious if the most singular part is odd (then κ ˆ 0 (−n) = −ˆ problems with the stability of the collocation method (4.1) may occur. The condition (4.3) can actually be weakened to: there is an increasing sequence of natural numbers Nk , k ∈ N, for which κ ˆ (n + N p) (4.6) 0 k ≥ Ccn (−η, −θ) ∀n ∈ ZNk p∈Z with some C > 0. In this case one should use for discretization only spaces XNk , YNk with dimensions Nk , k ∈ N. The same convergence results will hold with Nk substituted for N in (4.4) and (4.5). For example, if κ0 is odd, then for even N the method does not work, but (4.6) may still hold for Nk = 2k + 1. Outline of proof of Theorem 4.1. We use a basic result in the theory of projection methods (see e.g. Theorems 13.6 and 13.7 in [7]): Theorem 4.4. Let X and Y be Banach spaces, S : X → Y a bounded linear operator with a bounded inverse and B : X → Y a compact operator with S + B injective. Let f ∈ (S + B)(X) be given. Let XN ⊂ X and YN ⊂ Y with dim XN = dim YN = N be given and let QN : Y → YN be a projection operator. Assume that the finite dimensional operators SN := QN S : XN → YN −1 are invertible for N large enough and SN QN S : X → X are uniformly bounded. Then for N large enough the discrete equation QN (S+B)uN = QN f is uniquely solvable and its solution uN satisfies
uN − u X ≤ C inf u − v X , v∈XN
where u is the unique solution of (S + B)u = f .
274
Kangro
IEOT
k We take X = Aλ,α , Y = Aλ+η,α+θ , S = A0 and B = j=1 Aj . Compactness of Aj , j = 1, . . . , k is proved in Section 5. Boundedness of the interpolation projector in Aλ+η,α+θ is proved in Section 6. We also have to show that for N large enough the operators A0,N = QN A0 : XN → YN are invertible and A−1 0,N QN A0 : Aλ,α → Aλ,α are uniformly bounded. This is also done in Section 6. To obtain the rate of convergence, one has to compute the number inf v∈XN u − v λ,α , which is estimated in Section 7. The convergence of AuN immediately follows from the fact that A ∈ L(Aλ,α , Aλ+η,α+θ ).
5. Compactness of Aj In this section we will prove that under the assumptions of Theorem 3.2, Aj , j = 1, . . . , k are compact operators from Aλ,α to Aα+θ,λ+η . First we will prove a compact embedding result for the spaces. Lemma 5.1. If (λ, α) ≺ (γ, β), then the space Aγ,β is compactly embedded into the space Aλ,α . Proof. Let I : Aγ,β → Aλ,α be the natural injection operator (i.e. it preserves the Fourier coefficients). For N ∈ N define IN : Aγ,β → Aλ,α by IN v =
N
vˆ(n)eint .
n=−N
Clearly IN are compact operators. Since |ˆ v (n)e|n|λ |n|α |2
(I − I N )v 2λ,α = |n|>N
≤
sup e|n|(λ−γ) |n|(α−β)
n>N
2
|ˆ v (n)e|n|γ |n|β |2 ,
|n|>N
we have
(I − I N ) ≤ sup e|n|(λ−γ) |n|(α−β) → 0 n>N
for N → ∞,
hence I is a compact operator.
To prove compactness of Aj , j = 1, . . . , k we show that it maps the space Aλ,α into a space of “smooth enough” functions. First a result on boundedness of the multiplication operator is needed. ∞
Lemma 5.2. Let v and q be such that (ˆ v (n)cn (|λ|, |α|))n=−∞ ∈ lq with q = 1 1 if α = 0, q = 2 if |α| > 1/2 and q ∈ 1, 1−|α| otherwise. Then multiplication by v is a bounded linear operator in Aλ,α .
Vol. 66 (2010)
Collocation Method with Delta Functions
275
Proof. Let u ∈ Aλ,α . Assume first that α ≥ 0. Then for all m, n ∈ Z we have eλ|n| ≤ eλ|n−m| e|λm| and max{|n|, 1}α ≤ C(α) (max{|n − m|, 1}α + max{|m|, 1}α ) , cn (λ, α) ≤ C(α) e|λm| cn−m (λ, α) + eλ|n−m| cm (|λ|, α)
hence and
uv 2λ,α
=
c2n (λ, α)
n∈Z
⎛
m∈Z
2 u ˆ(n − m)ˆ v (m)
2 ≤ 2C 2 (α) ⎝ cn−m (λ, α)ˆ u(n − m)e|λm| vˆ(m) n∈Z m∈Z 2 ⎞ λ|n−m| e u ˆ(n − m)cm (|λ|, α)ˆ v (m) ⎠ . + n∈Z m∈Z
To estimate these two sums we use a variant of Young’s inequality, namely ∞ fn−m gm ≤ (fn ) lp (gn ) lp , m=−∞ l2
1 p
1 p
3 2.
To estimate the first sum above we use where p, p ∈ [1, 2] with + = p = 2, p = 1 and note that |λn| v (n)) lq . e vˆ(n) 1 ≤ C (cn (|λ|, |α|)ˆ l
For α > 12 the second sum can be estimated similarly, exchanging the roles of u and v and using the fact that λ|n| ˆ(n) 1 ≤ max{1, |n|}−α l2 (cn (λ, α)ˆ u(n)) l2 ≤ C u λ,α . e u l
For α = 0 the second sum is the same as the first sum so the same estimate holds. For 0 < α ≤ 12 we use Young’s inequality with p = q and the corresponding p and note that in this case by H¨ older’s inequality we have p λ|n| ˆ(n) = max{1, |n|}−pα (cn (λ, α)ˆ u(n))p e u lp
n∈Z
≤
n∈Z
p/2 2
(cn (λ, α)ˆ u(n)) p
pα − 1−p/2
1− p2
max{1, |n|}
n∈Z
= C u λ,α because mate
pα > 1 by assumption. In all cases we have obtained the esti1 − p/2
uv λ,α ≤ C (cn (|λ|, |α|)ˆ v (n)) lq u λ,α .
276
Kangro
IEOT
For α < 0 note that for any u ∈ Aλ,α and w ∈ A−λ,−α we have uv, w = u, vw, where ·, · denotes the duality pairing between Aλ,α and A−λ,−α . Hence the multiplication operator is bounded in Aλ,α if and only if it is bounded in A−λ,−α . This completes the proof of the lemma. Corollary 5.3. If v satisfies |ˆ v (n)| ≤ Ccn (−γ, −β), then multiplication by vis 1 for all λ, α which satisfy |λ|, max{ , |α|} ≺ a bounded linear operator in A λ,α 2 1 γ, β − 2 . Proof. For |λ| < γ the assumptions of Lemma 5.2 are obviously satisfied. For |λ| = γ and α = 0 or |α| > 12 they are also easy to check. For |λ| = γ and 0 < |α| ≤ 12 note that the assumption of the corollary reduces to β > 1 and 1 1 therefore one can find q ≥ 1 such that β−|α| < q < 1−|α| . With such q, the assumptions of Lemma 5.2 are satisfied. Lemma 5.4. Let the assumptions (3.4)-(3.5) together with i) and ii) of Theorem 3.1 be satisfied. Let ε1 be such that 0 < ε1 ≤ ε, and if |λ + η| = λ1 , then additionally ε1 < α1 − 12 − α − θ. Then Aj , j = 1, . . . , k are bounded linear operators from Aλ,α to Aλ+η,α+θ+ε1 and hence compact operators from Aλ,α to Aλ+η,α+θ . Proof. From ii) it follows that if |λ + η| = λ1 , then α1 − 12 − α − θ > 0, so that it is always possible to choose ε1 satisfying the assumptions. We can estimate the Fourier coefficients of Aj u as follows: κ ˆ j (n)ˆ a(k − n, n − l)ˆ u(l) |(Aj u)(k)| = l,n∈Z ≤C ck−n (−λ1 , −α1 )cn (−η, −θ − ε)cn−l (−λ2 , −α2 )|ˆ u(l)|. l,n∈Z
The last expression can be considered as the Fourier coefficients of the composition of the following three operators applied to |ˆ u(n)|eint : i) multiplication by
n∈Z
ii) convolution with
n∈Z
iii) multiplication by
n∈Z
n∈Z ins
cn (−λ2 , −α2 )e
,
cn (−η, −θ − ε)eins , and cn (−λ1 , −α1 )eint .
The convolution is a bounded linear operator from Aλ,α to Aλ+η,α+θ+ε , hence also to Aλ+η,α+θ+ε1 . By Corollary 5.3, under the assumptions i) and ii), the multiplication operators are bounded in Aλ,α and correspondingly 1 A . Here ε was chosen so that |λ + η|, max{ , |α + θ + ε1 |} ≺ 1 2 λ+η,α+θ+ε 1 λ1 , α1 − 12 to satisfy the assumptions of Corollary 5.3. It follows that Aj , j = 1, . . . , k are also bounded linear operators from Aλ,α to Aλ+η,α+θ+ε1 .
Vol. 66 (2010)
Collocation Method with Delta Functions
277
6. Properties of the interpolation operator and A0,N In this section first the uniform boundedness of QN in suitable spaces is proved. Then, for large enough N , invertibility of A0,N = QN A0 : XN → YN and uniform boundedness of A−1 0,N QN A0 : Aλ,α → Aλ,α is shown. First we need an auxiliary result. Lemma 6.1. Let (γ, β) ≺ 0, − 12 . Then c2n+N p (γ, β) ≤ Cc2N (γ, β)e−2γ|n| ∀N ≥ 1, ∀n ∈ ZN , p∈Z,p=0
with C depending only on γ and β. Proof. For n ∈ ZN we have
∞
c2n+N p (γ, β) =
p=1
p∈Z,p=0
≤N
2β 2γ(N −|n|)
e
e2γ(n+N p) (n + N p)2β + e2γ(N p−n) (N p − n)2β
∞
2γN (p−1)
e
2 max
p=1
1 p+ 2
2β 2β 1 , p− 2
≤ CN 2β e2γ(N −|n|) = Cc2N (γ, β)e−2γ|n| , since the last sum can be estimated by a constant not dependending on N . Next, the boundedness of QN in our spaces can be proved. Lemma 6.2. If (γ, β) 0, 12 , then the interpolation projectors QN are uniformly bounded in Aγ,β . Proof. Note that for p ∈ Z and n ∈ ZN we have hence for g ∈ Aγ,β
QN ei(n+N p)t = QN eint = eint , with (γ, β) 0, 12 we have Q g(n + N p). N g(n) = p∈Z
Let PN be the orthogonal projection onto YN . For any g ∈ Aγ,β we have 2
QN g − PN g 2γ,β = c2n (γ, β) g(n + N p) p∈Z,p=0 n∈ZN ⎞ ⎛ 1 2 ≤ c2n (γ, β) ⎝ c2n+N p (γ, β) | g (n + N p)| ⎠ c2n+N p (γ, β) n∈ZN p∈Z,p=0 p∈Z,p=0 ⎞ ⎛ 1 ⎠ g − PN g 2γ,β . ≤ max ⎝c2n (γ, β) n∈ZN c2n+N p (γ, β) p∈Z,p=0
278
Kangro
IEOT
1 = c2n+N p (−γ, −β), it follows from Lemma 6.1 that c2n+N p (γ, β) (we are using the lemma with (−γ, −β) instead of (γ, β)) Since
QN − PN 2γ,β ≤ Cc2N (−γ, −β) max e2γ|n| c2n (γ, β). n∈ZN
For N large enough the maximum here is obtained at |n| = [N/2]. By substituting it into the estimate above, we get an uniform bound on QN − PN γ,β and hence the interpolation projectors QN are uniformly bounded in Aγ,β . 6.3. Actually from the proof it follows that for g ∈ Aγ,β with (γ, β) Remark 0, 12 we have
QN g − g γ,β ≤ QN g − PN g γ,β + PN g − g γ,β ≤ C PN g − g γ,β , which implies convergence of QN g to g for any g ∈ Aγ,β . Now we prove invertibility of QN A0 in finite dimensional subspaces. Lemma 6.4. Assume that (4.3) and assumptions i) and ii) of Theorem 4.1 are satisfied. Then the operators A0,N = QN A0 : XN → YN are invertible and A−1 0,N QN A0 are uniformly bounded in Aλ,α . Proof. Since for w ∈ XN A0,N w(t) =
n∈ZN
=
n∈ZN
for g ∈ YN we have
which is defined if
⎝
⎞ κ ˆ 0 (n + N p)w(n ˆ + N p)⎠ eint
p∈Z
⎛ ⎝
⎞ int κ ˆ 0 (n + N p)⎠ w(n)e ˆ ,
p∈Z
⎞
⎛
A−1 0,N g(t) =
⎛
⎜ ⎟ i(n+N m)t ⎟e ⎜ gˆ(n) , ⎠ ⎝ κ ˆ 0 (n + N p) m∈Z n∈ZN p∈Z
κ ˆ 0 (n + N p) = 0 for n ∈ ZN (from this one can see that
p∈Z
some condition guaranteeing that the sum is not “too small” is needed, e.g. condition (4.3) or its weakened form (4.6) ). To show the uniform boundedness of A−1 0,N QN A0 first note that A0 is bounded from Aλ,α to Aλ+η,α+θ and, by Lemma 6.2, QN is bounded in Aλ+η,α+θ . We only have to prove that for any g ∈ YN we have A−1 0,N g λ,α ≤ κ0 (n)| for n ∈ ZN , and C g λ+η,α+θ . By (4.3) we have κ ˆ 0 (n + N p) > |ˆ p∈Z
Vol. 66 (2010)
hence 2
A−1 0,N g λ,α
Collocation Method with Delta Functions
279
2 gˆ(n) = c2n+mN (λ, α) κ ˆ 0 (n + N p) m∈Z n∈ZN p∈Z gˆ(n) 2 ≤ c2n+mN (λ, α) κ ˆ 0 (n) n∈ZN m∈Z ⎛ 2 ⎞ cn+N m (λ, α) gˆ(n) 2 ⎜ m∈Z ⎟ 2 ⎜ ⎟ ≤ max ⎝ cn (λ, α) ⎠ n∈ZN c2n (λ, α) κ ˆ 0 (n) n∈Z
≤ max
n∈ZN
1+
CN 2α e2λ(N −|n|) max{1, |n|}2α e2λ|n|
2 2
A−1 0 g λ,α ≤ C g λ+η,α+θ
(we used Lemma 6.1 and the fact that for N large, the maximum here is obtained at |n| = [N/2].
7. The approximation properties of XN The rate of convergence is determined by inf v∈XN u − v λ,α , which depends on the regularity of the solution. Under assumptions of Theorem 4.1 the solution u belongs to Aλ∗ ,α∗ . We can estimate the rate of convergence in spaces Aλ,α with (λ, α) ≺ (λ∗ , α∗ ). Lemma 7.1. Let u ∈ Aλ∗ ,α∗ be given and let (λ, α) ≺ (λ∗ , α∗ ) and (λ, α) ≺ (0 − 12 ). Then
inf u − v λ,α ≤ C u λ∗ ,α∗ max eλN N α , e(λ−λ∗ )N/2 N α−α∗ . v∈XN
Proof. Let PN denote the orthogonal projection to YN . Then inf u − v λ,α ≤ u − PN u λ,α + inf PN u − v λ,α .
v∈XN
v∈XN
Estimate these terms separately:
u − PN u 2λ,α = | u(n)|2 c2n (λ, α) n∈ZN
≤ sup
n∈ZN
c2n (λ, α) | u(n)|2 c2n (λ∗ , α∗ ) c2n (λ∗ , α∗ )
(λ−λ∗ )N
≤ Ce
n∈ZN
N
2(α−α∗ )
u − PN u 2λ∗ ,α∗ .
To estimate inf v∈XN PN u − v λ,α note that v for which the infimum is obtained, can be written down explicitly: its Fourier’ coefficients are u (n)c2n (λ, α) vˆ(n) = , c2n+N k (λ, α) k∈Z
n ∈ ZN ,
280
Kangro
IEOT
and vˆ(n + N p) = vˆ(n), n ∈ ZN , p ∈ Z. Hence inf PN u − v 2λ,α = PN u 2λ,α − v 2λ,α
v∈XN
=
| u(n)|2 c2n (λ, α) −
n∈ZN
=
| u(n)|2 c2n (λ, α)
n∈ZN
≤
| u(n)|2 c4n (λ, α) c2n+N k (λ, α) n∈ZN
k∈Z
c2n+N k (λ, α)
k∈Z,k=0
c2n+N k (λ, α)
k∈Z
| u(n)|2 c2N (λ, α)e−2λ|n|
n∈ZN
e−2λ|n| | u(n)|2 c2n (λ∗ , α∗ ) n∈ZN c2 n (λ∗ , α∗ ) n∈Z N
2 2λN 2α (λ−λ∗ )N 2(α−α∗ ) N ,e N ≤ C u λ∗ ,α∗ max e ≤ c2N (λ, α) max
(the last inequality follows from the fact that for large N the maximum over n is obtained at |n| = [N/2] or at a place, which does not depend on N and can therefore be absorbed in the constant). In the last estimate the first term under the maximum will be used when u is smooth enough to get the optimal convergence of the method in Aλ,α , i.e (λ∗ , α∗ ) (−λ, 0), the second term expresses the loss in the convergence rate if the solution is not smooth enough. This completes the proof of Theorem 4.1.
8. Conclusions We saw that even for integral equations of first kind with analytic kernels with the dominant part of convolution type it is possible to choose spaces in which the problem is well posed. The collocation method on uniform meshes and with δ-functions as the trial functions is then exponentially convergent in suitable norms. The choice of the particular method is motivated by simplicity of the discrete equations. Also, when solving differential equations by an interior source method, the resulting solution of the differential equation is of very simple form and easy to calculate at an arbitrary point. Exponential convergence of the method means that an accurate solution can be calculated with relatively few degrees of freedom in discrete equations. Since the problem is ill-posed in usual function spaces, the resulting discrete equations are usually very badly conditioned and the dependence of the condition number on the distance of the interior boundary and the physical boundary is exponential. On the other hand, the convergence rate also gets better exponentially (η being roughly a measure of the distance between the boundaries). Hence the distance between the boundaries is of
Vol. 66 (2010)
Collocation Method with Delta Functions
281
crucial importance. In our numerical experiments, when the interior boundary was chosen so that the solution of the exterior problem was analytically extendable to some distance across the interior boundary, then no additional conditioning was needed in solving the discrete equations. In case the singularities of the extension of the solution are not enclosed inside the interior boundary one can use FFT to solve the integral equtions and cut down the size of matrices representing Aj , j = 1, . . . , k. The discrete version of A0 is diagonal after FFT and therefore easy to invert. In case where the boundary is not analytic, it is very hard to find a suitable interior boundary for the collocation method to work. In this case we propose to use, instead of the collocation method, a least squares method, where one chooses more points on the physical boundary, and solves the resulting overdetermined system in the least squares sense. This method is not very sensitive on the choice of the interior boundary, nor on the choice of the supports of the δ-functions, and there is no need to transform the integral equation into form (3.1). The method is convergent on very mild conditions. On the other hand, it is very hard to obtain results on the rate of convergence of this method in cases where the collocation method is not convergent (it is easy to see that if the collocation method is convergent, the convergence rate is at least the same). Some numerical results will be presented in [4].
References [1] Am. Boag, Y Leviatan, Al. Boag, Analysis of electromagnetic scattering using a current model method. Comp. Phys. Comm., 68 (1991), 331-345. [2] A. Doicu, Y. A. Eremin, T. Wriedt, Acoustic & Electromagnetic Scattering Analysis Using Discrete Sources. Academic Press, 2000. [3] R. Kangro, U. Kangro, R.A. Nicolaides, Extendability of Solutions of Helmholtz’s Equation to the Interior of a Two Dimensional Scatterer. Quarterly of Applied Math., 58, 3 (2000), 591–600. [4] U. Kangro, Convergence of the interior source method with point matching for two dimensional scattering problems: the case of analytic boundary. To be submitted. [5] M. Katsurada, Charge Simulation Method Using Exterior Mapping Functions. Japan J. Indust. Appl. Math., 11 (1994), 47-61. [6] A. I. Kleev and A. B. Manenkov, The convergence of point matching techniques. IEEE Trans. Ant. and Prop., 37, 1, (1989), 50–54. [7] R. Kress, Linear Integral Equations. Springer-Verlag 1999. [8] R. Kress, A. Mohsen, On the simulation source technique for exterior problems in acoustics. Math. Meth. in the Appl. Sci., 8, (1986), 585-597. [9] Y. Leviatan, Analytic continuation considerations when using generalized formulations for scattering problems. IEEE Trans. Ant. Prop., 38, (1990), 1259– 1263. [10] K. Ruotsalainen, J. Saranen, Some boundary element methods using Dirac’s distributions as trial functions. SIAM J. Numer. Anal., 24, 4 (1987), 816-827.
282
Kangro
IEOT
[11] J. Saranen, G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer Monographs in Mathematics, Springer-Verlag 2002. Urve Kangro University of Tartu J. Liivi 2 50409 Tartu Estonia e-mail: [email protected] Received: December 19, 2008.
Integr. Equ. Oper. Theory 66 (2010), 283–292 DOI 10.1007/s00020-010-1745-3 Published online January 26, 2010 © Birkhäuser Verlag Basel/Switzerland 2010
Integral Equations and Operator Theory
New Formulae for the Wave Operators for a Rank One Interaction Serge Richard and Rafael Tiedra de Aldecoa Abstract. We prove new formulae for the wave operators for a Friedrichs scattering system with a rank one perturbation, and we derive a topological version of Levinson’s theorem for this model. Mathematics Subject Classification (2010). Primary 47A40; Secondary 81U15. Keywords. Friedrichs model, wave operators, Levinson’s theorem.
1. Introduction and main results Let us consider the Hilbert space H := L2 (R) with norm · and scalar product ·, ·, and let H0 ≡ X be the self-adjoint operator of multiplication by the variable, i.e. (H0 f )(x) = xf (x) for any f ∈ D(H0 ) ≡ L2 R, (1+x2 ) dx . For u ∈ H, we also consider the rank one perturbation of H0 defined by Hu f := H0 f + u, f u,
f ∈ D(H0 ).
It is well known that the wave operators Ω± := s- limt→±∞ eiHu t e−iH0 t exist and are asymptotically complete, and that the scattering operator S := Ω∗+ Ω− is a unitary operator in H. In fact, S ≡ S(X) is simply an operator of multiplication by a function S : R → T, with T the set of complex numbers of modulus 1. A rather explicit formula for the wave operators for this model was proposed in [8]. Its expression involves singular integrals that have to be manipulated with some care. In this Note, we propose a simpler formula for the wave operators, and put into light a straightforward consequence of it. However, we stress that contrary to [8], our formula and its corollary hold only if some additional (but weak) hypotheses on u are imposed. To state our results, and in particular to have an explicit formula of the Hilbert transform in terms of the generator of dilations (see Equation (2.3) below), we need to introduce the even / odd representation of H. Given any function m on R, we write me and mo for the even part and the odd part of
284
Richard and Tiedra de Aldecoa
IEOT
2 2 m. We also set H := L + ; C ) and introduce the unitary map U : H → H f(R 1 given on any f ∈ H, f2 ∈ H , x ∈ R by √ f 1 f U f := 2 e and U ∗ 1 (x) := √ [f1 (|x|) + sgn(x)f2 (|x|)] . fo f2 2
Now, observe that if m is a function on R and A the generator of dilations in H, then we have me (X+ ) mo (X+ ) A+ 0 ∗ ∗ U m(X)U = and U AU = , mo (X+ ) me (X+ ) 0 A+ where X+ is the operator of multiplication by the variable in L2 (R+ ), and A+ the generator of dilations in L2 (R+ ), namely (eitA+ f )(x) := et/2 f (et x) for f ∈ L2 (R+ ), x ∈ R+ . In the sequel we assume that the vector u satisfies the following assumption. Assumption 1.1. The function u ∈ H is H¨ older continuous with exponent α > 0. Furthermore, if x0 ∈ R satisfies u(x0 ) = 0 and 1 − R dy |u(y)|2 (x0 − y)−1 = 0, then there exists an exponent α > 1/2 such that
|u(x) − u(y)| ≤ Const. |x − y|α for all x, y near x0 .
This assumption implies that u is bounded and satisfies lim|x|→∞ u(x) = 0. Moreover, it is known that under Assumption 1.1, the operator Hu has at most a finite number of eigenvalues of multiplicity one [1, Sec. 2] (see also [6, 11]). Clearly, Assumption 1.1 is satisfied if u ∈ H is H¨older continuous with exponent α > 1/2. Our main result is the following representation of the wave operator Ω− in H . Theorem 1.2. Let u satisfy Assumption 1.1. Then, one has 1 0 1 −φ(A+ ) ∗ 1 U Ω− U = +2 0 1 1 −φ(A+ ) So (X+ ) Se (X+ ) − 1 + K, × Se (X+ ) − 1 So (X+ )
(1.1)
where φ(A+ ) := tanh(πA+ ) + i[cosh(πA+ )]−1 and K is a compact operator in H . Let us immediately mention that a similar formula holds for Ω+ . Indeed, by using Ω+ = Ω− S(X) one gets 1 0 φ(A+ ) 1 ∗ 1 +2 U Ω+ U = 0 1 1 φ(A+ ) Se (X+ ) − 1 So (X+ ) × + K , So (X+ ) Se (X+ ) − 1
Vol. 66 (2010)
New Formulae for the Wave Operators
285
where K is a compact operator in H . We also note that the proof of Theorem 1.2 will make clear why the minimal hypothesis u ∈ H is not sufficient in order to prove the claim. We now state a corollary of the above theorem which corresponds to a topological version of Levinson’s Theorem: Corollary 1.3. Let u satisfy Assumption 1.1. Then S(±∞) = 1 and the following equality holds: ω(S) = − number of eigenvalues of Hu , where ω(S) is the winding number of the continuous map S : R → T. Such a result was already known for more general perturbations but under stronger regularity conditions [2, 4] (see also [6, 14] for general information on the Friedrichs model). Our result does require neither the differentiability of the scattering matrix nor the differentiability of u. Nonetheless, if S is continuously differentiable, then the winding number can also be expressed in terms of an integral involving the (on-shell) time delay operator, which is the logarithmic derivative of the scattering matrix [13]. Remark 1.4. The authors emphasize that Corollary 1.3 is a straightforward consequence of formula (1.1), even if its proof requires the algebraic framework presented in Section 3. They do not doubt that for smooth u this result can also be obtained via more analytical technics, but one of their motivations was to show that Levinson’s Theorem can directly be inferred from the explicit formula (1.1), without any further analysis (see [10] and references therein for similar proofs of Levinson’s Theorem for other scattering systems). The content of this Note is the following. In Section 2 we prove Formula (1.1) and derive some auxiliary results. In Section 3 we give a description of the algebraic framework involved in the proof of the Corollary 1.3, which is proved at the end of the section.
2. Derivation of the new formula We start by recalling some notations and results borrowed from [1] and [8]. We shall always suppose that u satisfies Assumption 1.1. For x ∈ R and ε > 0 we set
|u(y)|2 ε I± (x) := dy . x − y ± iε R ε By Privalov’s theorems, the limit I± (x) := limε0 I± (x) exists for all x ∈ R. Furthermore, the set of x such that I± (x) = 1 is finite [1, p. 396]. In consequence the expression [1 − I± (x)]−1 is well defined for almost every x ∈ R, and the domain D± of [1 − I± (X)]−1 in H is dense.
286
Richard and Tiedra de Aldecoa
IEOT
Let F denote the Fourier transform in H, namely
1 (F f )(x) := √ dy e−ixy f (y), f ∈ H ∩ L1 (R). 2π R Given a Borel function m on R, we set m(D) := F ∗ m(X)F . Finally, χ(−∞,0) stands for the characteristic function for the half-line (−∞, 0). We are now in a position to recall the formula [8, eq. (56b)] for Ω− . One has Ω− = 1 − 2πiu(X)χ(−∞,0) (D)u(X)[1 − I+ (X)]−1 on D+ (note that we use a convention for the sign ± of the wave operators Ω± which differs from the one of [8]). So, if K := −2πi[u(X), χ(−∞,0) (D)]u(X)[1− I+ (X)]−1 , then one gets on D+ Ω− − 1 = −2πiu(X)χ(−∞,0) (D)u(X)[1 − I+ (X)]−1 = χ(−∞,0) (D) − 2πi|u(X)|2 [1 − I+ (X)]−1 + K = χ(−∞,0) (D){S(X) − 1} + K,
(2.1)
by using [8, Eq. (66b)] in the last equality. In the next lemma, we determine the action of χ(−∞,0) (D) in H . For that purpose, we define φ ∈ C([−∞, ∞]; T) by φ(x) := tanh(πx) + i[cosh(πx)]−1 for all x ∈ R. Lemma 2.1. One has U χ(−∞,0) (D)U ∗ = Φ(A+ ), where 1 −φ(A+ ) Φ(A+ ) := 12 . 1 −φ(A+ ) Proof. The usual Hilbert transform H on R satisfies sgn(D) = iH. Thus (2.2) χ(−∞,0) (D) = 12 1 − sgn(D) = 12 (1 − iH). Using the expression for iH in terms of the generator of dilations in H given in [10, Lemma 3], one gets U iHU ∗ 0 = tanh(πA+ ) + i[cosh(πA+ )]−1
(2.3) tanh(πA+ ) − i[cosh(πA+ )]−1 . 0
The claim follows then from (2.2) and (2.3).
We now recall some results on the scattering matrix. Lemma 2.2. Let u satisfy Assumption 1.1. Then the map S belongs to C([−∞, ∞]; T) and satisfies S(±∞) = 1. Proof. The continuity of S follows from [1, Thm. 1.(i)]. The equalities S(±∞) = 1 follow from the formula S(x) − 1 = −2πi|u(x)|2 [1 − I+ (x)]−1 together with Lemma 1.(a) of [1] and the fact that lim|x|→∞ |u(x)|2 = 0. The last lemma deals with the remainder term K of Formula (1.1).
Vol. 66 (2010)
New Formulae for the Wave Operators
287
Lemma 2.3. Let u satisfy Assumption 1.1. Then the operator [u(X), χ(−∞,0) (D)]u(X)[1 − I+ (X)]−1 , defined on D+ , extends to a compact operator in H. Proof. (i) Define for all x ∈ R the function ψ(x) := u(x)[1 − I+ (x)]−1 . We know that u is bounded and that lim|x|→∞ u(x) = 0. We also know from [1, Lemma 1.(a)] that I+ is H¨older continuous and that lim|x|→∞ I+ (x) = 0. So, outside any neighbourhood of the finite set of points where I+ equals 1, the function ψ is bounded. Furthermore, Assumption 1.1 and [1, Lemma 1.(c)], imply that ψ is locally square integrable (see also [8, p. 2423]). Therefore, ψ can be written as ψ = ψ∞ + ψ2 , with ψ∞ ∈ L∞ (R) and ψ2 ∈ L2 (R) with support in a small neighbourhood of the points where I+ equals 1. We now show the compacity of the operator [u(X), χ(−∞,0) (D)]ψ∞ (X) and of the operator [u(X), χ(−∞,0) (D)]ψ2 (X). (ii) Choose a function ϕ1 ∈ C ∞ (R) and a function ϕ2 ∈ L∞ (R) with compact support such that ϕ1 + ϕ2 = χ(−∞,0) . Then [u(X), ϕ1 (D)] is compact due to [3, Thm. C], and [u(X), ϕ2 (D)] is Hilbert-Schmidt due to [12, Thm. 4.1]. So [u(X), χ(−∞,0) (D)]ψ∞ (X) = [u(X), ϕ1 (D)]ψ∞ (X) + [u(X), ϕ2 (D)]ψ∞ (X) is a compact operator. (iii) For each f ∈ H and almost every x ∈ R, define the operator
i u(x) − u(y) ψ2 (y)f (y). dy (K0 f )(x) := 2π R y−x From the Assumption 1.1 we know that
|u(y + x) − u(y)| ≤ Const. |x|α ,
α > 1/2
for each y ∈ supp(ψ2 ) and each x ∈ R with |x| small enough. In particular, there exists δ > 0 such that
2
i u(x) − u(y)
2
ψ2 (y)
dx dy 4π 2π y−x R2
|u(y + x) − u(y)|2 = dy dx |ψ2 (y)|2 x2 R R
|u(y + x) − u(y)|2 = dy dx |ψ2 (y)|2 x2 R R\[−δ,δ]
δ |u(y + x) − u(y)|2 + dy dx |ψ2 (y)|2 2 x −δ R
δ 4u2∞ 2 ≤ dy dx |ψ2 (y)| + Const. dy dx |x|2(α −1) |ψ2 (y)|2 2 x R R\[−δ,δ] R −δ < ∞.
288
Richard and Tiedra de Aldecoa
IEOT
Thus, K0 is a Hilbert-Schmidt operator. Furthermore, we have for f ∈ D+ and almost every x ∈ R i [u(X), χ(−∞,0) (D)]ψ2 (X)f (x) = − [u(X), H]ψ2 (X)f (x) 2
u(x) − u(y) i ψ2 (y)f (y) dy =− 2π R x−y = (K0 f )(x). Therefore, the operator [u(X), χ(−∞,0) (D)]ψ2 (X) extends to a HilbertSchmidt operator. Proof of Theorem 1.2. The operator K extends to a compact operator due to Lemma 2.3. So Equation (2.1) holds on H, and the claim follows from Lemma 2.1. Remark 2.4. The proof of Corollary 1.3 relies on the fact that the range of the wave operators is the orthocomplement of the subspace spanned by the eigenvectors of Hu . Since the wave operators are complete, such a property holds if and only if Hu has no singularly continuous spectrum. Now, by using the characterization of the singular spectrum recalled in [5, p. 299] and by taking into account Lemmas 1 and 2 of [1] (which are valid since u satisfies Assumption 1.1), one easily gets that the singular spectrum of Hu only consists of a finite set. So Assumption 1.1 implies the absence of singularly continuous spectrum for Hu .
3. Algebraic framework This section is dedicated to the presentation of the algebraic framework leading to Corollary 1.3. Since a similar construction already appears in [10] for the proof of Levinson’s theorem in one dimensional potential scattering, we only sketch the construction and refer to this reference for more details. We start by giving the definition of the Mellin transform associated with the generator of dilations A+ in L2 (R+ ) (see [9, Sec. 2] for a general presentation when the operator acts in L2 (Rn )). Let V : L2 (R+ ) → L2 (R) be defined by (V f )(x) := ex/2 f (ex ) for x ∈ R, and remark that V is a unitary map with adjoint V ∗ given by (V ∗ g)(x) = x−1/2 g(ln x) for x ∈ R+ . Then, the Mellin transform M : L2 (R+ ) → L2 (R) is defined by M := F V . Its main property is that it diagonalizes the generator of dilations, namely, M A+ M ∗ = X. Formally, one also has M ln(X+ )M ∗ = −D. Let us now recall from Remark 2.4 that under the Assumption 1.1 the wave operators Ω± are isometries with range projection 1 − Pp , where Pp is the projection onto the subspace spanned by the finite number N of eigenvectors of Hu . In particular, Ω− is a Fredholm operator with index(Ω− ) = − Tr(Pp ) = −N . Furthermore, we recall that any Fredholm operator F in H is invertible modulo a compact operator, that is, its image q(F ) in the Calkin algebra B(H)/K(H) is invertible.
Vol. 66 (2010)
New Formulae for the Wave Operators
289
Now, assume that Ω− belongs to a norm-closed subalgebra E of B(H) containing K(H). Moreover, assume that E/K(H) is isomorphic to C S; M2 (C) , the algebra of continuous functions over the circle with values in the 2 × 2 matrices. Then, viewing q(Ω− ) as such a function, we can take pointwise its determinant to obtain a non-vanishing function over the circle. The winding number of that latter function can be related to the index of Ω− ; this is essentially the content of Corollary 1.3. The choice of E is suggested by the formula obtained in Theorem 1.2. Indeed, we consider the closure E in B(H ) of the algebra generated by elements of the form ϕ(A+ )ψ(X+ ), where ϕ is a continuous function on R with values in M2 (C) which converges at ±∞, and ψ is a continuous function on Stated differR+ with values in M2 (C) which converges at 0 and at +∞. ently, ϕ ∈ C R; M2 (C) with R = [−∞, ∞], and ψ ∈ C R+ ; M2 (C) with R+ = [0, ∞]. Let J be the norm closed algebra generated by ϕ(A+ )ψ(X+ ) with functions ϕ and ψ for which the above limits vanish. Then, J is an ideal in E , and the are obtained ifψ(X+ ) is replaced by η(ln(X+ )) same algebras with η ∈ C R; M2 (C) or η ∈ C0 R; M2 (C) , respectively. These algebras have already been studied in [7] in a different context. The authors constructed them in terms of the operators X and D on L2 (R, E), with E an auxiliary Hilbert space, possibly of infinite dimension. In that situation, ϕ and η are norm continuous functions on R with values in K(E). The isomorphism between our algebras and the algebras introduced in [7, Sec. 3.5], with E = C2 , is given by the Mellin transform M , or more precisely by M ⊗ 1, where 1 is identity operator in M2 (C). For that reason, we shall freely use the results obtained in that reference, and refer to it for the proofs. For instance, it is proved that J = K(H ), and an explicit description of the quotient E /J is given, which we specify now in our context. To describe the quotient E /J , we consider the square := R+ × R, whose boundary ∂ is the union of four parts: ∂ ≡ B1 ∪ B2 ∪ B3 ∪ B4 , with It B1 := {0} × R, B2 := R+ × {∞}, B3 := {∞} × R, and B4 := R+ × {−∞}. is proved in [7, Thm. 3.20] that E /J is isomorphic to C ∂; M2 (C) . This algebra can be seen as the subalgebra of C R; M2 (C) ⊕ C R+ ; M2 (C) ⊕ C R; M2 (C) ⊕ C R+ ; M2 (C) (3.1) given by elements γ ≡ (γ1 , γ2 , γ3 , γ4 ) which coincide at the corresponding end points, that is, γ1 (∞) = γ2 (0), γ2 (∞) = γ3 (∞), γ3 (−∞) = γ4 (∞), and γ4 (0) = γ1 (−∞). Furthermore, for any ϕ ∈ C R; M2 (C) and ψ ∈ C R+ ; M2 (C) , the image of the operator ϕ(A+ )ψ(X+ ) through the quotient map q : E → C ∂; M2 (C) is given by γ1 = ϕψ(0), γ2 = ϕ(∞)ψ, γ3 = ϕψ(∞) and γ4 = ϕ(−∞)ψ. From what precedes we deduce that the subalgebra E of B(H), defined operators by E := U ∗ E U , contains the ideal of compact on H and that the quotient E/K(H) is isomorphic to C ∂; M2 (C) ∼ = C S; M2 (C) . We are thus in the framework defined above, and the for any invertible element
290
Richard and Tiedra de Aldecoa
IEOT
γ of C ∂; M2 (C) , the winding number of its pointwise determinant is a well-defined quantity. So we are ready to give the proof of Corollary 1.3. Proof of Corollary 1.3. We know from Theorem 1.2 and Lemma 2.2 that U Ω− U ∗ ∈ E , or equivalently that Ω− ∈ E. Due to Formula (1.1), the element γ belonging to (3.1) and associated with q(Ω− ) is given by suitable restrictions of the function Γ : R+ × R → M2 (C), where s (x) − 1 so (x) Γ(x, y) := 1 + Φ(y) e se (x) − 1 so (x) se (x) − φ(y)so (x) + 1 so (x) − φ(y)[se (x) − 1] . = 12 se (x) − φ(y)so (x) + 1 so (x) − φ(y)[se (x) − 1] Namely, γ1 = Γ(0, · ), γ2 = Γ( · , +∞), γ3 = Γ(+∞, · ), and γ4 = Γ( · , −∞). The pointwise determinants of these functions are easily calculated by using the identity φ(±∞) = ±1: one gets det γ1 (y) = se (0), det γ2 (x) = s(−x), det γ3 (y) = 1 and det γ4 (x) = s(x). The precise relation between the winding number of the map det γ : ∂ → T and the index of Ω− has been described in [10, Prop. 7]. However, the algebra corresponding to E in that reference was defined in terms of the operators A+ and B+ = 12 ln (D2 )+ which satisfy the relation [iA+ , B+ ] = −1. In our case, the algebra E has been constructed with the operators A+ and ln(X+ ) which satisfy the relation [iA+ , ln(X+ )] = 1. Therefore, in order to apply [10, Prop. 7] in our setting, one previously needs to apply the au tomorphism of C R; M2 (C) defined by η(x) := η(−x) for all x ∈ R, or := ψ(x−1 ) equivalently the automorphism of C R+ ; M2 (C) defined by ψ(x) j assofor all x ∈ R+ . Therefore the pointwise determinants of the function γ 1 (y) = 1, det γ 2 (x) = s(−x−1 ), det γ 3 (y) = se (0) ciated with q(Ω− ) are det γ and det γ 4 (x) = s(x−1 ). Now, [10, Prop. 7] reads ω(det γ ) = index(Ω− ) = −N , where N is the number of eigenvalues of Hu . The convention used in that reference for the calculation of the winding number implies that the contribution of x → 4 (x) is det γ 2 (x) is from x = 0 to x = +∞ and the contribution of x → det γ from x = +∞ to x= 0. This corresponds to the calculation of the winding number of x → det S(x) , from x = −∞ to x = +∞. Since the contributions 3 are null because these terms are constant, the claim is of det γ 1 and det γ proved. Acknowledgment S. Richard is supported by the Swiss National Science Foundation. R. Tiedra de Aldecoa is partially supported by N´ ucleo Cient´ıfico ICM P07-027-F “Mathematical Theory of Quantum and Classical Magnetic Systems” and by the Chilean Science Foundation Fondecyt under the Grant 1090008.
Vol. 66 (2010)
New Formulae for the Wave Operators
291
References [1] P. Alsholm, Inverse scattering theory for perturbations of rank one, Duke Math. J. 47 no. 2 (1980), 391–398. [2] V. S. Buslaev, Spectral identities and the trace formula in the Friedrichs model, in Spectral theory and wave processes, 43–54. Consultants Bureau Plenum Publishing Corporation, New York, 1971. [3] H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal. 18 (1975), 115–131. [4] T. Dreyfus, The determinant of the scattering matrix and its relation to the number of eigenvalues, J. Math. Anal. Appl. 64 no. 1 (1978), 114–134. [5] E. M. Dyn kin, S. N. Naboko, S. I. Yakovlev, A finiteness bound for the singular spectrum in a selfadjoint Friedrichs model, St. Petersburg Math. J. 3 no. 2 (1992), 299–313. [6] L. D. Faddeev, On a model of Friedrichs in the theory of perturbations of the continuous spectrum, in American Mathematical Society Translations. Series 2, Vol. 62: Five papers on functional analysis, 177–203, American Mathematical Society, Providence, R.I. 1967. [7] V. Georgescu, A. Iftimovici, C ∗ -algebras of quantum Hamiltonians, in Operator Algebras and Mathematical Physics, Conference Proceedings: Constant¸a (Romania) July 2001, 123–167, Theta Foundation, 2003. [8] M. A. Grubb, D. B. Pearson, Derivation of the wave and scattering operators for an interaction of rank one, J. Mathematical Phys. 11 (1970), 2415–2424. [9] A. Jensen, Time-delay in potential scattering theory, some ”geometric” results, Comm. Math. Phys. 82 no. 3 (1981/82), 435–456. [10] J. Kellendonk, S. Richard, On the structure of the wave operators in one dimensional potential scattering, Mathematical Physics Electronic Journal 14 (2008), 1–21. [11] B. S. Pavlov, S. V. Petras, The singular spectrum of a weakly perturbed multiplication operator, Funkcional. Anal. i Priloˇzen. 4 1970 no. 2, 54–61. Translation in Functional Anal. Appl. 4 (1970), 136–142. [12] B. Simon, Trace ideals and their applications, Mathematical Surveys and Monographs 120, American Mathematical Society, Providence, RI, second edition, 2005. [13] R. Tiedra de Aldecoa, Time delay for dispersive systems in quantum scattering theory, Rev. Math. Phys. 21 no. 5 (2009), 675-708. [14] D. R. Yafaev, Mathematical scattering theory. General theory, Translations of Mathematical Monographs, 105. American Mathematical Society, Providence, RI, 1992.
292
Richard and Tiedra de Aldecoa
Serge Richard Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Cambridge CB3 0WB United Kingdom On leave from : Universit´e de Lyon Universit´e Lyon 1, CNRS, UMR5208 Institut Camille Jordan 43 blvd du 11 novembre 1918 69622 Villeurbanne-Cedex France e-mail: [email protected] Rafael Tiedra de Aldecoa Facultad de Matem´ aticas Pontificia Universidad Cat´ olica de Chile Av. Vicu˜ na Mackenna 4860 Santiago Chile e-mail: [email protected] Submitted: February 22, 2009. Revised: October 6, 2009.
IEOT
Integr. Equ. Oper. Theory 66 (2010), 293–325 DOI 10.1007/s00020-010-1750-6 Published online February 16, 2010 © Birkhäuser/Springer Basel AG 2010
Integral Equations and Operator Theory
Holomorphic Semi-almost Periodic Functions Alexander Brudnyi and Damir Kinzebulatov Abstract. We study the Banach algebras of bounded holomorphic functions on the unit disk whose boundary values, having, in a sense, the weakest possible discontinuities, belong to the algebra of semi-almost periodic functions on the unit circle. The latter algebra contains as a special case an algebra introduced by Sarason in connection with some problems in the theory of Toeplitz operators. Mathematics Subject Classification (2010). Primary 30H05, Secondary 46J20. Keywords. Approximation property, semi-almost periodic functions, maximal ideal space.
1. Introduction We study the Banach algebras of holomorphic semi-almost periodic functions, i.e., bounded holomorphic functions on the unit disk D ⊂ C whose boundary values belong to the algebra SAP (∂D) ⊂ L∞ (∂D) of semi-almost periodic functions on the unit circle ∂D. A function f ∈ L∞ (∂D) is called semi-almost periodic if for any s ∈ ∂D and any ε > 0 there exist functions fk : ∂D → C (k ∈ {−1, 1}) and arcs γk with s being their right (if k = −1) or left (if k = 1) endpoint with respect tox the counterclockwise orientation of ∂D such that the functions x → fk seike , −∞ < x < 0, k ∈ {−1, 1}, are restrictions of Bohr’s almost periodic functions on R (see Definition 2.1 below) and sup |f (z) − fk (z)| < ε,
z∈γk
k ∈ {−1, 1}.
The graph of a real-valued semi-almost periodic function discontinuous at a single point has a form as depicted in Figure 1. The Algebra SAP (∂D) contains as a special case an algebra introduced by Sarason [23] in connection with some problems in the theory of Toeplitz operators. Our primary interest in holomorphic semi-almost periodic functions Research of the first author was supported in part by NSERC.
294
Brudnyi and Kinzebulatov
IEOT
Figure 1
was motivated by the problem of description of the weakest possible boundary discontinuities of functions in H ∞ (D), the Hardy algebra of bounded holomorphic functions on D. (Recall that a function f ∈ H ∞ (D) has radial limits almost everywhere on ∂D, the limit function f |∂D ∈ L∞ (∂D), and f can be recovered from f |∂D by means of the Cauchy integral formula.) In the general form this problem is as follows (see also [7]): Given a continuous function Φ : C → C to describe the minimal Banach subalgebra HΦ∞ (D) ⊂ H ∞ (D) containing all elements f ∈ H ∞ (D)∗ such that Φ(f )|∂D is piecewise Lipschitz having finitely many first-kind discontinuities. Here H ∞ (D)∗ is the group of invertible elements of H ∞ (D). Clearly, each HΦ∞ (D) contains the disk-algebra A(D) (i.e., the algebra of holomorphic functions continuous up to the boundary). Moreover, if Φ(z) = z, Re(z) or Im(z), then the Lindel¨ of theorem, see, e.g., [13], implies that HΦ∞ (D) = A(D). In contrast, if Φ is constant on a closed simple curve which does not encompass 0 ∈ C, then HΦ∞ (D) = H ∞ (D). (This result is obtained by consequent applications of the Carath´eodory conformal mapping theorem, the Mergelyan theorem and the Marshall theorem, see, e.g., [13].) In [6] we studied the case of Φ(z) = |z| and showed that HΦ∞ (D) coincides with the algebra of holomorphic semi-almost periodic functions SAP (∂D) ∩ H ∞ (D). In the present paper we continue the investigation started in [6]. Despite the fact that our results concern the particular choice of Φ(z) = |z|, the methods developed here and in [6] can be applied further to a more general class of functions Φ. Let bD be the maximal ideal space of the algebra SAP (∂D) ∩ H ∞ (D), i.e., the set of all nonzero homomorphisms SAP (∂D)∩H ∞ (D) → C equipped with the Gelfand topology. The disk D is naturally embedded into bD. In [6] we proved that D is dense in bD (the so-called corona theorem for SAP (∂D)∩ H ∞ (D)). We also described the topological structure of bD. In the present
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
295
paper we refine and extend some of these results. In particular, we introduce Bohr-Fourier coefficients and spectra of functions from SAP (∂D), describe ˇ Cech cohomology groups of bD and establish projective freeness of certain subalgebras of SAP (∂D) ∩ H ∞ (D). Recall that a commutative ring R with identity is called projective free if every finitely generated projective R-module is free. Equivalently, R is projective free iff every square idempotent matrix F with entries in R (i.e., such that F 2 = F ) is conjugate over R to a matrix of the form Ik 0 , 0 0 where Ik stands for the k × k identity matrix. Every field F is trivially projective free. Quillen and Suslin proved that if R is projective free, then the rings of polynomials R[x] and formal power series R[[x]] over R are projective free as well (see, e.g., [18]). Grauert proved that the ring O(Dn ) of holomorphic functions on the unit polydisk Dn is projective free [14]. In turn, it was shown in [8] that the triviality of any complex vector bundle of finite rank over the connected maximal ideal space of a unital semi-simple commutative complex Banach algebra is sufficient for its projective freeness. We employ this result to show that subalgebras of SAP (∂D) ∩ H ∞ (D) whose elements have their spectra in non-negative or non-positive semi-groups are projective free. Note that if a unital semi-simple commutative complex Banach algebra A is projective free, then it is Hermite, i.e., every finitely generated stably free A-module is free. Equivalently, A is Hermite iff any k × n matrix, k < n, with entries in A having rank k at each point of the maximal ideal space of A can be extended to an invertible n × n matrix with entries in A, see [9]. (Here the values of elements of A at points of the maximal ideal space are defined by means of the Gelfand transform.) Finally, we prove that SAP (∂D) ∩ H ∞ (D) has the approximation property. (This result strengthen the approximation theorem of [6].) Recall that a Banach space B is said to have the approximation property if for every compact set K ⊂ B and every ε > 0 there is an operator T : B → B of finite rank so that T x − xB < ε
for every
x ∈ K.
(Throughout this paper all Banach spaces are assumed to be complex.) Although it is strongly believed that the class of spaces with the approximation property includes practically all spaces which appear naturally in analysis, it is not known yet even for the space H ∞ (D) (see, e.g., the paper of Bourgain and Reinov [3] for some results in this direction). The first example of a space which fails to have the approximation property was constructed by Enflo [10]. Since Enflo’s work several other examples of such spaces were constructed, for the references see, e.g., [20]. Many problems of Banach space theory admit especially simple solutions if one of the spaces under consideration has the approximation property. One of such problems is the problem of determination whether given two Banach algebras A ⊂ C(X), B ⊂ C(Y ) (X and Y
296
Brudnyi and Kinzebulatov
IEOT
are compact Hausdorff spaces) their slice algebra S(A, B) := {f ∈ C(X × Y ) : f (·, y) ∈ A for all y ∈ Y, f (x, ·) ∈ B for all x ∈ X} coincides with A⊗B, the closure in C(X ×Y ) of the symmetric tensor product of A and B. For instance, this is true if either A or B have the approximation property. The latter is an immediate consequence of the following result of Grothendieck. Let A ⊂ C(X) be a closed subspace, B be a Banach space and AB ⊂ CB (X) := C(X, B) be the Banach space of all continuous B-valued functions f such that ϕ(f ) ∈ A for any ϕ ∈ B ∗ . By A ⊗ B we denote the completion of the symmetric tensor product of A and B with respect to norm m m ak ⊗ bk := sup ak (x)bk with ak ∈ A, bk ∈ B. (1.1) x∈X k=1
k=1
B
Theorem 1.1 ([15]). The following statements are equivalent: 1) A has the approximation property; 2) A ⊗ B = AB for every Banach space B. Our proof of the approximation property for SAP (S) ∩ H ∞ (D) is based on Theorem 1.1 and on a variant of the approximation theorem in [6] for Banach-valued analogues of algebra SAP (S) ∩ H ∞ (D). The paper is organized as follows. Section 2 is devoted to the algebra of semi-almost periodic functions SAP (∂D). In Section 3 we formulate our main results on the algebra of holomorphic semi-almost periodic functions SAP (∂D) ∩ H ∞ (D). All proofs are presented in Section 4. The results of the present paper have been announced in [7].
2. Preliminaries on semi-almost periodic functions We first recall the definition of a Bohr almost periodic function on R. In what follows, by Cb (R) we denote the algebra of bounded continuous functions on R endowed with sup-norm. Definition 2.1 (see, e.g., [2]). A function f ∈ Cb (R) is said to be almost periodic if the family of its translates {Sτ f }τ ∈R , Sτ f (x) := f (x + τ ), x ∈ R, is relatively compact in Cb (R). The basic example of an almost periodic function is given by the formula m x → cl eiλl x , cl ∈ C, λl ∈ R. l=1
Let AP (R) be the Banach algebra of almost periodic functions endowed with sup-norm. The main characteristics of an almost periodic function f ∈ AP (R) are its Bohr-Fourier coefficients aλ (f ) and the spectrum spec(f ) defined in terms of the mean value T 1 M (f ) := lim f (x)dx. (2.1) T →+∞ 2T −T
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
297
Specifically, aλ (f ) := M (f e−iλx ),
λ ∈ R.
(2.2)
[2]. These Then aλ (f ) = 0 for at most countably many values of λ, see, e.g., ∞ iλl x values constitute the spectrum spec(f ) of f . In particular, if f = l=1 cl e ∞ (cl = 0 and l=1 |cl | < ∞), then spec(f ) = {λ1 , λ2 , . . . }. One of the main results of the theory of almost periodic functions states that each function m f ∈ AP (R) can be uniformly approximated by functions of the form l=1 cl eiλl x with λl ∈ spec(f ). Let Γ ⊂ R be a unital additive semi-group (i.e., 0 ∈ Γ). It follows easily from the cited approximation result that the space APΓ (R) of almost periodic functions with spectra in Γ forms a unital Banach subalgebra of AP (R). We will use the following result. Theorem 2.2. APΓ (R) has the approximation property. Next, we recall the definition of a semi-almost periodic function on ∂D introduced in [6]. In what follows, we consider ∂D with the counterclockwise orientation. For s := eit , t ∈ [0, 2π), let γsk (δ) := {seikx : 0 ≤ x < δ < 2π},
k ∈ {−1, 1},
(2.3)
be two open arcs having s as the right and the left endpoints (with respect to the orientation), respectively. Definition 2.3 ([6]). A function f ∈ L∞ (∂D) is called semi-almost periodic if for any s ∈ ∂D, and any ε > 0 there exist a number δ = δ(s, ε) ∈ (0, π) and functions fk : γsk (δ) → C, k ∈ {−1, 1}, such that functions x f˜k (x) := fk seikδe , −∞ < x < 0, k ∈ {−1, 1}, are restrictions of some almost periodic functions from AP (R), and sup |f (z) − fk (z)| < ε,
z∈γsk (δ)
k ∈ {−1, 1}.
By SAP (∂D) we denote the Banach algebra of semi-almost periodic functions on ∂D endowed with sup-norm. It is easy to see that the set of points of discontinuity of a function in SAP (∂D) is at most countable. For S being a closed subset of ∂D we denote by SAP (S) the Banach algebra of semialmost periodic functions on ∂D that are continuous on ∂D \ S. (Note that the Sarason algebra introduced in [23] is isomorphic to SAP ({z0 }), z0 ∈ ∂D.) Example ([6]). A function g defined on R (R + iπ) is said to belong to the space AP (R (R + iπ)) if the functions g(x) and g(x + iπ), x ∈ R, belong to AP (R). The space AP (R (R + iπ)) is a function algebra (with respect to sup-norm). Given s ∈ ∂D consider the map ϕs : ∂D \ {−s} → R, ϕs (z) := 2i(s−z) s+z , and define a linear isometric embedding Ls : AP (R (R + iπ)) → L∞ (∂D) by the formula (Ls g)(z) := (g ◦ Log ◦ ϕs )(z), (2.4)
298
Brudnyi and Kinzebulatov
IEOT
where Log(z) := ln |z| + iArg(z), z ∈ C \ R− , and Arg : C \ R− → (−π, π) stands for the principal branch of the multi-function arg. Then the range of Ls is a subspace of SAP ({−s, s}). Theorem 2.4 ([6]). For every s ∈ ∂D there exists a homomorphism of Banach algebras Es : SAP (∂D) → AP (R (R + iπ)) of norm 1 such that for each f ∈ SAP (∂D) the function f − Ls (Es f ) ∈ SAP (∂D) is continuous and equal to 0 at s. Moreover, any bounded linear operator SAP (∂D) → AP (R (R + iπ)) satisfying this property coincides with Es . The functions f−1,s (x) := (Es f )(x) and f1,s (x) := (Es f )(x+iπ), x ∈ R, are used to define the left (k = −1) and the right (k = 1) mean values Msk (f ) of a function f ∈ SAP (∂D) over s (cf. Remark 2.6 below). Precisely, we put Msk (f ) := M (fk,s ). Similarly, we define the left (k = −1) and the right (k = 1) Bohr-Fourier coefficients and spectra of f over s by the formulas akλ (f, s) := aλ (fk,s ) and specks (f ) := {λ ∈ R : akλ (f, s) = 0}. It follows immediately from the properties of the spectrum of an almost periodic function on R that specks (f ) is at most countable. Let Σ : S × {−1, 1} → 2R be a set-valued map which associates with each s ∈ S, k ∈ {−1, 1} a unital semi-group Σ(s, k) ⊂ R. By SAP Σ (S) ⊂ SAP (S) we denote the Banach algebra of semi-almost periodic functions f with specks (f ) ⊂ Σ(s, k) for all s ∈ S, k ∈ {−1, 1}. By the definition homomorphism Es of Theorem 2.4 sends each f ∈ SAPΣ (S) to the pair of functions f−1,s , f1,s such that fk,s ∈ APΣ(s,k) , k ∈ {−1, 1}. For a unital semi-group Γ ⊂ R by bΓ (R) we denote the maximal ideal space of algebra APΓ (R). (E.g., for Γ = R the space bR := bR (R), commonly called the Bohr compactification of R, is a compact abelian topological group viewed as the inverse limit of compact finite-dimensional tori. The group R admits a canonical embedding into bR as a dense subgroup.) Let bSΣ (∂D) be the maximal ideal space of algebra SAP Σ (S) and S : bSΣ (∂D) → ∂D rΣ
be the map transpose to the embedding C(∂D) → SAP Σ (S). The proof of the next statement is analogous to the proof of Theorem 1.7 in [6]. Theorem 2.5. (1) The map transpose to the restriction of homomorphism Es to SAPΣ (S) determines an embedding hsΣ : bΣ(s,−1) (R) bΣ(s,1) (R) → bSΣ (∂D) S −1 whose image coincides with (rΣ ) (s).
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
299
Figure 2
S S −1 (2) The restriction rΣ : bSΣ (∂D) \ (rΣ ) (S) → ∂D \ S is a homeomorphism. (For an m point set S and each Σ(s, k), s ∈ S, k ∈ {−1, 1}, being a group, the maximal ideal space bSΣ (∂D) is the union of ∂D \ S and 2m Bohr compactifications bΣ(s,k) (R) that can be viewed as (finite or infinite dimensional) tori; see Figure 2).
Remark 2.6. There is an equivalent way to define the mean value of a semialmost periodic function. Specifically, it is easily seen that for a semi-almost periodic function f ∈ SAP (∂D), k ∈ {−1, 1}, and a point s ∈ S the left (k = −1) and the right (k = 1) mean values of f over s are given by the formulas bn
t 1 k Ms (f ) := lim f seike dt, n→∞ bn − an a n where {an }, {bn } are arbitrary sequences of real numbers converging to −∞ such that lim (bn − an ) = +∞. n→∞
The Bohr-Fourier coefficients of f over s can be then defined by the formulas k akλ (f, s) := Msk (f e−iλ logs ), where logks (seikx ) := ln x,
0 < x < 2π,
k ∈ {−1, 1}.
The next result encompasses the basic properties of the mean value and the spectrum of a semi-almost periodic function.
300
Brudnyi and Kinzebulatov
IEOT
Theorem 2.7. (1) For each s ∈ S, k ∈ {−1, 1} the mean value Msk is a complex continuous linear functional on SAP (∂D) of norm 1. (2) A function f ∈ SAP Σ (S) if and only if for each s ∈ S and k ∈ {−1, 1} the almost periodic functions f˜k in Definition 2.3 can be chosen from AP Σ(s,k) (R). (3) The “total spectrum“ s∈S, k=±1 specks (f ) of a function f ∈ SAP (S) is at most countable.
3. Holomorphic semi-almost periodic functions: Main results 3.1. Let Cb (T ) denote the complex Banach space of bounded continuous functions on the strip T := {z ∈ C : Im(z) ∈ [0, π]} endowed with sup- norm. Definition 3.1 (see, e.g., [2]). A function f ∈ Cb (T ) is called holomorphic almost periodic if it is holomorphic in the interior of T and the family of its translates {Sx f }x∈R , Sx f (z) := f (z + x), z ∈ T , is relatively compact in Cb (T ). We denote by AP H(T ) the Banach algebra of holomorphic almost periodic functions endowed with sup-norm. Any function in AP H(T ) is uniformly continuous on T . The mean value of a function f ∈ AP H(T ) is defined by the formula T 1 f (x + iy)dx ∈ C (3.1) M (f ) := lim T →+∞ 2T −T (M (f ) does not depend on y, see, e.g., [2]). Further, the Bohr-Fourier coefficients of f are defined by aλ (f ) := M (f e−iλz ),
λ ∈ R.
(3.2)
Then aλ (f ) = 0 for at most countably many values these values form ∞of λ, iλ lz c e , with cl = 0, the spectrum spec(f ) of f . For instance, if f = l=1 l ∞ |c | < ∞, then spec(f ) = {λ , λ , . . . }. Similarly to the case of func1 2 l=1 l tions from AP (R) each f ∈ AP H(T ) can be uniformly approximated by m functions of the form l=1 cl eiλl z with λl ∈ spec(f ). Let Γ ⊂ R be a unital additive semi-group. The space AP HΓ (T ) of holomorphic almost periodic functions with spectra in Γ forms a unital Banach algebra. Analogously to Theorem 2.2 one has Theorem 3.2. AP HΓ (T ) has the approximation property. The functions in SAPΣ (S)∩H ∞ (D) are called holomorphic semi-almost periodic. Example ([6]). For s ∈ ∂D consider the map 2i(s − z) . s+z Here H+ is the upper half-plane. Then ϕs maps D conformally onto H+ and ∂D \ {−s} diffeomorphically onto R (the boundary of H+ ) so that ϕs (s) = 0. ¯ \ {−s} → H ¯ +, ϕs : D
ϕs (z) :=
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
301
Let T0 be the interior of the strip T . Consider the conformal map Log : H+ → T0 , z → Log(z) := ln |z|+iArg(z), where Arg : C\R− → (−π, π) is the principal branch of the multi-function arg. The function Log is extended to a homeomorphism of H+ \{0} onto T . Let g ∈ AP H(T ). Then the function (Ls g)(z) := (g ◦ Log ◦ ϕs )(z),
z ∈ D,
∞
belongs to SAP ({−s, s}) ∩ H (D). Proposition 3.3. Suppose that f ∈ SAP (S) ∩ H ∞ (D). Then 1 spec−1 s (f ) = specs (f ) =: specs (f )
and, moreover, λπ 1 a−1 λ (f, s) = e aλ (f, s)
λ ∈ specs (f ).
for each
(3.3)
(Recall that the choice of the upper indices ±1 is determined by the orientation of ∂D.) Using Proposition 3.3 and the Lindel¨ of theorem (see, e.g., [13]) one obtains that SAPΣ (S) ∩ H ∞ (D) = SAPΣ (S ) ∩ H ∞ (D), where S := {s ∈ S : Σ(s, −1) ∩ Σ(s, 1) = {0}} and
Σ (s, k) := Σ(s, −1) ∩ Σ(s, 1) for k = −1, 1, s ∈ S . In what follows we assume that Σ(s, −1) = Σ(s, 1) =: Σ(s) and each Σ(s), s ∈ S, is non-trivial. iλ
Example. If g(z) := e π z , z ∈ T , then Ls g = eλh , where h is a holomorphic functions whose real part Re(h) is the characteristic function of the closed arc going in the counterclockwise direction from the initial point at s to the 2 endpoint at −s, and such that h(0) = 12 + i ln π . Thus specs (eλh ) = {λ/π}. Indeed, in this case the restriction of g ◦ Log to R is equal to x → eλ(χR+ (x)+
i ln |x| ) π
,
where χR+ is the characteristic function of R+ . In turn, the restriction of the λ(χR+ + pullback eλh ◦ ϕ−1 s to R coincides with e required result.
i ln |x| ) π
as well. This implies the
3.2. The main result of this section is Theorem 3.4. SAPΣ (S) ∩ H ∞ (D) has the approximation property. Our proof of Theorem 3.4 is based on the equivalence established in Theorem 1.1 and on an approximation result for Banach-valued analogues of algebra SAP (S) ∩ H ∞ (D) formulated below. Specifically, for a Banach space B we define SAPΣB (S) := SAPΣ (S) ⊗ B. Using the Poisson integral formula we can extend each function from SAPΣB (S) to a bounded B-valued harmonic function on D having the same
302
Brudnyi and Kinzebulatov
IEOT
∞ sup-norm. We identify SAPΣB (S) with its harmonic extension. Let HB (D) be the Banach space of bounded B-valued holomorphic functions on D equipped with sup-norm. By (SAPΣ (S) ∩ H ∞ (D))B we denote the Banach space of all continuous B-valued functions f on the maximal ideal space bS (D) of algebra SAP (S) ∩ H ∞ (D) such that ϕ(f ) ∈ SAPΣ (S) ∩ H ∞ (D) for any ϕ ∈ B ∗ . In what follows we naturally identify D with a subset of bS (D).
Proposition 3.5. Let f ∈ (SAPΣ (S) ∩ H ∞ (D))B . Then ∞ f |D ∈ SAPΣB (S) ∩ HB (D).
Let ASΣ be the closed subalgebra of H ∞ (D) generated by the disk-algebra A(D) and the functions of the form geλh , where Re(h)|∂D is the characteristic function of the closed arc going in the counterclockwise direction from the initial point at s to the endpoint at −s such that s ∈ S, πλ ∈ Σ(s) and g(z) := z + s, z ∈ D (in particular, geλh has discontinuity at s only). The next result combined with Proposition 3.5 and Theorem 1.1 implies Theorem 3.4. ∞ (D) = ASΣ ⊗ B. Theorem 3.6. SAPΣB (S) ∩ HB
As a corollary we obtain Corollary 3.7. SAPΣ (S) ∩ H ∞ (D) = ASΣ . This immediately implies the following result. Theorem 3.8. SAPΣ (S) ∩ H ∞ (D) is generated by algebras SAPΣ|F (F ) ∩ H ∞ (D) for all possible finite subsets F of S. 3.3. The algebras SAPΣ (S) ∩ H ∞ (D) are preserved under the action of the group Aut(D) of biholomorphic automorphisms D → D. More precisely, each ¯ →D ¯ (denoted by the same κ ∈ Aut(D) is extended to a diffeomorphism D symbol). We denote by κ∗ : H ∞ (D) → H ∞ (D) the pullback by κ, and put κ∗ S := κ(S), (κ∗ Σ)(s, ·) := Σ(κ(s), ·). Theorem 3.9. The linear operator κ∗ maps SAPκ∗ Σ (κ∗ S) ∩ H ∞ (D) isometrically onto SAPΣ (S) ∩ H ∞ (D). We conclude this section with a result on the tangential behavior of functions from SAP (∂D) ∩ H ∞ (D). Theorem 3.10. Let {zn }n∈N ⊂ D and {sn }n∈N ⊂ ∂D converge to a point s0 ∈ ∂D. Assume that |zn − sn | = 0. (3.4) lim n→∞ |s0 − sn | Then for every f ∈ SAP (∂D) ∩ H ∞ (D) the limits lim f (zn ) and lim f (sn ) n→∞ n→∞ do not exist or exist simultaneously and in the latter case they are equal.
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
303
Remark 3.11. This result implies that the extension (by means of the Gelfand transform) of each f ∈ SAP (∂D) ∩ H ∞ (D) to the maximal ideal space of H ∞ (D) is constant on a nontrivial Gleason part containing a limit point of a net in D converging tangentially to ∂D. In turn, one can easily show that if s ∈ S and the minimal subgroup of R containing Σ(s) is not isomorphic to Z, then the algebra SAPΣ (S) ∩ H ∞ (D) separates points of each nontrivial Gleason part containing a limit point of a net in D converging non-tangentially to s (we refer to [13] for the corresponding definitions). In the next two sections we formulate some topological results about the maximal ideal spaces of algebras SAPΣ (S) ∩ H ∞ (D). 3.4. Let bSΣ (D) denote the maximal ideal space of SAPΣ (S) ∩ H ∞ (D). The inclusion SAPΣ|F1 (F1 ) ∩ H ∞ (D) ⊂ SAPΣ|F2 (F2 ) ∩ H ∞ (D)
if
F1 ⊂ F2
determines a continuous map of maximal ideal spaces F1 2 ωFF12 : bF Σ|F (D) → bΣ|F (D). 2
1
{bF Σ|F (D) ;
ω}F ⊂S ; #F <∞ forms the inverse limiting system. From The family Theorem 3.8 we obtain Theorem 3.12. bSΣ (D) is the inverse limit of {bF Σ|F (D) ; ω}F ⊂S ; #F <∞ . Let ¯ aSΣ : bSΣ (D) → D
(3.5)
be the continuous surjective map of maximal ideal spaces transpose to the embedding A(D) → SAPΣ (S) ∩ H ∞ (D). (Recall that the maximal ideal ¯ By bΓ (T ) we denote space of the disk-algebra A(D) is homeomorphic to D.) the maximal ideal space of algebra AP HΓ (T ) and by ιΓ : T → bΓ (T ) the continuous map determined by evaluations at points of T . (Observe that ιΓ is not necessarily an embedding.) The proof of the next statement can be obtained by following closely the arguments in the proof of Theorem 1.14 in [6]. In its formulation we assume that the corresponding algebras are defined on their maximal ideal spaces by means of the Gelfand transforms. Theorem 3.13. (1) For each s ∈ S there exists an embedding isΣ : bΣ(s) (T ) → bSΣ (D) whose image is (aSΣ )−1 (s) such that the pullback (isΣ )∗ maps algebra SAPΣ (S) ∩ H ∞ (D) surjectively onto AP HΣ(s) (T ). Moreover, the composition of the restriction map to R (R + iπ) and (isΣ ◦ ιΣ(s) )∗ coincides with the restriction of homomorphism Es to SAPΣ (S) ∩ H ∞ (D) (cf. Theorem 2.4). ¯ \ S is a homeomorphism. (2) The restriction aSΣ : bSΣ (D) \ (aSΣ )−1 (S) → D
304
Brudnyi and Kinzebulatov
IEOT
Figure 3
Since SAPΣ (S) ∩ H ∞ (D) separates the points on D, the evaluation at points of D determines a natural embedding ι : D → bSΣ (D). One has the following commutative diagram of maximal ideal spaces considered in the present paper, where the ‘dashed’ arrows stand for embeddings in the case Σ(s, −1) = Σ(s, 1) are (non-trivial) groups for all s ∈ S, and for continuous maps otherwise. Theorem 3.14 (Corona Theorem). ι(D) is dense in bSΣ (D) iff each Σ(s), s ∈ S, is a group. Recall that the corona theorem is equivalent to the following statement: for any collection of functions f1 , . . . , fm ∈ SAPΣ (S) ∩ H ∞ (D) such that max |fk (z)| ≥ δ > 0
1≤k≤m
for all z ∈ D
there exist functions g1 , . . . , gm ∈ SAPΣ (S) ∩ H ∞ (D) such that f1 g 1 + · · · + fm g m = 1 Our next result shows that
bSΣ (D),
on D.
S = ∅, is not arcwise connected.
Theorem 3.15. Assume that F : [0, 1] → bSΣ (D) is continuous. Then either ¯ \S F ([0, 1]) ⊂ D or there exists s ∈ S such that F ([0, 1]) ⊂ (aSΣ )−1 (s). Remark 3.16. From Theorem 3.15 one obtains straightforwardly a similar statement with [0, 1] replaced by an arcwise connected topological space. 3.5. Let KΣS be the Shilov boundary of algebra SAPΣ (S)∩H ∞ (D), that is, the minimal closed subset of bSΣ (D) such that for every f ∈ SAPΣ (S) ∩ H ∞ (D) sup |f (z)| = max |f (ϕ)|, z∈D
S ϕ∈KΣ
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
305
where f is assumed to be extended to bSΣ (D) by means of the Gelfand transform. For a non-trivial semigroup Γ ⊂ R by clΓ (R + iπ) and clΓ (R) we denote closures of ιΓ (R + iπ) and ιΓ (R) in bΓ (T ) (the maximal ideal space of AP HΓ (T )). One can easily show that these closures are homeomorphic to is the minimal subgroup of R containing Γ. bΓ (R), where Γ We retain notation of Theorem 3.13. Theorem 3.17. KΣS
=
isΣ
clΣ(s) (R) ∪ clΣ(s) (R + iπ) ∪ ∂D \ S.
s∈S
Remark 3.18. If each Σ(s), s ∈ S, is a group, then the Shilov boundary KΣS is naturally homeomorphic to the maximal ideal space bSΣ (∂D) of algebra SAPΣ (S), cf. Theorem 2.5. ˇ Next, we formulate a result on the Cech cohomology groups of bSΣ (D). Let bT (S) := s∈S bΣ(s) (T ). According to Theorem 3.13 there exists a natural embedding isΣ : bΣ(s) (T ) → bSΣ (D) whose image is (aSΣ )−1 (s). Then the map I : bT (S) → (aSΣ )−1 (S),
I(ξ) := isΣ (ξ)
for
ξ ∈ bΣ(s) (T ),
is a bijection. Theorem 3.19. (1) The map transpose to the composition I
bT (S) −→ (aSΣ )−1 (S) → bSΣ (D) induces an isomorphism of groups k H k bSΣ (D), Z ∼ H bΣ(s) (T ), Z , =
k ≥ 1.
s∈S
(2) Suppose that each Σ(s) is a subset of R+ or R− . Then H k bSΣ (D), Z = 0, k ≥ 1, and SAPΣ (S) ∩ H ∞ (D) is projective free. ˆ is its dual, If G is a compact connected abelian topological group and G k k ˆ ∼ then H (G, Z) = ∧Z G, k ≥ 1 (see, e.g., [16]). Using Pontryagin duality one obtains Corollary 3.20. Assume that each Σ(s) is a group. Then H k (bSΣ (D), Z) ∼ ∧kZ Σ(s), k ≥ 1. = s∈S
In particular, if for a fixed n ∈ N each Σ(s) is isomorphic to a subgroup of Qn , then H k (bSΣ (D), Z) = 0 for all k ≥ n + 1. Finally, we describe the set of connected components of the group of invertible matrices with entries in SAPΣ (S) ∩ H ∞ (D). Let GLn (A) denote the group of invertible n × n matrices with entries in a unital Banach algebra A. By [GLn (A)] we denote the group of connected
306
Brudnyi and Kinzebulatov
IEOT
components of GLn (A), i.e., the quotient of GLn (A) by the connected component containing the unit In ∈ GLn (A) (this is clearly a normal subgroup of GLn (A)). We set n GΣ(s) (T ) := GLn AP HΣ(s) (T ) and GnΣ (S) := GLn SAPΣ (S)∩H ∞ (D) . Theorem 3.21. The map transpose to the composition I
bT (S) −→ (aSΣ )−1 (S) → bSΣ (D) induces an isomorphism of groups [GnΣ (S)] ∼ =
[GnΣ(s) (T )]. s∈S
In particular, if each Σ(s), s ∈ S, is a subset of R+ or R− , then GnΣ (S) is connected. Remark 3.22. According to a result of Arens [1], [GnΣ(s) (T )], s ∈ S, can be identified with the group [bΣ(s) (T ), GLn (C)] of homotopy classes of continuous maps bΣ(s) (T ) → GLn (C). Moreover, if Σ(s) is a group, then bΣ(s) (T ) is homotopically equivalent to bΣ(s) (R), the maximal ideal space of algebra APΣ(s) (R) (see the proof of Corollary 3.20). In this case bΣ(s) (R) is the inverse limit of a family of finite-dimensional tori. Then [bΣ(s) (T ), GLn (C)] is isomorphic to the direct limit of groups of homotopy classes of maps from tori to GLn (C). As follows from the classical results of Fox [12], the latter can be expressed as a direct sum of certain homotopy groups of the unitary group Un ⊂ GLn (C).
4. Proofs 4.1. Proofs of Theorems 2.2 and 3.2 We will prove Theorem 3.2 only (the proof of Theorem 2.2 is similar). We refer to the book of Besicovich [2] for the corresponding definitions and facts from the theory of almost periodic functions. Proof. Let K ⊂ AP HΓ (T ) be a compact subset. For a given ε > 0 consider an 3ε -net {f1 , . . . , fl } ⊂ K. Let
ν νr ν1 νr −i n1 β1 +···+ n β t r r 1 K(t) := 1− ... 1 − e n1 nr |ν1 |≤n1 ,...,|νr |≤nr
be a Bochner-Fejer kernel such that for all 1 ≤ k ≤ l sup |fk (z) − Mt {fk (z + t)K(t)}| ≤
z∈Σ
ε . 3
(4.1)
Here β1 , . . . , βr are linearly independent over Q and belong to the union of spectra of functions f1 , . . . , fl , ν1 , . . . , νr ∈ Z, n1 , . . . , nr ∈ N, and T 1 Mt {fk (z + t)K(t)} := lim fk (z + t)K(t) dt T →∞ 2T −T
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
307
are the corresponding Bochner-Fejer polynomials belonging to AP HΓ (T ) as well (clearly, the spectrum of the function z → Mt {fk (z +t)K(t)} is contained in spec(fk )). We define a linear operator T : AP HΓ (T ) → AP HΓ (T ) from the definition of the approximation property by the formula (T f )(z) := Mt {f (z + t)K(t)},
f ∈ AP HΓ (T ).
(4.2)
Then T is a bounded linear projection
onto a finite-dimensional subspace i
ν1
β +···+ νr β
z
nr r generated by functions e n1 1 , |ν1 | ≤ n1 , . . . , |νr | ≤ nr . Moreover, since K(t) ≥ 0 for all t ∈ R and Mt {K(t)} = 1, the norm of T is 1. Finally, given f ∈ K choose k such that f − fk AP HΓ (T ) ≤ ε3 . Then we have by (4.1)
T f − f AP HΓ (T ) ≤ T (f − fk )AP HΓ (T ) + T fk − fk AP HΓ (T ) + fk − f AP HΓ (T ) < ε.
4.2. Proof of Theorem 2.7 (1) The result follows directly from Remark 2.6. (2) The fact that for f ∈ SAPΣ (S) the functions f˜k in Definition 2.3 can be chosen from APΣ(s,k) (R) follows from Theorem 2.4 and the definition of spectra of elements of SAP (∂D). Let us show the validity of the converse statement. Let f ∈ SAP (S) and s ∈ S. Assume that for any ε > 0 the functions f˜k in Definition 2.3 (for f and s) can be chosen in APΣ(s,k) (R). Let ρ be a smooth cut-off function equals 1 in a neighbourhood of s contained in the open set γs−1 (δ)∪{s} ∪γs1(δ) and 0 outside of this set. Let us consider a function fˆ on ∂D \ {s} that coincides with ρf1 on γs1 (δ) and with ρf−1 on γs−1 (δ) and equals 0 outside of these arcs. By the definition of spectra of functions in SAP (∂D) the function fˆ belongs to SAPΣ (S). Next, Theorem 2.7 (1) implies that |akλ (fˆ, s) − akλ (f, s)| = |akλ (fˆ, s) − akλ (ρf, s)| < ε,
k ∈ {−1, 1}.
Since ε > 0 is arbitrary, the latter inequality shows that if akλ (f, s) = 0, then λ ∈ Σ(s, k), as required. (3) First, let us show that the set T (f ) of points of discontinuity of a function f ∈ SAP (∂D) is at most countable. For each s ∈ ∂D define ⎛ ⎞ cs (f ) := lim ⎝ ε→0+
sup
s ,s ∈ ei(t−ε) ,ei(t+ε)
⎠ |f (s ) − f (s )| ,
(f ). For n ∈ N, we put where s = eit . One has cs (f ) = 0 if and only if s ∈ T ∞ Tn (f ) := {s ∈ ∂D : cs (f ) ≥ n1 }, so that T (f ) = n=1 Tn (f ). Suppose that T (f ) is uncountable, then there exists n ∈ N such that Tn (f ) is infinite. Since 1 ∂D is compact, Tn (f ) has a limit point eit0 . Choosing ε < 2n in Definition it0 it0 2.3 (for f and e ) from the fact that e is a limit point of Tn (f ) one obtains that the required functions fk do not exist, a contradiction. According to statement (2) fk,s ≡ const for all points s ∈ ∂D at which f is continuous. Therefore due to the previous statement specks (f ) is {0} or
308
Brudnyi and Kinzebulatov
IEOT
∅ for all but at most countably many values of s ∈ S. Since for each s ∈ S the spectrum specks (f ) is at most countable, the required result follows. 4.3. Proof of Proposition 3.3 Let g ∈ AP H(T ), put g1 (x) := g(x), g2 (x) := g(x + iπ), x ∈ R. It follows easily from the approximation result for algebra AP H(T ) cited in Section 3.1 that spec(g1 ) = spec(g2 ) and for each λ ∈ spec(g1 ) aλ (g1 ) = eλπ aλ (g2 ). Suppose that f ∈ SAP (S) ∩ H ∞ (D). Then Theorem 3.13 (1) implies that for each s ∈ S and k ∈ {−1, 1} the functions fk,s are the boundary values of the function (isΣ ◦ ιΣ(s) )∗ (f ) ∈ AP H(T ). 4.4. Proofs of Proposition 3.5 and Theorem 3.6 Our proof of Theorem 3.6 is based on the equivalence established in Theorem 1.1. We first formulate the B-valued analogues of the definitions of almost periodic and semi-almost periodic functions for B being a complex Banach space. Let CbB (R) and CbB (T ) denote the Banach spaces of B-valued bounded continuous functions on R and T , respectively, with norms f := sup f (x)B . x
CbB (R)
is said to be almost periodic if the Definition 4.1. 1) A function f ∈ family of its translates {Sτ f }τ ∈R , Sτ f (x) := f (x + τ ), x ∈ R, is relatively compact in CbB (R). 2) A function f ∈ CbB (T ) is called holomorphic almost periodic if it is holomorphic in the interior of T and the family of its translates {Sx f }x∈R is relatively compact in CbB (T ). Let AP B (R) and AP H B (T ) denote the Banach spaces of almost periodic and holomorphic almost periodic functions on R and T , respectively, endowed with sup-norms. Remark 4.2. Since AP (R) and AP H(T ) have the approximation property, it m iλl x b e (x ∈ R, follows from Theorem 1.1 that the functions of the form l l=1 iλl z B bl ∈ B, λl ∈ R) and m b e (z ∈ T, b ∈ B, λ ∈ R) are dense in AP (R) l l l l=1 B and AP H (T ), respectively. As in the case of scalar almost periodic functions, a Banach-valued almost periodic function f ∈ APB (R) is characterized by its Bohr-Fourier coefficients aλ (f ) and the spectrum spec(f ), defined in terms of the mean value T 1 M (f ) := lim f (x)dx ∈ B. (4.3) T →+∞ 2T −T Namely, we define aλ (f ) := M (f e−iλx ), λ ∈ R.
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
309
It follows from the above remark and the properties of scalar almost periodic functions that aλ (f ) = 0 in B for at most countably many values of λ. ∞ iλl x These values constitute the spectrum spec(f ) of f . E.g., if f = l=1 bl e ∞ ( l=1 bl < ∞ and all bl = 0), then spec(f ) = {λ1 , λ2 , . . . }. Similarly, one defines the mean-values and the spectra for functions from AP H B (T ). Let L∞ B (∂D) be the Banach space of B-valued bounded measurable functions on ∂D equipped with sup-norm. Definition 4.3. A function f ∈ L∞ B (∂D) is called semi-almost periodic if for any s ∈ ∂D and any ε > 0 there exist a number δ = δ(s, ε) ∈ (0, π) and functions fk : γsk (δ) → B, γsk (δ) := {seikx : 0 ≤ x < δ < 2π}, k ∈ {−1, 1}, such that functions x x → fk seikδe , −∞ < x < 0, k ∈ {−1, 1}, are restrictions of B-valued almost periodic functions from AP B (R) and sup f (z) − fk (z)B < ε,
z∈γsk (δ)
k ∈ {−1, 1}.
Analogously to the scalar case, for a closed subset S ⊂ ∂D, we denote by SAP B (S) ⊂ L∞ B (∂D) the Banach space of semi-almost periodic functions that are continuous on ∂D \ S, so that SAP (S) := SAP C (S). Let SAP (S) ⊗ B denote the completion in L∞ B (∂D) of the symmetric tensor product of SAP (S) and B. Proposition 4.4. SAP B (S) = SAP (S) ⊗ B. The statement is an immediate consequence of Theorem 1.1 and the following two facts: each function f ∈ SAP B (S) admits a norm preserving extension to the maximal ideal space bS (∂D) of the algebra SAP (S) as a continuous B-valued function, and C(bS (∂D)) has the approximation property. The first fact follows straightforwardly from the definitions of SAP B (S) and bS (∂D) (see [6]) and the existence of analogous extensions of functions in AP B (R) to bR, while the second fact is valid for any algebra C(X) on a compact Hausdorff topological space X (it can be proved using finite partitions of unity of X). Next, we introduce the Banach space AP B (R (R + iπ)) := AP (R (R + iπ)) ⊗ B of B-valued almost periodic functions on R (R + iπ), see Example 2. Also, B ∞ for each s ∈ S we define a linear isometry LB s : AP (R(R+iπ)) → LB (∂D) by the formula (cf. (2.4)) (LB s g)(z) := (g ◦ Log ◦ ϕs )(z),
g ∈ AP B (R (R + iπ)).
(4.4)
Now, using Proposition 4.4 we prove a B-valued analog of Theorem 2.4. Theorem 4.5. For every point s ∈ ∂D there exists a bounded linear operator EsB : SAP B (∂D) → AP B (R (R + iπ))
310
Brudnyi and Kinzebulatov
IEOT
B of norm 1 such that for each f ∈ SAP B (∂D) the function f − LB s (Es f ) ∈ B SAP (∂D) is continuous and equal to 0 at s. Moreover, any bounded linear operator SAP B (∂D) → AP B (R (R + iπ)) satisfying this property coincides with EsB .
Proof. According to Proposition 4.4, it suffices to define m the required operator EsB on the space of functions of the form f = l=1 bl fl , where bl ∈ B, fl ∈ SAP (∂D). In this case we set EsB (f ) :=
m
bl Es (fl ),
l=1
where Es is the operator from Theorem 2.4. Let B1 denote the unit ball in B ∗ . Then according to Theorem 2.4 we have sup |ϕ(EsB (f )(z))| EsB (f ) = sup EsB (f )(z)B = z∈∂D z∈∂D, ϕ∈B1 m = sup ϕ(bl )Es (fl )(z) z∈∂D, ϕ∈B1 l=1 m ≤ sup ϕ(bl )fl (z) = f . z∈∂D, ϕ∈B1 l=1
Therefore, the operator EsB is continuous and of norm 1 on a dense subspace of SAP B (∂D). Moreover, for any function f from this subspace we have B B (by Theorem 2.4), f − LB s (Es (f )) ∈ SAP (∂D) is continuous and equal to B 0 at s. Extending Es by continuity to SAP B (∂D) we obtain the operator satisfying the required properties. Its uniqueness follows from the uniqueness of operator Es . B B (x) := (EsB f )(x) and f1,s (x) := (EsB f )(x+iπ), Using the functions f−1,s x ∈ R, belonging to APB (R) we define the left (k = −1) and the right (k = 1) mean values of f ∈ SAP B (∂D) over s:
Msk (f ) := M (fk,s ) ∈ B. Then using formulas similar to those of the scalar case we define the BohrFourier coefficients akλ (f, s) ∈ B and the spectrum specks (f ) of f over s. It follows straightforwardly from the properties of the spectrum of a B-valued almost periodic function on R that specks (f ) is at most countable. B By SAP Σ (S) ⊂ SAP B (S) we denote the Banach algebra of semi-almost periodic functions f with specks (f ) ⊂ Σ(s, k) for all s ∈ S, k ∈ {−1, 1}. Note that SAPΣB (S) = SAPΣ (S) ⊗ B, i.e., this definition is equivalent to the one used in Section 3.2 (the proof is obtained easily from Definition 4.3, using an appropriate partition of unity on ∂D and Theorems 1.1 and 3.2, see [6] for similar arguments). Also, a statement analogous to Theorem 3.3 holds for SAPΣB (S) ∩ ∞ ∞ HB (D). Namely, if f ∈ SAPΣB (S) ∩ HB (D), then spec1s (f ) = spec−1 s (f ) =: specs (f ).
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
311
Proof of Proposition 3.5. We must show that if f ∈ (SAPΣ (S) ∩ H ∞ (D))B on the maximal ideal space bS (D) of algebra SAP (S) ∩ H ∞ (D), then the ∞ (D). restriction f |D ∈ SAPΣB (S) ∩ HB B S Indeed, since f ∈ C (b (D)) and C(bS (D)) has the approximation property, f ∈ C(bS (D)) ⊗ B by Theorem 1.1. Next, C(bS (D)) is generated by algebra SAP (S) ∩ H ∞ (D) and its conjugate. Therefore f can be uniformly approximated on bS (D) by a sequence of B-valued polynomials in variables from algebras SAP (S) ∩ H ∞ (D) and its conjugate. This easily implies that f |∂D is well defined and belongs to SAP B (S). In fact, f |∂D ∈ SAPΣB (S) because k φ(f )k ∈ SAP Σ (S) and the Bohr-Fourier coefficients of ∗f satisfy aλ (ϕ(f ), s) = ϕ aλ (f, s) for any s ∈ S, k ∈ {−1, 1} and φ ∈ B . Further, by the definition f |D is such that ϕ(f ) ∈ H ∞ (D) for any ϕ ∈ B ∗ . This shows that ∞ f ∈ HB (D). For the proof of Theorem 3.6 we require some auxiliary results. Let AP C(T ) be the Banach algebra of functions f : T → C uniformly continuous on T and almost periodic on each horizontal line. We define AP C B (T ) := AP C(T ) ⊗ B. The proof of the next statement is analogous to the proof of Lemma 4.3 in [6]. Lemma 4.6. Suppose that f1 ∈ AP B (R), f2 ∈ AP B (R+iπ). Then there exists a function F ∈ AP C B (T ) harmonic in the interior of Σ whose boundary values are f1 and f2 . Moreover, F admits a continuous extension to the maximal ideal space bT of AP H(T ). The proof of the next statement uses Lemma 4.6 and is very similar to the proof of Lemma 4.2 (for B = C) in [6], so we omit it as well. Suppose that s := eit and γsk (δ) are arcs defined in (2.3). For δ ∈ (0, π) we set γ1 (s, δ) := Log(ϕs (γs1 (δ))) ⊂ R and γ−1 (s, δ) := Log(ϕs (γs−1 (δ))) ⊂ R + iπ (see Example 3). Lemma 4.7. Let s ∈ S. Suppose that f ∈ SAP B ({−s, s}). We put fk =f |γsk (π) and define on γk (s, π) −1 hk := fk ◦ ϕ−1 , s ◦ Log
k ∈ {−1, 1}.
Then for any ε > 0 there exist δε ∈ (0, π) and a function H ∈ AP C B (T ) harmonic in the interior T0 of T such that sup z∈γk (s,δε )
hk (z) − H(z)B < ε,
k ∈ {−1, 1}.
Let s ∈ ∂D and Us be the intersection of an open disk of some radius ¯ \ s. We call such Us a circular neighbourhood of s. ≤ 1 centered at s with D Definition 4.8. We say that a bounded continuous function f : D → B is almost-periodic near s if there exist a circular neighbourhood Us , and a function f¯ ∈ AP C B (T ) such that ¯ ¯ f (z) = (LB z ∈ Us . (4.5) s f )(z) := f ◦ Log ◦ ϕs (z),
312
Brudnyi and Kinzebulatov
IEOT
In what follows for Σ : S × {−1, 1} → 2R such that Σ(s) = R for each s ∈ S we omit writing Σ in aSΣ , isΣ , bSΣ (D) etc., see Section 3.4 for the corresponding definitions. In the proof of Theorem 1.8 of [6] (see Lemmas 4.4, 4.6 there) we established, cf. Theorem 3.13, (1) Any scalar harmonic function f on D almost periodic near s admits a ¯s ) ⊂ bS (D) for some circular neighcontinuous extension fs to (aS )−1 (U bourhood Us . (2) For any s ∈ S and any g ∈ AP H(T ) the holomorphic function g˜ := Ls g on D almost periodic near s is such that g˜s ◦ is coincides with the extension of g to bT . More generally, Lemma 4.6, statements (1), (2) and the fact that AP B (R) = AP (R) ⊗ B (see Theorems 1.1 and 2.2) imply (3) Any B-valued harmonic function f on D almost periodic near s admits ¯s ) ⊂ bS (D) for some circular a continuous extension fsB to (aS )−1 (U neighbourhood Us . (4) For any s ∈ S and any g ∈ AP H B (T ) the B-valued holomorphic function g˜ := LB ˜sB ◦ is coincides s g on D almost periodic near s is such that g with the extension of g to bT . ∞ (D) and s ∈ ∂D. There is a bounded Lemma 4.9. Let f ∈ SAPΣB (S) ∩ HB ˆ B-valued holomorphic function f on D almost periodic near s such that for any ε > 0 there is a circular neighbourhood Us;ε of s so that
sup ||f (z) − fˆ(z)||B < ε.
z∈Us;ε
B ¯ ¯ Moreover, fˆ = LB s f for some f ∈ AP HΣ(s) (T ).
Proof. Assume, first, that s ∈ S. By Lemma 4.7, for any n ∈ N there exist a number δn ∈ (0, π) and a function Hn ∈ AP C B (T ) harmonic on T0 such that 1 (4.6) sup fk (z) − Hn (z)B < , k ∈ {−1, 1}. n z∈γk (s,δn ) Using the Poisson integral formula for the bounded B-valued harmonic function f − hn , hn := LB s Hn := Hn ◦ Log ◦ ϕs , on D we easily obtain from (4.6) that there is a circular neighbourhood Vs;n of s such that 2 (4.7) sup f (z) − hn (z)B < . n z∈Vs;n ˆ n to (aS )−1 (s) ∼ According to (3) each hn admits a continuous extension h = ˆ n }n∈N to bT . Moreover, (4.7) implies that the restriction of the sequence {h ˆ ∈ C B ((aS )−1 (s)) (aS )−1 (s) forms a Cauchy sequence in C B ((aS )−1 (s)). Let h be the limit of this sequence. Further, for any functional φ ∈ B ∗ the function φ ◦ f belongs to the admits a continuous extension fφ algebra SAP (S) ∩ H ∞ (D) and therefore to (aS )−1 (s) such that on (is )−1 (aS )−1 (s) the function fφ ◦ is belongs to
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
313
ˆ for any φ ∈ B ∗ . Then it follows AP H(T ). Now, (4.7) implies that fφ = φ ◦ h s ˆ ◦ i ∈ AP H B (T ). Therefore by (4) we find from Theorems 1.1 and 3.2 that h ˆ a bounded B-valued holomorphic function fˆ on D of the same sup-norm as h S −1 ˆ almost periodic near s such that its extension to (a ) (s) coincides with h. Next, by the definition of the topology of bS (D), see [6], Lemma 4.4 (a), we obtain that for any ε > 0 there is a number N ∈ N such that for all n ≥ N , ε sup fˆ(z) − hn (z)B < . 2 z∈Vs;n Now, choose n ≥ N in (4.7) such that the right-hand side there is < ε2 . For this n we set Us;ε := Vs;n . Then the previous inequality and (4.7) imply the required inequality sup ||f (z) − fˆ(z)||B < ε. z∈Us;ε
Further, if s ∈ S, then, by definition, f |∂D is continuous at s. In this case as the function fˆ we can choose the constant B-valued function equal to f (s) on D. Then the required result follows from the Poisson integral formula for f − fˆ. By definition, fˆ is determined by formula (4.5) with an f¯ ∈ AP H B (T ). B (T ). To this end it suffices to prove that Let us show that f¯ ∈ AP HΣ(s) ¯ ϕ(f ) ∈ AP HΣ(s) (T ) for any ϕ ∈ B ∗ . Indeed, it follows from the last inequality that the extension of ϕ(f ) ∈ SAPΣ (S) ∩ H ∞ (D) to (aSΣ )−1 (s) coincides with ϕ(f¯). By the definition of spectrum of a semi-almost periodic function, this implies that specs (ϕ(f¯)) ⊂ Σ(s). Now, we are ready to prove Theorem 3.6. The inclusion ASΣ ⊂ SAPΣ (S)∩H ∞ (D) follows from Example 3. Indeed, for s ∈ S assume that the holomorphic function eλh ∈ H ∞ (D) is such that Re(h)|∂D is the characteristic function of the closed arc going in the counterclockwise direction from the initial point at s to the endpoint at −s and such that πλ ∈ Σ(s). Then Example 3 implies that eλh ∈ SAP ({s, −s}) ∩ H ∞ (D) λ and specs (eλh ) = { πλ }. In particular, (z + s)e π h ∈ SAPΣ|{s} ({s}) ∩ H ∞ (D), as required. Let us prove the opposite inclusion. (A) Consider first the case S = F , where F = {si }m i=1 is a finite subset of ∞ ∂D. Let f ∈ SAPΣB (F ) ∩ HB (D). Then according to Lemma 4.9 there exists B a function fs1 ∈ AP HΣ(s (T ) such that the bounded B-valued holomorphic 1) function gs1 − f , where gs1 := fs1 ◦ Log ◦ ϕs1 , on D is continuous and equals 0 at s1 . B Let us show that gs1 ∈ A{s1 ,−s1 } ⊗ B. Since fs1 ∈ AP HΣ(s (T ), by 1) B Theorems 1.1 and 3.2 it can be approximated in AP HΣ(s1 ) (T ) by finite sums of functions of the form beiλz , b ∈ B, λ ∈ Σ(s1 ), z ∈ T . In turn, gs1 can be approximated by finite sums of functions of the form beiλLog◦ϕs1 , b ∈ B.
314
Brudnyi and Kinzebulatov
IEOT
As was shown in [6], eiλLog◦ϕs1 ∈ A{s1 ,−s1 } . Hence, gs1 ∈ A{s1 ,−s1 } ⊗ B. We define gs (z)(z + s1 ) . gˆs1 = 1 2s1 Then, since the function z → (z + s1 )/(2s1 ) ∈ A(D) and equals 0 at −s1 , and gs1 ∈ A{s1 ,−s1 } ⊗ B, the function gˆs1 ∈ A{s1 } ⊗ B. Moreover, by the {s } construction of gˆ1 and the definition of the spectrum gˆs1 ∈ AΣ(s1 1 ) ⊗ B. By definition, the difference gˆs1 − f is continuous and equal to zero at z1 . Thus, B ∞ (F \ {s1 }) ∩ HB (D). gˆs1 − f ∈ SAPΣ| F \{s } 1
{s }
We proceed in this way to get functions gˆsk ∈ AΣ(sk k ) ⊗ B, 1 ≤ k ≤ m, such that m gˆsk ∈ AB (D), f− k=1
where AB (D) is the Banach space of B-valued bounded holomorphic functions on D continuous up to the boundary. As in the scalar case using the Taylor expansion at 0 of functions from AB (D) one can easily show that AB (D) = A(D) ⊗ B. Therefore, f ∈ AF Σ ⊗ B. (B) Let us consider the general case of S ⊂ ∂D being an arbitrary ∞ (D). As follows from Lemma 4.9 and the closed set. Let f ∈ SAPΣB (S) ∩ HB arguments presented in part (A), given an ε > 0 there exist points sk ∈ ∂D, {s } functions fk ∈ AΣ(sk k ) ⊗ B and circular neighbourhoods Usk (1 ≤ k ≤ m) m such that {Usk }m k=1 forms an open cover of ∂D \ {sk }k=1 and f (z) − fk (z)B < ε on Usk ,
1 ≤ k ≤ m.
(4.8)
¯s . Since S is closed, for sk ∈ S we may assume that fk is continuous in U k m Let us define a B-valued 1-cocycle {ckj }k,j=1 on intersections of the sets in {Usk }m k=1 by the formula ckj (z) := fk (z) − fj (z),
z ∈ Usk ∩ Usj .
(4.9)
Then (4.8) implies supk,j,z ||ckj (z)||B < 2ε. Let A ⊂⊂ ∪m k=1 Usk be an open annulus with outer boundary ∂D. Using the argument from the proof of Lemma 4.7 in [6] one obtains that if the width of the annulus is sufficiently small, then there exist B-valued functions ci holomorphic on Usi ∩ A and ¯si ∩ A¯ satisfying continuous on U sup z∈Usi ∩A
||ci (z)||B ≤ 3ε
(4.10)
and such that ci (z) − cj (z) = cij (z),
z ∈ Usi ∩ Usj ∩ A.
For such A let us define a function fε on A¯ \ {si }m i=1 by formulas fε (z) := fi (z) − ci (z),
¯ z ∈ Usi ∩ A.
(4.11)
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
315
According to (4.9) and (4.11), fε is a bounded continuous B-valued function on A¯ \ {si }m i=1 holomorphic in A. Furthermore, since ci is continuous on ¯si ∩ A, ¯ and fi ∈ A{si } ⊗ B for si ∈ S, and fi ∈ AB (D) otherwise, fε |∂D ∈ U Σ(si ) B SAPΣ| (F ), where F = {si }m i=1 ∩ S. Also, from inequalities (4.8) and (4.10) F we obtain (4.12) sup f (z) − fε (z)B < 4ε. z∈A
Next, as in [6] we consider a 1-cocylce subordinate to a cover of the unit disk D consisting of an open annulus having the same interior boundary as A and the outer boundary {z ∈ C : |z| = 2}, and of an open disk centered at 0 not containing A but intersecting it by a nonempty set. Resolving this cocycle1 one obtains a B-valued holomorphic function Fε on D such that for an absolute constant Cˆ > 0 ˆ sup f (z) − Fε (z)B < Cε z∈D
B ∞ and by definition Fε ∈ SAPΣ| (F ) ∩ HB (D), where F = {s1 , . . . , sm } ∩ S. F The latter inequality and part (A) of the proof show that the complex vector space generated by spaces AF Σ|F ⊗ B for all possible finite subsets F ⊂ S ∞ (D). Since by definition the closure of all such is dense in SAPΣB (S) ∩ HB S B ∞ S AF Σ|F ⊗ B is AΣ ⊗ B, we obtain the required: SAPΣ (S) ∩ HB (D) = AΣ ⊗ B.
4.5. Proofs of Theorems 3.9 and 3.10 Proof of Theorem 3.9. Corollary 1.6 in [6] states that κ∗ |∂D : C(∂D) → C(∂D), the pullback by κ|∂D , maps SAP (κ∗ S) isomorphically onto SAP (S). Following closely the arguments in its proof, one obtains even more: κ∗ maps SAPκ∗ Σ (κ∗ S) isomorphically onto SAPΣ (S). Since κ∗ preserves H ∞ (D), the required result follows. Proof of Theorem 3.10. Let f ∈ SAP (∂D) ∩ H ∞ (D). According to Lemma 4.9 there exists a function fs ∈ AP H(T ) such that the difference h := f − Fs , where Fs := fs ◦Log◦ϕs , see (4.5), is continuous and equal to 0 at s. Therefore, it suffices to prove the assertion of the theorem for Fs . Let {zn } ⊂ T0 and {sn } ⊂ R ∪ (R + iπ) be the images of sequences {zn } and {sn } under the mapping Log◦ϕs (see Example 3). By the hypotheses of the theorem we have zn , sn → ∞ and |zn − sn | → 0 as n → ∞ (this follows from condition (3.4)). Since any function in AP H(T ) is uniformly continuous (see Section 3), the latter implies the required result. 1 There
is a misprint in [6] at this place: the inequality maxi ∇ρi L∞ (C) ≤ Cw(B∩A) < Cε
for smooth radial functions ρ1 , ρ2 , must be replaced by maxi ∇ρi L∞ (C) ≤
C . w(B∩A)
316
Brudnyi and Kinzebulatov
IEOT
4.6. Proofs of Theorems 3.14 and 3.15 Proof of Theorem 3.14. Below we identify ι(D) ⊂ bSΣ (D) with D, see Section 3.4. By Theorem 3.13, the maximal ideal space bSΣ (D) is ¯ \ S) s∈S is (bΣ(s) (T )) (D Σ (here isΣ : bΣ(s) (T ) → (aSΣ )−1 (s) is a homeomorphism). For each s ∈ S one has the natural map ιΣ(s) : T → bΣ(s) (T ) (determined by evaluations at points of T ). Also, the argument of the proof of Theorem 1.12 in [6] implies that the ¯ closure of D in bSΣ (D) contains (as a dense subset) (D\S) s∈S isΣ (ιΣ(s) (T )) . Thus in order to prove the theorem, it suffices to show that ιΣ(s) (T ) is dense in bΣ(s) (T ) if and only if Σ(s) is a group. We will use the following result. Theorem 4.10 ([21]). Suppose that Γ is the intersection of an additive subgroup of R and R+ . Then the image of the upper half-plane H+ in the maximal ideal space bΓ (T ) is dense. Observe that in this case each element of AP HΓ (T ) is extended to a holomorphic almost periodic function on H+ by means of the Poisson integral. Therefore the evaluations at points of H+ of the extended algebra determine the map H+ → bΓ (T ) of the theorem. First, assume that Σ(s) is a group. We have to show that ιΣ(s) (T ) is dense in bΣ(s) (T ). Assume the opposite. Then there exists ξ ∈ bΣ(s) (T ) and a neighbourhood of ξ U (λ1 , . . . , λm , ξ, ε) := {η ∈ bΣ(s) (T ) : |η(eiλk z ) − ck | < ε, 1 ≤ k ≤ m}, where λ1 , . . . , λm ∈ Σ(s), ck := ξ(eiλk z ), such that U (λ1 , . . . , λm , ξ, ε) ∩ cl ιΣ(s) (T ) = ∅, cf. the proof of Theorem 2.4 in [6]. Therefore, max |eiλk z − ck | ≥ ε > 0
1≤k≤m
for all z ∈ T.
(4.13)
Without loss of generality we may assume that ck = 0 and λk > 0, i.e., eiλk z − ck ∈ AP HΣ(s)∩R+ (T ). (For otherwise we replace eiλk z − ck with −iλk z e−iλk z −c−1 −c−1 k . Here e k ∈ AP HΣ(s)∩R+ (T ) since Σ(s) is a group. Also, (4.13) will be satisfied, possibly with a different ε > 0.) Note that eiλk z − ck is not invertible in AP HΣ(s) (T ), since ξ(eiλk z − ck ) = 0. Therefore, since each function eiλk z − ck is periodic (with period λ2πk ), it has a zero in T . Since solutions of the equation eiλk z = ck are of the form zk = −
i ln |ck | Argck + 2πl + , λk λk
l ∈ Z,
all zeros of eiλk z − ck belong to T . Hence, in virtue of inequality (4.13), one has max |eiλk z − ck | ≥ ε˜ > 0 for all z ∈ H+ . 1≤k≤m
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
317
Hence, by Theorem 4.10, there exist functions g1 , . . . , gm ∈ AP HΣ(s)∩R+ (H+ ) such that m gk (z)(eiλk z − ck ) = 1 for all z ∈ H+ . k=1
In particular, the above identity holds on T . This gives a contradiction with the assumption ξ(eiλk z − ck ) = 0, 1 ≤ k ≤ m. Now, assume that Σ(s) is not a group, i.e., it contains a non-invertible element λ0 . Suppose that ιΣ(s) (T ) is dense in bΣ(s) T . Then, since the modulus of f1 := eiλ0 z is bounded from below on T by a positive number, there exists g1 ∈ AP HΣ(s) (T ) such that f1 g1 ≡ 1. Therefore, g1 = e−iλ0 z ∈ AP HΣ(s) (T ), i.e., −λ0 ∈ Σ(s), a contradiction. Proof of Theorem 3.15. For the proof we will need the following auxiliary result. Let Γ ⊂ R be a nontrivial additive semi-group. For a subset X ⊂ T by X∞ we denote the set of limit points of ιΓ (X) in bΓ (T ) \ ιΓ (T ). Lemma 4.11. Let G ∈ C([0, 1), T ) be such that the closure of G([0, 1)) in T is non-compact. Then the set G∞ contains more than one element. Proof. If there exists a horizontal line R + ic, 0 ≤ c ≤ π, such that lim distT G(t), R + ic = 0, t→1−
then clearly (R + ic)∞ = G∞ . Moreover, (R + ic)∞ is infinite (e.g., it contains a subset homeomorphic to interval [0, 1]). In the case that such a line does not exist, one can find two closed substrips T1 , T2 ⊂ T , T1 ∩ T2 = ∅, such that the closures in T of both G([0, 1)) ∩ T1 and G([0, 1)) ∩ T2 are noncompact. Then (G([0, 1)) ∩ T1 )∞ and (G([0, 1)) ∩ T2 )∞ are nonempty, while (T1 )∞ ∩ (T2 )∞ = ∅. This implies the required statement. Now, we are ready to prove the theorem. Suppose on the contrary that for a continuous map F : [0, 1] → bSΣ (D) the conclusion of the theorem is not ¯ \S valid. First, assume that there exists a point c ∈ [0, 1) such that F (c) ∈ D ¯ \ S. Then, because bS (D) \ (D ¯ \ S) is a compact set (here but F ([0, 1]) ⊂ D Σ ¯ \ S with a subset of bS (D)), passing to a subinterval, we naturally identify D Σ ¯ \S if necessary, we may assume without loss of generality that F [0, 1) ⊂ D and F (1) ∈ (aSΣ )−1 (s) for a certain s ∈ S. Define G(t) := (Log ◦ ϕs ) F (t) ⊂ T, t ∈ [0, 1) (cf. Example 2). Then G satisfies conditions of Lemma 4.11 for Γ = Σ(s). Next, consider an f ∈ SAPΣ (S). According to Lemma 4.9 there exists a (unique) function fs ∈ AP HΣ(s) (T ) such that the difference f − Fs , where Fs := fs ◦ Log ◦ ϕs , is continuous and equal to 0 at s. This yields lim fs (G(t)) − f (F (t)) = 0. t→1−
¯ The latter implies that the set of limit points of F ([0, 1)) in bSΣ (D)\(D\S) is in one-to-one correspondence with the set of limit points G∞ of ιΣ(s) (G([0, 1)))
318
Brudnyi and Kinzebulatov
IEOT
in bΣ(s) (T ) \ ιΣ(s) (T ). By our assumption the set of limit points of F ([0, 1)) ¯ \ S) consists of the point F (1). This contradicts the assertion in bSΣ (D) \ (D ¯ \ S. of Lemma 4.11. Hence, in this case F ([0, 1]) ⊂ D ¯ \ S). Let s ⊂ S be such that In the second case, F ([0, 1]) ⊂ bSΣ (D) \ (D F ([0, 1]) ∩ (aSΣ )−1 (s) = ∅. Consider the continuous map {s}
ωs : bSΣ (D) → bΣ|{s} (D), transpose to the embedding SAPΣ|{s} ({s}) ∩ H ∞ (D) ⊂ SAPΣ (S) ∩ H ∞ (D). {s}
According to the case considered above, if ωs ◦ F : [0, 1] → bΣ|{s} (D) is such ¯ \ {s} for some c ∈ [0, 1), then (ωs ◦ F )([0, 1]) ⊂ D ¯ \ {s} that (ωs ◦ F )(c) ∈ D S −1 which contradicts the assumption F ([0, 1]) ∩ (aΣ ) (s) = ∅. Thus {s} ¯ \ {s}) = (a{s} )−1 (s). (ωs ◦ F )([0, 1]) ⊂ bΣ|{s} (D) \ (D Σ|{s}
−1 (s). This implies that F ([0, 1]) ⊂ aSΣ
4.7. Proof of Theorem 3.17 Since SAPΣ (S) ∩ H ∞ (D) is generated by algebras SAPΣ|F (F ) ∩ H ∞ (D) for F all possible finite subsets F of S, the inverse limit of {KΣ| ; ω}F ⊂S ; #F <∞ F of the corresponding Shilov boundaries coincides with KΣS (see Section 3.4 for the corresponding notation). Therefore to establish the result it suffices to prove that
F s iΣ|F clΣ(s) (R) ∪ clΣ(s) (R + iπ) ∪ ∂D \ F. (4.14) KΣ|F = s∈F
Since each point of ∂D \ F is a peak point for A(D) (which is a subset F of SAPΣ |F (F ) ∩ H ∞ (D)), ∂D \ F ⊂ KΣ| . Next, the closure of ∂D \ F in F ∞ bF (D) (the maximal ideal space of SAP Σ |F (F ) ∩ H (D)) coincides with Σ|F the right-hand side of (4.14), see the proof of Theorem 1.14 in [6]. Thus the F right-hand side of (4.14) is a subset of KΣ| . Finally, Theorem 1.14 of [6] F ∞ implies that for each f ∈ SAPΣ |F (F ) ∩ H (D), |f | attains its maximum on the set in the right-hand side of (4.14). This produces the required identity. One can easily show that the inverse limit of the family of sets in the right-hand sides of equations (4.14) coincides with
s iΣ clΣ(s) (R) ∪ clΣ(s) (R + iπ) ∪ ∂D \ S. s∈S
The proof of the theorem is complete. 4.8. Proofs of Theorems 3.19, 3.21 and Corollary 3.20 Proof of Theorem 3.19. (1) Consider first the case of S being a finite subset of ∂D. For s ∈ S we define U1 := bSΣ (D) \ (aSΣ )−1 (s).
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
319
Let U2 be the union of (aSΣ )−1 (s) and a circular neighbourhood of s whose ¯ Both U1 , U2 are open in bS (D) and U1 ∩ U2 = closure is a proper subset of D. Σ S −1 U2 \(aΣ ) (s) is the circular neighbourhood of s. Since U1 ∩U2 is contractible, H k (U1 ∩ U2 , Z) = 0, k ≥ 1. Let us show that for any k ∈ Z, H k (U2 , Z) ∼ = H k (bΣ(s) (T ), Z).
(4.15)
To this end consider a sequence V1 ⊃ V2 ⊃ . . . of circular neigbourhoods of ¯ s such that ∩∞ k=1 Vk = {s} and V1 = U1 ∩ U2 . We set ˆk := Vk ∪ (aS )−1 (s). U Σ S −1 ˆ ˆ (s) Let ιm l : Um → Ul , m ≥ l, be the corresponding embedding. Then (aΣ ) ˆ is the inverse limit of the family {Uj ; ι}j∈N . It is well known (see, e.g., [4], ˇ Chapter II, Corollary 14.6) that the direct limit of Cech cohomology groups k ˆ k S −1 H (Ul , Z) with respect to this family gives H ((aΣ ) (s), Z). Note also that ˆ1 := U2 . Thus the maps ιl1 induce ˆl is a deformation retract of U each U ˆl , Z), l ∈ N. Since (aS )−1 (s) ∼ isomorphisms H k (U2 , Z) ∼ = H k (U = bΣ(s) (T ), Σ these facts imply (4.15). Next, consider the Mayer-Vietoris sequence corresponding to the cover {U1 , U2 } of bSΣ (D):
· · · → H k−1 (bSΣ (D), Z) → H k (U1 ∩ U2 , Z) → H k (U1 , Z) ⊕ H k (U2 , Z) → H k (bSΣ (D), Z) → . . . . By the above results H k (U1 ∩ U2 , Z) = 0 and H k (U2 , Z) ∼ = H k (bΣ(s) T, Z), k ≥ 1. Therefore, H k (bSΣ (D), Z) = H k (U1 , Z) ⊕ H k bΣ(s) (T ), Z , k ≥ 1. Proceeding further inductively (i.e., applying similar arguments to U1 etc.) and using the fact that H k (bSΣ (D) \ S, Z) = 0, k ≥ 1, we obtain that H k (bSΣ (D), Z) = H k bΣ(s) (T ), Z . s∈S
Now, if S ⊂ ∂D is an arbitrary closed subset, then since bSΣ (D) is the inverse limit of bF Σ|F (D) for all possible finite subsets F ⊂ S, by the cited result in [4] H k (bSΣ (D), Z) is the direct limit of H k (bF Σ|F (D), Z). Based on the case considered above we obtain that this limit is isomorphic to k b H (T ), Z . Σ(s) s∈S This proves the first part of the theorem. (2) As is shown in [5], if Γ ⊂ R+ or Γ ⊂ R− , then bΓ (T ) is contractible. Therefore under hypotheses of the theorem H k bΣ(s) (T ), Z = 0 for all s ∈ S. The required result now follows from (1), i.e., H k (bSΣ (D), Z) = 0 for all k ≥ 1. Further, according to [8] the connectedness of bSΣ (D) and the topological triviality of any complex vector bundle of a finite rank over bSΣ (D) are sufficient for projective freeness of SAPΣ (S) ∩ H ∞ (D).
320
Brudnyi and Kinzebulatov
IEOT
Clearly bSΣ (D) is connected. For otherwise, according to the Shilov theorem on idempotents, see [24], SAPΣ (S) ∩ H ∞ (D) contains a function f not equal identically to 0 or 1 on D such that f 2 = f , a contradiction. Next, we show that any finite rank complex vector bundle ξ over bSΣ (D) is topologically trivial. Since bSΣ (D) is the inverse limit of the system {bF Σ|F (D) ; ω}F ⊂S ; #F <∞ , see Section 3.4, ξ is isomorphic (as a topological bundle) to pullback to bSΣ (D) of a bundle on some bF Σ|F (D) with F ⊂ S finite, see, e.g., [11] and [17]. Therefore it suffices to prove the statement for S ⊂ ∂D being a finite subset. In this case, for each s ∈ S by the contractibility of (aSΣ )−1 (s) ∼ = bΣ(s) (T ) (see [5]) we have that the restriction of ξ to (aSΣ )−1 (s) is topologically trivial. Using a finite open cover {Ui }1≤i≤m of (aSΣ )−1 (s) such that ξ|Ui ∼ = U i × Cn , n = rankC ξ, for each i, we extend (by the Urysohn lemma) global continuous sections tj : (aSΣ )−1 (s) → ξ, 1 ≤ j ≤ n, determining the trivialization of ξ over (aSΣ )−1 (s) to each Ui . Then using a continuous partition of unity subordinate to a finite refinement of {Ui }1≤i≤m we glue together these extensions to get global continuous sections t˜j , 1 ≤ j ≤ n, of ξ on a neighbourhood Us of (aSΣ )−1 (s) in bSΣ (D) such that t˜j |(aSΣ )−1 (s) = tj for each j. Since sections tj , 1 ≤ j ≤ n, are linearly independent at each point of (aSΣ )−1 (s), diminishing, if necessary, Us we obtain that sections t˜j , 1 ≤ j ≤ n, are linearly independent at each point of Us . Thus ξ is topologically trivial on Us . Also, by the definition of the topology on bSΣ (D) without loss of generality we may assume that Us \ (aSΣ )−1 (s) is a circular neighbourhood of s. Suppose that S = {s1 , . . . , sk }. We cover bSΣ (D) by sets Uj := Usj , ¯ \ V , where V ⊂ ∪k Us and 1 ≤ j ≤ k, described above and by U0 := D j j=1 V ∩ Usj is a circular neighbourhood of sj distinct from Usj \ (aSΣ )−1 (sj ), 1 ≤ j ≤ k. Since U0 is contractible, ξ|U0 is topologically trivial. Using trivializations of ξ on Uj , 0 ≤ j ≤ k, we obtain that ξ is defined by a 1-cocycle {cij } with values in GLn (C) defined on intersections Ui ∩Uj , 0 ≤ i < j ≤ k. In turn, ˜j }k of D ¯ such that by the definition of sets Uj , there is an acyclic cover {U j=0 S −1 ˜ ˜ j }k (aΣ ) (Uj ) = Uj , 0 ≤ j ≤ k. Thus there exists a cocycle {˜ cij } on {U j=0 S such that c˜ij ◦ aΣ = cij for all i, j. This cocycle determines a continuous vec¯ trivial on each U ˜i , 0 ≤ i ≤ k, such that (aS )∗ ξ˜ = ξ. Since tor bundle ξ˜ on D Σ ¯ is contractible, ξ˜ is topologically trivial. Hence ξ is topologically trivial as D well. Proof of Corollary 3.20. Let G ⊂ R be an additive subgroup and let AP CG (T ) ⊂ AP C(T ) be the algebra of uniformly continuous almost periodic functions on T having their spectrum in G. Here the spectrum of a function in AP C(T ) is the union of the spectra of its restrictions to each horizontal k line in T (see [2]). The vector space of functions j=1 cj (y)eiλj x , x + iy ∈ T , cj ∈ C([0, π]), λj ∈ G, k ∈ N, is dense in AP CG (T ) and, hence, the maximal ideal space M (AP CG (T )) of AP CG (T ) is homeomorphic to bG (R) × [0, π]. On the other hand, AP HG (T ) ⊂ AP CG (T ) and the extension of AP HG (T )
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
321
to M (AP CG (T )) separates the points of M (AP CG (T )). Since the image of T in bG (T ) is dense (see the proof of Theorem 3.14), the latter implies that bG (T ) ∼ = M (AP CG (T )). Hence, taking G := Σ(s), s ∈ S, we obtain ∼ H k (bΣ(s) (R), Z). H k (bΣ(s) (T ), Z) = Since bΣ(s) (R) is a compact connected abelian group, the required statements follow from the remark before the formulation of the corollary and Theorem 3.19 (1). Proof of Theorem 3.21. In what follows we assume that uniform algebras are defined on their maximal ideal spaces via the Gelfand transforms. We will require the following auxiliary result. Lemma 4.12. Assume that a set-valued map Σ as in Section 3.1 is defined on {−s, s} and f ∈ SAPΣ ({−s, s}) ∩ H ∞ (D). Consider the function z+s Hs f (z) := f , z ∈ D. 2 Then Hs f ∈ SAPΣ|{s} ({s}) ∩ H ∞ (D) and [(isΣ|{s} ◦ ιΣ(s) )∗ Hs f ](z) = [(isΣ ◦ ιΣ(s) )∗ f ](z − ln 2),
z∈T
(4.16)
(see Theorem 3.13 for notations). This result states that the map Hs : SAPΣ ({−s, s}) ∩ H ∞ (D) → SAPΣ|{s} ({s}) ∩ H ∞ (D) is a bounded linear operator which induces under the identification of the {−s,s} −1 fibre (aΣ ) (s) with bΣ(s) (T ) by isΣ the map AP HΣ(s) (T ) → AP HΣ(s) (T ) defined by h(z) → h(z − ln 2), z ∈ T , h ∈ AP HΣ(s) (T ). Proof. Clearly Hs f is holomorphic on D and continuous on ∂D \ {s}. Let us consider the function g(z) := [(Hs f ) ◦ (Log ◦ ϕs )−1 ](z) − [f ◦ (Log ◦ ϕs )−1 ](z − ln 2),
z ∈ T.
Next, we have (Log ◦ ϕs )−1 (z) + s se2z − (Log ◦ ϕs )−1 (z − ln 2) = →0 2 (2i + ez )(4i + ez ) as Re(z) → −∞. Since by the definition of SAPΣ ({−s, s}) ∩ H ∞ (D) the function f ◦ (Log ◦ ϕs )−1 is uniformly continuous on T , the last expression implies that g(z) → 0 as Re(z) → −∞. But Re(z) → −∞ if and only if (Log ◦ ϕs )−1 (z) → s. Therefore the function g ◦ Log ◦ ϕ is continuous in a circular neighbourhood of s and equals 0 at s. Since the pullback of the function [f ◦ (Log ◦ ϕs )−1 ](z − ln 2), z ∈ T , by (Log ◦ ϕs )−1 belongs to SAPΣ ({−s, s}) ∩ H ∞ (D) (it is obtained as the composition of f with a M¨ obius transformation preserving points −s and s), the function Hs f ∈ SAPΣ|{s} ({s}) ∩ H ∞ (D). Now, the identity (4.16) follows from the fact that (g ◦ Log ◦ ϕ)(s) = 0 by the definition of isΣ .
322
Brudnyi and Kinzebulatov
IEOT
Corollary 4.13. Let f ∈ GnΣ(s) (T ) (see Section 3.5 for notations). Consider the function F := Hs [Kf (Log ◦ ϕs )], Then F ∈
GnΣ|{s} ({s})
Kf (z) := f (z − ln 2),
where
z ∈ T.
and (isΣ|{s} ◦ ιΣ(s) )∗ F = f.
Proof. The implication F ∈ GnΣ|{s} ({s}) follows from the proof of Lemma 4.12 because the pullback by Log ◦ ϕs maps the space AP HΣ(s) (T ) isometrically into SAP ({−s, s}) ∩ H ∞ (D) so that specs of each of the pulled back function is a subset of Σ(s). The second statement of the corollary follows directly from (4.16) because (isΣ|{s} ◦ ιΣ(s) )∗ (h ◦ Log ◦ ϕs ) = h for any h ∈ AP HΣ(s) (T ) by Theorem 3.13 (1).
We are ready to prove the theorem. First we will consider the case when S = {s1 , . . . , sm } is a finite subset of ∂D. By the definition of connected components of GLn (A), where A is a Banach algebra, the map f → ((isΣ|1 {s
◦ ιΣ(s1 ) )∗ f, . . . , (isΣ|m{s
1}
induces a homomorphism ΨS : [GnΣ (S)] →
m}
si ∈S
◦ ιΣ(sm ) )∗ f ),
f ∈ GnΣ (S),
[GnΣ(si ) (T )].
We will show that ΨS is an isomorphism. n Suppose that (g1 , . . . , gm ) ∈ si ∈S GΣ(si ) (T ) represents an element n [g] ∈ si ∈S [GΣ(si ) (T )]. Then according to Corollary 4.13 for an element g˜ := Hs1 [K(Log ◦ ϕs1 )∗ g1 ] · · · Hsm [K(Log ◦ ϕsm )∗ gm ] ∈ GnΣ (S) and each l ∈ {1, . . . , m} we have (isΣ|l {s } ◦ ιΣ(sl ) )∗ g˜ = c1l · · · cl−1l · gl · cl+1l · · · cml , l
where every cjl is an invertible matrix. Since the matrix-function on the righthand side is homotopic to gl , for the element [˜ g] ∈ [GnΣ (S)] representing g˜, g ]) = [g]. Hence ΨS is a surjection. we obtain ΨS ([˜ To prove that ΨS is an injection, we require a modification of the construction of Corollary 4.13. So suppose that Fsl = Hsl [K(Log◦ϕsl )∗ f ], where f ∈ GnΣ(sl ) (T ). By the definition, Fsl (sj ), j = l, are well-defined invertible matrices. Let M be a matrix-function with entries from A(D) such that M (sj ) = Log(Fsl (sj )), j = l, and M (sl ) = 0. (Here the logarithm of an invertible matrix c is a matrix c˜ such that exp(˜ c) = c.) Then we have n (1) F˜sl := Fsl · exp(−M ) ∈ GΣ|{s } ({sl }) and satisfies l
(isΣ|l {s } ◦ ιΣ(sl ) )∗ F˜sl = f l
and F˜sl (sj ) = In ,
(here In is the unit n × n matrix);
j = l
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
323
(2) F˜sl is homotopic to Fsl . Statement (2) follows from the fact that exp(−M ) belongs to the connected component containing In . Now, suppose that f ∈ GnΣ (S) is such that every matrix-function gl := (isΣ|l {s } ◦ ιΣ(sl ) )∗ f , l ∈ {1, . . . , m}, belongs to the connected compol nent of GnΣ(sl ) (T ) containing the unit matrix In , (i.e., [f ] ∈ Ker(ΨS )). We set ˜s , ˜ := G Gs := Hs [K(Log ◦ ϕs )∗ gl ], G := Gs , G l
l
l
l
1≤l≤m
l
1≤l≤m
˜ s is constructed from Gs as F˜s from Fs . where each G l l l l According to property (1), ˜ = gl , (isΣ|l {s } ◦ ιΣ(sl ) )∗ G l
for l ∈ {1, . . . , m}.
˜ is homotopic to G. Observe also that Moreover, property (2) implies that G each Gsl is homotopic to In (because gl satisfies this property and so the required homotopy is defined as the image of the homotopy between gl and In under the continuous map Hsl ◦ K ◦ (Log ◦ ϕsl )∗ ) and therefore G and are homotopic to In . Finally, according to our construction f · G ˜ −1 is an G ¯ invertible matrix with entries from A(D). Since D is contractible, each such a matrix is homotopic to In . These facts imply that f is homotopic to In , that is [f ] = 1 ∈ [GnΣ (S)], where [f ] stands for the connected component containing f ∈ GnΣ (S). So ΨS is an injection, which completes the proof of the theorem in the case of a finite S. To prove the result in the general case we require the following lemma. Lemma 4.14. For every f ∈ GnΣ (S) there exists f¯ ∈ GnΣ|F (F ), where F ⊂ S is finite, such that f¯ ∈ [f ]. Proof. Let MΣn (S) be the Banach algebra of n × n matrix-functions with entries in SAPΣ (S)∩H ∞ (D) equipped with the norm h := supz∈D h(z)2 , h ∈ MΣn (S), where · 2 is the 2 operator norm on the complex vector space Mn (C) of n × n matrices. According to Corollary 3.7, f can be approximated n (F ) for some finite subsets F ⊂ S. Since in MΣn (S) by functions from MΣ| F the connected component [f ] is open (because GnΣ (S) ⊂ MΣn (S) is open), the latter implies the required statement: there exists f¯ ∈ GnΣ|F (F ), where F ⊂ S is finite, such that f¯ ∈ [f ]. This lemma implies that [GnΣ (S)] is the direct limit of the family F ⊂ S, #F < ∞}. Therefore we can define a homomorphism [GnΣ(s) (T )] ΨS : [GnΣ (S)] →
{[GnΣ|F (F )];
s∈S
as the direct limit of homomorphisms ΨF described above. Then ΨS is an isomorphism because each ΨF is an isomorphism on each image. This proves the first statement of the theorem.
324
Brudnyi and Kinzebulatov
IEOT
The second statement follows from the fact that if Σ(s) ⊂ R+ or R− , then the maximal ideal space bΣ(s) (T ) of Banach algebra AP HΣ(s) (T ) is contractible [5]. Then the result of Arens [1] implies that GnΣ(s) (T ) is connected and therefore [GnΣ(s) (T )] is trivial. From here and the first statement of the theorem we obtain that [GnΣ (S)] is trivial, or equivalently, that GnΣ (S) is connected. Acknowledgment We are grateful to S. Favorov, S. Kislyakov and O. Reinov for useful discussions.
References [1] R. Arens, To what extent does the space of maximal ideals determine the algebra? in “Function Algebras“ (Birtel, ed.), Scott-Foresman, Chicago, 1966. [2] A. S. Besicovich, Almost periodic functions. Dover Publications, 1958. [3] J. Bourgain and O. Reinov, On the approximation properties for the space H ∞ . Matemathische Nachrichten 122 (1983), 19–27. [4] G. E. Bredon, Sheaf theory. Second edition. Graduate Texts in Mathematics 170, Springer-Verlag, New York, 1997. [5] A. Brudnyi, Contractibility of maximal ideal spaces of certain algebras of almost periodic functions. Integral Equations and Operator Theory 52 (2005), 595–598. [6] A. Brudnyi and D. Kinzebulatov, On uniform subalgebras of L∞ on the unit circle generated by almost periodic functions. Algebra and Analysis 19 (2007), 1–33. [7] A. Brudnyi and D. Kinzebulatov, On algebras of holomorphic functions with semi-almost periodic boundary values. C. R. Math. Rep. Acad. Sci. Canada, in press. [8] A. Brudnyi and A. Sasane, Sufficient conditions for projective freeness of Banach algebras, J. Funct. Anal. 257 (2009), 4003–4014. [9] P.M. Cohn, From Hermite rings to Sylvester domains. Proc. Amer. Math. Soc. 128 (2000), 1899–1904. [10] P. Enflo, A counterexample to the approximation property in Banach spaces. Acta Math. 130 (1973), 309–317. [11] S. Eilenberg and N. Steenrod, Foundations of algebraic topology. Princeton University Press, Princeton, New Jersey, 1952. [12] R. Fox, Homotopy groups and torus homotopy groups. Ann. Math. 49 (1948), 471–510. [13] J. Garnett, Bounded analytic functions. Academic Press, 1981. [14] H. Grauert, Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen. (German) Mathematische Annalen, 133 (1957), 450–472. [15] A. Grothendieck, Products tensoriels toplogiques et espaces nucl´eaires. Memoirs Amer. Math. Society 16, 1955. [16] K. Hofmann and O. Mostert, Cohomology Theories for Compact Abelian Groups. Springer-Verlag, 1973.
Vol. 66 (2010)
Holomorphic Semi-almost Periodic Functions
325
[17] D. Husemoller, Fibre bundles. Springer-Verlag, New York, 1994. [18] T.Y. Lam, Serre’s Conjecture. Lecture Notes in Mathematics 635, SpringerVerlag, 1978. [19] V. Ya. Lin, Holomorphic fiberings and multivalued functions of elements of a Banach algebra. Func. Anal. Appl., 7, 1973, English translation. [20] J. Lindenstrauss, Some open problems in Banach space theory. S´eminaire Choquet 18 (1975), 1–9. [21] L. Rodman and I. Spitkovsky, Almost periodic factorization and corona problem. Indiana Univ. Math. J. 47 (1998), 243–281. [22] W. Rudin, Fourier Analysis on Groups. Interscience Publishers, 1962. [23] D. Sarason, Toeplitz operators with semi-almost periodic kernels. Duke Math J. 44 (1977), 357–364. [24] G. E. Shilov, On decomposition of a commutative normed ring in a direct sums of ideals. Mat. Sb. 32(74):2 (1953), 353–364. Alexander Brudnyi Department of Mathematics and Statistics Math Science Building, Room 476 612 Campus Place University of Calgary 2500 University Drive NW Calgary, AB Canada T2N 1N4 e-mail: [email protected] Damir Kinzebulatov Department of Mathematics University of Toronto Bahen Centre 40 St. George St. Toronto, ON Canada M5S 2E4 e-mail: [email protected] Submitted: November 09, 2009. Revised: January 01, 2010.
Integr. Equ. Oper. Theory 66 (2010), 327–365 DOI 10.1007/s00020-010-1753-3 Published online February 23, 2010 © Birkhäuser/Springer Basel AG 2010
Integral Equations and Operator Theory
Estimates for Solutions of a Parameter-Elliptic Multi-Order System of Differential Equations R. Denk and M. Faierman Abstract. This paper is concerned with a boundary value problem defined over a bounded region of Euclidean space, and in particular it is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi-order system of differential equations under limited smoothness assumptions. In this endeavour we extend the results of Agranovich, Denk, and Faierman pertaining to a priori estimates for solutions associated with a parameter-elliptic scalar problem, as well as the results of various other authors who have extended the results of Agranovich et. al. from the scalar case to parameter-elliptic systems of operators which are either of homogeneous type or have the property that the diagonal operators are all of the same order. In addition, we extend some results of Kozhevnikov and Denk and Volevich who have also dealt with sytems of the kind under consideration here, in that one of the works of Kozhevnikov deals only with 2 × 2 systems, while the other, as well as the work of the last two authors, do not cover Dirichlet boundary conditions. Mathematics Subject Classification (2010). Primary 35J55; Secondary 35S15. Keywords. Parameter-ellipticity, multi-order systems, a priori estimates.
1. Introduction This paper is concerned with a boundary problem defined over a bounded region in Rn , and in particular is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi-order system of differential equations under limited smoothness assumptions. To elaborate on what was just said, let us now explain in more detail our two main objectives.
328
Denk and Faierman
IEOT
With regards to the first, we point out that very general results were derived in [ADF] pertaining to the spectral theory for scalar, non-selfadjoint elliptic boundary problems involving differential operators under limited smoothness assumptions and under a parameter-ellipticity condition. In particular, a method was developed there for deriving results pertaining to the eigenvalue asymptotics even under the limited conditions imposed. In [DFM] the techniques of [ADF] were used to extend the results for the scalar problem to that for a problem involving a parameter-elliptic system of differential operators of homogeneous type, and subsequently this result was extended in [F] to the case where only the diagonal operators of the system all had to be of the same order. It is important to mention at this point that all the spectral results derived in the above works depended fundamentally upon the establishment of a priori estimates for solutions of the boundary problem under consideration. Furthermore, in the above works, the a priori estimates could be established using standard methods, but when one attempts to extend the results of [ADF] to fully parameter-elliptic multi-order systems of differential operators, one finds that the standard methods are no longer adequate, and new techniques must be introduced. Accordingly, with all of this in mind, let us turn to the problem under N consideration here. Let N ∈ N with N > 1 and let {sj }N 1 and {tj }1 denote sequences of integers satisfying s1 ≥ s2 ≥ ... ≥ sN , t1 ≥ t2 ≥ ... ≥ tN ≥ 0, and put mj = sj + tj for j = 1, ..., N . We suppose that m1 = m2 = ... = mk1 > mk1 +1 = ... = mkd−1 > mkd−1 +1 = ... = mkd > 0, where kd = N , put m j = mkj for j = 1, ..., d, and let Ir denote the (kr − kr−1 ) × (kr − kr−1 ) identity matrix for r = 1, ..., d, where k0 = 0. In the sequel we will use the notation I to denote the × unit matrix for ∈ N and also impose kr conditions which will ensure that for r = 1, ..., d, the sum j=1 mj is even; N we henceforth denote this sum by 2Nr . Then with {σj } , N = Nd , denoting
1
a sequence of integers such that max{σj }N 1 < sN , we shall be concerned here with the boundary problem A(x, D)u(x) − λu(x) = f (x) in Ω, , Bj (x, D)u(x) = gj (x) on Γ for j = 1, . . . , N
(1.1) (1.2)
where Ω is a bounded region in Rn , n ≥ 2, with boundary Γ, u(x) = T (u1 (x), . . . , uN (x))T , and f (x) = (f1 (x), . . . , fN (x)) are N × 1 matrix T denotes transpose, the gj (x) are scalar functions functions defined in Ω, defined on Γ, A(x, D) is an N ×N matrix operator whose entries Ajk (x, D) are linear differential operators defined on Ω of order not exceeding sj + tk , is a 1 × N and defined to be 0 if sj + tk < 0, and Bj (x, D), 1 ≤ j ≤ N matrix operator whose entries Bj,k (x, D) are linear differential operators defined on Γ of order not exceeding σj + tk , and defined to be 0 if σj + tk < 0. Our assumptions concerning the problem (1.1), (1.2) will be made precise in Section 2.
Vol. 66 (2010)
Estimates for Solutions
329
To motivate the second objective of this paper, let us point out that the first investigation into the spectral theory for a fully parameter-elliptic multi-order system of operators was investigated by Kozhevnikov [K1]. In this paper the author deals with a system of pseudodifferential operators acting over a compact manifold without boundary; and by introducing the so-called Kozhevnikov conditions, the author is able to establish a priori estimates for solutions as well as various other spectral results. In subsequent works [K2], [K3] the author deals with a genuine boundary problem under infinitely smooth conditions involving a parameter-elliptic multi-order system of differential operators acting over a bounded region in Rn and a system of differential operators defined on the boundary. In particular, in [K2] the author restricts himself to a 2 × 2 system of differential operators and a system of boundary operators which can be expressed in the form a a lower triangular block matrix, that is, in the terminology of this paper, we now 2 have N = 2, A(x, D) = (Ajk (x, D))j,k=1 , B(x, D) = (Bjk (x, D))j=1,...,µ/2 = k=1,2 2 2 ord A (x, D). By (Tjk (x, D))j,k=1 , with T12 (x, D) = 0, where µ = jj j=1 introducing various conditions related to parameter-ellipticity, a result pertaining to the resolvent operator is established. In [K3] the author removes the restriction of [K2] that N = 2, but now requires that the system of boundary operators admits a representation in the form of an upper triangular block matrix. Furthermore, he requires that A(x, D) be of Petrovskii type. Then under various conditions, including those of [K1], the author derives some results pertaining to the resolvent operator. The problem considered in [K1] was also dealt with by Denk, Mennicken, and Volevich [DMV], but now in more detail. And by introducing conditions equivalent to those of Kozhevnikov and using the method of Newton polygons, the authors establish a priori estimates for solutions as well as various spectral results. In a subsequent work [DV], Denk and Volevich deal with a genuine boundary problem of the form (1.1), (1.2), but under the assumption that the operators Bjk (x, D), if not identically zero, contain only top order terms and that these orders must satisfy a special condition. Then by appealing to the Kozhevnikov conditions as well as to the conditions of VishikLyusternik [VL], the authors establish a priori estimates for solutions of the boundary problem (1.1), (1.2) for the case where Ω = Rn+ and Γ = Rn−1 . What is novel in this latter work is the introduction of the Vishik-Lyusternik conditions, as this enables one to deal with multi-order systems of operators in the traditional way, that is, by means of contour integration. Unfortunately, because of the restrictions imposed, both papers [K3] and [DV] are not even able to deal with the important problem where sj = tj = tj for j = 1, . . . , N, (here {tj }N 1 denotes a monotonic decreasing sequence of positive integers) and the boundary conditions are of Dirichlet type (see [ADN, Section 2], [G, p.448]). Of course the paper [K2] does include Dirichlet boundary conditions, but the requirement that N = 2 is very restrictive.
330
Denk and Faierman
IEOT
Thus the first objective of this paper is to extend the results of [ADF] concerning solutions of the boundary problem for the scalar case to the nonscalar problem under consideration here, while our second objective is to extend the results of [K3] and [DV] by establishing a priori estimates for solutions of the boundary problem (1.1), (1.2) under boundary conditions which do include those of Dirichlet. And in this endeavour we shall make use of the conditions of both Kozhevnikov and Vishik-Lyusternik (see Definitions 2.4 and 2.6 below). Our main result is then given in Theorem 2.8 below. Let us also mention at this point that from a consideration of length, we have limited ourselves in this paper solely to the proof of the sufficiency part of Theorem 2.8, that is to say, that the conditions cited in Definition 2.6 ensure the validity of this theorem. The converse of this result, that is, the proof of the necessity part of the theorem, will be left for a later work. Finally, let us outline the contents of the paper. In Section 2 we introduce some terminology, definitions, and assumptions concerning the boundary problem (1.2), (1.2) which we require for our work and then present the main result of this work, namely Theorem 2.8 below. In Section 3 we restrict ourselves to the case where all the operators involved have constant coefficients and which act either over Rn , without boundary conditions, or over Rn+ , with boundary conditions defined on Rn−1 . All the preliminary results required for the proof of Theorem 2.8 are established here, with the main result being Proposition 3.6. These results are then used in Section 4 to prove Theorem 2.8.
2. Preliminaries In this section we are going to introduce some terminology, definitions, and assumptions concerning the boundary problem (1.1), (1.2) which we require for our work and then state the main result of this paper, namely Theorem 2.8 below. Accordingly, we let x = (x1 , . . . , xn ) = (x , xn ) denote a generic point n in R and use the notation Dj = −i∂/∂xj , D = (D1 , . . . .Dn ), Dα = D1α1 · · · Dnαn = D α Dnαn , and ξ = ξ1α1 · · · ξnαn for ξ = (ξ1 , . . . , ξn ) = (ξ ,nξn ) ∈ n R , where α = (α1 , . . . , αn ) = (α , αn ) is a multi-index whose length j=1 αj is denoted by |α|. Differentiation with respect to another variable, say y ∈ Rn , instead of x will be indicated by replacing D, Dα , Dα , and Dnαn by Dy , Dyα , Dyα , and Dyαnn , respectively. For 1 < p < ∞, s ∈ N0 = N ∪ { 0 }, and G an open set in R , ∈ N, we let Wps (G) denote the Sobolev space of order s related to Lp (G) and denote the norm in this space by · s,p,G , where 1/p α p us,p,G = |D u(x)| dx for u ∈ Wps (G). In addition we shall |α|≤s G
use norms depending upon a parameter λ ∈ C\{ 0 }, namely for 1 ≤ j ≤ d, we let (j) j u0,p,G for u ∈ Wps (G). |||u|||s,p,G = us,p,G + |λ|s/m
Vol. 66 (2010)
Estimates for Solutions
331
˚ s (G) denote the closure of C ∞ (G) in W s (G). We also let W p 0 p In the sequel we shall also at times deal with the Bessel-potential space Hps (G) for 0 ≥ s ∈ Z and equipped with norms depending upon the parameter (j)
λ. Namely for u ∈ Hps (G) and 1 ≤ j ≤ d, we introduce the norms |||u|||s,p,G =
(j) (j) F −1 ξ, λsj F u0,p,Rn if G = Rn and |||u|||s,p,G = inf |||v|||s,p,Rn otherwise, where the infimum is taken over all v ∈ Hps (Rn ) for which u = v G, F denotes
1 2 2 j the Fourier transformation in Rn (x → ξ) and ξ, λj = |ξ|2 + |λ| m (see [GK, Section 1], [T, p. 177]). Lastly, let Rn+ = { x ∈ Rn |xn > 0 }, R+ = { t ∈ R|t > 0 }, and let Rn− and R− be defined similarly. Assume for the moment that the boundary Γ of Ω (see (1.1), (1.2)) is of class C m−1,1 for some m ∈ N , and let s be an integer satisfying 1 ≤ s ≤ m. Then for G = Ω or G = Rn+ , the vectors u ∈ Wps (G) have boundary values s−1/p (∂G) v = u∂G and we denote the space of these boundary values by Wp and denote by · s−1/p,p,∂G the norm in this space, where vs−1/p,p,∂G
s−1/p
= inf us,p,G for v ∈ Wp (∂G) and the infimum is taken over those u ∈ Wps (G) for which u∂G = v (see also [ADF, Section 2] and [Gr, p.20] for s−1/p
further definitions of Wp (∂G)). In addition we shall use norms depending upon a parameter λ ∈ C\{ 0 }, namely for 1 ≤ j ≤ d, (j)
j |||v|||s−1/p,p,∂G = vs−1/p,p,∂G +|λ|(s−1/p)/m v0,p,∂G
for v ∈ Wps−1/p (∂G),
where · 0,p,∂G denotes the norm in Lp (∂G). Turning now to the boundary problem (1.1), (1.2), let us write α Ajk (x, D) = ajk for x ∈ Ω and 1 ≤ j, k ≤ N, α (x)D |α|≤sj +tk
Bjk (x, D) =
α . (2.1) bjk for x ∈ Γ and k = 1, . . . , N, j = 1, . . . , N α (x)D
|α|≤σj +tk
Observing that the orders of the operators Ajk (x, D), Bjk (x, D) remain N N unchanged if we replace the sequences {sj }N 1 , {tj }1 , and {σj }1 by {sj − N † N † N † σ † }N 1 , {tj + σ }1 , and {σj − σ }1 , respectively, where σ = max{σj }1 + 1, we see that by making such substitutions if necessary, there is no loss of . Likewise, by generality in henceforth supposing that σj < 0 for j = 1, . . . , N N N N N replacing the sequences {sj }1 , {tj }1 , and {σj }1 by {sj − sN }N 1 , {tj + sN }1 , and {σj − sN }N 1 }, respectively, if necessary, we may also henceforth suppose that sN ≥ 0. Assumption 2.1. It will henceforth be supposed that tj ≥ 0 and sj ≥ 0 for . j = 1, . . . , N , and that σj < 0 for j = 1, . . . , N Assumption 2.2. It will henceforth be supposed that: (1) Γ is of class C κ0 −1,1 ∩
jk sj C s1 , where κ0 = max t1 , max{−σj }N 1 ; (2) for each pair j, k, aα ∈ C (Ω) jk for |α| ≤ sj + tk if sj > 0, while if sj = 0, then aα ∈ L∞ (Ω) for |α| < sj + tk
332
Denk and Faierman
IEOT
0 jk −σj −1,1 and ajk (Γ) α ∈ C (Ω) for |α| = sj + tk ; (3) for each pair j, k, bα ∈ C for |α| ≤ σj + tk .
For ξ ∈ Rn let
N ˚ ξ) = A ˚jk (x, ξ) A(x, for x ∈ Ω, j,k
˚jk (x, ξ) ˚ ξ) = B B(x, for x ∈ Γ, j=1,...,N k=1,...,N
˚jk (x, ξ) (resp. B ˚jk (x, ξ)) consists of those terms in Ajk (x, ξ) (resp. where A ˚j (x, ξ) Bjk (x, ξ)), which are just of order sj +tk (resp. σj +tk ). We denote by B ˚ ξ). Then in the sequel we shall also require the following the j-th row of B(x, notation. For x ∈ Ω and ξ ∈ Rn , let kr (r) ˚jk (x, ξ) for 1 ≤ r ≤ d, A11 (x, ξ) = A (r) A12 (x, ξ)
˚jk (x, ξ) = A
j,k=1
j=1,...,kr k=kr +1,...,N
,
(r) A21 (x, ξ)
N (r) ˚jk (x, ξ) and A22 (x, ξ) = A
˚jk (x, ξ) = A j=kr +1,...,N , k=1,...,kr
j,k=kr +1
for 1 ≤ r ≤ d − 1.
Also for x ∈ Γ, ξ ∈ Rn , and 1 ≤ r ≤ 1 , ≤ d, let ˚jk (x, ξ) B (r,) (x, ξ) = B j=N−1 (1−δr, )+1,...,N , (r,) B1 ,1 (x, ξ)
k=1,...,N
˚jk (x, ξ) = B j=N−1 (1−δr, )+1,...,N , k=1,...,k1
and
(r,) ˚jk (x, ξ) B1 ,2 (x, ξ) = B j=N−1 (1−δr, )+1,...,N , k=k1 +1,...,N
where δr, is the Kronecker delta. In addition we let Ir,0 = diag(0 · I1 , . . . , 0 · Ir−1 , Ir ) for r = 2, . . . , d and I1,0 = I1 . Note that when x0 ∈ Γ we can rewrite the boundary problem (1.1), (1.2) in terms of a local coordinate system at x0 wherein x0 → 0 and ν → en , where ν denotes the exterior normal to Γ at x0 and (e1 , . . . , en ) denotes the standard basis in Rn . Then supposing that this has been done, we shall in the sequel be concerned with the boundary problem ˚ D)u(x) − λ u(x) = f (x) for x ∈ Rn , A(0, + ˚j (0, D)u(x) = gj (x ) at xn = 0 for j = 1, . . . , N, B and corresponding to this boundary problem we define the associated sym˚j,k (0, ξ), A(r) (0, ξ), B (r,) (0, ξ), and B (r,) (0, ξ) in pre˚ ξ), B ˚j,k (0, ξ), A bols B(0, j,k 1 ,j cisely the same way their analogues were defined in the original coordinate system.
Vol. 66 (2010)
Estimates for Solutions
333
Definition 2.3. Let L be a closed sector in the complex plane with vertex at the origin. Then the operator A(x, D) − λ IN will be called parameter (r) elliptic in L if det A11 (x, ξ) − λIr,0 = 0 for x ∈ Ω, ξ ∈ Rn \ {0}, and λ ∈ L, r = 1, . . . , d. In the sequel we let C± = {z ∈ C, Im z > < 0}. Definition 2.4. Suppose that the operator A(x, D)−λ IN is parameter-elliptic in the sector L introduced above. Let x0 be an arbitrary point of Γ and assume that the boundary problem (1.1), (1.2) has been rewritten in a local coordinate system associated with x0 in the manner just explained. Then the operator A(x, D) − λ IN will be called properly parameter-elliptic in L if the following conditions are satisfied. (r) (1) The polynomial det A11 (0, ξ , z) − λIr,0 has precisely Nr zeros lying in C+ for ξ ∈ Rn−1 \ {0} and λ ∈ L, r = 1, . . . , d. (r) (2) The polynomial det A11 (0, 0, z) − λIr,0 has precisely Nr − Nr−1 zeros lying in C+ for λ ∈ L \ {0}, r = 2, . . . , d. Remark 2.5. Referring to Condition (1) of Definition 2.4, we know from [AV, (r) Section 2] that det A11 (0, ξ , z) − λIr,0 has precisely Nr zeros in C+ if r = 1 or if r > 1 and n > 2. In the sequel, when proper parameter-ellipticity is supposed, it will be assumed that this is also the case when r > 1 and n = 2. Turning next to Condition (2) of the definition, it is clear that the number of zeros of the determinant in C+ (resp. C− ) does not depend upon λ. Hence it follows from an expansion of the determinant in powers of z and λ that r is odd, kr − kr−1 is even, Condition (2) always holds if m r is even or if m and there is a λ ∈ L \ {0} such that −λ ∈ L. Lastly we mention at this point that it is also clear from what was said above that Condition (2) is always satisfied if the operator A(x, D) is essentially upper triangular at x0 (see Definition 2.7 below). Definition 2.6. Let L denote the sector introduced in Definition 2.3 above. Then we say that the boundary problem (1.1), (1.2) is parameter-elliptic in L if A(x, D) − λ IN is properly parameter-elliptic in L and the following conditions are satisfied. Let x0 be an arbitrary point of Γ and suppose that the boundary problem (1.1), (1.2) has been rewritten in a local coordinate system associated with x0 , as explained above. According to the notation introduced above, let (r,r) ˚jk (0, ξ , Dn ) 1 ≤ r ≤ d, Br,1 (0, ξ , Dn ) = B j=1,...,Nr , (r,) Br, (0, ξ , Dn )
k=1,...,k
r ˚jk (0, ξ , Dn ) = B j=N−1 +1,...,N ,
k=1,...,k
Then
1 ≤ r < d, r < ≤ d.
334
Denk and Faierman
IEOT
(1) the boundary problem on the half-line A11 (0, ξ , Dn )v(xn ) − λIr,0 v(xn ) = 0 for xn > 0, (r)
(r,r)
Br,1 (0, ξ , Dn )v(xn ) = 0 at xn = 0, |v(xn )| → 0 as xn → ∞, has only the trivial solution for ξ ∈ Rn−1 \ {0}, λ ∈ L and 1 ≤ r ≤ d; (2) the boundary problem on the half-line () A11 (0, 0, Dn )v(xn ) − λ I,0 v(xn ) = 0 for xn > 0, (r,)
B,1 (0, 0, Dn )v(xn ) = 0 at xn = 0, |v(xn )| → 0 as xn → ∞, has only the trivial solution for λ ∈ L \ {0}, 1 ≤ r < d and r < ≤ d. For our purposes we need to introduce some further terminology. To this end we henceforth let π1 (j) = r if 1 ≤ j ≤ N and kr−1 < j ≤ kr , and π2 (j) = r if 1 ≤ j ≤ Nd and Nr−1 < j ≤ Nr , where N0 = 0. In addition 1 2 2 1 1 j we let ξ = 1 + |ξ|2 2 , ξ = 1 + |ξ |2 2 , and ξ , λj = |ξ |2 + |λ| m for 1 ≤ j ≤ d. We also require the following definition. Definition 2.7. Let x0 ∈ Γ. Then we say that the operator A(x, D) is essen0 tially upper triangular at x0 if ajk α (x ) = 0 for |α| = sj + tk , k−1 < j ≤ k , 1 ≤ k ≤ k−1 , = 2, . . . , d. Likewise we say that the operator B(x, D) = 0 (Bjk (x, D))j=1,...,N is essentially upper triangular at x0 if bjk α (x ) = 0 for k=1...,N
|α| = σj + tk , N−1 < j ≤ N , 1 ≤ k ≤ k−1 , = 2, . . . , d. We are now in a position to state the main result of this paper, namely Theorem 2.8 below, and which will be proved in Section 4. In this theorem we will require the further assumption, which will be made precise in Definition 3.13 below, that the operators A(x, D) and B(x, D) are compatible at each point of Γ. Hence for the moment let us state that this condition will always be satisfied if B(x, D) is of Dirichlet type on Γ or if the operators A(x, D) and B(x, D) are essentially upper triangular at every point of Γ. Theorem 2.8. Suppose that the boundary problem (1.1), (1.2) is parameterelliptic in L. Suppose also that the operators A(x, D)and B(x, D) are compatible at every point of Γ. In addition, suppose that B(x, D) is essentially upper triangular at every point of Γ. Then there exists a constant λ0 = λ0 (p) > 1 such that for λ ∈ L with |λ| ≥ λ0 , the boundary problem (1.1), (1.2) has
N
N t −s a unique solution u ∈ j=1 Wp j (Ω) for every f ∈ j=1 Hp j (Ω) and g = 1 T N −σj − p g1 , . . . , gN ∈ j=1 Wp (Γ), and the a priori estimate ⎞ ⎛ N N N (π1 (j)) (π (j)) (π (j)) (2.2) |||uj |||tj ,p,Ω ≤C⎝ |||fj |||−s1j ,p,Ω + |||gj |||−σ2j − 1 ,p,Γ ⎠ j=1
j=1
j=1
p
holds, where the constant C does not depend upon the fj , gj , and λ.
Vol. 66 (2010)
Estimates for Solutions
335
Remark 2.9. The proof of Theorem 2.8 will depend upon the results of Section 3 and those of [AV], and as a consequence of these results it will also follow that the estimate (2.2) is 2-sided, i.e., an estimate reverse to (2.2) holds. Indeed, we know from [AV] that we can cover Ω by a finite number of open sets {Uk }n1 1 , where Uk ∩ Γ = ∅ if k ≤ n0 for some n0 < n1 , and Uk ⊂ Ω 1 for k > n0 . If {φk }N 1 denotes a partition of unity subordinate to the covering n1 {Uk }1 such that supp φk ∩ Γ = ∅ for k ≤ n0 and supp φk ∩ Γ = ∅ otherwise, (π (j)) where supp denotes support, then a norm equivalent to the norm |||fj |||−s1j ,p,Ω defined above is given by n0 n1 (π (j)) (π (j)) |||φk fj |||−s1j ,p,Rn + |||φk fj |||−s1j ,p,Rn , +
k=1
where the norm
(π (j)) |||φk fj |||−s1j ,p,Rn +
k=n0 +1
is taken in local coordinates. Since similar (π (j))
(π (j)) , 1 j − p ,p,Γ
1 statements can be made for both |||uj |||tj ,p,Ω and |||gj |||−σ2
the asser-
tion concerning the 2-sidedness of (2.2) follows directly from the results of Section 3.
3. The Constant Coefficient Case In this section we are going to establish some results concerning the existence of and a priori estimates for solutions of a boundary problem involving constant coefficient systems which is related to (1.1), (1.2). These results will then be used in Section 4 to prove Theorem 2.8. To this end, let x0 ∈ Ω and let us fix our attention upon the differential equation ˚ 0 , D)u(x) − λ u(x) = f (x) for x ∈ Rn and λ ∈ L \ {0}. A(x (3.1) Then we have the following two results, where here and below we let uj and fj , 1 ≤ j ≤ N , denote the components of u and f , respectively.
N t Proposition 3.1. Suppose that u ∈ j=1 Wp j (Rn ) and that (3.1) holds. Then
N N −s (π1 (j)) (π1 (j)) N f ∈ j=1 Hp j (Rn ) and j=1 |||fj |||−sj ,p,Rn ≤ C j=1 |||uj |||tj ,p,Rn , where the constant C does not depend upon u and λ. Proposition 3.2. Suppose that the operator A(x, D) − λ IN is parameter −sj elliptic in L and that f ∈ N (Rn ). Then there exists the constant j=1 Hp 0 λ > 0 such that for λ ∈ L with |λ| ≥ λ0 , the differential equation (3.1) has
N t a unique solution u ∈ j=1 Wp j (Rn ) and N j=1
(π (j))
1 |||uj |||tj ,p,R n ≤ C
N
(π (j))
|||fj |||−s1j ,p,Rn ,
j=1
where the constant C does not depend upon f and λ. We will only prove Proposition 3.2 as the proof of Proposition 3.1 follows directly from the definition and the Mikhlin-Lizorkin multiplier theorem.
336
Denk and Faierman
IEOT
Proof of Proposition 3.2. Under our assumptions we know from [DMV] and ˚ 0 , ξ) − λ IN is invertible [K1] that for ξ ∈ Rn and λ ∈ L with |λ| ≥ λ0 , A(x and d 2(N −N ) ˚ 0 , ξ) − λ IN ≥ C ξ, λ j j−1 , det A(x j
j=1
where the constant C does not depend upon ξ and λ. Furthermore, if we −1 N ˚ 0 , ξ) − λ IN = ( aj,k (ξ, λ))j,k=1 , then the aj,k (ξ, λ) are rational put A(x functions of their arguments, while it also follows from the references just cited that for any multi-index α whose entries are either 0 or 1, −m
j k aj,k (ξ, λ)| ≤ Cξsj +tk ξ, λπ1 (j) ξ, λ−m |ξ α Dξα π1 (k)
for all ξ ∈ Rn whose components are all non-zero, where the constant C does not depend upon ξ and λ. Now observe that under a Fourier transformation (3.1) becomes ˚ 0 , ξ)F u(ξ) − λ F u(ξ) = F f (ξ). A(x Furthermore, in light of what was said above, we conclude that this equation has a unique solution in the space of tempered distributions on Rn given −1 ˚ 0 , ξ) − λ IN F f (ξ). Hence all of the assertions of the by F u(ξ) = A(x proposition follow immediately from this last result, the definitions of the terms involved, and the Mikhlin-Lizorkin multiplier theorem. Let us suppose from now on that x0 ∈ Γ. Then assuming that the boundary problem (1.1), (1.2) has been rewritten in terms of the local coordinates at x0 as explained in the text preceding Definition 2.3, let us fix our attention upon the problem in the half-space ˚ D)u(x) − λ u(x) = f (x) for x ∈ Rn and λ ∈ L \ {0}. A(0, +
(3.2)
−s
Then from a consideration of the pairing between Hp j (Rn+ ), equipped with (π (j)) ˚ sj (Rn+ ), equipped with the norm the norm ||| · |||−s1j ,p,Rn , and its dual W p (π (j))
+
1 ||| · |||sj ,p = p/(p − 1) (see [GK, Theorem 1.1]), we can ,Rn , 1 ≤ j ≤ d, p + easily derive the following analogue of Proposition 3.1.
N t Proposition 3.3. Suppose that u ∈ j=1 Wp j (Rn+ ) and that (3.2) holds. Then
N N N −s (π1 (j)) (π1 (j)) f ∈ j=1 Hp j (Rn+ ) and ≤ C j=1 |||uj |||tj ,p,R n , where j=1 |||fj |||−sj ,p,Rn + + the constant C does not depend upon u and λ.
Proposition 3.4. Suppose that the operator A(x, D) − λ IN is parameter N −s elliptic in L and that f ∈ j=1 Hp j (Rn+ ). Then there exists a λ0 > 0 such that for λ ∈ L with |λ| ≥ λ0 , the differential equation (3.2) has a solution N
tj (π1 (j)) (π1 (j)) n ≤C N , where u∈ N j=1 Wp (R+ ) and j=1 |||uj |||tj ,p,Rn j=1 |||fj |||−sj ,p,Rn + + the constant C does not depend upon f and λ.
Vol. 66 (2010)
Estimates for Solutions
337
Proof. It follows from [T, Lemma 2.9.3, p.218] and [GK, Eqn.(1.27)] that
N −s there is a f ∈ j=1 Hp j (Rn ) such that fRn+ = f and N
|||fj |||−s1j ,p,Rn ≤ C (π (j))
j=1
N
(π (j))
|||fj |||−s1j ,p,Rn , +
j=1
where the constant C does not depend upon f and λ. Hence if u denotes n the solution of (3.1) when f there is replaced by f and u = u R+ , then the assertion of the proposition follows directly from Proposition 3.2. Let us next fix our attention upon the boundary problem ˚ D)u(x) − λ u(x) = 0 for x ∈ Rn , A(0, +
(3.3)
˚j (0, D)u(x) = gj (x ) at xn = 0, j = 1, . . . , N , B
(3.4)
with λ ∈ C \ {0}.
N t Proposition 3.5. Suppose that u ∈ j=1 Wp j (Rn+ ) and that (3.4) holds. Then 1 T N −σj − p ∈ j=1 Wp (Rn−1 ) and g = g1 , . . . , gN N j=1
(π (j)) |||gj |||−σ2j − 1 ,p,Rn−1 p
≤C
N N
(π (j))
1 |||uk |||tk ,p,R n,
(3.5)
+
j=1 k=1
where the constant C does not depend upon u and λ. Furthermore, if B(x, D) is essentially upper triangular at x0 , then we may replace (3.5) by N
(π (j)) 1 n−1 j − p ,p,R
|||gj |||−σ2
j=1
≤C
N
(π (j))
1 |||uj |||tj ,p,R n. +
j=1
and let µ(j) = π2 (j) if B(x, D) is essentially upper Proof. Let 1 ≤ j ≤ N 0 triangular at x and let µ(j) = 1 otherwise. Then it follows from [ADF, Proposition 2.2] that (π (j)) 1 n−1 j − p ,p,R
|||gj |||−σ2
N ≤ C1
(π2 (j)) 0 α bjk (x )D u k α
−σj ,p,Rn +
k=1 |α|=σj +tk N
≤ C2
−σj
π (j) 2 Dα uk −σj ,p,Rn+ + |λ| m Dα uk 0,p,Rn+
k=µ(j) |α|=σj +tk N (π2 (j)) uk tk ,p,Rn+ + |||uk |||tk ,p,R n
≤ C3
+
k=µ(j)
≤ 2 C3
N k=µ(j)
(π (j))
2 |||uk |||tk ,p,R n,
338
Denk and Faierman
IEOT
where the constants Cj do not depend upon u and λ. Hence all the assertions of the proposition follow from this last result. We now come to the main result of this section. Proposition 3.6. Suppose that the boundary problem (1.1), (1.2) is parameterelliptic in L. Suppose also that the operators A(x, D) and B(x, D) are compatible at x0 . Then there exists a constant λ0 = λ0 (p) > 1 such that for λ ∈ L with |λ| ≥ λ0 , the boundary problem (3.3), (3.4) has a unique solution 1
N −σj − p tj n T u∈ N (Rn−1 ), and ) ∈ j=1 Wp j=1 Wp (R+ ) for every g = (g1 , . . . , gN the a priori estimate N
(π (j))
1 |||uj |||tj ,p,R n ≤ C
N
+
j=1
(π (j))
|||gj |||−σ2j − 1 ,p,Rn−1 p
j=1
holds, where the constant C does not depend upon the gj and λ. As a consequence of Propositions 3.4, 3.5, and 3.6 as well as from a standard argument, we obtain the following result. Proposition 3.7. Suppose that the hypotheses of Proposition 3.6 hold. Suppose also that the operator B(x, D) is essentially upper triangular at x0 . Then the exists a constant λ0 = λ0 (p) > 1 such that for λ ∈ L with |λ| ≥ λ0 , the
tj n boundary problem (3.2), (3.4) has a unique solution u ∈ N j=1 Wp (R+ ) for 1
N
−σj − p −s N every f ∈ j=1 Hp j (Rn+ ) and g ∈ j=1 Wp (Rn−1 ), and the a priori estimate ⎞ ⎛ N N N (π1 (j)) (π (j)) (π (j)) |||uj |||tj ,p,Rn ≤ C ⎝ |||fj |||−s1j ,p,Rn + |||gj |||−σ2j − 1 ,j,p,Rn−1 ⎠ +
j=1
+
j=1
j=1
p
holds, where the constant C does not depend upon f, g, and λ. In order to prove Proposition 3.6, some preliminary results are required. Accordingly, let us fix λ ∈ L with |λ| > 1 sufficiently large and let { j }d1 denote a sequence of numbers satisfying 0 < j < 1, j = 1, . . . , d (the magnitudes of λ and the j will be specified below). Also for r = 0, . . . , d, let us introduce functions ψr ∈ C ∞ (Rn−1 ) satisfying 0 ≤ ψr (ξ ) ≤ 1,
1 supp ψ0 ∈ ξ ∈ Rn−1 : |ξ | ≤ 34 1 |λ|1/m 1 and ψ0 (ξ ) = 1 for |ξ | ≤ 14 1 |λ|1/m ,
n−1 1 1/m r r+1 supp ψr ∈ ξ ∈ R : 4 r |λ| ≤ |ξ | ≤ 34 r+1 |λ|1/m r r+1 and ψr (ξ ) = 1 for 34 r |λ|1/m ≤ |ξ | ≤ 14 r+1 |λ|1/m , 1 ≤ r < d,
n−1 1/m d 1 supp ψd ∈ ξ ∈ R : |ξ | ≥ 4 d |λ| d , and ψd (ξ ) = 1 for |ξ | ≥ 34 d |λ|1/m d while in addition r=0 ψr (ξ ) = 1 and each ψr is a Fourier multiplier of type (p, p) whose norm is bounded by a constant not depending upon λ and the j . Then in the sequel we will require the following three results.
Vol. 66 (2010)
Estimates for Solutions
339
Proposition 3.8. Suppose that the boundary problem (1.1),(1.2) is parameterelliptic in L. Suppose also that λ ∈ L with |λ| > 1 and that 0 ≤ |ξ | ≤ 1 7 0 m 1 . Then we can choose the numbers 0 sufficiently small and λ suf8 1 |λ| ˚ ξ , z) − λ IN has ficiently large so that for 1 ≤ 0 and |λ| ≥ λ0 , det A(0,
) zeros, say z (0) (ξ , λ) Nd , lying in C+ , and satisfying precisely Nd (= N j j=1 (0)
(0)
Im zj (ξ , λ) ≥ C1 ξ , λ1 , |zj (ξ , λ)| ≤ C2 ξ , λ1 , j = 1, . . . , N1 , (0)
(0)
Im zj (ξ , λ) ≥ C1 ξ , λ ,
|zj (ξ , λ)| ≤ C2 ξ , λ , j = N−1 + 1, . . . , N
for = 2, . . . , d, and where C2 ξ , λ < C1 ξ , λ+1 for = 1 . . . , d − 1, and the Cj denote constants not depending upon ξ and λ. Proposition 3.9. Suppose that the boundary problem (1.1), (1.2) is parameterelliptic in L. Suppose also that 1 ≤ r < d, that λ ∈ L with |λ| > 1, and that 1
1 1 m r 8 r |λ| 0
r+1 ≤ |ξ | ≤ 78 r+1 |λ| m . Then for fixed r we can choose the numbers 0 sufficiently small and λ sufficiently large so that for r+1 ≤ 0 and |λ| ≥
˚ ξ , z) − λ IN has precisely Nd zeros, say z (r) (ξ , λ) Nd , lying λ0 , det A(0, j j=1 in C+ and satisfying
(r)
Im zj (ξ , λ) ≥ C1 ξ , λr , (r)
Im zj (ξ , λ)| ≥ C1 ξ , λ ,
(r)
|zj (ξ , λ)| ≤ C2 ξ , λr , j = 1, . . . , Nr , (r)
|zj (ξ , λ)| ≤ C2 ξ , λ , j = N−1 , . . . , N
for = r + 1, . . . , d, and where C2 ξ , λ < C1 ξ , λ+1 for = r, . . . , d − 1, and the Cj denote positive constants not depending upon ξ and λ. Proposition 3.10. Suppose that the boundary problem (1.1), (1.2) is para1 d meter-elliptic in L. Suppose also that λ ∈ L with |λ| > 1 and that 18 d |λ| m ≤ 0 number λ sufficiently large |ξ | < ∞. Then for fixed d we can choose the ˚ ξ , z) − λ IN has precisely Nd zeros, say so that for |λ| ≥ λ0 , det A(0, (d) Nd zj (ξ , λ) j=1 , lying in C+ and satisfying (d)
Im zj (ξ , λ) ≥ C1 ξ , λd ,
(d)
|zj (ξ , λ)| ≤ C2 ξ , λd for j = 1, . . . , Nd ,
where the Cj denote constants not depending upon ξ and λ. Since the proofs of Propositions 3.8 and 3.10 are similar to that of Proposition 3.9, we will only prove this latter proposition. Proof of Proposition 3.9. To begin with let us observe that (r) (r) A11 (0, ξ , z) − λ Ikr A12 (0, ξ , z) ˚ A(0, ξ , z) − λ IN = (r) (r) A21 (0, ξ , z) A22 (0, ξ , z) − λ IN −kr and that A11 (0, ξ , z) − λ Ikr = A11 (0, ξ , z) − λIr,0 − λ(Ikr − Ir,0 ). (r)
(r)
340
Denk and Faierman
IEOT
(r) Then as a consequence of our hypotheses we know that det A11 (0, ξ , z) − λIr,0 has precisely Nr zeros lying in C+ and that there is a closed contour γr+ (ξ , λ) ⊂ C+ containing all these zeros in its interior such that for z ∈ γr+ (ξ , λ), Im z ≥ C1 ξ , λr , |z| ≤ C2 ξ , λr , and (r) r r ≤ det A11 (0, ξ , z) − λIr,0 ≤ C4 ξ , λ2N , C3 ξ , λ2N r r
where the constants Cj do not depend upon ξ , z, and λ. Furthermore, it is easy to show that for z ∈ γr+ (ξ , λ), (r) (r) det A11 (0, ξ , z) − λ Ikr − det A11 (0, ξ , z) − λIr,0
≤C
mi(k) |λ|1/mi(k)
kr−1
ξ , λr
=1 1≤i(1)<...
r ξ , λ2N , r
where the constant C does not depend upon ξ , z, and λ, while by employing the Laplace method of expanding a determinant, we can alsoshow that, apart ˚ ξ , z) − from a constant not depending upon ξ , z, and λ, the term det A(0, (r) N −k r is bounded by the sum λ IN - det A11 (0, ξ , z) − λ Ikr λ r −1 N −k N −kr −µ r+1 det A(r) (0, ξ , z) − λ Ikr |λ|µ ξ , zm
11
+
µ=0
⎛ q
⎝ξ , z
µ=1
q
⎛
miµ
⎞ ν
ξ , z
µ=1
j1 <...<jν q
× ⎝ξ , z
µ=1
miµ
ν
ξ , z
µ=1
mjµ
|λ|kr −q−ν ⎠
(3.6)
⎞ mjµ
|λ|N −kr −q−ν ⎠ .
j1 <...< jν
1 Here ξ , z = |ξ |2 + |z|2 2 , q indicates that the summation is over those q satisfying 1 ≤ q ≤ kr (resp. 1 ≤ q ≤ N − kr ) if 2kr ≤ N (resp. 2kr > N ), and {iµ }qµ=1 and {jµ }νµ=1 denote distinct sequences of integers satisfying 1 ≤ i1 < . . . < iq ≤ kr and 1 ≤ j1 < . . . < jν ≤ kr , 1 ≤ ν ≤ kr − q, respectively, while {iµ }qµ=1 and { jµ }νµ=1 denote distinct sequences of integers satisfying j1< . . . < jν ≤ N, 1 ≤ ν ≤ N −kr −q, kr +1 ≤ i1 < . . . < iq ≤ N and kr +1 ≤ respectively, and where the summation j1 <...<jν in (3.6) is to be replaced by 1 if q = kr and the summation j1 <...<jν in (3.6) is to be replaced by 1 if q = N − kr . Hence it follows from Rouch´e’s theorem that if r+1 is chosen ˚ ξ , z) − λ IN has sufficiently small and |λ| sufficiently large, then det A(0, precisely Nr zeros contained in γr+ (ξ , λ). Suppose next that > r. Then we can write () () A12 (0, ξ , z) A11 (0, ξ , z) − λ Ik ˚ A(0, ξ , z) − λ IN = (3.7) () () A22 (0, ξ , z) − λ IN −k A21 (0, ξ , z)
Vol. 66 (2010)
Estimates for Solutions
341
if < d, and where (3.7) must be modified in an obvious way if = d. In addition we have () () () A11 (0, ξ , z) − λ Ik = A11 (0, 0, z) − λ I,0 + A11 (0, ξ , z)
− A11 (0, 0, z) − λ(Ik − I,0 ). () Then as a consequence of our hypotheses we know that det A11 (0, 0, z) − λI,0 has precisely N − N−1 zeros lying in C+ and that there is a closed + contour γr, (λ) ⊂ C+ containing all these zeros in its interior such that for z ∈ 2N () 1 1 + , |z| ≤ C2 |λ| m and C3 |λ| m ≤ | det A11 (0, 0, z) − γr, (λ), Im z ≥ C1 |λ| m 2N λ I,0 | ≤ C4 |λ| m , where the constants Cj do not depend upon λ. Further+ (λ), more, we can show that for z ∈ γr, () () |det A11 (0, ξ , z) − λ Ik − det A11 (0, 0, z) − λI,0 | 2N 1 1 −m −1 ≤ C |λ| m + δ,r+1 r+1 |λ| m , where δ,r+1 denotes the Kronecker delta and the constant C does not depend upon ξ , z,and λ, and that, apart from upon a constant not depending ˚ ξ , z) − λ IN − det A() (0, ξ , z) − λ Ik λN −k | is ξ , z and λ, |det A(0, 11 bounded by the expression (3.6) if we replace r there by . It follows again from Rouch´e’s theorem that if r+1 is chosen sufficiently small and |λ| suf ˚ ξ , z) − λ IN has precisely N − N−1 zeros ficiently large, then det A(0, + contained in γr, (λ). Hence since was chosen arbitrary, this completes the proof of the proposition. Turning again to Propositions 3.8-3.10 and supposing henceforth that the boundary problem (1.1), (1.2) is parameter-elliptic in L and that 0 has been chosen sufficiently small and λ0 sufficiently large so that the conclusions of these propositions hold, let us suppose that 0 ≤ r ≤ d and that 0 ≤ 1 1 1 m r+1 1 r ≤ |ξ | ≤ 7 |ξ | ≤ 78 1 |λ| m if r = 0, 18 r |λ| m if 1 ≤ r < d, 8 r+1 |λ| 1
d and 18 d |λ| m ≤ |ξ | < ∞ if r = d. Then it follows from [V] that the set of solutions of the differential equation
˚ ξ , Dn )u(xn ) − λ u(xn ) = 0 for xn > 0 A(0,
(3.8)
which decay exponentially at ∞ form a vector space of dimension Nd . Furthermore, this vector space is precisely the vector space spanned by the columns of the matrix −1 ˚ ξ , z) − λ IN IN , zIN , . . . , z m1 −1 IN dz, (3.9) eixn z A(0, γr+ (ξ ,λ)
if r = d, while if r < d, then it is precisely the direct sum of the vector space (of dimension Nr ) spanned by the columns of (3.9) and the N − r vector
342
Denk and Faierman
IEOT
spaces (of dimension N − N−1 , = r + 1, . . . , d) spanned by the columns of each of the matrices + γr, (λ)
−1 ˚ ξ , z) − λ IN IN , zIN , . . . , z m1 −1 IN dz eixn z A(0,
for = r + 1, . . . , d. Note that here the contours γ0+ (ξ , λ) (resp. γd+ (ξ , λ)) + and the γ0, (λ) are defined in an analogous fashion to the way they were defined in the proof of Proposition 3.9 for the case 0 < r < d. Proposition 3.11. Suppose that the boundary problem (1.1),(1.2) is parameterelliptic in L and that λ ∈ L. Then we can choose the constants 0 and λ0 of Propositions 3.8–3.10 sufficiently small and sufficiently large, respectively, so that the following results hold. 1 1 , then If 0 ≤ |ξ | ≤ 78 1 |λ| m rank
−1 ˚ ξ , z) − λ IN B (1,1) (0, ξ , z) A(0,
γ + (ξ ,λ)
0 × IN , zIN , . . . , z m1 −1 IN dz = N1 , rank
and
−1 ˚ ξ , z) − λ IN B (1,)(0, ξ , z) A(0,
(3.10)
+ γ0, (λ)
× IN , zIN , . . . , z m1 −1 IN dz = N − N−1
for = 2, . . . , d;
1
r ≤ |ξ | ≤ if 1 ≤ r ≤ d and 18 r |λ| m |ξ | < ∞ if r = d, then
rank
1
7 m r+1 8 r+1 |λ|
if r < d or
1
1 m d 8 d |λ|
≤
−1 ˚ ξ , z) − λ IN B (r,r)(0, ξ , z) A(0,
γr+ (ξ ,λ)
× IN , zIN , . . . , z m1 −1 IN dz = Nr , rank
and B
(r,)
+ γr, (λ)
−1 ˚ ξ , z) − λ IN (0, ξ , z) A(0,
(3.11)
× IN , zIN , . . . , z m1 −1 IN dz = N − N−1
for = r + 1, . . . , d if r < d.
Proof. We will only prove the proposition for the case 1 ≤ r < d; the remaining cases can be similarly treated. Accordingly, referring to the proof of Proposition 3.9 for notation, we see that for z ∈ γr+ (ξ , λ) we have the
Vol. 66 (2010)
Estimates for Solutions
343
representation ˚ ξ , z) − λ IN = A(0,
(r) 0 A11 (0, ξ , z) − λ Ir,0 0 −λ IN −kr (r) λ(Ir,0 − Ikr ) A12 (0, ξ , z) + (r) (r) A21 (0, ξ , z) A22 (0, ξ , z)
(3.12)
Furthermore, it follows from the proof of Proposition 3.9 that for each fixed (r) triple (ξ , z, λ), the operator A11 (0, ξ , z) − λ Ir,0 : Ckr → Ckr is bounded and invertible, while if we denote by ajk (ξ , z, λ) the entry in the j-th row and (r) −1 k-th column of A11 − λIr,0 , then ajk (ξ , z, λ) is a rational function of its r arguments and satisfies the quasi-homogeneity condition ajk (ρξ , ρ z, ρm λ) = ρ−tj −sk ajk (ξ , z, λ) for ρ > 0. From this it is clear that | ajk (ξ , z, λ)| ≤ −t −s Cξ , λr j k , where here and for the remainder of this proof C denotes a generic constant that does not depend upon the variables indicated, and that (r) the entry in the j-th row and k-th column of λ(Ir,0 − Ikr ) A11 (0, ξ , z) − −1 λ Ir,0 is λ ajk (ξ , z, λ) if j ≤ kr−1 and is 0 otherwise. Thus if we let r λ, and assume henceforth that ρ = ξ , λr , η = ρ−1 ξ , ζ = ρ−1 z, µ = ρ−m |λ| has been chosen sufficiently large, it is now a simple matter to show that if we put −1 (r) H11 (ξ , z, λ) = λ Ir,0 − Ikr A11 (0, ξ , z) − λIr,0
= H11 (η , ζ, µ, ρ) kr r ajk (η , ζ, µ) diag ρ−t1 , . . . , ρ−tkr−1 , 0Ir = µρm j,k=1 × diag ρ−s1 , . . . , ρ−skr , then the matrix Ikr + H11 (ξ , z, λ) is invertible and admits the representation −1 11 (ξ , z, λ), where (Ikr + H11 (ξ , z, λ)) = Ikr + H 11 (ξ , z, λ) = H 11 (η , ζ, µ, ρ) H
kr jk (η , ζ, µ, ρ) = diag(ρs1 , . . . , ρskr ) H 11
diag ρ−s1 , . . . , ρ−skr
j,k=1
jk (η , ζ, µ, ρ) is a rational function of its arguments which satisfies the and H 11 m j jk (η , ζ, µ, ρ) ≤ C|λ|1− m r estimate H if j ≤ kr−1 and equals 0 otherwise. 11
Next let us put H(ξ , z, λ) =
H11 (ξ , z, λ) −H12 (ξ , z, λ) H21 (ξ , z, λ) −H22 (ξ , z, λ),
(r)
where H12 (ξ , z, λ) = λ−1 A12 (0, ξ , z),
−1 (r) (r) , H21 (ξ , z, λ) = A21 (0, ξ , z) A11 (0, ξ , z) − λIr,0
344
Denk and Faierman
IEOT
(r)
and H22 (ξ , z, λ) = λ−1 A22 (0, ξ , z). Thus it follows from the foregoing results and (3.12) that ˚ ξ , z) − λ IN A(0, = (IN + H(ξ , z, λ))
(r) A11 (0, ξ , z) − λ Ir,0 0
0
−λ IN −kr ,
,
(3.13)
while a factorization in the sense of Schur (see [G, Theorem 3.16, p.302]) gives Ikr 0 IN + H(ξ , z, λ) = −1 IN −kr H21 (ξ , z, λ) Ikr + H11 (ξ , z, λ) 0 Ikr + H11 (ξ , z, λ) × 0 IN −kr − H † (ξ , z, λ) −1 Ikr − Ikr + H11 (ξ , z, λ) H12 (ξ , z, λ) × , 0 IN −kr where −1
H † (ξ , z, λ) = H22 (ξ , z, λ) − H21 (ξ , z, λ) (Ikr + H11 (ξ , z, λ))
H12 (ξ , z, λ)
= H † (η , ζ, µ, ρ)
N † (η , ζ, µ, ρ) = diag(ρskr +1 , . . . , ρsN ) Hjk
j,k=kr +1
diag(ρ−skr +1 , . . . , ρ−sN ),
† and where Hjk (η , ζ, µ, ρ) is a rational function of its arguments and m r+1 † r+1 −1 H (η , ζ, µ, ρ) ≤ C m m r . r+1 + |λ| jk
Thus if we henceforth suppose that 0 has been chosen sufficiently small and λ0 sufficiently large, then IN −kr − H † (ξ , z, λ) is invertible and we have −1 † (ξ , z, λ), where IN −kr − H † (ξ , z, λ) = IN −kr + H † (ξ , z, λ) = H † (η , ζ, µ, ρ) H N † (η , ζ, µ, ρ) = diag (ρskr +1 , . . . , ρsN ) H jk
j,k=kr +1
diag ρ−skr +1 , . . . , ρ−sN
† (η , ζ, µ, ρ) is a rational function of its arguments and and H jk m r+1 r+1 −1 † (η , ζ, µ, ρ)| ≤ C m m r + |λ| |H . r+1 jk ˚ ξ , z) − λ IN is An immediate consequence of this last result is that A(0, invertible and −1 −1 (r) A (0, ξ , z) − λ I 0 r,0 ˚ ξ , z) − λ IN 11 A(0, = 0 −λ−1 IN −kr (3.14) × IN + H (ξ , z, λ) ,
Vol. 66 (2010)
Estimates for Solutions
345
where
H11 (ξ , z, λ) H12 (ξ , z, λ) , (ξ , z, λ) H22 (ξ , z, λ) H21 11 (ξ , z, λ) + Ikr + H 11 (ξ , z, λ) H12 (ξ , z, λ)H (ξ , z, λ), (ξ , z, λ) = H H11 21 † 11 (ξ , z, λ) H12 (ξ , z, λ) IN −kr + H (ξ , z, λ) , H12 (ξ , z, λ) = Ikr + H † (ξ , z, λ) H21 (ξ , z, λ) Ikr + H 11 (ξ , z, λ) , H21 (ξ , z, λ) = − IN −kr + H H (ξ , z, λ) =
† (ξ , z, λ). (ξ , z, λ) = H H22
Furthermore, we can write H11 (ξ , z, λ) = H11 (η , ζ, µ, ρ) jk = diag (ρs1 , . . . , ρskr ) H11 (η , ζ, µ, ρ)
diag ρ−s1 , . . . , ρ−skr , j,k=1,...,kr
jk (η , ζ, µ, ρ) is a rational function of its arguments and where H11 m r+1 m r−1 m r+1 jk r r , |H11 (η , ζ, µ, ρ)| ≤ C r+1 + |λ|−1+ m + |λ|1− m H21 (ξ , z, λ) = H21 (η , ζ, µ, ρ)
jk (η , ζ, µ, ρ) j=kr +1,...,N = diag (ρskr +1 , . . . , ρsN ) H21 × diag ρ−s1 , . . . , ρ−skr ,
k=1,...,kr
jk (η , ζ, µ, ρ) is a rational function of its arguments and where H21 jk (η , ζ, µ, ρ)| ≤ C. |H21
Let us next observe that under our assumptions concerning ξ and λ, (η , µ) ∈ Σr , where
2 1
r = 1, | r Σr = ( η ,µ ) ∈ Rn−1 × L : | η |2 + | µ| m η | ≥ 18 r | µ| m − 12 r = ( ⊂Σ η , µ ) ∈ Rn−1 × L : 1 + 64 ≤ | η | ≤ 1, 0 ≤ | µ | ≤ 1 .
2 r
r , det A(r) (0, η , ζ) − It also follows from Definition 2.6 that for ( η , µ ) ∈ Σ 11 µ Ir,0 has precisely Nr zeros lying in C+ and that there is a closed contour γr+ ⊂ C+ , not depending upon η and µ , containing all these zeros in its interior. And it is to be understood henceforth that the contour γr+ (ξ , λ) introduced above is obtained from γr+ by dilation by the factor ρ. Furthermore, as a consequence of Definition 2.6 we also know from [V] that if we fix our r , then there exists matrices J ) ∈ Σ attention upon a particular pair ( η , µ and Z(ζ) depending upon η and µ , such that the matrix −1 (r,r) (r) η , µ ) = Br,1 (0, η , ζ) A11 (0, η , ζ) − µ Ir,0 JZ(ζ)dζ (3.15) K1 ( γr+
346
Denk and Faierman
IEOT
has rank Nr . Here J denotes a kr × Nr matrix with the property that each of its columns has precisely one non-zero component, namely 1, and Z(ζ) = diag ζ q(1) , . . . , ζ q(Nr ) , where the q(j) denote non-negative integers not exceeding m1 − 1. Thus det K1 ( η , µ ) > c (3.16) for some c > 0. As a consequence of (3.15), (3.16) we see that for the values of ξ and λ under consideration here, the rank of the matrix −1 (r,r) (r) Br,1 (ξ , z) A11 (0, ξ , z) − λ Ir,0 JZ(z)dz (3.17) γr+ (ξ ,λ)
is Nr and that (3.17) can be written in the form ρ diag (ρσ1 , . . . , ρσNr ) K1 (η , µ) diag ρq(1)−sk(1) , . . . , ρq(Nr )−sk(Nr ) , (3.18) where 1 ≤ k(j) ≤ kr for 1 ≤ j ≤ Nr . Hence referring to (3.10), let us now use this fact to show that if 0 is chosen sufficiently small and λ0 sufficiently large, then the rank of −1 J (r,r) ˚ B (0, ξ , z) A(0, ξ , z) − λ IN Z(z)dz (3.19) 0 γr+ (ξ ,λ) is Nr . Indeed, it follows from (3.14) that the expression (3.19) can be expressed as the sum of the expression (3.18) plus σ1 σNr K2 (η , ζ, µ, ρ)JZ(ζ)dζ ) ρ diag (ρ , . . . , ρ γr+ × diag ρq(1)−sk(1) , . . . , ρq(Nr )−sk(Nr ) , where 2 (η , ζ, µ, ρ) K
−1 (r,r) (r) = B11 (0, η , ζ) A11 (0, η , ζ) − µ Ir,0 H11 (η , ζ, µ, ρ) (r,r)
r −1 ) B11 (0, η , ζ) diag (ρmkr +1 , . . . , ρmN ) H21 (η , ζ, µ, ρ) − (µ ρm
is an Nr × Nr matrix function whose entry in the j-th row and k-th column, kjk (η , ζ, µ, ρ) is a rational function of its arguments and m r+1 m r−1 m r+1 −1 1− m m r r |kjk (η , ζ, µ, ρ)| ≤ C r+1 + |λ| + |λ| . (3.20) It follows immediately from these results that if we suppose that 0 is chosen sufficiently small and λ0 sufficiently large, then the matrix (3.19) has rank Nr , which proves the proposition for the case under consideration. + Suppose next that r < ≤ d and that z ∈ γr, (λ). Then as a consequence () of our hypotheses we know that A11 (0, 0, z) − λ I,0 is invertible and if we −1 k () () () let A11 (0, 0, z) − λ I,0 = ajk (z, λ) , then ajk (z, λ) is a rational j,k=1
Vol. 66 (2010)
Estimates for Solutions
347
functional function of its arguments and satisfies the quasi-homogeneity con() () dition ajk (ρ z, ρm λ) = ρ−tj −sk ajk (z, λ) for ρ > 0. Let us now introduce the 2 () matrix function H () (ξ , z, λ) = Hjk (ξ , z, λ) , where j,k=1
() () () H11 (ξ , z, λ) = A11 (0, ξ , z) − A11 (0, 0, z) + λ(I,0 − Ik −1 () , × A11 (0, 0, z) − λ I,0
()
and if < d, then H12 (ξ , z, λ) = λ−1 A12 (ξ , z),
−1 () () () , H21 ((ξ , z, λ) = A21 (ξ , z) A11 (0, 0, z) − λ I,0 ()
()
and H22 (ξ , z, λ) = λ−1 A22 (ξ , z). Then proceeding as in the previous case, we can write ()
()
H11 (ξ , z, λ) = H11 (η , ζ , µ , ρ ) k sk () jk = diag ρs 1 , . . . , ρ H11 (η , ζ , µ , ρ )
j,k=1
−sk 1 diag ρ−s , . . . , ρ ,
1
−m −1 where ρ = |λ| m , η = ρ−1 λ, and the coefficient ξ , ζ = ρ z, µ = ρ () jk H11 (η , ζ , µ , ρ ) is a rational functional of its arguments satisfying () jk m −m −1 H , (3.21) (η , ζ , µ , ρ ) ≤ C δ + ρ r+1, r+1 11
where here and below C denotes a generic constant which does not depend upon any of the variables indicated. Hence it follows that for 0 sufficiently () small and λ0 sufficiently large, Ik + H11 (ξ , z, λ) is invertible and we have
−1 () () (η , ζ , µ , ρ ) Ik + H11 (ξ , z, λ) = diag (ρs 1 , . . . , ρs ) Ik + H −sk 1 , , . . . , ρ × diag ρ−s
() (η , ζ , µ , ρ ) = where H 11
k () jk (η , ζ , µ , ρ ) H 11
and the coefficient
j,k=1
() jk (η , ζ , µ , ρ ) is a rational functional function of its arguments and H 11 is bounded in modulus by the expression on the right side of (3.21). Suppose next that < d. Then proceeding as we did in our previous case, let us now put ()
H † () (ξ , z, λ) = H22 (ξ , z, λ)
−1 () () () H12 (ξ , z, λ). − H21 (ξ , z, λ) Ik + H11 (ξ , z, λ)
348
Denk and Faierman
IEOT
Then we can write H † () (ξ , z, λ) = H † () (η , ζ , µ , ρ ) s N k +1 † () Hjk (η , ζ , µ , ρ ) = diag ρ , . . . , ρs N j,k=k +1 −s k +1 −sN , × diag ρ , . . . , ρ † ()
where it follows from its definition that Hjk (η , ζ , µ , ρ ) is a rational func−1+
m +1
m tion of its arguments and is bounded in modulus by C|λ| . Hence if we 0 † () suppose that λ is sufficiently large, then IN −k − H (ξ , z, λ) is invertible, and if we write −1 † () (ξ , z, λ), = IN −k + H IN −k − H † () (ξ , z, λ)
and † () (ξ , z, λ) = H † () (η , ζ , µ , ρ ) H N s k +1 † () (η , ζ , µ , ρ ) H = diag ρ , . . . , ρs N jk j,k=k +1 −s k +1 −sN , × diag ρ , . . . , ρ then H jk (η , ζ , µ , ρ ) is a rational function of its arguments and is bounded † ()
−1+
m +1
s −s
m in modulus by C|λ| ρ j k . In light of these definitions, we can now proceed as in the first part of the proof to show that for < d, −1 −1 () A11 (0, 0, z) − λ I,0 0 ˚ = A(0, ξ , z) − λ IN 0 −λ−1 IN −k (3.22) () × IN + H (ξ , z, λ) ,
where
2 () , H () (ξ , z, λ) = Hjk (ξ , z, λ) j,k=1 () () (ξ , z, λ) + Ik + H () (ξ , z, λ) H11 (ξ , z, λ) = H 11 11 ()
()
× H12 (ξ , z, λ)H21 (ξ , z, λ), () () (ξ , z, λ) H () (ξ , z, λ) IN −k + H † () (ξ , z, λ) , H12 (ξ , z, λ) = Ik + H 11 12 () () † () H21 (ξ , z, λ) = − IN −k + H (ξ , z, λ) H21 (ξ , z, λ) () (ξ , z, λ) , × Ik + H 11 () † () (ξ , z, λ), H22 (ξ , z, λ) = IN −k + H
Vol. 66 (2010)
Estimates for Solutions
349
and where these results are to be modified in an obvious way if = d. Note also that ()
()
H11 (ξ , z, λ) = H11 (η , ζ , µ , ρ ) k sk () H11,jk (η , ζ , µ , ρ ) = diag ρs 1 , . . . , ρ
j,k=1
−sk 1 diag ρ−s , . . . , ρ ,
()
where H11,jk (η , ζ , µ , ρ ) is a rational functional function of its arguments m +1 m −1 −1+ m 1− m and is bounded in modulus by C |λ| + |λ| + r+1 , and if < d, then ()
()
H21 (ξ , z, λ) = H21 (η , ζ , µ , ρ ) s () k +1 = diag ρ , . . . , ρs N H21,jk (η , ζ , µ , ρ ) j=k +1,...,N −sk 1 × diag ρ−s , . . . , ρ ,
k=1,...,k
()
where H21,jk (η , ζ , µ , ρ ) is a rational function of its arguments and is bounded in modulus by C. Let us next our assumptions concerning ξ and observe that under
λ, µ ∈ L1 = µ ∈ L | | µ| = 1 . It also follows from Definition 2.4 that () I,0 has precisely N − N−1 zeros lying in for µ ∈ L1 , det A (0, 0, ζ) − µ 11
+ C+ and that there is a closed contour γr, ⊂ C+ , not depending upon µ , containing all these zeros in its interior. And it is to be understood henceforth + + that the contour γr, (λ) introduced above is obtained from γr, by dilation by the factor ρ . Furthermore, as consequence of Definition 2.6 it also follows from [V] that if we fix our attention upon a particular µ ∈ L1 , then there exist matrices J () and Z () (ζ ), depending upon µ , such that −1 () (r,) () K1 (0, µ ) = B,1 (0, 0, ζ ) A11 (0, 0, ζ ) − µ I,0 J () Z () (ζ )dζ (3.23) + γr,
has rank N − N−1 . Here J () is a k × (N − N−1 )matrix having the same q(N −N ) q(1) properties as the matrix J of (3.15), Z () (ζ ) = diag ζ , . . . , ζ −1 , and the q(j) are defined as in the text following (3.15). Then det K () (0, µ ) > c() (3.24) 1 for some constant c() > 0. As a consequence of (3.23) and (3.24) we see that for the values of ξ and λ under consideration here the rank of the matrix −1 (r,) () B,1 (0, z) A11 (0, 0, z) − λ I,0 J () Z () (z)dz (3.25) + γr, (λ)
350
Denk and Faierman
is N − N−1 and that (3.25) can be written in the form σ σN −N N +1 () ρ diag ρ −1 , . . . , ρ −1 K1 (0, µ ) q(N −N−1 )−sk(N −N−1 ) q(1)−sk(1) × diag ρ , . . . , ρ ,
IEOT
(3.26)
where 1 ≤ k(j) ≤ k for 1 ≤ j ≤ N − N−1 . Hence referring to (3.11), let us now use these facts to show that if 0 is sufficiently small and λ0 is sufficiently large, then the rank of the matrix −1 () J ˚ ξ , z) − λ IN Z () (z)dz B (r,) (0, ξ , z) A(0, (3.27) + 0 γ (λ) r,
is N − N−1 . Indeed it follows from (3.22) that the expression (3.27) can be expressed as the sum of the expression (3.26) plus the expression σ σN N−1 +1 () () () ρ diag ρ K2 (η , ζ , µ , ρ )J Z (ζ )dζ , . . . , ρ + γr,
q(N −N−1 )−sk(N −N−1 ) q(1)−sk(1) , . . . , ρ × diag ρ where () (η , ζ , µ , ρ ) K 2 −1 (r,) (r,) () = B,1 (0, η , ζ ) − B11 )(0, 0, ζ ) A11 (0, 0, ζ ) − µ I,0 −1 (r,) () () + B,1 (0, η , ζ ) A11 (0, 0, ζ ) − µ I,0 H11 (η , ζ , µ , ρ ) (r,)
()
−1 − (1 − δ,d )(µ ρm B,2 (0, η , ζ )H21 (η , ζ , µ , ρ ) )
is an (N − N−1 ) × (N − N−1 ) matrix function whose entry in the j-th row () and k-th column, kjk (η , ζ , µ , ρ ), is a rational function of its arguments and 1 1 −m () −1 |kjk (η , ζ , µ , ρ )| ≤ C δr+1, r+1 + |λ| m (3.28) m −1 m +1 1− m m −1 . + |λ| + (1 − δ,d )|λ| It follows immediately from these results that if we suppose that 0 and λ0 are sufficiently large and small, respectively, then the matrix (3.27) has rank N − N−1 , and this completes the proof of the proposition. In light of Proposition 3.11 and from what was said in the text preceding that proposition, we are now in a position to present some results pertaining to the solutions of (3.8). To this end let us remark for later use that the contour γr+ and the set Σr for r = 0 and r = d are defined in an analogous manner to the way they were in the proof of Proposition 3.11, except that
2 1 = 1 . A similar remark we now take Σ0 = (ξ , λ) ∈ Rn−1 × L : |ξ |2 + |λ| m + holds for γ0, .
Vol. 66 (2010)
Estimates for Solutions
351
Proposition 3.12. Suppose that the hypotheses of Proposition 3.11 hold and that 0 and λ0 have been chosen small enough and large enough, respectively, so that the conclusions of that proposition hold. Suppose also that either 1 , or (i) r = 0 and |ξ | ≤ 78 1 |λ|1/m 1 1/m r r+1 (ii) 1 ≤ r < d and 8 r |λ| ≤ |ξ | ≤ 78 r+1 |λ|1/m , or 1 1/m d ≤ |ξ | < ∞. (iii) r = d and 8 d |λ| linearly independent solutions, Then the differential equation (3.8) has N
{w(r,ν) (ξ , xn , λ)}N ν=1 , which decay exponentially at ∞ and satisfy B (1,) (0, ξ , Dn )W (0,) (ξ , 0, λ) = IN −N−1
for = 1, . . . , d,
if r = 0, B (r,) (0, ξ , Dn )W (r,) (ξ , 0, λ) = IN −N †
for = r, . . . , d,
(3.29)
r,−1
if 1 ≤ r < d,
B (d,d)(0, ξ , Dn )W (d,d) (ξ , 0, λ) = INd ,
† where Nr,−1 = (1 − δmax{r,1}, )N−1 , W (0,) (ξ , xn , λ) denotes the N × (N − N−1 ) matrix function whose columns are precisely the w(0,ν) (ξ , xn , λ) for ν = N−1 + 1, . . . , N , while for r > 0, W (r,) (ξ , xn , λ) denotes the N × (N − † Nr,−1 ) matrix function whose columns are precisely the w(r,ν) (ξ , xn , λ) for
† ν = Nr,−1 + 1, . . . , N . Furthermore, we have the representations (here we let r† = max{r, 1}) † † ˚ ξ , z) − λIN −1 G(r,r ) (ξ , z, λ)dz, W (r,r ) (ξ , xn , λ) = eixn z A(0, + γ (ξ ,λ) r ˚ ξ , z) − λIN −1 G(r,) (ξ , z, λ)dz, W (r,) (ξ , xn , λ) = eixn z A(0, + γr, (λ)
for = r† + 1, . . . , d if r < d,
(3.30)
where for r† ≤ ≤ d (r,)
G
(r,) G (ηr, , ζr, , µr, , ρr, ) (ξ , z, λ) = , 0 IN −k
(r,) (η , ζr, , µr, , ρr, ) = ρ−1 diag(ρs1 , . . . , ρsk )K (r,) (η , ζr, , µr, , ρr, ) G r, r, r, r, r, −σ
× diag(ρr,
1/m
ρr,r† = ξ , λr† , ρr, = |λ| m µr, = ρ− r, λ, and
N
† +1 r,−1
†
for > r ,
−σN
, . . . , ρr, ηr,
=
(r,) , ζr, , µr, , ρr, ) = Kjk (ηr, , ζr, , µr, , ρr, ) K (r,) (ηr, (r,)
),
ρ−1 r, ξ ,
ζr, = ρ−1 r, z,
, j=1,...,k † k=Nr,−1 +1,...,N
, ζr, , µr, , ρr, ) is a finite sum of a such that for each pair j, k, Kjk (ηr, product of a power of ζr, and an expression of rational type in ηr, , ζr, , µr, and ρr, (i.e., a rational function of terms which are integrals over
352
Denk and Faierman
IEOT
+ δr† , γr+ +(1−δr†, )γr, of rational functions of the components of ηr, , ζr, , µr, and ρr, ) and is bounded in modulus by a constant depending only upon ηr,r † † † and µr,r† (∈ Σr ) if = r and only upon ρr, (∈ L1 ) if > r , and lastly the block 0 IN −k is to be omitted when = d.
Proof. We will only prove the proposition for the case 1 ≤ r < d as the proofs of the other cases are similar. Accordingly, returning to the proof of Proposition 3.11 and employing the notation of that proposition, let Λ(ξ , λ) denote the expression (3.19). Then we know from the proof of Proposition 3.11 that Λ(ξ , λ) = ρ diag ρσ1 , . . . , ρσNr K1 (η , µ) + K2 (η , µ, ρ) × diag ρq(1)−sk(1) , . . . , ρq(Nr )−sk(Nr ) , 2 (η , ζ, µ, ρ)JZ(ζ)dζ and where we refer to the where K2 (η , µ, ρ) = γr+ K text following (3.15) for notation. Hence bearing in mind (3.16) and (3.20), it is an immediate consequence of Proposition 3.11 that Λ(ξ , λ) is invertible and Λ(ξ , λ)−1 = ρ−1 diag ρsk(1) −q(1) , . . . , ρsk(Nr ) −qN (r) × (INr + K3 (η , µ, ρ)K1 (η , µ)−1 diag ρ−σ1 , . . . , ρ−σNr , where K3 (η , µ, ρ) is an Nr × Nr matrix function defined by −1 INr + K3 (η , µ, ρ) = INr + K1 (η , µ)−1 K2 (η , µ, ρ) and where each entry of K3 (η , µ, ρ) is a function of its arguments of rational type and is bounded in modulus by the expression on the right-hand side of (3.20). Hence it follows that ˚ ξ , z) − λIN −1 B (r,r)(0, ξ , z) A(0, γr+ (ξ ,λ)
×
(1) (η , ζ, µ, ρ) + G (2) (η , ζ, µ, ρ) G dz = INr , 0IN −kr
where (1) (η , ζ, µ, ρ) = ρ−1 diag(ρs1 , . . . , ρskr )JZ(ζ)K1 (η , µ)−1 G × diag(ρ−σ1 , . . . , ρ−σNr ), (2) (η , ζ, µ, ρ) = ρ−1 diag(ρs1 , . . . , ρskr )JZ(ζ)K3 (η , µ, ρ)K1 (η , µ)−1 G × diag(ρ−σ1 , . . . , ρ−σNr ).
(3.31)
Suppose next that r < ≤ d and let Λ() (ξ , λ) denote the expression (3.27). Then we know that σN +1 σN () () Λ() (ξ , λ) = ρ diag ρ −1 , . . . , ρ K1 (µ ) + K2 (η , µ , ρ ) q(1)−sk(1) q(N −N−1 )−sk(N −N−1 ) , , . . . , ρ × diag ρ
Vol. 66 (2010)
Estimates for Solutions
()
where K2 (η , µ , ρ ) =
+ γr,
353
() (η , ζ , µ , ρ )J () Z () (ζ )dζ . Hence bearing K 2
in mind (3.24) and (3.28), it follows that Λ() (ξ , λ) is invertible and Λ() (ξ , λ)−1 sk(N −N ) −q(N −N−1 ) s(1)−q(1) = ρ−1 , . . . , ρ −1 diag ρ () −σN +1 () × IN −N−1 + K3 (η , µ , ρ ) K1 (µ )−1 diag ρ −1 , . . . , ρ−σN , ()
where K3 (η , µ , ρ ) is an (N −N−1 )×(N −N−1 ) matrix function defined by −1 () () () IN −N−1 + K3 (η , µ , ρ ) = IN −N−1 + K1 (µ )−1 K2 (η , µ , ρ ) ()
and each entry of K3 (η , µ , ρ ) is a function of its arguments of rational type and is bounded in modulus by the expression on the right-hand side of (3.28). Hence it follows that ˚ ξ , z) − λIN −1 B (r,) (0, ξ , z) A(0, + γr, (λ)
(1,) (2,) (η , ζ , µ , ρ ) dz = IN −N , × G (ζ , µ , ρ ) + G −1 where (1,) (ζ , µ , ρ ) = ρ−1 diag(ρs1 , . . . , ρsk )J () Z () (ζ )K () (µ )−1 G 1 −σN−1 +1
× diag(ρ (2,) (η , ζ , µ , ρ ) G
−σN
, . . . , ρ
),
sk () s1 () () = ρ−1 Z (ζ )K3 (η , ζ , µ , ρ ) diag(ρ , . . . , ρ )J −σN −σN +1 () × K1 (µ )−1 diag(ρ −1 , . . . , ρ ).
, ζr,r , and µr,r , respectively, Finally if we rewrite ρ, η , ζ, and µ as ρr,r , ηr,r and put K r,r (ηr,r , ζr,r , µr,r , ρr,r ) = JZ(ζr,r )(INr + K3 (ηr,r , µr,r , ρr,r ))K1 (ηr,r , µr,r )−1 , , ζr, , and µr, , while if for > r we rewrite ρ , η , ζ , and µ as ρr, , ηr, respectively, and put , ζr, , µr, , ρr, ) K (r,) (ηr, ()
()
= J () Z () (ζr, )(IN −N−1 + K3 (ηr, , µr, , ρr, ))K1 (µr, )−1 ,
then all the assertions of the proposition for the case under consideration here follow from the foregoing results. Let us fix r, 0 ≤ r < d, and referring again to Proposition 3.12 for notation, let us denote by W (r) (ξ , xn , λ) the N × Nd matrix function whose
354
Denk and Faierman
IEOT
columns are precisely the w(r,ν) (ξ , xn , λ). For r† ≤ j ≤ d, let † d Ir,j (ξ , λ) = B (r ,) (0, ξ , Dn )W (r,1 ) (ξ , 0, λ)
,1 =j d
† (r,1 ) (ξ , 0, λ) Ir,j (ξ , λ) = B(r ,) (0, ξ , Dn )W
,1 =j
, ,
(3.32)
where −σN † +1 † † −σN B(r ,) (0, ξ , Dn ) = diag ρr, r,−1 , . . . , ρr, B (r ,) (0, ξ , Dn ), σN † +1 σ (r,1 ) (ξ , 0, λ) = W (r,1 ) (ξ , 0, λ) diag ρ r,1 −1 , . . . , ρ N1 . W r,1 r,1 Then we can write ˚ ξ , Dn )W (r) (ξ , 0, λ) = Ir,r† (ξ , λ) B(0, σN −σN 1 = diag ρσ1 1 , . . . , ρNdd Ir,r† (ξ , λ) diag ρ−σ , . . . , ρNd d , 1
where ρν =
(3.33)
for 1 ≤ ν ≤ Nr† , for N−1 < ν ≤ N , = r† + 1, . . . , d.
ρr,r† ρr,
We remark at this point that as a consequence of Proposition 3.14 be† (r,1 ) (ξ , 0, λ) is a low, it will be seen that for = 1 , B(r ,) (0, ξ , Dn )W † † (N − Nr,−1 ) × (N1 − Nr,1 −1 ) matrix function whose entries are products of powers of ρr, /ρr,1 and an expression of rational type in the components of ηr, , µr,1 , and ρr,1 . 1 Definition 3.13. Suppose that the boundary problem (1.1), (1.2) is parameterelliptic in L and that λ ∈ L. Suppose also that x0 ∈ Γ and that the boundary problem has been rewritten in terms of a fixed local coordinate system at x0 (see the paragraph preceding Definition 2.3 and [AV, p. 63]). In addition, suppose that with respect to this boundary problem all hypotheses of Proposition 3.12 hold. Then bearing in mind the definitions of the various terms introduced above, we say that the operators A(x, D) and B(x, D) = (B1 (x, D), . . . , BN (x, D))T are compatible at x0 if for each r satisfying 0 ≤ r < d and for each pair of integers , 1 satisfying r† ≤ 1 < d, (r,1 ) (ξ , 0, λ) is 1 + 1 ≤ ≤ d, each entry of the matrix B(r,) (0, ξ , Dn )W bounded in modulus by a constant depending only upon (ηr,r † , µr,r † ) ∈ Σr and µr, ∈ L1 for > r. Proposition 3.14. Suppose that the hypotheses of Proposition 3.12 hold and that the operators A(x, D) and B(x, D) are compatible at x0 . Suppose also that 0 ≤ r < d. Then for 0 sufficiently small and λ0 sufficiently large the matrix function Ir,r† (ξ , λ) of (3.33), and hence the matrix function Ir,r† (ξ , λ) are invertible and we have σN −σNd , Ir,r† (ξ , λ)−1 = diag ρσ1 , . . . , ρ d Ir,r† (ξ , λ)−1 diag ρ−σ1 , . . . , ρ 1
Nd
1
Nd
where | det Ir,r† (ξ , λ)| > 12 , while the entries of Ir,r† (ξ , λ)−1 are rational functions of the ρr, and expressions of rational type in ηr, , µr, , and ρr,
Vol. 66 (2010)
Estimates for Solutions
355
for ≥ r† and are bounded in modulus by a constant depending only upon (ηr,r † , µr,r † ) ∈ Σr and µr, ∈ L1 for > r. Lastly, the operators A(x, D) and B(x, D) are always compatible at x0 if (i) the boundary conditions (1.2) are of Dirichlet type at x0 or if (ii) the operators A(x, D) and B(x, D) are both essentially upper triangular at x0 . Proof. We will only prove the proposition for the case 1 ≤ r < d; the case r = 0 can be similarly treated. Accordingly, let us fix our attention upon (3.32) and suppose that = 1 . Then it follows from the proofs of Proposition 3.11 and 3.12 that (r,) (ξ , 0, λ) B(r,)(0, ξ , Dn )W ρ −σ † ρ −σN +1 r, N r, r,−1 = diag ,..., ρr,1 ρr,1 (r,1 ) k1 (r,) × , ζr,1 ) ajk (ηr,1 , ζr,1 , µr,1 ) j,k=1 B1 ,1 (ηr, 1 γ r,1
(r, ) × Ik1 + H11 1 (ηr, , ζr,1 , µr,1 , ρr,1 ) 1 mk +1 m (r,) N , ζr,1 ) diag ρr,11 , . . . , ρm − (µr,1 ρr,11 )−1 B1 ,2 (ηr, r,1 1 (r,1 )
× H21
(3.34)
(ηr, , ζr,1 , µr,1 , ρr,1 ) K (r,1 ) (ηr, , ζr,1 , µr,1 , ρr,1 ) dζr,1 , 1 1
+ where, referring to Proposition 3.11 for notation, γ r, denotes γr+ if 1 = r 1 (r,1 )
+ and γr, otherwise, ajk 1
(ηr, , ζr,1 , µr,1 ) denotes ajk (η , ζ, µ) if 1 = r 1
( ) and ajk1 (ζ1 , µ1 ) otherwise, notes Hjk (η , ζ, µ, ρ) if 1 = r
(r,1 )
(ηr, , ζr,1 , µr,1 , ρr,1 ) 1 (1 ) Hjk (η1 , ζ1 , µ1 , ρ1 ) otherwise. (r,)
and finally Hjk and
de-
(0, ξ , Dn ) Suppose that > 1 . Then by hypothesis the entries of B (r,1 ) (ξ , 0, λ) are bounded in modulus by a constant depending only on W , µr, ) ∈ Σr and µr, ∈ L1 for > r. Let us now show that this bound(ηr,r edness condition is always satisfied if the boundary conditions (1.2) are of Dirichlet type at x0 . Indeed, if this latter condition holds, then every entry (r,) (r,) of B1 ,1 (ηr, , ζr,1 ) is 0, while the only non-zero entries of B1 ,2 (ηr, , ζr,1 ) 1 1 are those lying in rows N − ν, ν = 0, . . . , N − N−1 − 1, and columns k − ν, ν = 0, . . . , k − k−1 − 1. Then recalling from Section 1 that we now have sj = tj = tj for j = 1, . . . , N , it follows from the foregoing results and (3.34) that the entry in the (N − ν)-th row and the (k − ν)-th column of (r,1 ) (ξ , 0, λ) is bounded in modulus by B(r,)(0, ξ , Dn )W tk
− 12 − t
C|λ|
k 1
†
− ν2 ( t1 − t 1 ) k
k 1
|λ|
tk t k 1
+ δr,1 r+1 ,
356
Denk and Faierman
IEOT
where 0 ≤ ν † ≤ tk −1 and the constant C depends only upon (ηr,r , µr,r ) ∈ Σr and µr, ∈ L1 . Our claim concerning Dirichlet boundary conditions are an immediate consequence of this last result. Note also from (3.34) that when A(x, D) and B(x, D) are essentially (r,) , ζr,1 ) (resp. every entry upper triangular at x0 , then every entry of B1 ,1 (ηr, 1 (r,1 ) (η , ζr,1 , µr,1 , ρr,1 )) is 0, and hence the entries of B(r,)(0, ξ , Dn ) of H 21
r,1
(r,1 ) (ξ , 0, λ) are all 0. Thus the boundedness condition also holds under W the cited conditions concerning A(x, D) and B(x, D). Let us now consider the case 1 > . Then it is a simple matter to deduce that for this case each entry of (3.34) is bounded in modulus by m +1 1 1 1 −m −1 m 1 + δ 1 , (3.35) C |λ| m r+1,1 r+1 + (1 − δ1 ,d )|λ| where the constant C depends only upon µr, ∈ L1 . Returning again to (3.32), let us next introduce the block row and column matrices d (1,2) (r,1 ) (ξ , 0, λ) Ir,j (ξ , λ) = B(r,j) (0, ξ , Dn )W , 1 =j+1
d (2,1) (r,j) (ξ , 0, λ) Ir,j (ξ , λ) = B(r,) (0, ξ , Dn )W
,
=j+1
respectively. Then a factorization in the sense of Schur gives 0 INj −N † r,j−1 Ir,j (ξ , λ) = (2,1) Ir,j (ξ , λ) INd −Nj 0 INj −N † r,j−1 × (2,1) (1,2) 0 Ir,j+1 (ξ , λ) − Ir,j (ξ , λ)Ir,j (ξ , λ) (1,2) INj −N † Ir,j (ξ , λ) r,j−1 . × 0 INd −Nj (2,1) Since we already know that each entry of Ir,j (ξ , λ) is bounded in mod ulus by a constant depending only upon (ηr,r , µr,r ) ∈ Σr , that each en(1,2) try of Ir,j (ξ , λ) is bounded in modulus by the quantity (3.35), and that Ir,d (ξ , λ) = I , an inductive argument involving Schur factoriza† Nd −Nr,d−1
tions shows that we may choose 0 sufficiently small and λ0 sufficiently large so that all the assertions of the proposition concerning Ir,r (ξ , λ) hold. This completes the proof of the proposition for the case under consideration. As a consequence of Proposition 3.14, we are now in a position to present the last result required for the proof of Proposition 3.6. To this end we require some further notation. Accordingly, for 0 ≤ r ≤ d and λ ∈ C \ {0} let (r) (r) (r) χλ ∈ C ∞ (Rn−1 ) such that 0 ≤ χλ (ξ ) ≤ 1, χλ = 1 for 0 ≤ |ξ | ≤ (r) 3 1/m 1 1 and vanishes for |ξ | ≥ 78 1 |λ|1/m if r = 0, χλ (ξ ) = 1 for 4 1 |λ| 1 1/m r r+1 r ≤ |ξ | ≤ 34 r+1 |λ|1/m and vanishes for |ξ | ≤ 18 r |λ|1/m and for 4 r |λ|
Vol. 66 (2010)
Estimates for Solutions
357
(r)
r+1 d |ξ | ≥ 78 r+1 |λ|1/m if 1 ≤ r < d, χλ (ξ ) = 1 for 14 d |λ|1/m ≤ |ξ | < ∞ and (r) 1 d vanishes for |ξ | ≤ 8 d |λ|1/m if r = d, while χλ (ξ ) is a Fourier multiplier of type (p, p) whose norm is bounded by a constant not depending upon λ, r , and r+1 . We also let S(Rn−1 ) denote the Schwartz space of rapidly decreasing functions on Rn−1 and F the Fourier transform in Rn−1 (x → ξ ).
Proposition 3.15. Suppose that the boundary problem (1.1)–(1.2) is parameter-elliptic in L. Suppose that x0 ∈ Γ and that the boundary problem (1.1), (1.2) is rewritten in terms of the local coordinates at x0 . In addition, suppose that the operators A(x, D) and B(x, D) are compatible at x0 and for −1 g ∈ S(Rn−1 )N and 0 ≤ r ≤ d, let g (r) (x ) = (F ψr (ξ )F g)(x ). Then there exists a λ0 = λ0 (p) > 0 such that for λ ∈ L with |λ| ≥ λ0 , the boundary problem ˚ D)u(x) − λu(x) = 0 A(0, ˚ D)u(x) = g B(0, (r)
for xn > 0, , (r)
(x )
(r)
has a solution u(r) (x) = (u1 (x), . . . , uN (x))T ∈ priori estimate N
(r) (π (j))
1 |||uj |||tj ,p,R n ≤ C +
j=1
N j=1
(3.36)
at xn = 0,
N j=1
(3.37)
t
Wp j (Rn+ ) and the a
(r) (π (j)) 1 n−1 j − p ,p,R
|||gj |||−σ2
(3.38)
holds, where the constant C does not depend upon g and λ. Proof. We will only prove the proposition for the case 1 ≤ r < d; the remaining cases can be treated similarly. Accordingly, under Fourier transform with respect to x , the boundary problem (3.36), (3.37) is transformed into the boundary problem: (3.8), ˚ ξ , Dn )F u(ξ , xn ) = F g (r) (ξ ) B(0,
at xn = 0.
(3.39)
Then assuming that 0 and λ0 have been chosen so that the conclusions of Propositions 3.12 and 3.14 hold, it follows from these propositions that for a fixed λ ∈ L with |λ| ≥ λ0 , this boundary problem has a solution in the space of vector functions decaying exponentially at ∞, u (r) (ξ , xn , λ), which 1 7 1/m r 1/m r+1 is unique for 8 r |λ| ≤ |ξ | ≤ 8 r+1 |λ| and is given in block row matrix form by u (r) (ξ , xn , λ) =
d
−tN 1 diag(ρ−t r, , . . . , ρr, )
=r
×
+ γr,
eiρr, xn ζr, G (r,) (η r , ζr, , µr , ρr )dζr,
−σN 1 × diag ρ−σ , . . . , ρNd d g (r) (ξ ), 1
(3.40)
358
Denk and Faierman
IEOT
where η r = (ηr,r , . . . , ηr,d ), µr = (µr,r , . . . , µr,d ), ρr = (ρr,r , . . . , ρr,d ),
G
(r,)
(η r , ζr, , µr , ρr )
=
k (r,) (r,) , ζr, , µr, ) j,k=1 (Ik + H11 (·)K (r,) (·) aj,k (ηr, −λ−1 H21
(r,)
(·)K (r,) (·)
() × Ir,r (η r , µr , ρr )−1 , () † Ir,r (η r , µr , ρr )−1 denotes the matrix whose rows are precisely rows Nr,−1 + −1 1 to Nr, of Ir,r (η , µ , ρ ) , and for brevity we have written (·) for the r
r
r
, ζr, , µr, , ρr, ). arguments (ηr,
Referring again to the texts preceding (3.15) and (3.23), respectively, for r for µ η ∈ Rn−1 | ( ηr , µ ) ∈ Σ ∈ L1 } and let ( η0 , µ0 ) notation, let Sr ( r ) = { denote an arbitrary point of Sr ( r ) × L1 . Then it follows from Definition 2.6 that there is a neighbourhood of ( η0 , µ0 ) in Rn−1 × C, say U ( η0 , µ0 ), such η0 , µ0 ) ∩ (Sr ( r ) × L1 ), ( η ,µ ) ∈ Σr and all assertions that for ( η , µ) ∈ U ( r /2 concerning (3.15)–(3.18) hold with µ = (1 − | η |2 )m µ, ξ = ρr,r η , z = ρr,r ζ m r , while for r < ≤ d, all assertions concerning (3.23)–(3.26) and µ = ρr,r µ hold with µ = µ, z = ρr, ζ, and λ = ρr, µ. Since Sr ( r ) × L1 is compact, it is covered by a finite number of such opens sets, say {Uj }M j=1 , and we let {φj ( η , µ)}M denote a partition of unity subordinate to this covering of j=1 Sr ( r ) × L1 . r ( r ) = { | Let S η ∈ S ( ) η | < 1} and Σ(λ0 ) = {(ξ , λ) ∈ Rn−1 × r r 7 0 1 1/m r r+1 L |λ| ≥ λ , 8 r |λ| ≤ |ξ | ≤ 8 r+1 |λ|1/m }. Then we observe that each 0 ηr,r , µ) ∈ Uj ∩(Sr ( r )∩L1 ) pair (ξ , λ) ∈ Σ(λ ) uniquely determines the point ( = ρ−1 for some j, where ηr,r r,r ξ ,
7 8 r+1 |≤ | ηr,r , 7 r −2/m r+1 1/2 ( 8 r+1 )2 + |λ|2/m
µ =
λ |λ| ,
and ρr,r is defined as before. Conversely, the set of all such pairs M j , where (ξ , λ) is contained in j=1 U
r r /2 j = (ξ , λ) ∈ Rn−1 × L ξ = τ η , |λ| = τ m U (1 − | η |2 )m µ 1/m r λ0 for τ ≥ , and ( η , µ) ∈ Uj ∩ (Sr ( r ) × L1 ) . 2 m /2 (1 − | η| ) r An immediate consequence of these considerations is that if we henceforth (r,) j , agree to replace G (r,) (·) in (3.40) by Gj (·) whenever (ξ , λ) ∈ Σ(λ0 ) ∩ U (r,) (r) M (r,) and let G (·) = j=1 φj (ηr,r , µr,r+1 )Gj (·)χλ (ξ ), where for brevity we
Vol. 66 (2010)
Estimates for Solutions
359
have written (·) for (η r , ζr, , µr , ρr ), then we obtain a solution u(r) (ξ , xn , λ) =
d
−tN 1 diag(ρ−t r, , . . . , ρr, )
=r
×
+ γr,
(3.41)
−σN 1 ρ−σ eiρr, xn ζr, G(r,) (η r , ζr, µr , ρr )dζr, diag( , . . . , ρNd d )g (r) (ξ ) 1
of (3.8), (3.39) for all values of ξ ∈ Rn−1 and λ ∈ L1 with |λ| ≥ λ0 with the property that each entry of (r,) G(r,) (η r , ζr, , µr , ρr ) = Gjk (η r , ζr, , µr , ρr ) j=1,...,N k=1,...,Nd
has derivatives of all orders with respect to the components of ξ which are continuous in ξ and λ, and, in addition, for any multi-index α whose entries are either 0 or 1,
α
|ξ α D ξ Gj,k (η r , ζr, , µr , ρr )| ≤ C
(resp. |λ|−1 C) if j ≤ k (resp. j > k ),
where the constant C does not depend upon ξ , λ, and α . In light of (3.41) we see that the proposition will be proved for the case (r) (π1 (j)) under consideration here if we can show that |||uj,k, |||tj ,p,R n is bounded by + the expression on the right-hand side of (3.38) for every j, k, and satisfying 1 ≤ j ≤ N , 1 ≤ k ≤ Nd , and r ≤ ≤ d with π2 (k) = , where −tj (r) (r,) k (r) uj,k, (ξ , xn , λ) = ρr, eiρr, xn ζr, Gj,k (η r , ζr, , µr , ρr )dζr, ρ−σ r, gk (ξ ). + γr,
To this end let us observe that (r)
(π (j))
1 |||uj,k, |||tj ,p,R n ≤ C +
tj −1 m F ξ , λπ1 (j) ρ−σk −m m=0
×
γ+
(r,) (r) eixn ρζ ζ tj −m Gj,k (η r , ζ, µr , ρr )dζ F gk (ξ )
0,p,Rn +
,
+ , and where C where for brevity we have put ρ = ρr, , ζ = ζr, , γ + = γr, (r) denotes a constant not depending upon g and λ. Hence if we put Λ(ξ , λ) = ξ if = r, Λ(ξ , λ) = ξ , λ if > r, and −1 (r) h(x, λ) = F e−Λ(ξ ,λ)xn F gk (x) for x ∈ Rn+
(see [ADF, Proposition 2.3]), argue as in [V, Section 5], and appeal to Minkowski’s inequality, then we deduce that 2 ∞ ∞ p 1/p (r) (π1 (j)) |||uj,k, |||tj ,p,R dxn Jν (xn , τ, λ)dτ , (3.42) n ≤ C +
ν=1
xn =0
τ =0
360
Denk and Faierman
IEOT
where J1 (xn , τ, λ) = (xn + τ )−1 F
−1 −σk
ρ
F h(s , τ )J1 (xn , τ, ξ , λ)0,p,Rn−1 ,
J2 (xn , τ, λ) = (xn + τ )−1 F
−1 −σk −1
ρ
F Dτ h(s , τ )J2 (xn , τ, ξ , λ)0,p,Rn−1 ,
(3.43) (3.44)
J1 (xn , τ, ξ , λ) −m (r,) = ρζ(xn + τ )eiρζ(xn +τ ) ξ , λm Gjk (η r , ζ, µr , ρr )ζ tj −m dζ, π1 (j) ρ γ+
J2 (xn , τ, ξ , λ) −m (r,) = ρζ(xn + τ )eiρζ(xn +τ ) ξ , λm Gjk (η r , ζ, µr , ρr )ζ tj −1−m dζ, π1 (j) ρ γ+
and the constant C does not depend upon g (r) and λ. Let us firstly fix our attention upon the case where π2 (k) ≤ r, so that now we have = r in (3.42)–(3.44). Then with the use of the Mikhlin-Lizorkin multiplier theorem we can show that for 1 ≤ ν ≤ 2, ρ −σk −ν+1 (r) Jν (xn , τ, ξ , λ)χλ (ξ ) ξ is a Fourier multiplier of type (p, p) with norm bounded by a constant not depending upon g (r) , xn , τ , and λ. Hence it follows from (3.42)–(3.44) and [ADF, Proposition 2.3] that (r)
π (j)
(r)
(r) π (k) 1 n−1 , k − p ,p,R
2 |||uj,k, |||tj1,p,Rn ≤ Cr gk −σk − p1 ,p,Rn−1 ≤ Cr |||gk |||−σ +
where the constant Cr does not depend upon g (r) and λ. Turning next to the case where π2 (k) = > r, we can again show that for 1 ≤ ν ≤ 2 −σk −ν+1 ρ (r) Jν (xn , τ, ξ , λ)χλ (ξ ) ξ , λπ2 (k) is a Fourier multiplier of type (p, p) with norm bounded by a constant not depending upon g (r) , xn , τ , and λ. Thus it follows from (3.42)–(3.44) and [ADF, Proposition 2.3] that (r)
π (j)
(r) π (k) 1 n−1 , k − p ,p,R
2 |||uj,k, |||tj1,p,Rn ≤ C |||gk |||−σ +
and the constant C has the same properties as the constant C above; and this completes the proof of the proposition. Proof of Proposition 3.6. In proving the proposition there is no loss of gen erality in supposing that g ∈ S(Rn−1 )N since the result for general g ∈ 1
N −σj − p (Rn−1 ) will then follow from a standard approximation procej=1 Wp dure. Accordingly, under a Fourier transformation with respect to x (x → ξ ), the boundary problem (3.3), (3.4) is transformed into the boundary
Vol. 66 (2010)
Estimates for Solutions
361
problem (3.8), (3.39). Then assuming that 0 and λ0 have been chosen so that the conclusions of Propositions 3.12 and 3.14 hold, it follows from [DV, Theorem 3.5] and the above considerations that for a fixed λ ∈ L with |λ| ≥ λ0 this boundary problem has a unique solution u(ξ , xn , λ) in the space of vector functions decaying exponentially at ∞, and hence d u(ξ , xn , λ) = r=0 u(r) (ξ , xn , λ). The assertions of the proposition follow directly from the results of Proposition 3.15 and the Mikhlin-Lizorkin multiplier theorem.
4. Proof of Theorem 2.8 We suppose henceforth that the hypotheses of Theorem 2.8 hold. Moreover, by employing a standard extension procedure we can also assume that the jk n ajk α (x) and bα (x) of (2.1) are defined on all of R , are compactly supported, and satisfy the same smoothness assumptions as asserted in Assumption 2.2 for Ω and Γ, respectively. Lastly for x0 ∈ Ω and δ > 0, let Bδ (x0 ) denote the open ball in Rn with centre x0 and radius δ. Then in order to prove the theorem we require some further results. Proposition 4.1. For any > 0 and x0 ∈ Ω there exists a δ > 0 and a λ0 > 0 such that for λ ∈ L with |λ| ≥ λ0 N N N (π1 (j)) ˚jk (x0 , D) uk Ajk (x, D) − A ≤ ||uj ||tj ,p,Ω (4.1) j=1
k=1
for every u ∈
−sj ,p,Ω
N
tj j=1 Wp (Ω)
j=1 0
such that supp u ⊂ Bδ (x ) ∩ Ω.
Proof. Let us fix a j, 1 ≤ j ≤ N , and suppose firstly that sj > 0. Then we have two cases to consider: (1) x0 ∈ Ω and (2) x0 ∈ Γ. Let x0 ∈ Ω and suppose that Bδ (x0 ) ⊂ Ω . Then for 1 ≤ k ≤ N we have α jk 0 ˚jk (x0 , D) uk = Ajk (x, D) − A ajk α (x) − aα (x ) D uk |α|=sj +tk
+
α ajk α (x)D uk .
(4.2)
|α|<sj +tk
α jk 0 jk α It is clear that for a fixed α, ajk α (x) − aα (x ) D uk resp. aα (x)D uk is a distribution on Rn of order sj (resp. ≤ sj ) whose support is contained in that of uk . Hence it follows from [GK, Section 1], [T, Theorem 2.6.1, p.198] that for |α| = sj + tk , α (π1 (j)) jk 0 ||| ajk α (x) − aα (x ) D uk |||−sj ,p,Ω (π (j))
jk 0 α 1 ≤ |||ajk α (x) − aα (x )D uk |||−sj ,p,Rn jk 0 α = sup aα (x) − ajk α (x ) D uk , ζ ζ
jk 0 = sup Dα−β uk , Dβ ajk α (x) − aα (x ) ζ , ζ
(4.3)
362
Denk and Faierman
IEOT
−s
where ·, · denotes the pairing between Hp j (Rn ) and its dual, the supre(π1 (j)) mum is taken over all ζ ∈ C0∞ (Rn ) satisfying |||ζ|||sj ,p ,Rn ≤ 1, and |β| = sj . Note that here duality between the spaces indicated is to be understood in the sense that the spaces are now equipped not with their standard norms, but with equivalent norms, namely their parameter dependent norms (π (j)) (π1 (j)) ||| · |||−s1j ,p,Rn and ||| · |||sj ,p ,Rn , respectively (see [GK, Eqn. (1.9)]). We conclude from (4.3) that the expression on the left side of this equation does not exceed uk tk ,p,Rn ρζsj ,p ,Rn + Cζsj −1,p ,Rn , where ρ = supx∈Bδ (x0 ) |ajk α (x) − jk 0 aα (x )|, and here and below C denotes a generic constant that does not depend upon uk , λ, and ζ.Thus it follows that the expression on the left side of (4.3) does not exceed
−m
C|λ|
1 π1 (j)
+ ρ uk tk ,p,Rn . On the other hand
if |α| < sj + tk , then similar arguments show that (π (j))
−m
α 1 |||ajk α (x)D uk |||−sj ,p,Ω ≤ C|λ|
1 π1 (j)
uk tk ,p,Rn .
Suppose next that x0 ∈ Γ and let {U, φ} be a chart on Γ such that x ∈ U, φ(x0 ) = 0, and φ is a diffeomorphism of class C κ0 −1,1 ∩ C s1 mapping U onto an open set in Rn with φ(U ∩ Ω) ⊂ Rn+ and φ(U ∩ Γ) ⊂ Rn−1 . We suppose henceforth that B2δ (x0 ) ⊂ U . Then for 1 ≤ k ≤ N we have the representation from which it follows that for a fixed α (4.2), jk jk 0 α α, ajk α (x) − aα (x ) D uk resp. aα (x)D uk is a distribution on Ω of order sj (resp. ≤ sj ) whose support is contained in that of uk . Further(π (j)) (π (j)) more, it is clear from the definition that |||v|||−s1j ,p,Ω ≥ |||v|||−s1j ,p,Ω∩U for both jk 0 α jk α (1) v = aα (x) − ajk α (x ) D uk and (2) v = aα (x)D uk . To obtain a reverse estimate, let us firstly fix our attention upon the v of case (1), choose −s (π (j)) (π (j)) w ∈ Hp j (Rn ) such that wΩ∩U = v and |||w|||−s1j ,p,Rn ≤ 2|||v|||−s1j ,p,Ω∩U , and let ψ ∈ C0∞ (Rn ) such that ψ(x) = 1 for x ∈ Bδ (x0 ) and supp ψ ⊂ U . The −s ψ w ∈ Hp j (Rn ), ψ wΩ = v, and hence 0
(π (j))
(π (j))
(π (j))
(π (j))
|||v|||−s1j ,p,Ω ≤ |||ψ w|||−s1j ,p,Rn ≤ C|||w|||−s1j ,p,Rn ≤ 2C|||v|||−s1j ,p,Ω∩U , where we have used an argument similar to that used in (4.3). Likewise we can show that (π (j))
(π (j))
1 1 α jk α |||ajk α (x)D uk |||−sj ,p,Ω ≤ C|||aα (x)D uk |||−sj ,p,Ω∩U .
By means of the mapping φ we pass to the local coordinates at can now jk 0 α x0 , and in these local coordinates ajk α (x) − aα (x ) D uk can be written as α jk α jk 0 aα (y) − φ∗ ajk ajk k α (x) − aα (x ) D uk = α (0) Dy u β + ajk k , α (y)Dy u 0<|β|<|α|
where y = φ(x). Furthermore, from the results given in [GK, Section 1], [T, Theorem 4.8.1, p.332] and the proofs of Lemma 4.2 and Theorems 4.7.2
Vol. 66 (2010)
Estimates for Solutions
363
and 4.8.2 of [T, pp.310-311 and 332-333] it is not difficult to deduce that α (π1 (j)) jk 0 ||| ajk α (x) − aα (x ) D uk |||−sj ,p,Ω∩U α (π1 (j)) hjk 0 ≤ C|||φ∗ ajk α (x) − aα (x ) D uk |||−sj ,p,φ(Ω∩U) α (π1 (j)) jk 0 ≤ C|||φ∗ ajk α (x) − aα (x ) D uk |||−sj ,p,Rn + (4.4) α jk jk ≤ C sup Dy u k , aα (y) − aα (0) ψζ ζ
sup Dyβ u k , ajk β (y)ψζ ,
+
0<|β|<|α|
ζ
where the constant C does not depend upon uk , ·, · denotes the pairing −s ˚ sj (Rn ), equipped with their paramebetween Hp j (Rn+ ) and its dual W + p (π (j))
(π (j))
1 ter dependent norms ||| · |||−s1j ,p,Rn and ||| · |||sj ,p ,Rn , respectively, the supre + ∞ n (π+1 (j))
mum is taken over the set ζ ∈ C0 (R+ )|||ζ|||sj ,p ,Rn ≤ 1 , ψ = φ∗ (ψ), +
and ψ ∈ C0∞ (Rn ) such that ψ(x) = 1 for x ∈ Bδ (x0 ), supp ψ ⊂ B2δ (x0 ). From this last result it is not difficult the expression on the to deduce that −
1
π (j) 1 left side of (4.4) does not exceed C ρ + C1 |λ| m uk tk ,p,Ω∩U , where jk jk 0 ρ = supx∈B2δ (x0 ) aα (x) − aα (x ) and the constant C1 does not depend
(π (j))
1 α upon uk and λ. Since a similar result holds for |||ajk α (x)D uk |||−sj ,p,Ω∩U for |α| < sj + tk , the proof of the proposition is complete for the case sj > 0; the proof for the case sj = 0 follows from a standard argument.
Arguments similar to those used in the proof of Proposition 4.1 can also be used to prove the following two results. Proposition 4.2. For any > 0 and x0 ∈ Ω there exits a δ, 0 < δ 0< jk 0 dist{x0 , Γ}, and a λ0 > 0 such that if supp ajk (x) − a (x ) ⊂ Bδ (x ) α α 0 for |α| = sj + tk , j, k = 1, . . . , N , and λ ∈ L with |λ| ≥ λ , then the estimate N N N ˚jk (x, D) − A ˚jk (x0 , D) uk |||(π1 (j)) n ≤ ||| A uk tk ,p,Rn −sj ,p,R j=1 k=1
holds for every u ∈
j=1
N
tj n j=1 Wp (R ).
We wish now to consider the analogue of Proposition 4.2 for the case x0 ∈ Γ. Accordingly, let us turn again to the chart {U, φ} introduced in the proof of Proposition 4.1. Then by means of the mapping φ we can now pass to local coordinates at x0 , and in the local coordinates each of the operators α jk (y, D) = Ajk (x, D) can be written as A ajk α (y)Dy for y ∈ φ(U ). |α|≤sj +tk We are interested in the case where for some ball Bδ (0) ⊂ φ(U ), ajk α (y) = jk aα (0) for y ∈ φ(U ) \ Bδ (0) and for all α satisfying |α| = sj + tk . Because o
jk (y, Dy ), the principal of this assumption we can now extend the operator A
364
Denk and Faierman
IEOT
jk (y, Dy ), to an operator on all of Rn by putting part of A ajk ajk α (y) = α (0) n for y ∈ R \ φ(U ) and |α| = sj + tk . Proposition 4.3. For any > 0 there exists a δ, 0 < δ < dist{0, ∂φ(U )}, ajk ajk and a λ0 > 0 such that if supp α (y) − α (0) ⊂ Bδ (0) for |α| = sj + tk , 0 j, k = 1, . . . , N and λ ∈ L with |λ| ≥ λ , then the estimate N N N o o Dy ) − A(0, Dy ) uk |||(π1 (j)) n ≤ ||| A(y, uk tk ,p,Rn+ −sj ,p,R +
j=1 k=1
holds for every u ∈
N
j=1
tj n j=1 Wp (R+ ).
Finally, the proof of the theorem follows directly from the results of Section 3, Propositions 4.1 - 4.3, and the arguments used in the proofs of Theorems 4.1 and 5.1 of [AV].
References [ADN]
[ADF]
[AV]
[DFM]
[DMV] [DV]
[F] [Gr] [G] [GK]
[K1]
S. Agmon, R. Douglis, and L. Nirenberg: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17 (1964), 35–92. M. S. Agranovich, R. Denk, and M. Faierman: Weakly smooth nonselfadjoint elliptic boundary problems. In: Advances in partial differential equations: Spectral theory, microlocal analysis, singular manifolds, Math. Top. 14 (1997), 138–199. M. S. Agranovich and M.I. Vishik: Elliptic problems with a parameter and parabolic problems of general form. Russ. Math. Surveys 19 (1964), 53–157. R. Denk, M. Faierman, and M. M¨ oller: An elliptic boundary problem for a system involving a discontinuous weight. Manuscripta Math. 108 (2001), 289–317. R. Denk, R. Mennicken, and L. Volevich: The Newton polygon and elliptic problems with parameter. Math. Nachr. 192 (1998), 125–157. R. Denk and L. Volevich: Elliptic boundary value problems with large parameter for mixed order systems. Amer. Math. Soc. Transl.(2) 206 (2002), 29–64. M. Faierman: Eigenvalue asymptotics for a boundary problem involving an elliptic system. Math. Nachr. 279 (2006), 1159–1184. P. Grisvard: Elliptic problems in nonsmooth domains.Pitman, London, 1992. G. Grubb: Functional calculus of pseudodifferential boundary problems 2nd edn.. Birkh¨ auser, Boston, 1996. G. Grubb and N.J. Kokholm: A global calculus of parameter-dependent pseudodifferential boundary problems in Lp Sobolev spaces. Acta Math. 171 (1993), 1–100. A. N. Kozhevnikov: Spectral problems for pseudodifferential systems elliptic in the Douglis-Nirenberg sense, and their applications. Math. USSR Sobornik 21 (1973), 63–90.
Vol. 66 (2010) [K2] [K3] [T] [VL]
[V]
Estimates for Solutions
365
A. N. Kozhevnikov: Parameter-ellipticity for mixed order boundary problems. C. R. Acad. Sci. Paris S´ er. I Math. 324 (1997), 1361–1366. A. N. Kozhevnikov: Parameter-ellipticity for mixed order systems in the sense of Petrovskii. Commun. Appl. Anal. 5 (2001), 277–291. H. Triebel: Interpolation theory, function spaces, differential operators. North-Holland, Amsterdam, 1978. M. I. Vishik and L. A. Lyusternik: Regular degeneration and boundary layer for linear differential equations with small parameter. Amer. Math. Soc. Transl.(2) 20 (1962), 239–264. L. R. Volevich: Solvability of boundary value problems for general elliptic systems. Amer. Math. Soc. Transl.(2), 67 (1968), 182–225.
R. Denk Department of Mathematics and Statistics University of Konstanz D-78457 Konstanz Germany e-mail: [email protected] M. Faierman School of Mathematics and Statistics The University of New South Wales UNSW Sydney NSW 2052 Australia e-mail: [email protected] Submitted: September 17, 2009. Revised: January 8, 2010.
Integr. Equ. Oper. Theory 66 (2010), 367–395 DOI 10.1007/s00020-010-1756-0 Published online March 10, 2010 © Birkhäuser / Springer Basel AG 2010
Integral Equations and Operator Theory
Products of Toeplitz Operators on a Vector Valued Bergman Space Robert Kerr Abstract. We give a necessary and a sufficient condition for the boundedness of the Toeplitz product TF TG∗ on the vector valued Bergman space L2a (Cn ), where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [13]. We also characterize boundedness and invertibility of Toeplitz products TF TG∗ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoff for the scalar valued Bergman space [17]. Mathematics Subject Classification (2010). Primary 47B35. Keywords. Bergman space, Vector valued functions, Toeplitz operator.
1. Introduction 1.1. Notation 1 For a measurable function f : D → Cn with ( D f (z)pCn dA(z)) p < ∞, we say that f ∈ Lp (D, Cn ). The vector-valued Bergman space Lpa (D, Cn ) is the intersection of Lp (D, Cn ), with the analytic Cn - valued functions on D with the usual identification of functions which only differ on sets of measure 1 0. The norm is given by f Lpa(Cn ) = ( D f (z)pCn dA(z)) p , where dA is normalized Lebesgue measure on the unit disk D. In the case p = 2 this the inner product given by f, g = space becomes a Hilbert pspace with 2 f (z), g(z) dA(z). L and L are Banach spaces for 1 ≤ p < ∞. For n a C D details see for example [2]. On the scalar valued Bergman space L2a , the Toeplitz operator with symbol f ∈ L2 is the densely defined operator Tf v = P (f v), where P is the orthogonal projection from L2 into L2a and v is a polynomial. The Toeplitz This work was completed with the support of the EPSRC. Part of this work was carried out while visiting the Fields Institute as part of the Thematic Program on New Trends in Harmonic Analysis.
368
Robert Kerr
IEOT
operator is a multiplication operator composed with an orthogonal projection. The Bergman projection is explicitly given by the following integral; P f (w) = f, Kw =
D
f (w) dA(w), (1 − zw)2
1 where Kw (z) = (1−zw) 2 is the reproducing kernel of the Bergman space 2 L2 (D). So using this explicit form we can define a Toeplitz operator on a dense subset of L2a , the polynomials, with symbol in L2 rather than L∞ . We can also see that with a symbol f ∈ L2 and v ∈ L2a , Tf v(w) is well defined point-wise for each w ∈ D. In [11] Sarason conjectured that a product of Toeplitz operators (defined densely in an appropriate way for analytic functions f and g) Tf Tg on 2 (w) was uniformly |2 (w)|g| the Hardy Space H 2 is bounded if and only if |f bounded on the disc, f(w) being the Poisson integral of f . This turned out to be false, [8], but a sufficient condition is found in [21]. Another conjecture by Sarason dealt with in [9, 10, 13, 14, 15, 16] and [17] was as to when the densely defined operator Tf Tg is bounded on L2a for f, g ∈ L2a ? The question was originaly posed by Sarason in [11] and a conjecture in section 8 of [13] more explicitly resembles Sarason’s Hardy space case conjecture. This time it is conjectured that the Toeplitz product Tf Tg is bounded for analytic f and g on the Bergman space L2a if and 2 (w) is uniformly bounded, where f is the Berezin transonly if |f |2 (w)|g| form of f . The question is investigated in various different cases, such as the weighted Bergman space with standard weights and the Bergman space on the unit ball and polydisk. These papers prove results that approximate to the Bergman space version of Sarason’s conjecture as stated in section 8 of [13]. The purpose of this paper is to investigate products of Toeplitz operators on a Bergman space of vector-valued functions. In the case of the vector valued Bergman space L2a (Cn ), we define the Toeplitz operator to be the densely defined composition of multiplication with a matrix valued function and the orthogonal projection from L2 (Cn ) into L2a (Cn ). So in this case the symbol F will be a matrix of L2 functions and TF v = P (F v), where v is a bounded analytic Cn valued function. If
⎛
f11
⎜ F =⎜ ⎝f21 .. .
f12 .. .
⎞ ... ⎟ ⎟ ⎠
and v = (v1 , v2 , . . . , vn ), where fij ∈ L2 and vi ∈ H ∞ , then
TF v = P (F v) = P
n i=1
f1i vi ,
n i=1
f2i vi , . . . ,
n i=1
fni vi
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
⎛
Tf11
⎜ =⎜ ⎝Tf21 .. .
Tf12 .. .
369
⎞ ... ⎟ ⎟ v, ⎠
where each Tfij is a densely defined Toeplitz operator on the scalar Bergman space L2a . When looking at products of these Toeplitz operators analagous to the treatment in [13] we have products of the form TF TG∗ , where F and G are square matrices of scalar valued Bergman space L2a functions. 1.2. Main Theorems The first two main theorems follow, one giving a sufficient condition for the Toeplitz product TF TG∗ to be bounded and the other a necessary condition. Both are conditions involving the Berezin transform; 2 Definition 1.1. The Berezin transform of a matrix A with L entries is the matrix-valued function B(A), where B(A)(w) = (A ◦ φw )(z)dA(z),w ∈ D, composition here being composition with each matrix entry. Here, φw is the w−z M¨ obius transform z → 1−wz . We should also note here that B(A)(w) = 2 2 (1−|w| ) A(z) |1−wz|4 dA(z) by a change of variables. Defining the normalized re-
producing kernel kw (z) to be
Kw (z) Kw ,
we obtain |kw (z)|2 =
(1−|w|2 )2 |1−wz|4 .
Here is our first main result: Theorem 1.2. If for some > 0 the trace of the matrix B((F ∗ F )
2+ 2
)(w)B((G∗ G)
2+ 2
)(w)
is uniformly bounded for all w ∈ D, then the Toeplitz product TF TG∗ is bounded L2a (Cn ) → L2a (Cn ). We also have the following condition: If there exists > 0 such that 1 2+ ∗ ∗ 2+ 2 2 2 (tr(G(z)F (x) F (x)G(z) )) |kw (z)| dA(z) |kw (x)| dA(x) D
D
is uniformly bounded, then the Toeplitz product TF TG∗ : L2a (Cn ) → L2a (Cn ) is bounded. Here is the necessary condition: Theorem 1.3. If the product of Toeplitz operators TF TG∗ is bounded, then the trace of the matrix B(F ∗ F )(w)B(G∗ G)(w) is uniformly bounded for w ∈ D. The next theorem is the other main result presented here, involving a characterization of bounded and invertible Toeplitz products. Theorem 1.4. The Toeplitz product TF TG∗ is bounded and invertible if and only if the trace of the matrix B(F ∗ F )(w)B(G∗ G)(w) is uniformly bounded and there exists η > 0 with (F G∗ GF ∗ )(z) > ηI for all z ∈ D. This last inequality is a matrix inequality.
370
Robert Kerr
IEOT
2. Bounded Toeplitz Products 2.1. A Sufficient Condition (Proof of Theorem 1.2) The technique in [13] for showing a sufficient condition on the boundedness of a Toeplitz product involves an inner product formula that easily generalizes to the vector valued case. So for g, f ∈ L2a (Cn ) f (z), g(z)Cn dA(z) = 3 (1 − |z|2 )2 f (z), g(z)Cn dA(z) f, gL2a (Cn ) = D D 1 (1 − |z|2 )2 f (z), g (z)Cn dA(z) + 2 D 1 + (1 − |z|2 )3 f (z), g (z)Cn dA(z). 3 D So to estimate the norm of TG TF∗ , we will look at the inner product TG TF∗ u, vL2 (Cn ) with v, u ∈ L2a (Cn ) in the form just given. a Let us start by estimating the term TF ∗ (u)(w), TG∗ (v)(w) Cn . Definition 2.1. For f, g ∈ L2 (D) , define the rank 1 operator f ⊗ g : L2 (D) → L2 (D) by (f ⊗ g)h = h, g f for h ∈ L2 (D). Also for F, G ∈ Mn×n (L2 (D)), define the operator F ⊗ G : L2 (D, Cn ) → L2 (D, Cn ) by ⎞ ⎛ ... i f1i ⊗ g2i i f1i ⊗ gni i f1i ⊗ g1i ⎜ ⎟ ... ⎟ ⎜ i f2i ⊗ g1i ⎟ ⎜ . . .. .. (F ⊗ G)h = ⎜ ⎟h ⎟ ⎜ ⎠ ⎝ f ⊗ g f ⊗ g . . . f ⊗ g ni 1i ni 2i ni ni i i i for h ∈ L2 (D, Cn ). Theorem 2.2. TF ∗ (u)(w), TG∗ (v)(w) Cn 1 = (Gkw ⊗ F kw )u(z), v(z)Cn dA(z), (1 − |w|2 )2 D where kw is the normalized reproducing kernel. Proof. TF ∗ (u)(w), TG∗ (v)(w)Cn ∗ ∗ = F (z)u(z)Kw (z)dA(z), G (ζ)v(ζ)Kw (ζ)dA(ζ) D
D
Cn
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
371
∗ G(ζ)Kw (ζ) (F (z)Kw (z)) u(z), v(ζ) Cn dA(z)dA(ζ) D D 1 ∗ = G(ζ)kw (ζ) (F (z)kw (z)) u(z), v(ζ) Cn dA(z)dA(ζ) 2 2 (1 − |w| ) D D 1 (Gkw ⊗ F kw u)(ζ), v(ζ)Cn dA(ζ). = (1 − |w|2 )2 D =
Lemma 2.3. (F ⊗ G)(F ⊗ G)∗ op ∼ tr{(F ⊗ G)(G ⊗ F )} n n n n
=
fqr , fql L2 gml , gmr L2 .
q=1 m=1 r=1 l=1
Proof. Firstly note that (F ⊗ G)∗ = G ⊗ F . As (F ⊗ G)(G ⊗ F ) is of finite rank the trace of (F ⊗ G)(G ⊗ F ) will be an equivalent norm. We can express F⊗ G as a matrix of operators on the scalar Bergman space with the entries n [ l=1 fil ⊗ gjl ]i,j . We can then express (F ⊗ G)(G ⊗ F ) in a similar manner;
n
n
m=1
=
fil ⊗ gml
l=1
n
m=1
n
gml ⊗ fjl
l=1
n n
i,j
·, fjl L2 gml , gmr L2 fir
r=1 l=1
. i,j
√ Noting that we have as an orthonormal basis el,m = (0, . . . , 0, z l l + 1, 0, . . .) i.e. a vector with each coordinate 0 apart from the mth entry which is the lth orthonormal basis element of the scalar valued Bergman space. So the trace of the operator (F ⊗ G)(G ⊗ F ) will be (F ⊗ G)(G ⊗ F )ep,q , ep,q p,q
=
∞ n z p 1 + p, fql q=1 p=1 m,r,l
L
gml , gmr L2 2
D
fqr (z)z p
1 + pdA(z).
√ We can write each fij as a power series s=1 as,ij z s 1 + s and thus this trace becomes
n n ∞ n n ap,ql gml , gmr L2 ap,qr ∞
q=1 p=1 m=1
r=1 l=1
and thus by Parseval’s identity the expression for the trace becomes;
n n n n fqr , fql L2 gml , gmr L2 . q=1 m=1
r=1 l=1 1
Theorem 2.4. Gkw ⊗ F kw op ≈ (tr(B(G∗ G)(w)B(F ∗ F )(w))) 2 .
372
Robert Kerr
IEOT
Proof. We can see that Gkw ⊗ F kw op is equivalent to the square root of the trace of the operator (Gkw ⊗ F kw )(Gkw ⊗ F kw )∗ and hence Lemma 2.3 implies that this is equal to n n
q=1 m=1
n n fqr , fql |kw |2 L2 gml , gmr |kw |2 L2
r=1 l=1 ∗
= tr(B(G G)(w)B(F ∗ F )(w)).
Definition 2.5. The operator P0 defined on Lp (D) is the operator that sends f (z) f ∈ L2 to the function given by (P0 f )(w) = D |1−wz| 2 dA(z). Elements from the following two theorems are borrowed from Theorem 3.2 in [13]. Lemma 2.6. If we have a scalar valued integrable function h and a scalar valued Bergman space function v then for each w ∈ D, 1 2+ 2 xh(x)|v(x)| 2+ |kw (x)| (P0 |v|δ )(w) |h(x)| dA(x) dA(x) ≤ 2 3 2 (1 − xw) 1 − |w| D D Here, for > 0, δ =
1 δ
.
2+ 1+ .
Proof. By H¨older’s inequality, |h(x)1 − xwv(x)| xh(x)|v(x)| dA(x) dA(x) ≤ 3 (1 − xw) |1 − xw|4 D D 1 2+ 1δ |h(x)|2+ |1 − xw|δ |v(x)|δ dA(x) dA(x) ≤ 4 |1 − xw|4 D |1 − xw| D " 1δ 1 ! 2+ 2 2 1+ δ |k (x)| |1 − |w| | |v(x)| w = |h(x)|2+ dA(x) , dA(x) 2 1 − |w|2 D D |1 − xw| |1 − xw| +1 and our result follows from the fact that (1 − |w|)(1 + |w|) (1 − |w|)(1 + |w|) 1 − |w|2 ≤ ≤ |1 − wz| |1 − wz| |1 − |wz (1 − |w|)(1 + |w|) ≤ ≤ 1 + |w| < 2. |1 − |w Let us now take a look at TF ∗ (u) (w), TG∗ (v) (w)Cn .
Vol. 66 (2010)
Toeplitz Products on L2a(Cn)
373
Theorem 2.7. Let w ∈ D. Then |TF ∗ (u) (w), TG∗ (v) (w)Cn | 1 2+ |kw (z)|2 |kw (x)|2 ∗ ∗ 2+ 2 (tr(G(z)F (x) F (x)G(z) )) dA(z) dA(x) ≤C 1 − |w|2 1 − |w|2 D D 1 1 × (P0 uδCn )(w) δ (P0 vδCn )(w) δ # 1 $ 2+ 2+ 2+ 1 ∗ ∗ tr(B((F F ) 2 )(w)B((G G) 2 )(w)) ≤C (1 − |w|2 )2 1 1 × (P0 uδCn )(w) δ (P0 vδCn )(w) δ where C is a constant, > 0 and
1 δ
=1−
1 2+ .
Proof. First note that for a function u ∈ L2a (D, Cn ) u, Kw = u (w),
so that | TF ∗ (u) (w), TG∗ (v) (w)Cn | (z), G∗ (x)v(x)K (x) = F ∗ (z)u(z)Kw dA(z)dA(x) w Cn Kw (z) Kw (x) dA(z)dA(x) . G(z)F ∗ (x)Cn op u(z)v(x) ≤ 1 − wz 1 − wx w (x) Then, using Lemma 2.6 with h(z) = G(z)F ∗ (x)Cn op v(x) K 1−wx , we arrive at the following inequality, | TF ∗ (u) (w), TG∗ (v) (w)Cn | 1 ! " 2+ Kw (x) 2+ |kw (z)|2 ∗ ≤ 2 dA(z) G(z)F (x)Cn op v(x) D 1 − wx 1 − |w|2 D 1 × (P0 uδ )(w) δ dA(x) 1 2+ 2 Kw (x) 2+ |kw (z)| ∗ dA(x) n (G(z)F (x)C op ) dA(z) v(x) = D 1 − |w|2 1 − wx D 1 δ δ × (P0 u )(w) , 1 . where > 0 and 1δ = 1 − 2+ Again, using Lemma 2.6, but this time with 1 2+ 2 2+ |kw (z)| ∗ h(x) = (G(z)F (x)Cn op ) dA(z) , 1 − |w|2 D
374
Robert Kerr
IEOT
we see that | TF ∗ (u) (w), TG∗ (v) (w)Cn | 1 2+ 2 |kw (x)|2 2+ |kw (z)| ∗ n (G(z)F (x)C op ) dA(z) dA(x) ≤ 4 D 1 − |w|2 1 − |w|2 D 1 1 δ δ δ δ (P0 u )(w) × (P0 v )(w) |kw (z)|2 ∗ ∗ 2+ 2 ≤4 (C tr(G(x)F (z) F (z)G(x) )) dA(z) 1 − |w|2 D D 1 1 1 |kw (x)|2 × dA(x) 2+ (P0 uδCn )(w) δ (P0 vδCn )(w) δ . 2 1 − |w| (This is what we want but we can go a step further and get something that looks even more similar to the analogous result in the scalar case.) Letting the 4 be absorbed into the constant C and using the inequality on matrix norms from [1], theorem IX.2.10 on page 258, we see that | TF ∗ (u) (w), TG∗ (v) (w)Cn | 2 2+ |kw (z)| ∗ ∗ 2 n ≤ (C(G(x)F (z) F (z)G(x) )C op ) dA(z) 1 − |w|2 D D 1 1 1 |kw (x)|2 δ δ δ δ 2+ (P (P dA(x) u v × n )(w) n )(w) 0 0 C C 1 − |w|2 2 2+ |kw (z)| 1 1 ∗ ∗ ∗ 2 2 2 ≤ (C((G G(x)) F F (z)(G G(x)) )Cn op ) dA(z) 1 − |w|2 D D 1 1 1 |kw (x)|2 δ δ δ δ 2+ × (P (P dA(x) u v n )(w) n )(w) 0 0 C C 1 − |w|2 2+ 2+ 2+ |kw (z)|2 ∗ ∗ ∗ 4 2 4 ≤ C(G G(x)) (F F (z)) (G G(x)) Cn op dA(z) 1 − |w|2 D D 1 1 1 |kw (x)|2 × dA(x) 2+ (P0 uδCn )(w) δ (P0 vδCn )(w) δ 2 1 − |w| 1 & % & & 2+ % % 2+ 2+ 1 ∗ ∗ ≤ C tr B (F F ) 2 (w)B (G G) 2 (w) (1 − |w|2 )2 1 1 × (P0 uδCn )(w) δ × (P0 vδCn )(w) δ where B is the Berezin transform and C is a constant that is possibly different from line to line. Now let us use the estimates from theorems 2.2 and 2.7 in the inner product formula. Taking our inner product formula
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
375
TF ∗ (u), TG∗ (v)L2 (Cn ) = a
D
TF ∗ (u), TG∗ (v)Cn dA(z)
(1 − |z|2 )2 TF ∗ (u), TG∗ (v)Cn dA(z) 1 + (1 − |z|2 )2 TF ∗ (u), TG ∗ (v)Cn dA(z) 2 D 1 + (1 − |z|2 )3 TF ∗ (u), TG ∗ (v)Cn dA(z), 3 D
=3
D
let us take the term 12 D (1 − |z|2 )2 TF ∗ (u), TG ∗ (v)Cn dA(z) and estimate its modulus; 1 (1 − |z|2 )2 TF ∗ (u), TG ∗ (v)Cn dA(z) 2 D % 1 % % & % & && 2+ 2+ 2+ 1 (1 − |w|2 )2 C tr B (F ∗ F ) 2 (w)B (G∗ G) 2 (w) ≤ 2 D (1 − |w|2 )2 1 1 × P0 uδCn (w) δ P0 vδCn (w) δ dA(w)| 1 % % % & % & && 2+ 2+ 2+ 1 ≤ sup C tr B (F ∗ F ) 2 (w)B (G∗ G) 2 (w) 2 w∈D 1 1 P0 uδCn (w) δ P0 vδCn (w) δ dA(w). × D
By Cauchy-Schwarz, this expression will be less than or equal to 1 % % % & % & && 2+ 2+ 2+ 1 sup C tr B (F ∗ F ) 2 (w)B (G∗ G) 2 (w) 2 w∈D 12 12 2 2 δ δ δ δ P0 uCn (w) dA(w) P0 vCn (w) dA(w) × .
D
D
Now, as the operator P0 is Lp bounded for p > 1, [6], this expression will be less than or equal to half the supremum over all w ∈ D of %
1 & % & && 2+ % % 2+ 2+ C tr B (F ∗ F ) 2 (w)B (G∗ G) 2 (w) )uL2 (Cn ) vL2 (Cn ) .
Estimating the term 1 (1 − |z|2 )3 TF ∗ (u), TG ∗ (v)Cn dA(z) 3 D from the inner product formula is similar. Finally, let us estimate 3 D (1 − |z|2 )2 TF ∗ (u), TG∗ (v)Cn dA(z). We can see from 2.2 that
376
Robert Kerr
IEOT
2 2 (1 − |z| ) TF ∗ (u), TG∗ (v)Cn dA(z) D (1 − |w|2 )2 Gk ⊗ F k u, v dA(z)dA(w) = n w w C D (1 − |w|2 )2 D ≤ Gkw ⊗ F kw op dA(w)uL2a Cn vL2a Cn D 1
≤ sup (tr B(G∗ G)(w)B(F ∗ F )(w)) 2 uL2aCn vL2a Cn . w∈D
Now we just use H¨older’s inequality to get an expression similar to the one in the previous estimate. tr(B(G∗ G)(w)B(F ∗ F )(w)) ∗ 2 = tr( G(x) G(x)|kw (x)| dA(x) F (z)∗ F (z)|kw (z)|2 dA(z)) D D ∗ 2 = tr(G(x) G(x)|kw (x)| F (z)∗ F (z)|kw (z)|2 )dA(x)dA(z) D D = tr {G(x)∗ G(x)F (z)∗ F (z)} |kw (z)|2 |kw (x)|2 dA(x)dA(z) D
D
≤
D
D
∗
∗
(tr {G(x) G(x)F (z) F (z)})
2+ 2
2 2+ |kw (z)| |kw (x)| dA(x)dA(z)
2
2
This is then less than or equal to % # $& 2+ 2+ C tr (G(x)∗ G(x)) 2 (F (z)∗ F (z)) 2 D
D
2 $ 2+ × |kw (z)|2 |kw (x)|2 dA(x)dA(z)
by Theorem IX.2.10 on page 258 of [1] and similar steps as before. This final expression is then equal to 2+ 2+ ∗ ∗ 2 2 ((G(x) G(x)) (F (z) F (z)) tr D
D
2 $ 2+ × |kw (z)|2 |kw (x)|2 dA(x)dA(z)) 2 # # $ # $ $ 2+ 2+ 2+ = tr(B (G(x)∗ G(x)) 2 (w)B (F (z)∗ F (z)) 2 (w)) .
Note that here we can use the same reasoning if 2+ 1 (C tr(G(z)F (x)∗ F (x)G(z)∗ )) 2 |kw (z)|2 dA(z) |kw (x)|2 dA(x) 2+ D
D
is uniformly bounded for some > 0; then our Toeplitz product TF TG∗ will be bounded. This condition is seemingly stronger and less aesthetic than the other one but it will be used later on when dealing with Toeplitz products that
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
377
are also invertible. Note that these last inequalities show that the sufficient condition is stronger than the necessary condition. 2.2. A Necessary Condition Proof of Theorem 1.3. In [9] (see also [13] for a different approach) Park shows that for functions f and g in the scalar Bergman space L2a the operator f ⊗ g defined by f ⊗ gh = h, g f with h ∈ L2a is equal to Tf Tg − 2Tz Tf Tg Tz + Tz2 Tf Tg Tz2 . Using this result in the vector valued case, we can see that ⎞ ⎛ f1i ⊗ g1i f1i ⊗ g2i . . . ⎟ ⎜ .. ⎟ . F ⊗G=⎜ ⎠ ⎝ f2i ⊗ g1i .. . = TF TG∗ − 2Tz TF TG∗ Tz + Tz2 TF TG∗ Tz2 . Let us estimate the norm of the operator (F ◦ φw ) ⊗ (G ◦ φw ), where F ◦ φw is the matrix-valued function ⎛ ⎞ f11 ◦ φw f12 ◦ φw . . . ⎜ ⎟ .. ⎜f21 ◦ φw ⎟. . ⎝ ⎠ .. . Noting that the operator (F ◦ φw ) ⊗ (G ◦ φw ) is of finite rank we can take as an equivalent norm the square root of the trace of the operator (F ◦ φw ⊗ G ◦ φw )(G ◦ φw ⊗ F ◦ φw ) by Lemma 2.3. By the same Lemma we can see that this will be equal to
n n n n fqr ◦ φw , fql ◦ φw L2 gml ◦ φw , gmr ◦ φw L2 q=1 m=1
=
r=1 l=1
n n
n n
q=1 m=1
r=1 l=1
B(fqr fql )(w)B(gml gmr )(w) ,
which is equal to the trace of the matrix B(F ∗ F )(w)B(G∗ G)(w). Let Uw be the unitary operator on our vector valued L2 space given by Uw f = (f ◦ φw )kw . It is well known that TF ◦φw Uw = Uw TF . Hence TF ◦φw = Uw TF Uw∗ , and thus 1
{tr (B(F ∗ F )(w)B(G∗ G)(w))} 2 = CF ◦ φw ⊗ G ◦ φw op = TF ◦φw TG∗ ◦φw − 2Tz TF ◦φw TG∗ ◦φw Tz + Tz2 TF ◦φw TG∗ ◦φw Tz2 op = Uw TF Uw∗ Uw TG∗ Uw∗ − 2Tz Uw TF Uw∗ Uw TG∗ Uw∗ Tz + Tz2 Uw TF Uw∗ Uw TG∗ Uw∗ Tz2 op = (Uw TF TG∗ Uw∗ − 2Tz Uw TF TG∗ Uw∗ Tz + Tz2 Uw TF TG∗ Uw∗ Tz2 )op = (Uw TF TG∗ Uw∗ − 2Uw Tφw TF TG∗ Tφ Uw∗ + Uw Tφ2w TF TG∗ Tφ2 Uw∗ )op . w
w
378
Robert Kerr
IEOT
We can now use the triangle inequality on the operator Uw (TF TG∗ − 2Tφw TF TG∗ Tφw + Tφ2w TF TG∗ Tφ2 )Uw∗ as in [13] to get our result, w using that Tφw ≤ 1. In the following we will be working with square matrices F and G with entries from the scalar valued Bergman space L2a (D). Where it is not explicitly stated otherwise, this will be the case. When we refer to a matrix being less than another matrix, F < G, we mean in the sense of L¨owner partial ordering of matrices. See [1], [19] and [20] for details on this.
3. Bounded and Invertible Toeplitz Products 3.1. A reverse H¨ older inequality We will now develop some of the theory needed to show a reverse H¨older inequality used to characterize the matrices of analytic functions, F and G, such that the Toeplitz product TF TG∗ is bounded and invertible on the vector valued Bergman space. Compare this next lemma with Lemma 4.3 in [17]. Lemma 3.1. If F (z) is invertible for all z ∈ D, then (F ∗ (w)F (w)) ≤ B(F ∗ F )(w), (F ∗ (z)F (z)) ≤ ηs B(F ∗ F )(w) and (F −1 (w)F ∗−1 (w)) ≤ B(F −1 F ∗−1 )(w) (F −1 (z)F −1∗ (z)) ≤ ηs B(F −1 F ∗−1 )(w), when z ∈ D(w, s) the pseudohyperbolic disk with radius 0 < s < 1 and centre w and ηs is a constant dependent only on s. Proof. Let e be an arbitrary vector. Then for F ∈ L2a (Cn ), F (u)e, F (u)e = F (z)Ku (z)dA(z)e, F (z)Ku (z)dA(z)e ' '2 ' ' ' = ' F (z)Ku (z)dA(z)e' ' n C 2 ∗ ≤ F e, F e dA(z)Ku L2 = F (z)F (z)dA(z)e, e Ku 2L2 . So if u ∈ D(0, s), then ∗ ∗ 2 F (u)F (u) ≤ F (z)F (z)dA(z)Ku 2 ≤ F ∗ (x)F (x)dA(z)
1 . (1 − s2 )2
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
379
If z ∈ D(w, s), then z = φw (u) for some u ∈ D(0, s). Thus F ∗ (z)F (z) = F ∗ (φw (u))F (φw (u)) ≤ F ∗ (φw (x))F (φw (x))dA(z) = B(F ∗ F )(w)
1 (1 − s2 )2
1 . (1 − s2 )2
Now let us show that F −1 (w)F ∗−1 (w) ≤ B(F −1 F ∗−1 )(w) : F ∗−1 (w)e, F ∗−1 (w)e = F ∗−1 (φw (0))e, F ∗−1 (φw (0))e −1 ∗−1 (φw (z))dA(z)e, e = F (φw (z))dA(z) F
and we arrive at the conclusion that F −1 (w)F ∗−1 (w) ≤ B(F −1 F ∗−1 )(w) in a similar manner as before. So for z ∈ D(w, s) we know that F −1 (w)F ∗−1 (w) ≤ B(F −1 F ∗−1 )(w) 1 and F ∗ (z)F (z) ≤ B(F ∗ F )(w) (1−s 2 )2 . The other inequalities follow from ap plying the same procedure to F −1 F ∗−1 instead of F ∗ F. Lemma 3.2. If there exists η such that F (z)G(z)∗ G(z)F (z)∗ > ηI for all z ∈ D and tr(B(G∗ G)(w)B(F ∗ F )(w)) is uniformly bounded on D, then 1
1
B(F −1 (F ∗ )−1 )(w)) 2 B(F ∗ F )(w) 2 is uniformly bounded on D. Proof. Let us suppose that F (w)G(w)∗ G(w)F (w)∗ > ηI for all w ∈ D. Then B(G∗ G)(w) ≥ G(w)∗ G(w) ≥ η(F (w)∗ F (w))−1 . The key inequality here is G(w)∗ G(w) ≥ η(F (w)∗ F (w))−1 , as this implies that B(G∗ G)(w) ≥ ηB((F ∗ F )−1 )(w) and so 1
1
(B(F ∗ F )(w)) 2 B(G∗ G)(w)(B(F ∗ F )(w)) 2 1
1
≥ η(B(F ∗ F )(w)) 2 B((F ∗ F )−1 (w))(B(F ∗ F )(w)) 2 . 1
1
Thus, as (B(F ∗ F )(w)) 2 B(G∗ G)(w)(B(F ∗ F )(w)) 2 < M for all w, tr(B(F −1 (F ∗ )−1 )(w)B(F ∗ F )(w)) 1 1 1 ≤ C(B(F ∗ F )(w)) 2 B(G∗ G)(w)(B(F ∗ F )(w)) 2 < ηCM η and so 1
1
B(F −1 (F ∗ )−1 )(w)) 2 B(F ∗ F )(w) 2 2 1
1
= (B(F −1 (F ∗ )−1 )(w)) 2 B(F ∗ F )(w)(B(F −1 (F ∗ )−1 )(w)) 2 ≤ C tr(B(F −1 (F ∗ )−1 )(w)B(F ∗ F )(w)) 1 1 1 ≤ C(B(F ∗ F )(w)) 2 B(G∗ G)(w)(B(F ∗ F )(w)) 2 , η where C and η are constants independent of w.
380
Robert Kerr
IEOT
Definition 3.3. A dyadic rectangle Qj,k,l is a subset of the unit disk of the form z = reiθ : (k − 1)2−j ≤ r ≤ k2−j , (l − 1)21−j π ≤ θ ≤ l21−j π , where j, k, l are non negative integers and k, l ≤ 2j .
Figure 1. Two nested dyadic rectangles in the unit disk. Lemma 3.4. There exists 0 < r < 1 such that for all dyadic rectangles Q with positive distance to the boundary Q ⊂ D(zQ , r). Here, D is the pseudohyperbolic disk and zQ is the centre of the dyadic rectangle Q.
Proof. This is just Proposition 4.4 in [17]. Compare this next lemma with Lemma 4.5 in [17]. Lemma 3.5. If 1
1
sup B(F −1 (F ∗ )−1 )(w)) 2 B(F ∗ F )(w) 2 < ∞, w∈D
then
sup
Q:dyadic
1 |Q|
12 12 1 −1 ∗−1 (F F )dA(z) (F F )dA(z) < ∞. |Q| Q Q
∗
Proof. If the dyadic rectangle Q is the whole disk, then as F ∗ F dA(z) = B(F ∗ F )(0) D
and
D
F −1 F ∗−1 dA(z) = B(F −1 F ∗−1 )(0),
Vol. 66 (2010) we see that
Toeplitz Products on L2a(Cn)
381
' 12 12 ' ' ' ' ' ∗ −1 ∗−1 F F dA(z) F F dA(z) ' ' ' D ' D 1
1
= B(F −1 (F ∗ )−1 )(0) 2 B(F ∗ F )(0) 2 . Now let us suppose that our dyadic rectangle Q has a positive distance from the boundary. By Lemma 3.4 our rectangle Q will be strictly contained in a pseudohyperbolic disk D(zQ , R), zQ being the centre of our dyadic rectangle and R being the same for each dyadic rectangle. Thus by Lemma 3.1 (F −1 (z)F −1∗ (z)) ≤ ηB(F −1 F ∗−1 )(zQ ) and (F ∗ (z)F (z)) ≤ ηB(F ∗ F )(zQ ) for all z in our pseudohyperbolic disk D(zQ , R). Here the constant η will only be dependent on R which is the same for all of these dyadic rectangles. Thus, using the fact that if A, B and C are positive matrices such that 1 1 1 1 1 1 1 1 A ≤ B implies C 2 AC 2 < C 2 BC 2 tr(C 2 AC 2 ) < tr(C 2 BC 2 ), we can deduce the following series of inequalities from our hypothesis: ' 12 12 ' ' 1 '2 1 ' ' ∗ −1 ∗−1 (F F )dA(z) (F F )dA(z) ' ' ' |Q| Q ' |Q| Q ' 1 ' 1 2 ' =' (F ∗ F )dA(z) ' |Q| Q 12 ' ' 1 1 ' −1 ∗−1 ∗ × (F F )dA(z) (F F )dA(z) ' ' |Q| Q |Q| Q
12 1 ≤ C tr (F ∗ F )dA(z) |Q| Q 12 1 1 ∗ −1 ∗ × (F F ) dA(z) (F F )dA(z) |Q| Q |Q| Q
12 1 ∗ (F F )dA(z) ≤ C tr |Q| Q 12 1 × ηB(F −1 F ∗−1 )(zQ ) (F ∗ F )dA(z) |Q| Q % 1 = C tr η(B(F −1 F ∗−1 )(zQ )) 2 1& 1 × (F ∗ F )dA(z) η(B(F −1 F ∗−1 )(zQ )) 2 |Q| Q % 1 1& ≤ C tr η(B(F −1 F ∗−1 )(zQ )) 2 {ηB(F ∗ F )(zQ )} (ηB(F −1 F ∗−1 )(zQ )) 2
382
Robert Kerr
IEOT
'% 1 ' ≤ C ' η(B(F −1 F ∗−1 )(zQ )) 2 {ηB(F ∗ F )(zQ )} η(B(F −1 F ∗−1 )(zQ )) ' ' 1 1 '2 ' ≤ C 2 η 4 ' (B(F −1 F ∗−1 )(zQ )) 2 {B(F ∗ F )(zQ )} 2 ' < M.
&' ' '
1 2
Note that C is a constant that possibly changes from line to line and is dependent on the dimension of Cn only. M will be dependent only on the uniform bound of B(F −1 F ∗−1 )(w)B(F ∗ F )(w), the dimension we are working in and the constant R which is the same for each dyadic rectangle not touching the boundary. What happens when we have a dyadic rectangle that touches the boundary but is not the whole disk? We can see that the centre of the rectangle zQ is at a distance of at least 1/2 from the centre, i.e. |zQ | ≥ 1/2. Then F ∗ (z)F (z)|kzQ |2 (z)dA(z) B(F ∗ F )(zQ ) = D c ∗ 2 ≥ F (z)F (z)|kzQ (z)| dA(z) ≥ F ∗ (z)F (z)dA(z) (1 − |zQ |)2 Q Q by Lemma 4.2 in [17]. We can also see in this case that |Q| = 8|zQ |(1 − |zQ |)2 and so 4c B(F ∗ F )(zQ ) ≥ F ∗ (z)F (z)dA(z). |Q| Q We can do the same for F −1 F ∗−1 to get that 4c −1 ∗−1 )(zQ ) ≥ F −1 (z)F ∗−1 (z)dA(z). B(F F |Q| Q We can then combine these and take the trace to see that
12 12 1 ∗ ∗ −1 ∗ tr F F (z)dA(z) (F F ) (z)dA(z) F F (z)dA(z) |Q|2 Q Q Q % 1 1 F ∗ (z)F (z)dA(z) ≤ 4c tr B(F −1 F ∗−1 )(zQ ) 2 |Q| Q 1& × B(F −1 F ∗−1 )(zQ ) 2 % 1 1& ≤ 16c2 tr B(F −1 F ∗−1 )(zQ ) 2 B(F ∗ F )(zQ ) B(F −1 F ∗−1 )(zQ ) 2 1 1 1 ≤ C B(F −1 F ∗−1 )(zQ ) 2 {B(F ∗ F )(zQ )} 2 ) 2 < M , where M is independent of Q.
If for all dyadic rectangles Q and some constant M ' 12 12 ' ' ' 1 1 ' ' ∗ −1 ∗−1 (F F )dA(z) (F F )dA(z) ' < M, ' ' ' |Q| Q |Q| Q we will say that F ∗ F has the matrix A2 condition. See [18] for a similar notion of matrix weights. We will now find a characterization of such functions F
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
383
in terms of the boundedness of certain averaging operators on the function space L2 (F ∗ F ). ∗ Theorem 3.6. If for F ∈ Mn×n (L2a ) the matrix F F has the A2 condition then 1 the averaging operators, f → χQ |Q| Q f (z)dA(z), are uniformly bounded on a dense subset L2 (Cn ) ∩ L2 (F ∗ F ), Q varying over all dyadic rectangles.
The proof here and of the next theorem follow the reasoning in Lemma 2.1 in [18]. Proof. Let R be the subspace 1 n . R = χQ 1 e : e ∈ C |Q| 2 We can see that the orthogonal projection from L2 (D, Cn ) onto R is given by 1 PQ : f → χQ f (z)dA(z). |Q| Q So we want to show that these projections are uniformly bounded with respect to the L2 (F ∗ F ) norm. Clearly, PQ f L2 (F ∗ F ) sup PQ L2 (F ∗ F ) = . f L2 (F ∗ F ) {f ∈L2 ∩L2 (F ∗ F ):f L2 (F ∗ F ) =0} If we let S denote the orthogonal complement of R in L2 , then f = f1 + f2 , where f1 ∈ R and f2 ∈ S = S ∩ L2 (F ∗ F ). Thus the expression for the norm of the projection will become f1 L2 (F ∗ F ) sup {f1 +f2 ∈L2 ∩L2 (F ∗ F ):f L2 (F ∗ F ) =0} f1 + f2 L2 (F ∗ F ) ⎧ ⎫ χQ 1 1 eL2 (F ∗ F ) ⎨ ⎬ |Q| 2 = sup 1 1 e + f2 L2 (F ∗ F ) ⎭ {e∈Cn : e=0} ⎩ inf {f2 ∈S} χQ |Q| 2 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ χQ 1 1 eL2 (F ∗ F ) ⎨ ⎬ |Q| 2 . = sup ⎪ {e∈Cn : e=0} ⎪ ⎪ ⎩ distL2 (F ∗ F ) χQ 1 1 e, S ⎪ ⎭ So let us take a look at distL2 (F ∗ F ) χQ distL2 (F ∗ F ) χQ =
sup
1
|Q| 2
1
e, S . 1
|Q| 2
1 1 1 ∗ 2 S e, (F F ) = distL2 (F ∗ F ) 2 χQ 1 |Q| 2 1 1 , (F ∗ F ) 2 χQ 1 e, h |Q| 2
1 e, S
% &⊥ 1 :h=1 h∈ (F ∗ F ) 2 S
|Q| 2
384
Robert Kerr
IEOT
(F ∗ F )−1 exists as we have the A2 condition. Note that % &⊥ % & 1 1 (F ∗ F ) 2 S = (F ∗ F )− 2 R .
Then we can see that 1 1 1 ∗ 2 distL2 (F ∗ F ) χQ =# % sup & (F F ) χQ 1 e, S 1 e, h $ 1 |Q| 2 |Q| 2 h∈ (F ∗ F )− 2 R :h=1 1 1 1 ∗ ∗ − 12 2 =! sup χQ (F F ) χQ 1 e, (F F ) 1 g " |Q| 2 |Q| 2 −1 1 g∈Cn :(F ∗ F )
2
χQ
1 |Q| 2
g≤1
1 1 1 ∗ ∗ − 12 2 = sup χQ (F F ) χQ 1 e, (F F ) 1 g 1 |Q| 2 |Q| 2 (F ∗ F )−1 g,g ≤1} {g∈Cn : |Q| Q 1 1 = sup e, χ g χ Q Q 1 1 1 |Q| 2 |Q| 2 (F ∗ F )−1 g,g ≤1} {g∈Cn : |Q| Q / − 12 0 1 −1 ∗−1 = sup F F χQ h e, |Q| {h∈Cn :h≤1} D ' − 12 ' ' ' 1 ' ' −1 ∗−1 F F e' . =' ' ' |Q| Q
Let us now put this equivalent expression for the distance back into our expression for the norm of the projection in L2 (F ∗ F );
PQ L2 (F ∗ F )→L2 (F ∗ F )
⎧ ⎪ ⎪ ⎪ ⎨
' ' 'χQ '
⎧ ⎪ ⎪ ⎪ ⎨
' ' 'χQ '
' ⎫ ' ⎪ ' ⎪ ⎪ 1 e ⎬ |Q| 2 'L2 (F ∗ F ) = sup ⎪ {e∈Cn :e=0} ⎪ 1 ⎪ ⎪ ⎪ ⎭ ⎩ distL2 (F ∗ F ) χQ 1 e, S ⎪ 1
' ' 1 ' e 1 |Q| 2 '
|Q| 2
2
∗
L (F F ) '# $− 12 ' 1 {e∈Cn :e=0} ⎪ ⎪ −1 ∗−1 ⎪ ⎩' ' |Q| Q F F ⎧ ' $ 12 ' '# 1 ' ⎪ ∗ ' ⎪ e' ⎨ ' |Q| Q F F ' '# = sup 1 $ − ' 1 2 {e∈Cn :e=0} ⎪ ⎪ −1 F ∗−1 ⎩' F ' |Q| Q
=
sup
⎫ ⎪ ⎪ ⎪ ⎬
' '⎪ ⎪ e' ⎭ '⎪ ⎫ ⎪ ⎪ ⎬ ' '⎪ ⎪ e' '⎭
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
385
'# $ 12 ' ' 1 ' ⎫ ⎪ ∗ ' e' ⎬ ' |Q| Q F F ' ⎪ '# ' = sup 1 $− 2 ' ⎪ ' 1 {e∈Cn :e=0} ⎪ ⎪ ⎪ −1 ∗−1 ⎩' e' '⎭ ' |Q| Q F F ' 12 12 ' ' 1 ' 1 ' ' ∗ −1 ∗−1 =' F F F F '. ' |Q| Q ' |Q| Q ⎧ ⎪ ⎪ ⎨
1 Lemma 3.7. If the averaging operators g → χQ |Q| g(z)dA(z) are uniformly Q 2 2 bounded on L (|f | ) over dyadic rectangles Q, then |f |2 has the scalar A2 condition. 1 Proof. Again we can see that the averaging operator g → χQ |Q| g(z)dA(z) Q is the projection P : L2 → χQ 1 1 C. We are working as before on the dense |Q| 2
subset L2 (C) ∩ L2 (|f |2 ). If we assume that |f1|2 is bounded then we can show as before that 1 2− 12 1 2− 12 1 1 1 1 1 distL2 (|f |2 ) (χQ χQ z = χQ . 1 z, S ) = D |f |2 D |f |2 |Q| |Q| |Q| 2 where |z| = 1. So if we drop this assumption on for > 0, then we can see that distL2 (|f |2 ) (χQ
1 1
|Q| 2
1 |f |2
but instead use
1 1 z, S ) |Q| 2 1 2− 12 1 1 = lim , →0 |Q| Q |f |2 +
z, S ) = lim distL2 (|f |2 +) (χQ →0
where S is the intersection of the orthogonal complement of χQ 2
1 |f |2 +
2
1
1
|Q| 2
C with
L (|f | ) and |z| = 1. As the norm of our bounded projection P is ' ' ⎫ ⎧ ' ' ⎪ ⎪ ' χQ 1 1 z ' ⎪ ⎪ ⎪ ⎪ ' |Q| 2 ' 2 2 ⎨ ⎬ L (|f | ) sup ⎪ z∈C:z=0,|z|=1 ⎪ ⎪ ⎪ 1 ⎪ ⎭ ⎩ distL2 (|f |2 ) χQ 1 z, S ⎪ |Q| 2
' ' ⎫ ' ' ⎪ ' χQ 1 1 ' ⎪ ⎪ ' |Q| 2 ' 2 2 ⎬ L (|f | ) , = sup ⎪ z∈C:z=0,|z|=1 ⎪ ⎪ ⎪ 1 ⎪ ⎭ ⎩ distL2 (|f |2 ) χQ 1 z, S ⎪ ⎧ ⎪ ⎪ ⎪ ⎨
|Q| 2
we know that distL2 (|f |2 ) χQ
1
1 z, S
|Q| 2
is nonzero for nonzero z and hence
1 2 1 1 <∞ lim 2 →0 |Q| Q |f | +
386
Robert Kerr
IEOT
and so by the Monotone Convergence Theorem 1 2 1 1 <∞ |Q| |f |2 Q
and
1 2− 12 1 1 1 distL2 (|f |2 ) χQ = , 1 z, S 2 |Q| |f | 2 |Q| Q
where |z| = 1. Thus
P L2 (|f |2 )
' ' 'χQ '
' ⎫ ' ⎪ ' ⎪ ⎪ 1 z ⎬ |Q| 2 'L2 (|f |2 ) = sup ⎪ z∈C:z=0,|z|=1 ⎪ ⎪ 1 ⎪ ⎪ ⎩ distL2 (|f |2 ) χQ 1 z, S ⎪ ⎭ ⎧ ⎪ ⎪ ⎪ ⎨
1
|Q| 2
3 4 1 ⎫ 1 2 2 ⎪ |f | ⎬ |Q| Q ⎪ 3 = sup 1 4 1 1 −2 ⎪ z∈C:z=0,|z|=1 ⎪ ⎪ ⎪ ⎩ |Q| Q |f |2 ⎭ !1 2 12 1 2 1 " 1 1 2 1 2 |f | = , |Q| Q |f |2 |Q| Q ⎧ ⎪ ⎪ ⎨
which is uniformly bounded as required.
Compare the next lemma with Lemma 3.6 in [18]. Lemma 3.8. If F ∗ F has the A2 condition, then tr(F ∗ F ) has the scalar A2 condition. Proof. We will show that each element on the diagonal of F ∗ F has the scalar A2 condition. We can then deduce that the sum of these will also have the A2 condition. Firstly we know that if F ∗ F has the A2 condition, then the 1 2 ∗ operators f → χQ |Q| Q f (z)dA(z) are uniformly bounded on L (F F ), f ∈ L2 (Cn ). So if we take g ∈ L2 (D) ∩ L2 (D F ∗ F (0, . . . , 1, . . . , 0), (0, . . . , 1, . . . , 0)), where F ∗ F (0, . . . , 1, . . . , 0), (0, . . . , 1, . . . , 0) is the scalar valued function z → F ∗ (z)F (z)(0, . . . , 1, . . . , 0), (0, . . . , 1, . . . , 0), then 1 g(z)(0, . . . , 1, . . . , 0)dA(z) g(0, . . . , 1, . . . , 0) → χQ |Q| Q is uniformly bounded between L2a (D, Cn ). This implies that 1 g(z)dA(z) g → χQ |Q| Q is uniformly bounded with respect to the scalar measure F ∗ F (0, . . . , 1, . . . , 0), (0, . . . , 1, . . . , 0)Cn ,
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
387
which will be whatever diagonal element of F ∗ F we want. Thus by the previous lemma the trace of F ∗ F will have the scalar A2 condition. Compare this next lemma with Lemma 4.6 in [17], Lemma 2.5 in [14] and also 1.7 on page 196 of [12]. Lemma 3.9. If a scalar valued function |f |2 has the A2 condition and for some 0 < δ < 1 then for each dyadic rectangle Q and E ⊂ Q such that |E| ≤ δ|Q| we have that µ(E) ≤ λµ(Q) for some 0 < λ < 1 where dµ = |f |2 dA and λ only depends on δ and the A2 constant of |f |2 . Proof. 2
!
"2 −1
|f f |
|Q/E| =
dA
! ≤
Q/E
" !
2
" −2
|f | dA Q/E
!
"
2
≤
|f |
|f | dA Q/E
!
"
−2
|f |
dA
Q
2
|f | dA C|Q|2
≤
dA
Q/E
Q/E
−1 |f |2 dA .
Q
By our A2 condition on |f |2 this equals −1 |f |2 dA − |f |2 dA C|Q|2 |f |2 dA Q E Q
=C
|f |2 dA
1− E
so we know that
−1
|Q|2 ,
Q
µ(E) |Q/E| ≤ C|Q| 1 − µ(Q) 2
and thus
2
|Q/E|2 µ(Q/E) . ≤C |Q|2 µ(Q)
Now we know that |Q/E| |Q|
|f |2 dA
|E| |Q|
≤ δ < 1 from our hypothesis, which implies that
≥ 1 − δ > 0. So we can now deduce that 0<
1 |Q/E|2 (1 − δ)2 µ(Q/E) ≤ . ≤ 2 C C |Q| µ(Q)
This lets us now see that µ(Q/E) + µ(E) µ(E) (1 − δ)2 µ(Q) = ≥ + 1= µ(Q) µ(Q) µ(Q) C and hence
µ(E) (1 − δ)2 ≤1− . µ(Q) C
388
Robert Kerr
IEOT
The following lemma will be crucial to our application of the A2 condition. Lemma 3.10. If F ∗ F has the A2 condition and J is a strictly positive matrix, then JF ∗ F J will have the A2 condition. The A2 constant of JF ∗ F J will depend on the A2 bound of F ∗ F and the dimension only. Proof. ' 12 12 ' '2 ' 1 1 ' ' ∗ ∗ −1 JF F J (JF F J) ' ' ' ' |I| I |I| I ' 1 12 ' ' ' 1 2 1 1 ' ' ∗ ∗ −1 ∗ =' JF F J (JF F J) JF F J ' ' ' |I| I |I| I |I| I
12 2 1 1 1 ≤ C tr JF ∗ F J (JF ∗ F J)−1 JF ∗ F J |I| I |I| I |I| I 1 1 (F ∗ F )−1 F ∗F = C tr |I| I |I| I 1 ' 1 12 ' ' 1 '2 2 1 ' ' ≤ C ' F ∗F (F ∗ F )−1 ' . ' |I| I ' |I| I C and C depend only on the dimension. Thus the result follows.
Definition 3.11. The dyadic maximal operator M∆ is defined by 1 |f (z)|dA(z), (M∆ f )(w) = sup w∈Q |Q| Q where the Q are dyadic rectangles and f ∈ L2 . Theorem 3.12 (The Calderon-Zygmund Decomposition Theorem). Let f ∈ L1 (D). If we have t > 0 such that the set Λ = {z ∈ D : M∆ f (z) > t} is not the whole of D, then we can decompose Λ into a disjoint union of dyadic intervals Qi such that t < |Q1i | Qi |f (z)|dA(z) < 8t. Proof. The proof of this is exactly as in [14] and [17].
Compare this next lemma with Proposition 4.11 in [17]. Lemma 3.13. The trace of F ∗ F satisies the following: 1. tr(F ∗ F ) ≤ M∆ tr(F ∗ F ) on D; 42 ∗ ∗ tr(F (z)F (z))dA(z) ≤ M∆ tr(F F )(0) ≤ tr(F ∗ (z)F (z))dA(z). 2. 3 D D Proof. 1. This follows from Proposition 4.11 in [17]. We just need to note that tr(F ∗ F ) is continuous and the proof works as it is.
Vol. 66 (2010)
Toeplitz Products on L2a(Cn)
389
2. D is a dyadic rectangle containing 0 so 1 M∆ tr(F ∗ F )(0) ≥ tr(F ∗ (z)F (z))dA(z) = tr(F ∗ (z)F (z))dA(z). |D| D D Let us take a dyadic rectangle Q containing 0 which is not the unit disk. We know that Q will be contained in the pseudohyperbolic disk D(0, 12 ). Let e ∈ Cn , then as F ∈ L2a (Cn ), 2 F (u)e, F (u)e = F (u)eCn = F (z)Ku (z)dA(z)e2Cn ≤
2 F (z)e|Ku (z)|dA(z)
F e, F e dA(z)Ku (z)2L2 ∗ = F (z)F (z)dA(z)e, e Ku (z)2L2 . ≤
So F ∗ (u)F (u) ≤ ≤ =
F ∗ (z)F (z)dA(z)Ku 22 F ∗ (x)F (x)dA(x)
1 (1 −
12 2 2 )
2 4 F ∗ (x)F (x)dA(x) 3
on each Q containing 0 which is not D. So
2 4 ∗ ∗ tr(F (u)F (u)) ≤ tr F (x)F (x)dA(x) 3 D for u ∈ Q. Hence 2 4 1 tr(F ∗ (z)F (z))dA(z) ≤ tr(F ∗ (x)F (x))dA(x) |Q| Q 3 D and so M∆ tr(F ∗ F )(0) ≤ (4/3)2
tr(F ∗ (z)F (z))dA(z).
D
The proof of the following theorem follows the lines of of Theorem 2.1 in [14] and Theorem 4.1 in [17]. It contains the key to the proof of Theorem 1.4 i.e. the reverse H¨ older property. Theorem 3.14. If F ∗ F satisfies A2 , then there exists > 0 such that (tr(F ∗ (z)F (z)))1+ dA(z) ≤ C (tr(F ∗ (z)F (z)))dA(z)1+ with C and dependent only on the A2 constant.
390
Robert Kerr
IEOT
Proof. For each k define ∗ 4k+1 ∗ Ek = z ∈ D : M∆ (tr(F F ))(z) > 2 (tr(F (z)F (z)))dA(z) . D
By Lemma 3.13 we can see that
M∆ tr(F ∗ (0)F (0)) ≤ (4/3)2 tr(F ∗ (z)F (z))dA(z) D 4k+1 <2 tr(F ∗ (z)F (z))dA(z) D
for all k. So we know that each Ek is not the whole disk (as 0 is not contained in it) and hence we can do a Calderon-Zygmund decomposition. So for each Ek we have a disjoint union of dyadic rectangles Qi whose union is equal to Ek and 1 4k+1 ∗ 2 (tr(F (z)F (z)))dA(z) < tr(F ∗ (z)F (z))dA(z) |Qi | Qi D 4(k+1) (tr(F ∗ (z)F (z)))dA(z). <2 D
Two inequalities we will use from this are; −1 |Qi | < 2−4k−1 (tr(F ∗ (z)F (z)))dA(z) D
and
tr(F ∗ (z)F (z))dA(z)
Qi
tr(F ∗ (z)F (z))dA(z) < |Qi |24(k+1) Qi
(tr(F ∗ (z)F (z)))dA(z).
D
We now take a maximal dyadic rectangle Q in Ek−1 (which is larger than Ek ) and note that |Ek ∩ Q| = |Qi | Qi ⊂Q
<
2
−4k−1
Qi ⊂Q
≤ 2−4k−1
−1 (tr(F (z)F (z)))dA(z) ∗
D
tr(F ∗ (z)F (z))dA(z)
Qi
−1 (tr(F ∗ (z)F (z)))dA(z) tr(F ∗ (z)F (z))dA(z),
D
Q
(where the Qi denote the maximal dyadic rectangles in Ek ) due to the dyadic decomposition of Ek . But as Q is also part of a Calderon-Zygmund decomposition (this time for Ek−1 ) we can also see that Q
tr(F ∗ (z)F (z))dA(z) < |Q|24k
(tr(F ∗ (z)F (z)))dA(z).
D
Putting the last two inequalities together we see that
Toeplitz Products on L2a(Cn)
Vol. 66 (2010) |Ek ∩ Q| <2
−4k−1
D
391
−1 4k (tr(F (z)F (z)))dA(z) |Q|2 (tr(F ∗ (z)F (z)))dA(z) ∗
D
1 = |Q|. 2 We are now in a position to use Lemma 3.9 as tr(F ∗ (z)F (z))) satisfies the scalar A2 condition and |Ek ∩ Q| ≤ 12 |Q|. So with 12 being our δ in 3.9, we can deduce that µ(Ek ∩ Q) < λµ(Q) for some 0 < λ < 1 independent of k, with dµ(z) = tr(F ∗ (z)F (z)))dA(z). We can now sum over all maximal dyadic rectangles in Ek−1 and see that µ(Ek ) = µ(Ek ∩ Q) < λ µ(Q) = λµ(Ek−1 ). Q
Q
Let us take a moment here to note that λ depends only on our A2 bound of tr(F ∗ (z)F (z)), (we can see this from Lemma 3.9), and that this A2 bound is controlled by the matrix A2 bound for F ∗ F and the dimension. We have established that for each k ≥ 1, µ(Ek ) < λµ(Ek−1 ) and so k k tr(F ∗ (z)F (z))dA(z) µ(Ek ) < λ µ(E0 ) = λ E0 k ≤λ tr(F ∗ (z)F (z))dA(z). D
Now let us move on and look at D tr(F ∗ (z)F (z))1+ dA(z) for some > 0. From Lemma 3.13 we know that tr(F ∗ F )(z) ≤ M∆ tr(F ∗ F )(z) on the disk, so tr(F ∗ (z)F (z))1+ dA(z) ≤ tr(F ∗ (z)F (z)) {M∆ tr(F ∗ F )(z)} dA(z) D D = tr(F ∗ (z)F (z)) {M∆ tr(F ∗ F )(z)} dA(z) x:M∆ tr(F ∗ F )(x)≤
+
≤
k
D
tr(F ∗ F (z))dA(z)
tr(F ∗ (z)F (z)) {M∆ tr(F ∗ F )(z)} dA(z)
Ek −Ek+1
1+ (4(k+1)+1) ∗ tr(F F )(z) + 2 tr(F F )(z)dA(z) µ(Ek ) ∗
D
k
D
1+ ∗ tr(F F )(z) ≤ D + 2(4(k+1)+1) tr(F ∗ F )(z)dA(z) λk tr(F ∗ F )(z)dA(z)
k
= D
D
D
1+ 1+ tr(F ∗ F )(z) + 2(4(k+1)+1) tr(F ∗ F )(z)dA(z) λk k
D
392
Robert Kerr
=
IEOT
1+ tr(F ∗ F )(z) (λ24 )k . 1 + 25
D
k
If we choose such that 0 < λ24 < 1, then this will become 1+ 1 tr(F ∗ F )(z) 1 + 25 . 1 − λ24 D older inequality will hold. Thus, for any 0 < ≤ , our reverse H¨
∗
Corollary 3.15. If F F satisfies A2 and J is a positive matrix then there exists > 0 such that (tr(JF ∗ (z)F (z)J))1+ dA(z) ≤ C (tr(JF ∗ (z)F (z)J))dA(z)1+ . The same and constant C hold for all positive matrices J and depends only on the dimension and the A2 constant of F ∗ F .
Proof. This follows from 3.14 and 3.10. 3.2. Proof of Theorem 1.3. Two easy lemmas follow before the proof of the Theorem 1.3.
Lemma 3.16. Let F and G be matrices consisting of Bergman space L2a (D) functions. If F G∗ GF ∗ > ηI and TF TG∗ is bounded, then the Toeplitz product TF TG∗ is invertible. Proof. F ∗ GG∗ F > ηI implies that G∗−1 F −1 F ∗−1 G−1 is bounded and so the operator TG∗−1 F −1 = TG∗−1 TF −1 is bounded. It remains to note that (TF TG∗ )TG∗−1 TF −1 F (kw , 0, 0, . . .) = F (kw , 0, 0, . . .) and TG∗−1 TF −1 (TF TG∗ )(kw , 0, 0, . . .) = (kw , 0, 0, . . .), and that these also hold for (0, . . . , kw , . . .). This implication holds because the linear spans of {F (0, . . . , kw , . . .)} and {(0, . . . , kw , . . .)} form dense subspaces. Lemma 3.17. If the trace of a positive matrix A is less than some constant λ > 0 then A < CI for some constant C > 0 depending only on λ and the dimension, I being the identity matrix.
Proof. Trivial. ∗
Proof of Theorem 1.3. “⇐” From Lemma 3.5 we know that F F satisfies our A2 condition. Then by Corollary 3.15, 1 1 (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))1+ dA(z) (3.1) 1 1 ≤ c (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))dA(z)1+ holds for all x ∈ D with some > 0 and a constant c independent of x. Note here that we need to use the fact that G∗ G is strictly positive.
Toeplitz Products on L2a(Cn)
Vol. 66 (2010)
393
We can also see that G∗ G satisfies our A2 condition, so a similar reverse H¨older inequality will hold; 1 1 &&1+ % % ∗ (F F )(z)dA(z) 2 (G∗ G)(x) (F ∗ F )(z)dA(z) 2 dA(x) tr &1+ % % 1 1& . ≤c tr (F ∗ F )(z)dA(z) 2 (G∗ G)(x) (F ∗ F )(z)dA(z) 2 dA(x) So let us set = min {, } . Thus integrating both sides of the reverse H¨older inequality (3.1) with respect to x, we get 1 1 (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))1+ dA(z)dA(x) $1+ # 1 1 ≤c (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))dA(z) dA(x) && % % 1+ 1 1 dA(x) =c tr ((G∗ G)(x)) 2 (F ∗ F )(z)dA(z)((G∗ G)(x)) 2 % % 61 5 6 1 &&1+ 5 ∗ (F F )(z)dA(z) 2 (G∗ G)(x) (F ∗ F )(z)dA(z) 2 dA(x) =c tr older and so as G∗ G also has the A2 condition, we can use our reverse H¨ inequality again to see that this last expression is less than or equal to # % $1+ 1 1& c tr (F ∗ F )(z)dA(z) 2 (G∗ G)(x) (F ∗ F )(z)dA(z) 2 dA(x) , where as usual c is a constant that possibly changes from line to line. By the M¨ obius invariance of the Berezin transform ([22] page 143) we see that 1 1 (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))1+ |kw (x)|2 |kw (z)|2 dA(z)dA(x) 1
1
≤ c(tr((B(G∗ G)(w)) 2 B(F ∗ F )(w)(B(G∗ G)(w)) 2 ))1+ < cM 1+ . Hence, by Theorem 1.2, we can see that the Toeplitz product TF TG∗ is bounded. The invertibility of this Toeplitz product follows from Lemma 3.16. “⇒” If TF TG∗ is bounded and invertible, we know from Theorem 1.3 that tr(B(F ∗ F )(w)B(G∗ G)(w)) is uniformly bounded and that TF TG∗ is bounded below. Thus, in particular, TF TG∗ kw e, TF TG∗ kw e dA(z) > η kw e, kw e dA(z) = η e, e for all vectors e ∈ Cn . We know that TF TG∗ kw = F (z)G∗ (w)kw (z) and so we deduce that G(w)B(F ∗ F )(w)G∗ (w) > ηI. From the fact that (TF TG∗ )∗ is also bounded below we can see that F (w)B(G∗ G)(w)F ∗ (w) > ηI. From these we deduce the following; B(G∗ G)(w) > ηF −1 (w)F ∗−1 (w) and B(F ∗ F )(w) > ηG−1 G∗−1 (w),
394
Robert Kerr
IEOT
which lets us see that 1 1 −1 ∗−1 G G (w) 2 B(G∗ G)(w) G−1 G∗−1 (w) 2 1 > η G−1 G∗−1 (w) 2 F −1 (w)F ∗−1 (w) G−1 G∗−1 (w)
1 2
and also 1
1
{B(G∗ G)(w)} 2 B(F ∗ F )(w) {B(G∗ G)(w)} 2 1
1
> η {B(G∗ G)(w)} 2 G−1 (w)G∗−1 (w) {B(G∗ G)(w)} 2 , thus tr(B(G∗ G)(w)B(F ∗ F )(w)) 1
1
= tr({B(G∗ G)(w)} 2 B(F ∗ F )(w) {B(G∗ G)(w)} 2 ) 1
1
> η tr({B(G∗ G)(w)} 2 G−1 G∗−1 {B(G∗ G)(w)} 2 )) 1 1 = η(tr( G−1 G∗−1 (w) 2 B(G∗ G)(w) G−1 G∗−1 (w) 2 )) 1 > η 2 (tr( G−1 G∗−1 (w) 2 F −1 (w)F ∗−1 (w) G−1 G∗−1 (w) ∗
1 2
)).
∗
Thus, as tr(B(G G)(w)B(F F )(w)) is uniformly bounded, tr(G∗−1 (w)F −1 (w)F ∗−1 (w)G−1 (w)) is uniformly bounded, by λ, say, and so G∗−1 (w)F −1 (w)F ∗−1 (w)G−1 (w) < λ I, which gives us that F (w)G∗ (w)G(w)F ∗ (w) > λ1 I. Acknowledgment I wish to thank S. Pott for suggesting this problem and for discussing various ideas relating to this work. I would also like to acknowledge the useful suggestions made by the referee.
References [1] R. Bhatia, Matrix analysis. Springer Verlag, New York, 1997. [2] O. Blasco, Introduction to vector valued Bergman spaces. Function spaces and operator theory 8 (2005), 9–30. [3] J. Duoandikoetxea, Fourier analysis. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29 American Mathematical Society. [4] P. Duren, A. Schuster, Bergman spaces. Mathematical Surveys and Monographs, 100. American Mathematical Society, 2004. [5] J. B. Garnett, Bounded analytic functions. Revised first edition. Graduate Texts in Mathematics, 236 Springer, New York, 2007. [6] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, 199. Springer-Verlag, New York, 2000. [7] J. Miao, Bounded Toeplitz products on the weighted Bergman spaces of the unit ball. J. Math. Anal. Appl. 346 (2008), no. 1, 305–313. [8] F. Nazarov, A counterexample to Sarason’s conjecture. Preprint
Vol. 66 (2010)
Toeplitz Products on L2a(Cn)
395
[9] J. Park, Bounded Toeplitz products on the Bergman space of the unit ball in Cn . Integral Equations Operator Theory 54 (2006), no. 4, 571–584. [10] S. Pott, E. Strouse, Products of Toeplitz operators on the Bergman spaces A2α . Algebra i Analiz 18 (2006), no. 1, 144–161. [11] D. Sarason, Products of Toeplitz operators, in ”Linear and Complex Analysis Problem Book 3,” Part I (V. P. Khavin and N. K. Nikol’skii, Eds.), Lecture Notes Math., 1573 318–319, Springer-Verlag, 1994. [12] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press. 1993. [13] K. Stroethoff, D. Zheng, Products of Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal. 169 (1999), 289–313. [14] K. Stroethoff, D. Zheng, Invertible Toeplitz Products. J. Funct. Anal. 195 (2002), 48–70. [15] K. Stroethoff, D. Zheng, Bounded Toeplitz products on the Bergman space of the polydisk. J. Math. Anal. Appl. 278 (2003), no. 1, 125–135. [16] K. Stroethoff, D. Zheng, Bounded Toeplitz products on Bergman spaces of the unit ball. J. Math. Anal. Appl. 325 (2007), no. 1, 114–129. [17] K. Stroethoff, D. Zheng, Bounded Toeplitz Products on Weighted Bergman Spaces. J. Operator Theory 59 (2008), no. 2, 277–308. [18] S. Treil, A. Volberg, Wavelets and the angle between past and future. J. Funct. Anal. 143 (1997), no. 2, 269–308. [19] X. Zhan, Matrix Inequalities. Lecture Notes in Mathematics. Springer, 2002. [20] F. Zhang, Matrix Theory: Basic Results and Techniques. Springer, 1999. [21] D. Zheng, The distribution function inequality and products of Toeplitz operators and Hankel operators. J. Funct. Anal. 138 (1996), no. 2, 477–501. [22] K. Zhu, Operator Theory in Function Spaces. Second edition. Mathematical Surveys and Monographs, 138 American Mathematical Society, 2007. Robert Kerr Department of Mathematics University of Glasgow University Gardens Glasgow G12 8QW UK e-mail: [email protected] Submitted: April 14, 2008. Revised: January 12, 2010.
Integr. Equ. Oper. Theory 66 (2010), 397–424 DOI 10.1007/s00020-010-1752-4 Published online February 16, 2010 © Birkhäuser/Springer Basel AG 2010
Integral Equations and Operator Theory
Abstract Hankel Operators in Kre˘ın Spaces S.A.M. Marcantognini and M.D. Mor´an Abstract. Hankel operators and their symbols, as generalized by V. Pt´ ak and P. Vrbov´ a, are considered in the Kre˘ın space setting. Under a generic assumption, without which the Krein space case may be untreatable, a necessary and sufficient condition for the existence of Hankel symbols for a given Hankel operator X is given. A parametric labeling of the Hankel symbols of X by means of Schur class functions is obtained. The proof is established by associating to the data of the problem an isometry V acting on a Kre˘ın space so that there is a bijective correspondence between the symbols of X and the minimal unitary Hilbert space extensions of V . The result includes uniqueness criteria and a Schur like formula. Mathematics Subject Classification (2010). Primary: 47B35; Secondary: 47B50, 47A20. Keywords. Hankel operators, Hankel symbols, Kre˘ın spaces.
1. Introduction The operators we deal with are the Kre˘ın space versions of the abstract Hankel operators and symbols introduced by V. Pt´ ak and P. Vrbov´ a in the Hilbert space setting [14, 15, 16]. Their starting point is to consider a pair of contractions T1 on H1 and T2 on H2 , H1 and H2 Hilbert spaces, with corresponding minimal isometric dilations U1 on K1 ⊇ H1 and U2 on K2 ⊇ H2 . With these data an abstract Hankel operator is a bounded linear map X from H1 to H2 satisfying the intertwining relation XT1∗ = T2 X. If K2 is the orthogonal projection form K2 onto H2 then the symbols of X PH 2 are the bounded linear operators Z from K1 to K2 which are solutions of the commutant lifting problem ZU1∗ = U2 Z
K2 and PH Z|H1 = X. 2
(1.1)
The classical Hankel operators and their symbols fit into the abstract scheme considered by Pt´ak and Vrbov´ a. Indeed, if S is the shift operator on the space of the L2 -functions on the unit circle of the complex plane, 2 H 2 is the Hardy space and H− is its orthogonal complement in L2 then, by
398
Marcantognini and Morán
IEOT
setting T1 := P S ∗ |H 2 and T2 := P− S|H−2 , with P and P− the orthogonal 2 projections form L2 onto H 2 and H− , respectively, we see that a Hankel 2 operator is a bounded linear map X : H 2 → H− such that XT1∗ = T2 X. The celebrated Nehari’s Theorem states that there exists an L∞ -function Φ, a symbol of X, such that Xf = P− Φf for all f ∈ H 2 . Hence the symbols are L∞ -functions with prescribed antianalytic part or, from another point of view, operators that commute with S and have the same fixed component 2 . Since the unitary operators U1 := S ∗ and U2 := S on L2 from H 2 into H− are the corresponding minimal isometric dilations of the contractions T1 and T2 , we can conclude that the symbols of the given Hankel operator X are those linear operators Z : L2 → L2 which are solutions of the commutant lifting problem (1.1) with the above defined data set. Unlike the classical case, in the general situation the commutant lifting problem (1.1) may have no solutions at all. The investigations carried out by Pt´ ak and Vrbov´ a indicate that the problem is solvable whenever X verifies certain boundedness condition that depends on the unitary parts of the WoldVon Neumann decompositions of U1 and U2 . More precisely, for j = 1, 2, let Kj = Rj ⊕Sj be the Wold-von Neumann decomposition of Kj , so that Rj and Sj reduce Uj , Uj |Rj is unitary and Uj |Sj is a unilateral shift with wandering K subspace Lj = Kj Uj Kj , and let PRjj be the orthogonal projection from Kj onto Rj . If X is an abstract Hankel operator for T1 and T2 then X is required to satisfy that there exists β ≥ 0 such that, for all h1 ∈ H1 and h2 ∈ H2 , K1 K2 h1 K1 PR h2 K 2 . |Xh1 , h2 H2 | ≤ β PR 1 2
(1.2)
This boundedness condition is necessary and sufficient for the existence of K2 intertwining liftings Z of X (ZU1∗ = U2 Z and PR Z = X) which altogether 2 verify Z = inf β, where β runs over all nonnegative numbers satisfying (1.2) (cfr. [14], [15], [16] and [5]). Since the Wold-Von Neumann decomposition is trivial in the classical case, for S being unitary, the result includes the classical situation. When carrying the notion of abstract Hankel operator over into the Kre˘ın space setting we must take into account that Kre˘ın space isometries present significant differences from their counterparts in the Hilbert space case. For instance, if U is a continuous isometry on a Kre˘ın space K, the subspace R on which U acts as a unitary operator need not satisfy the relation K = R ⊕ R⊥ . A sufficient condition for this situation not to occur is the uniform boundedness of the operators U, U 2 , U 3 , · · · . If this is the case then U has a Wold decomposition so that K can be written as the direct sum of two orthogonal subspaces R and S which reduce U such that U |R is unitary and U |S is a unilateral shift. In particular, if U is the minimal isometric dilation of a given continuous contraction T on a Kre˘ın space H, then U has a Wold decomposition when sup T n < ∞. Thus, subject to the assumption that sup T1n , sup T2n < ∞, the theory of Hankel operators on Kre˘ın spaces parallels the Hilbert space theory
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
399
by Pt´ ak and Vorbov´ a. In particular, the necessary and sufficient condition (1.2) finds to some extent its analogue in the Kre˘ın space setting. As counterpart of the Hilbert space case, we consider the problem of describing the symbols Z of any abstract Hankel operator X, for given continuous contractions T1 on a Kre˘ın space H1 and T2 on a Kre˘ın space H2 such that sup T1n , sup T2n < ∞. In dealing with the problem we show that there is a bijective correspondence between the symbols of X and the minimal unitary Hilbert space extensions of a Kre˘ın space isometry V determined by X, T1 and T2 . It turns out that the defect subspaces of the coupling isometry V underlying the data set {X, T1 , T2 } are Hilbert spaces. Since any isometry whose defect subspaces are Hilbert spaces has at least one minimal unitary Hilbert space extension, our approach provides a proof of the existence of symbols for the generalized Hankel operator on hand. The Kre˘ın space extension of the Arov-Grossman functional model yields a complete description of the abstractly indistinguishable minimal unitary Hilbert space extensions of V , as it associates to each minimal unitary Hilbert space extension U of V a function ϑU in a suitable Schur class of operator valued functions, and, to each function ϑ in the Schur class, a model operator Uϑ which gives rise to a minimal unitary Hilbert space extension of V, in such a way that the outlined correspondence is bijective. Then the coupling method combined with the Kre˘ın space extension of the Arov-Grossman model gives in return a bijective correspondence between the symbols of X and the Schur class. We show that the connection between the symbols and the Schur functions can be realized as a parametric description. We also include uniqueness criteria and a Schur like formula (see formula (4.4) in the proof of Theorem 4.1). In the Hilbert space case the methods were developed in [5]. The references given therein complement those listed in the bibliography. The paper is organized in five sections. Section 1, this section, serves as an introduction. In Section 2 we fix the notation and state some results needed in the rest of the paper. Our main results, along with some comments and remarks, are presented in Sections 3 and 4. We supplement this note with an Appendix containing a sketch of the proof of the Kre˘ın space extension of the Arov-Grossman model.
2. Notation and Preliminaries We follow the standard notation, so N, Z and C are, respectively, the set of natural, integral and complex numbers; D stands for the open unit disk and T for the unit circle, hence, D := {z ∈ C : |z| < 1} and T := ∂D. For any separable Hilbert space H (in the sequel all Hilbert spaces are supposed to be separable) we denote by L2 (H) the class of functions f : T → H which are measurable (strongly or weakly, which comes to be the same due to the separability of H) and such that 2π 1
f (eit ) 2H dt < ∞. 2π 0
400
Marcantognini and Morán
IEOT
With the pointwise linear operations and the scalar product 2π 1 f, gL2 (H) := f (eit ), g(eit )H dt (f, g ∈ L2 (H)) 2π 0 L2 (H) becomes a (separable) Hilbert space under the interpretation that two functions in L2 (H) are identical if they differ on a set of measure zero. Moreover, L2 (H) = n∈Z Gn (H) where, for each n ∈ Z, Gn (H) is the subspace of those functions f ∈ L2 (H) such that f (eit ) = eint x for some x ∈ H. The elements of H 2 (H) are all the analytic functions u : D → H whose Taylor coefficients are square summable, that is, if u ∼ {un }, in the sense n z un , z ∈ D and {un } ⊆ H, then that u(z) = ∞ n=0 ∞
un 2H < ∞.
n=0
We recall that H 2 (H) is a Hilbert space with the pointwise linear operations and the scalar product u, vH 2 (H) :=
∞
un , vn H
u, v ∈ H 2 (H), u ∼ {un }, v ∼ {vn } .
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 (H) into its radial limit provides an embedding of H 2 (H) into L2 (H) preserving the Hilbert space structures. Via the Poisson integral, ∞ it can be shown that the application maps H 2 (H) onto the subspace n=0 Gn (H) of L2 (H). Therefore we may consider that H 2 (H) and ∞ n=0 Gn (H) amount to the same Hilbert space. 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 supz∈D ϑ(z) ≤ 1. 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 ) −→ ϑ(eit )f (eit ) (f (eit ) ∈ L2 (N )) and a contraction operator from H 2 (N ) into H 2 (M) defined by u(z) −→ ϑ(z)u(z) (u(z) ∈ H 2 (N )). Due to the identification of H 2 (N ) (and H 2 (M)) with the subspace ∞ ∞ n=0 Gn (N ) (and n=0 Gn (M), respectively) the latter operator may be considered as a restriction of the former one. We denote both of them by ϑ. When N = M = H and ϑ(z) ≡ z (z times the identity operator on H) the associated operator is the (forward) shift S. Given ϑ ∈ S(N , M) we can likewise consider the positive square root ∆(eit ) of the positive selfadjoint operator 1 − ϑ(eit )∗ ϑ(eit ) (1 the identity operator on N ).
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
401
The basic reference for vector and operator valued analytic functions is [17]. We refer the reader to the detailed exposition given therein. Although familiarity with operator theory on Kre˘ın spaces is presumed, some background material is summarized in this section. We refer to [1], [4], [6], [9] and [13] as authoritative sources of information about indefinite inner product spaces and operators on them, and to [8] for the treatment of the Kre˘ın space extensions of the Hilbert space notions of defect and Julia operators, minimal isometric dilations and minimal unitary dilations. A Kre˘ın space is a linear space H equipped with an indefinite inner product (a hermitian sesquilinear form) ·, ·H such that there exist two linear subspaces H+ and H− with the following properties: 1) H is the algebraic direct sum of H+ and H− , 2) H+ , H− H = {0}, 3) (H+ , ·, ·H ) and (H− , −·, ·H ) are Hilbert spaces. The standard Hilbert space notation is carried over into the Kre˘ın space setting. So all algebraic and geometric notions are to be taken in their Kre˘ın space versions unless otherwise mentioned. In particular, in any Kre˘ın space (H, ·, ·H ) we use the notion of orthogonal direct sum in the same manner as for Hilbert spaces. A fundamental decomposition of a Kre˘ın space (H, ·, ·H ) is a representation of the form H = H+ ⊕ H− , where H+ and H− are subspaces as those in the above definition. A fundamental decomposition H = H+ ⊕ H− of a given Kre˘ın space (H, ·, ·H ) induces a Hilbert space inner product ·, ·|H| on H. Namely, if x, y ∈ H and x = x+ + x− , y = y+ + y− , then x, y|H| := x+ , y+ H − x− , y− H . In this situation the operator J defined on H by Jx := x+ − x− whenever x = x+ + x− is called a signature operator or fundamental symmetry for H. Clearly x, y|H| = Jx, yH and x, yH = Jx, y|H| for all x, y ∈ H. In general, a Kre˘ın space has infinitely many fundamental decompositions and therefore infinitely many associated Hilbert space inner products. However, the corresponding norms are equivalent. All topological notions on a Kre˘ın space are to be understood with respect to the norm topology associated with a fundamental decomposition. The symbol · |H| will be used to denote the quadratic norm associated with a fundamental decomposition of H. A vector x of a Kre˘ın space (H, ·, ·H ) is said to be negative if x, xH ≤ 0. A linear subspace E is negative if all its vectors are negative. A negative subspace E is said to be uniformly negative if, for a fixed norm · |H| , there exists a number β > 0 such that, for all x ∈ E, x, xH ≤ −β x 2|H| . A (uniformly) negative subspace E is said to be maximal (with respect to the respective property) if it is not a proper subspace of any subspace of the
402
Marcantognini and Morán
IEOT
same type. The analogous notions with “negative” replaced by “positive” are defined likewise. We point out that E is maximal (uniformly) negative if and only if E ⊥ is maximal (uniformly) positive. A regular subspace of a Kre˘ın space (H, ·, ·H ) is a closed subspace which is itself a Kre˘ın space in the inner product inherited from H. For a closed subspace E of H to be regular it is necessary and sufficient that H = E ⊕ E ⊥ . Therefore, E is regular if and only if E ⊥ is regular. Occasionally we write H E for E ⊥ if E is a regular subspace of H. Amongst the closed negative (positive) subspaces of H only those that are uniformly negative (positive) are regular. In particular, if E is a maximal uniformly negative subspace of H then E is regular and H E is a Hilbert subspace of H. Throughout this note, all Kre˘ın spaces are assumed to be complex and separable. If {Gι }ι∈I is a collection of linear subspaces of a Kre˘ın space H then ι∈I Gι is the smallest closed subspace of H containing all the subspaces Gι . If C, D are Kre˘ın spaces and E = C⊕ D, we shall write the elements of E c as sums c + d, pairs (c, d) or columns with no distinctness. d For the space of all everywhere defined continuous linear operators on the Kre˘ın space H to the Kre˘ın space K we keep the Hilbert space notation, so we denote it by L(H, K) and use L(H) instead of L(H, H). The space L(H, K) has the structure of a Banach space depending on choices of fundamental decompositions H = H+ ⊕ H− and K = K+ ⊕ K− and associated quadratic norms · |H| and · |K|. For any A ∈ L(H, K),
A =
sup x|H| =1
Ax |K| .
We call · an operator norm for L(H, K). Any two operator norms for L(H, K) are equivalent and, thus, define a unique uniform topology for L(H, K). The strong operator topology for L(H, K) is similarly defined. By 1 we indicate either the scalar unit or the identity operator, depending on context. For each A ∈ L(H, K) there is a unique A∗ ∈ L(K, H) such that Ax, yK = x, A∗ yH for all x ∈ H and y ∈ K. A projection P ∈ L(H) satisfies P = P 2 = P ∗ . The regular subspaces of H are those that are the ranges of projections. If E is a regular subspace of H we write PEH to indicate the orthogonal projection from H onto E.
∈ L(D,
H), By a defect operator for T ∈ L(H, K) we mean an operator D ∗
D
∗. where D is a Kre˘ın space, such that D has zero kernel and 1 − T T = D
The Kre˘ın space D is called a defect space for T . An operator T ∈ L(H, K) is a contraction if, for all x ∈ H, T x, T xK ≤ x, xH . Any defect space for a contraction T is a Hilbert space. The defect operators for a contraction T are abstractly indistinguishable in the sense that if
∈ L(D,
H) and D
∈ L(D
, H) are two defect operators for T then there D
=D
Φ.
→D
such that D exists a unitary operator Φ : D
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
403
We term isometry any linear operator V defined on a linear subspace D of a Kre˘ın space H to a Kre˘ın space K such that V x, V yK = x, yH for all x, y ∈ D. We call D the domain of V . Its range is R := V D. An isometry V is a weak isomorphism if D is dense in H and R is dense in K. An isometric isomorphism or unitary operator from H onto K is an isometry V such D = H and R = K. We remark that an isometry need not necessarily be continuous. An isometry with regular domain is continuous if and only if its range is regular. In particular, any unitary operator is continuous. As for a weak isomorphism, we point out that it can be extended to a unitary operator if either the domain or the range contains a maximal uniformly negative (positive) subspace. If V is an isometry acting on a Kre˘ın space H such that D and R are regular subspaces of H, the defect subspaces of V are N := H D and M := H R. Given a Kre˘ın space contraction T ∈ L(H), the 2 × 2 operator matrix T 0 H H UT := ∗ : →
D S H 2 (D) H 2 D) is a minimal isometric dilation of T . That is, UT is an isometry everywhere KT n 2
n defined on the Kre˘ın space ∞ KTn := H ⊕ H (D) such that T = PH UT |H , for all n ∈ N, and KT = n=0 UT H. If K is another Kre˘ın space containing H K n as regular subspace and U ∈ L(K) is an isometry such that T n = PH U |H , ∞ n for all n ∈ N, and K = n=0 U H, then the weak isomorphism defined by U n h :→ UTn h (h ∈ H, n ∈ N∪{0}) can be extended by continuity to a unitary operator τ ∈ L(K, KT ) satisfying τ |H = 1 and τ U = UT τ . Hence UT ∈ L(KT ) is the essentially unique minimal isometric dilation of T . In the above discussion a straightforward computation gives ⎤ ⎡ 0 Tn ⎦ , n ∈ N. k ∗ n−1−k UTn = ⎣n−1
T S D Sn k=0
We get that, for all n ∈ N and all h ∈ H, n−1 2
∗ T n−1−k h SkD k=0
=
H 2 (D)
n−1 k=0
=
∗ T n−1−k h 2
D D
n−1
D
∗ T n−1−k h, T n−1−k hH D
k=0
= h, hH − T n h, T n hH . Then we see that sup UTn < ∞ if and only if sup T n < ∞.
404
Marcantognini and Morán
IEOT
From [11] it follows that if sup T n < ∞ then limn→∞ UTn UT∗n exists (in the strong operator topology) and defines a projection with range ∞ RT := UTn KT . n=0 KT In particular, RT is a regular subspace of KT and PR = limn→∞ UTn UT∗n . T In this situation we get a Wold decomposition for UT so that RT reduces UT and UT |RT is unitary. KT Let ET be the closure of PR H. Since ET⊥ = (KT RT ) ⊕ (RT ∩ T
is regular, we get that ET is regular too. Besides, U ∗ ET ⊆ ET and H 2 (D)) T T := PEKTT UT |ET is a coisometric contraction on ET whose minimal isometric dilation is the unitary operator UT |RT . It can be seen that QT := (UT − T )ET is a Hilbert subspace of RT such that RT = ET ⊕ QT ⊕ UT QT ⊕ UT2 QT ⊕ · · · . Alternatively, PT := kernel (T ) is a Hilbert subspace of ET and RT = ET ⊕ UT PT ⊕ UT2 PT ⊕ UT3 PT ⊕ · · · . For more on Kre˘ın space isometries and their Wold decompositions we refer the reader to [10, 11, 12]. Let V be an isometry on a Kre˘ın space H whose defect subspaces N and M are Hilbert subspaces of H. Then there exist a Kre˘ın space F containing H as regular subspace and a unitary operator U ∈ L(F ) such that U |D = V , F H is a Hilbert space and F = n∈Z U n H. Under these conditions we say that U ∈ L(F ) is a minimal unitary Hilbert space extension of V . We identify two minimal unitary Hilbert space extensions of V , say U ∈ L(F ) and U ∈ L(F ), if there exists a unitary operator τ : F → F such that τ |H = 1 and τ U = U τ . Under the above identification, U(V ) is the set of the minimal unitary Hilbert space extensions of V . If V is an isometry on a Kre˘ın space H whose defect subspaces N and M are Hilbert subspaces of H, then with each minimal unitary Hilbert space extension U of V acting on F we may associate the L(N , M)-valued function ϑU (z) given by F ϑU (z) := PM U (1 − zPFF H U )−1 |N . It turns out that ϑU ∈ S(N , M). The S(N , M)-functions are relevant in the analysis of the minimal unitary Hilbert space extensions of V as they provide the parameters of the labeling of U(V ). The parametric description of U(V ) is the main tool in our treatment of the abstract Hankel operators in the Kre˘ın space setting. To conclude this section we state the result encompassing such a description. The model is a Kre˘ın space extension of the one given by D.Z. Arov and L.Z. Grossman in the Hilbert space framework [2, 3]. A more general result for continuous isometric operators with regular domain and range is given in [7].
Theorem 2.1. Let V be an isometry on a Kre˘ın space H with domain D, range R and defect subspaces N and M. Assume that N and M are Hilbert subspaces of H. Given ϑ ∈ S(N , M), set Eϑ := H 2 (M) ⊕ ∆L2 (N ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (N )}⊥ ,
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
405
where 1
∆(ζ) := (1 − ϑ(ζ)∗ ϑ(ζ)) 2 ,
|ζ| = 1.
Define Fϑ := H ⊕ Eϑ and Uϑ : Fϑ → Fϑ by ⎤ ⎡ ⎡ ⎤ H h + φ(0) V PDH h + ϑ(0)PN h ⎥ ⎢ H h) S ∗ (φ + ϑPN Uϑ ⎣ φ ⎦ := ⎣ ⎦ ψ ∗ H S (ψ + ∆P h)
φ h ∈ H, ∈ Eϑ ψ
N
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 Hilbert space extension of V such that Fϑ Uϑ (1 − zPEFϑϑ Uϑ )−1 |N = ϑ(z) PM
for all z ∈ D. (ii) For any minimal unitary Hilbert space extension U of V on F , the function F U (1 − zPFF H U )−1 |N ϑU (z) := PM
belongs to S(N , M). (iii) Two minimal unitary Hilbert space extensions of V , say U ∈ L(F ) and U ∈ L(F ), are identified under a unitary operator τ : F → F such that τ |H = 1 and τ U = U τ , if and only if
F F PM U (1 − zPFF H U )−1 |N = PM U (1 − zPFF H U )−1 |N
for all z ∈ D. Therefore, the map ϑ −→ 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). We prove the theorem in the Appendix.
3. Hankel symbols for a given Hankel operator Let H1 and H2 be two Kre˘ın spaces, and let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be two contractions with minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ), respectively. We recall from the Introduction that X ∈ L(H1 , H2 ) is a Hankel operator for T1 and T2 if and only if XT1∗ = T2 X, and that a Hankel K2 Z|H2 = X. symbol for X is any Z ∈ L(K1 , K2 ) such that ZU1∗ = U2 Z and PH 2 The next propostion gives a necessary and sufficient condition for a given Hankel operator to have Hankel symbols.
406
Marcantognini and Morán
IEOT
Proposition 3.1. Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be Kre˘ın space contractions with corresponding minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ), and let X ∈ L(H1 , H2 ) be a Hankel operator for T1 and T2 . Assume that sup T1n < ∞ and sup T2n < ∞. For j = 1, 2 consider the regular subspace Ej of Kj given by K
Ej := PRjj Hj where Rj :=
∞
Ujn Kj
n=0
is the regular subspace of Kj which reduces Uj such that Uj |Rj is unitary and K PRjj = limn→∞ Ujn Uj∗n . Then the following statements are equivalent: i) There exists a Z ∈ L(K1 , K2 ) such that Z|R1 is a contraction, ZU1∗ = U2 Z
and
K2 PH Z|H1 = X. 2
∈ L(E1 , E2 ) such that ii) There exists a contraction X K2
K1 |H1 . | )∗ XP X = (PR R1 2 H2
j ∈ L(D
j , Hj ) is a defect operator for Proof. In what follows, for j = 1, 2, D Tj , so that Tj 0 Hj Hj Uj := ∗ :
j )
j ) → H 2 D Dj S H 2 (D is the minimal isometric dilation of Tj and
j )). Kj Ej = (Kj Rj ) ⊕ (Rj ∩ H 2 (D i) ⇒ ii): Let Z ∈ L(K1 , K2 ) be given such that Z|R1 is a contraction, ZU1∗ = K2 Z|H1 = X. U2 Z and PH 2 Since U1 and U2 are isometries we also have that ZU1 = U2∗ Z. Furthermore, for all n ∈ N, Z = U2∗n U2n Z = U2∗n ZU1∗n = ZU1n U1∗n and Z = ZU1∗n U1n = U2n ZU1n = U2n U2∗n Z. Therefore, K1 K2 K2 K1 Z = ZPR = PR Z = PR ZPR . 1 2 2 1
Thus, for all h1 ∈ H1 and h2 ∈ H2 , Xh1 , h2 H2
K2 K1 = Zh1 , h2 K2 = PR ZPR h , h2 K2 2 1 1 K1 K2 K1 K2 = ZPR h , PR h = PEK22 ZPR h , PR h 1 1 2 2 K2 1 1 2 2 K2 K2 K1 = (PR | )∗ PEK22 ZPR h , h2 H2 . 2 H2 1 1
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
407
K1 |H1
:= P K2 Z|E1 then X
∈ L(E1 , E2 ), X = (P K2 |H2 )∗ XP Hence, if X E2 R2 R1 and, for all e1 ∈ E1 ,
1 K2 = P K2 Ze1 , P K2 Ze1 K2
1 , Xe Xe E2
E2
= Ze1 , Ze1 K2 − P K2
2) R2 ∩H 2 (D
Ze1 , P K2
2) R2 ∩H 2 (D
Ze1 K2
≤ Ze1 , Ze1 K2 ≤ e1 , e1 K1 ,
2 ) is a Hilbert subspace of K2 , and Z|R1 is contractive. since R2 ∩ H 2 (D
K1 |H1 .
∈ L(E1 , E2 ) be a contraction such that X= (P K2 |H2 )∗ XP ii) ⇒ i): Let X R2 R1 K
Set Tj := PEj j Uj |Ej for j = 1, 2. Then T1 ∈ L(E1 ) and T2 ∈ L(E2 ) are coisometric contractions whose minimal isometric dilations are the unitary
so
∗ = T X, operator U1 |R1 and U2 |R2 , respectively. It turns out that XT 1 2
∈ L(E1 , E2 ) is a Hankel operator for T and T . that X 1 2 In E1 × E2 define the hermitian sesquilinear form
1 , e2 K2 + e2 , Xe
1 K2 + e2 , e2 K2 . [(e1 , e2 ), (e1 , e2 )] := e1 , e1 K1 + Xe
∈ L(D
, E1 ) is a defect operator for X
(so that D
is a Hilbert If D X X X
=D
D
∗ ), set H := D
⊕ E2 with the standard inner
∗X space and 1 − X
X X X product (a, e2 ), (a , e2 )H = a, a D + e2 , e2 K2 . X
Then, for all (e1 , e2 ), (e1 , e2 ) ∈ E1 × E2 , [(e1 , e2 ), (e1 , e2 )]
1 , e K2 + e2 , Xe
K2 + e2 , e K2 = e1 , e1 K1 + Xe 2 1 2
∗ e1 , D
∗ e + Xe
1 + e2 , Xe
+ e K2 . = D 1 2
1 D X X X
Hence
∗ e1 , Xe
1 + e2 ) (e1 , e2 ) :−→ (D X
((e1 , e2 ) ∈ E1 × E2 )
gives an isometric map σ from (E1 × E2 , [·, ·]) into (H, ·, ·H ) such that σ(E1 × E2 ) is a dense subspace of H. Taking into account that Uj∗ |Rj is unitary and Uj∗ |Ej = Tj∗ , j = 1, 2,
we get that, for all
∗ = T X, and making use of the intertwining relation XT 1 2 (e1 , e2 ), (e1 , e2 ) ∈ E1 × E2 , σ(e1 , U2∗ e2 ), σ(e1 , U2∗ e2 )H = [(e1 , U2∗ e2 ), (e1 , U2∗ e2 )]
1 , U ∗ e K2 = e1 , e1 K1 + Xe 2 2
1 , K2 + U2∗ e2 , U2∗ e2 K2 + U2∗ e2 , Xe
1∗ e1 , e2 K2 = U1∗ e1 , U1∗ e1 K1 + XU
∗ e K2 + e2 , e K2 + e2 , XU 1 1 2 = σ(U1∗ e1 , e2 ), σ(U1∗ e1 , e2 )H . Hence, the operator V0 : σ(U1∗ E1 × E2 ) → σ(E1 × U2∗ E2 ) defined by V0 σ(U1∗ e1 , e2 ) := σ(e1 , U2∗ e2 ) (e1 ∈ E1 , e2 ∈ E2 )
408
Marcantognini and Morán
IEOT
is an isometry. Note that
∗ U ∗ E 1 ⊕ E2 σ(U1∗ H1 × E2 ) = D
1 X
is a regular subspace of H, thus a Kre˘ın space by itself. Viewing V0 as a weak
∗ U ∗ E1 ⊕ E2 to the Kre˘ın space H isomorphism from the Kre˘ın space D := D
1 X we can extend V0 to a continuous isometry V on H with domain D. Indeed, let E2 = E2+ ⊕ E2− be a fundamental decomposition of E2 . Then − E2 is a maximal uniformly negative subspace of E2 and, hence, of D. As E2 = σ ({0} × E2 ) is contained in the domain of V0 , so is E2− and the result follows. In addition, note that both E2− and V E2− = U2∗ E2− are maximal uniformly negative subspaces of H. From the above arguments we conclude that the defect subspaces of V are Hilbert subspaces of H. We can compute them to get
∗ U ∗ E1
D N =D
1 X X
and
a+X
∗ e2 = 0 . M = (a, e2 ) ∈ H : T2 e2 = 0 and D X
From the fundamental decomposition E2 = E2+ ⊕ E2− we get the fundamental decomposition
⊕ E +) ⊕ E − H = (D 2 2 X with corresponding norm · |H| , so that, for all e1 ∈ E1 ,
1 , P K−2 Xe
1 K2
σ(e1 , 0) 2|H| = σ(e1 , 0), σ(e1 , 0)H − 2PEK−2 Xe E 2
1 , P K−2 Xe
1 K2 . = e1 , e1 K1 − 2PEK−2 Xe E 2
2
2
Whence e1 −→ σ(e1 , 0) gives an isometry ϕ1 from E1 into H such that
1 , P K−2 Xe
1 K2
ϕ1 e1 2|H| = e1 , e1 K1 − 2PEK−2 Xe E 2
2
for all e1 ∈ E1 . In particular, ϕ1 ∈ L(E1 , H) and ϕ1 E1 is a regular subspace of H. Let U ∈ L(F ) be a minimal unitary Hilbert space extension of V . Notice that, for all e1 , e1 ∈ E1 and all n ∈ N ∪ {0}, U n σ(e1 , 0), σ(e1 , 0)F = σ(e1 , 0), U ∗n σ(e1 , 0)F = σ(e1 , 0), σ(U1∗n e1 , 0)H = e1 , U1∗n e1 K1 = U1n e1 , e1 K1 . Hence U n ϕ1 e1 , ϕ1 e1 F = U1n e1 , e1 K1 ,
e1 , e1 ∈ E1 , n ∈ N ∪ {0}.
(3.1)
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
409
If T := PϕF1 E1 U |ϕ1 E1 then T is a contraction on ϕ1 E1 since, according with (3.1), for all e1 ∈ E1 , T ϕ1 e1 , T ϕ1 e1 F = U ϕ1 e1 , T ϕ1 e1 F = U1 e1 , ϕ∗1 T ϕ1 e1 K1 = ϕ1 PEK11 U1 e1 , U ϕ1 e1 F = PEK11 U1 e1 , U1 e1 K1 = T1 e1 , T1 e1 K1 and, therefore, T ϕ1 e1 , T ϕ1 e1 F ≤ e1 , e1 K1 = ϕ1 e1 , ϕ1 e1 F . Besides, for all e1 ∈ E1 , (U − T )ϕ1 e1 , (U − T )ϕ1 e1 F = U ϕ1 e1 , U ϕ1 e1 F − T ϕ1 e1 , T ϕ1 e1 F = e1 , e1 K1 − T1 e1 , T1 e1 K1 = U1 e1 , U1 e1 K1 − PEK11 U1 e1 , PEK11 U1 e1 K1 = P K1
K1
1 ) U1 e1 , PR1 ∩H 2 (D
1 ) U1 e1 K1 . R1 ∩H 2 (D
Thus (U − T )ϕ1 e1 , (U − T )ϕ1 e1 F ≥ 0 for all e1 ∈ E1 and the equality holds if and only if U1 e1 ∈ E1 . If this is the case then T ϕ1 e1 = ϕ1 U1 e1 by (3.1). So (U − T )ϕ1 e1 = U ϕ1 U1∗ e1 − ϕ1 e1 , where e1 := U1 e1 ∈ E1 . Then U ϕ1 U1∗ e1 = U σ(U1∗ e1 , 0) = σ(e1 , 0) = ϕ1 e1 and (U − T )ϕ1 e1 = 0. Then Q := (U − T )ϕ1 E1 is a Hilbert subspace of F . Next we show that U k Q ⊥ U j Q for all nonnegative integers k = j. Indeed, by (3.1), for all e1 , e1 ∈ E1 and all m ∈ N, T m ϕ1 e1 , ϕ1 e1 F = U T m−1 ϕ1 e1 , ϕ1 e1 F = U1 ϕ∗1 T m−1 ϕ1 e1 , e1 K1 = T m−1 ϕ1 e1 , ϕ1 U1∗ e1 F = U1 ϕ∗1 T m−2 ϕ1 e1 , U1∗ e1 K1 = T m−2 ϕ1 e1 , ϕ1 U1∗2 e1 F = · · · = U1 e1 , U1∗m−1 e1 K1 = U1m e1 , e1 K1 = U m ϕ1 e1 , ϕ1 e1 F , so that U m (U − T )ϕ1 e1 , (U − T )ϕ1 e1 F = U m ϕ1 e1 , ϕ1 e1 F − U m−1 T ϕ1 e1 , ϕ1 e1 F − U m+1 ϕ1 e1 , T ϕ1 e1 F + U m T ϕ1 e1 , T ϕ1 e1 F = 0. Now set F1 := ϕ1 E1 ⊕ Q ⊕ U Q ⊕ U 2 Q ⊕ · · · . Clearly F1 is a regular subspace of F , hence a Kre˘ın space. Recall that Q1 = (U1 − T1 )E1 is a Hilbert subspace of R1 such that R1 = E1 ⊕ Q1 ⊕ U1 Q1 ⊕ U12 Q1 · · · , and extend ϕ1 to a unitary operator Φ1 ∈ L(R1 , F1 ) by setting Φ1 |E1 := ϕ1 and Φ1 U1n (U1 − T1 )e1 := U n (U − T )ϕ1 e1 Φ1 U1∗ |R1
(n ∈ N ∪ {0}, e1 ∈ E1 ).
= U ∗ Φ1 . Clearly, Φ1 U1 |R1 = U Φ1 so that ∞ We canas well define an isometric isomorphism Φ2 from R2= n=0 U2n E2 ∞ onto F2 := n=0 U ∗n E2 , satisfying Φ2 U2 |R2 = U ∗ Φ2 and Φ2 U2∗ |R2 = U Φ2 .
410
Marcantognini and Morán
IEOT
Now, view Φj as a linear operator from Rj into F ⊇ Fj , j = 1, 2, and K1 define Z : K1 → K2 by Z := Φ∗2 Φ1 PR . Then Z ∈ L(K1 , K2 ). 1 For all k1 ∈ K1 and k2 ∈ K2 , K1 ∗ K2 U1 k1 , Φ2 PR k ZU1∗ k1 , k2 K2 = Φ1 PR 1 2 2 F K1 K2 = Φ1 U1∗ PR k , Φ2 PR k 1 1 2 2 F K1 K2 = U ∗ Φ1 PR k1 , Φ2 PR k2 F 1 2 K1 K2 = Φ1 PR k1 , U Φ2 PR k2 F 1 2 K1 K2 ∗ = Φ1 PR k1 , Φ2 PR U2 k2 F 1 2
= U2 Zk1 , k2 K2 . ZU1∗
= U2 Z. Therefore, For all h1 ∈ H1 and h2 ∈ H2 , K1 K2 K1 K2 Zh1 , h2 K2 = Φ1 PR h , Φ2 PR h = σ(PR h1 , 0), σ(0, PR h )H 1 1 2 2 F 1 2 2 K1 K2
K1 h1 , P K2 h2 K2 = [(PR h , 0), (0, PR h )] = XP R1 R2 1 1 2 2 K2
K1 h1 , h2 K2 = Xh1 , h2 K2 . = (PR | )∗ XP R1 2 H2 K2 Thus PH Z|H1 = X. 2 The proof is complete if we show that Z|R1 is a contraction. To do that, note that E2− in any fundamental decomposition of E2 is maximal uniformly negative in R2 and in F as well. Since Φ2 E2− = E2− , it follows that Φ∗2 is a contraction. Therefore, for all r1 ∈ R1 , K1 K1 Zr1 , Zr1 K2 = Φ∗2 Φ1 PR r , Φ∗2 Φ1 PR r 1 1 1 1 F
= Φ∗2 Φ1 r1 , Φ∗2 Φ1 r1 F
≤ Φ1 r1 , Φ1 r1 F = r1 , r1 K1 .
As Propostion 3.1 clearly emphasizes, the relation XT1∗ = T2 X alone is not sufficient to grant the existence of symbols. In the Hilbert space case the difficulty is overtaken by assuming that: ii) There exists β ≥ 0 such that, for all h1 ∈ H1 and h2 ∈ H2 , K1 K2 |Xh1 , h2 H2 | ≤ β PR h PR h . 1 1 K1 2 2 K2
For a Hankel operator X for T1 and T2 satisfying the boundedness condition ii) , set X P V := inf β, where β runs over all nonnegative numbers satisfying
∈ L(E1 , E2 ) ii) . It turns out that ii) is equivalent to the the existence of an X K2 ∗ K1
such that X = (PR2 |H2 ) XPR1 |H1 and X ≤ β, and, in turn, is necessary and sufficient for the existence of a Z ∈ L(K1 , K2 ) such that ZU1∗ = U2 Z, K2 Z = X and Z = X P V (cfr. [14], [15], [16] and [5]). Therefore, when PR 2
X P V = 1, we can see that ii) is equivalent to i) and ii) in Proposition 3.1. The hypothesis X P V = 1 can be dropped provided we consider symbols Z of X such that Z = X P V . As a matter of fact, if X is a Hankel operator for T1 and T2 with X P V = b > 0, then X := 1b X is a Hankel operator for
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
411
T1 and T2 with X P V = 1 and, moreover, Z is a Hankel symbol for X if and only if bZ is a Hankel symbol for X. In the Kre˘ın space setting a boundedness condition like (ii) and equivalent to (ii) in Proposition 3.1 is given next. Proposition 3.2. Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be Kre˘ın space contractions with corresponding minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ), and let X ∈ L(H1 , H2 ) be a Hankel operator for T1 and T2 . Assume that sup T1n < ∞ and sup T2n < ∞. For j = 1, 2 consider the regular subspace Ej of Kj given by K
Ej := PRjj Hj where Rj :=
∞
Ujn Kj
n=0
is the regular subspace of Kj which reduces Uj such that Uj |Rj is unitary and K
∈ L(E1 , E2 ) such that PRjj = limn→∞ Ujn Uj∗n . Then there exists an X K2
K1 |H1 X = (PR |H2 )∗ XP R1 2
if and only if, for any fixed norms · |K1 | and · |K2 | , there exists α ≥ 0 such that K1 K2 h1 |K1 | PR h2 |K2 | |Xh1 , h2 H2 | ≤ α PR 1 2
for all h1 ∈ H1 and h2 ∈ H2 . K2
K1 |H1 for some X
∈ L(E1 , E2 ). Let Proof. Assume that X = (PR | )∗ XP R1 2 H2
· |K1 | and · |K2 | be fixed. Then, for all h1 ∈ H1 and h2 ∈ H2 , K2
K1 h1 , h2 K2 | = |(XP
K1 h1 , P K2 )h2 K2 | |Xh1 , h2 H2 | = |(PR |H2 )∗ XP R1 R1 R2 2
K1 h1 |K | P K2 h2 |K | ≤ XP 2 2 R1 R2 K1 K1
≤ X
P R1 h1 |K1 | PR2 h2 |K2 | ,
= supe =1 Xe
1 |K | . where X
2 1 |K1 | Conversely, suppose that, for fixed norms · |K1 | and · |K2 | , there exists α ≥ 0 such that, for all h1 ∈ H1 and h2 ∈ H2 , K1 K2 |Xh1 , h2 H2 | ≤ α PR h
PR h
. 1 1 |K1 | 2 2 |K2 |
We can assume that, for j = 1, 2, · |Kj | is the norm associated with a fundamental decomposition of the form Kj = [(Kj Hj ) ⊕ Hj+ ] ⊕ Hj− and corresponding signature operator Jj , so that · |Kj | |Hj = · |Hj | and Jj |Hj = Jj with · |Hj | and Jj the norm and the signature operator arisen from the fundamental decomposition Hj = Hj+ ⊕ Hj− .
412
Marcantognini and Morán
IEOT
K1 Define CPR h := Xh1 (h1 ∈ H1 ). Then, for all h1 ∈ H1 , 1 1 K1 h1 |H2 | =
CPR 1
sup h2 |H2 | =1
=
sup h2 |H2 | =1
=
sup h2 |H2 | =1
≤
sup h2 |H2 | =1
K1 |CPR h1 , h2 |H2 | | 1
|Xh1 , h2 |H2 | | |Xh1 , J2 h2 H2 | K1 K2 α PR h
PR J h
1 1 |K1 | 2 2 2 |K2 |
K2 K1 = α PR J |
PR h
. 2 2 H2 1 1 |K1 |
Hence C can be extended by continuity to all of E1 . K2 h2 := C ∗ h2 then, for all h2 ∈ H2 , In a similar fashion if DPR 2 K2
DPR h
= 2 2 |K1 |
≤
sup e1 |K1 | =1
sup e1 |K1 | =1
K2 |PR h2 , Ce1 |K2 | | 2 K2
PR h2 |K2 | Ce1 |H2 | 2
K2 = C
PR h2 |K2 | . 2
Whence D can be extended by continuity to get a continuous linear operator D : E2 → E1 .
:= D∗ . Then, for all h1 ∈ H1 and h2 ∈ H2 , Set X K2
K1 h1 , h2 H2 = Xh1 , h2 H2 . (PR |H2 )∗ XP R1 2
4. Labeling of all the Hankel symbols of a given Hankel operator We now turn our attention to the problem of describing the set of all Hankel symbols for a given Hankel operator. Theorem 4.1. Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be Kre˘ın space contractions with corresponding minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ). Assume that sup T1n < ∞ and sup T2n < ∞. For j = 1, 2 consider the regular subspace Ej of Kj given by K
Ej := PRjj Hj where Rj :=
∞
Ujn Kj
n=0
is the regular subspace of Kj which reduces Uj such that Uj |Rj is unitary K and PRjj = limn→∞ Ujn Uj∗n . Let X ∈ L(H1 , H2 ) be a Hankel operator for
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
413
K2
K1 |H1 for some contraction operator T1 and T2 such that X = (PR | )∗ XP R1 2 H2
∈ L(E1 , E2 ). Consider the Hilbert spaces X
∗ U ∗ E1
D N =D
1 X X and
X ⊕ E2 : P K2 U2 e2 = 0 and D
a+X
∗ e2 = 0 M = (a, e2 ) ∈ D E2 X
, E1 ) a defect operator for X.
Then there exists a bijection
∈ L(D with D X X between the Schur class S(N , M) and the set HS(X) of all Hankel symbols Z of X such that Z|R1 is a contraction. Before embarking in the proof of the theorem, let us mention that the map establishing the bijection between S(N , M) and HS(X) can be given by a close formula. Roughly speaking, there exist operator valued functions a, b, c, d, defined and analytic on a disc around 0, such that a Z ∈ HS(X) is fully determined from a given ϑ ∈ S(N , M) by the Schur like formula a + bϑ(1 − cϑ)−1 d in such a way that the map ϑ → Z is one to one and onto. The reader is referred to formula (4.4) and the discussion that follows. Proof. We proceed stepwise. STEP 1. As we did to prove Propostion 3.1, we build up a Kre˘ın space H and an isometry V acting on H whose defect subspaces are the Hilbert spaces N and M in the statement of the theorem so that each minimal unitary Hilbert space extension U of V gives rise to a Hankel symbol Z of X. It turns out that any Hankel symbol Z of X can be obtained as before from a minimal unitary Hilbert space extension U of the isometry V . We show that in the next step. K1 K2 K2 K1 = PR Z = PR ZPR STEP 2. Let Z ∈ HS(X) be given. Then Z = ZPR 1 2 2 1
Z ∈ L(D
Z , R1 ) and Z|R1 ∈ L(R1 , R2 ) is a contraction. In the sequel, let D stand for a defect operator for Z|R1 . A hermitian sesquilinear form on R1 × R2 is defined by setting
[(r1 , r2 ), (r1 , r2 )] := r1 , r1 K1 + Zr1 , r2 K2 + r2 , Zr1 R2 + r2 , r2 K2 . So, for all (r1 , r2 ), (r1 , r2 ) ∈ R1 × R2 , ∗ ∗
Z
Z r1 , D r1 D Z + Zr1 + r2 , Zr1 + r2 K2 . [(r1 , r2 ), (r1 , r2 )] = D
Z ⊕R2 , with the standard inner product, and τ is defined Therefore, if F := D on R1 × R2 as ∗
Z τ (r1 , r2 ) := (D r1 , Zr1 + r2 ) (r1 ∈ R1 , r2 ∈ R2 ) then τ is an isometry from (R1 × R2 , [·, ·]) to (F , ·, ·F ) with dense range in F .
414
Marcantognini and Morán
IEOT
K2 Since PH Z|H1 = X, it turns out that 2
PEK22 Z|E1 = X. Therefore, σ(e1 , e2 ), σ(e1 , e2 )H = τ (e1 , e2 ), τ (e1 , e2 ))F ,
e 1 ∈ E1 , e 2 ∈ E2 .
(4.1)
From (4.1) it follows that ρ0 σ := τ |E1 ×E2 is a densely defined isometry from H to F . Note that a fundamental decomposition E2 = E2− ⊕ E2+ of the
⊕E + )⊕E − Kre˘ın space E2 gives rise to fundamental decompositions H = (D 2 2 X + − − 2
and F = [DZ ⊕ (R2 ∩ H (D2 )) ⊕ E2 ] ⊕ E2 so that E2 is maximal uniformly negative in both H and F . Since E2− happens to be contained in the domain of ρ0 (E2 = σ({0} × E2 )) ρ0 can be extended to an isometry ρ defined on all of H. Therefore, via the isometry ρ, the Kre˘ın space H can be regarded as a regular subspace of the Kre˘ın space F . Moreover, as E2− is maximal uniformly negative in both H and F , and ρE2− = E2− , it turns out that F ρH is a Hilbert subspace of F . Set U0 τ (r1 , r2 ) := τ (U1 r1 , U2∗ r2 ),
r1 ∈ R1 , r2 ∈ R2 .
As ZU1∗ = U2 Z, the operator U0 is shown to be isometric. On the other hand, if r1 ∈ R1 and r2 ∈ R2 are given, then τ (r1 , r2 ) = τ (U1 U1∗ r1 , U2∗ U2 r2 ) = U0 τ (U1∗ r1 , U2 r2 ). It then turns out that U0 is a weak isomorphism on F . As in the proof of Proposition 3.1, when dealing with V0 , it can be seen that U0 can be extended to a unitary operator U on F . For all e1 ∈ E1 and e2 ∈ E2 , U ρσ(U1∗ e1 , e2 ) = U τ (U1∗ e1 , e2 ) = τ (U1 U1∗ e1 , U2∗ e2 ) = τ (e1 , U2∗ e2 ) = ρσ(e1 , U2∗ e2 ) = ρV σ(U1∗ e1 , e2 ).
Therefore, U ρ|D = ρV . In other words, by means of the isometry ρ, U can be interpreted as a unitary extension of V . If Φ1 : R1 → F and Φ2 : R2 → F are defined as Φ1 r1 := τ (r1 , 0) (r1 ∈ R1 ) and Φ2 r2 := τ (0, r2 ) (r2 ∈ R2 ), then Φ1 and Φ2 are isometries such that (i) Φ1 R1 ∨ Φ2 R2 = F and, for all k1 ∈ K1 and k2 ∈ K2 , K1 K2 k , Φ2 PR k = Zk1 , k2 K2 . (ii) Φ1 PR 1 1 2 2 F
Furthermore, for all e1 ∈ E1 , e2 ∈ E2 and n ∈ N ∪ {0}, (iii) Φ1 U1n e1 = U n Φ1 e1 = U n ρσ(e1 , 0) and (iv) Φ2 U2n e2 = U ∗n Φ2 e2 = U ∗n ρσ(0, e2 ).
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
415
From (iii) and (iv) we get that ∞
U n ρσ(E1 × {0}) =
n=0
and
∞
∞
U n Φ1 E 1 =
n=0
U ∗n ρσ({0} × E2 )
n=0
∞
∞
Φ1 U1n E1 = Φ1 R1 .
n=0
U ∗n Φ2 E2 =
n=0
∞
Φ2 U2n E2 = Φ2 R2 .
n=0
From the above relations and by applying (i), we conclude that F = n∈Z U n ρH, which shows that U is minimal. Finally, (ii) says that Z is given by U as in the correspondence we established in STEP 1. Therefore, we can conclude that the correspondence which associates to each minimal unitary extension U of V a Hankel symbol Z of X with Z|R1 contractive is surjective. STEP 3. Next we show that the correspondence is injective. Assume that U ∈ L(F ) and U ∈ L(F ) are two minimal unitary Hilbert space extensions of V and let Z and Z be the corresponding Hankel symbols. If Z = Z , then U n σ(e1 , 0), σ(0, e2 )F = ZU1n e1 , e2 K2
n
= Z U1n e1 , e2 K2 = U σ(e1 , 0), σ(0, e2 )F for all e1 ∈ E1 , e2 ∈ E2 and n ∈ N ∪ {0}. Besides, when n ∈ N ∪ {0}, U n σ(e1 , 0), σ(e1 , 0)F = U1n e1 , e1 K1 = U n σ(e1 , 0), σ(e1 , 0)F for all e1 , e1 ∈ E1 , and U n σ(0, e2 ) = σ(0, U2∗n e2 ) = U n σ(0, e2 ) for all e2 ∈ E2 . These relations, together with the minimality condition satisfied by both U and U and the fact that H = σ(E1 × E2 ), grants that U and U are weakly isomorphic under the map U n σ(e1 , e2 ) :→ U n σ(e1 , e2 ) ((e1 , e2 ) ∈ E1 × E2 , n ∈ Z) and also that
F n F PH U |σ(E1 ×E2 ) = PH U n |σ(E1 ×E2 ) ,
n ∈ N ∪ {0}.
As F H and F H are Hilbert subspaces of F and F , respectively, it turns out that, with respect to the Hilbert space inner products ·, ·|F | and ·, ·|F | determined by the fundamental decompositions F = [(F H) ⊕ H+ ] ⊕ H− and F = [(F H) ⊕ H+ ] ⊕ H− , U n σ(e1 , e2 ), U m σ(e1 , e2 )|F | = U n σ(e1 , e2 ), U m σ(e1 , e2 )F F n F m − 2PH σ(e1 , e2 )F − U σ(e1 , e2 ), PH− U
= U n σ(e1 , e2 ), U m σ(e1 , e2 )F
F n F m − 2PH σ(e1 , e2 ), PH σ(e1 , e2 )F −U −U
= U n σ(e1 , e2 ), U m σ(e1 , e2 )|F |
416
Marcantognini and Morán
IEOT
for all e1 , e1 ∈ E1 , e2 , e2 ∈ E2 and n, m ∈ Z. Then U and U can be identified. The identification is realized by the extension by continuity of the weak isomorphism, call it ω, which is a unitary operator from F onto F satisfying ωU = U ω and ω|H = 1. As for a summary of what we have achieved so far, let us point out that we have built up a Kre˘ın space H and an isometry V acting on H such that if U ∈ L(F ) is a minimal unitary Hilbert space extension of V , then there exist two isometries Φ1 : R1 → F and Φ2 : R2 → F such that the operator K1 is a symbol of X with Z|R1 a contraction, and that the Z := Φ∗2 Φ1 PR 1 mapping U → Z is a bijection between the family U(V ) of all minimal unitary Hilbert space extensions of V and the set HS(X) of all Hankel symbols Z of X such that Z|R1 is a contraction. At this point we are ready to consider the problem of labeling the set HS(X). STEP 4. Let Z ∈ HS(X) be given. Since ZU1 = U2∗ Z, then ZU1n e1 = U2∗n Ze1 for all e1 ∈ E1 and all n ∈ N. ∞ K2 K1 From this, as Z = PR ZPR and R1 = n=0 U1n E1 , it follows that Z is fully 2 1 K2 determined by PR Z|E1 . On the other hand, 2 R2 = E2 ⊕ U2 P2 ⊕ U22 P2 ⊕ U23 P2 ⊕ · · · , where P2 := kernel (T2 ) is a Hilbert subspace of E2 , and it turns out that K2
+ Z|E1 = X PR 2
∞ n=0
2
+ PUKn+1 Z|E1 = X P 2
2
∞ n=0
U2n+1 PPK22 U2∗n+1 Z|E1 .
Thus, Z is determined by the sequence of operators ∞ U2n+1 PPK22 U2∗n+1 Z|E1 . n=0
To Z ∈ HS(X) we associate an L(E1 , P2 )-valued function SZ (z) defined around 0 by the power series ∞ K2 ∗n+1 z n S S Z|E1 (n ≥ 0). SZ (z) := Z (n), Z (n) := PP2 U2 n=0
We get that, for all e1 ∈ E1 and e2 ∈ P2 , ∞ ∞ z n S (n)e , e = z n Ze1 , U2n+1 e2 K2 SZ (z)e1 , e2 K2 = Z 1 2 K2 =
n=0 ∞
n=0 ∞
z n Φ1 e1 , U ∗n+1 e2 F =
n=0
n=0
∗ e1 , e2 F z n U n+1 D X
∗ e1 , e2 F . = U (1 − zU )−1 D
X View P2 as a regular subspace of F and let Ψ denote the isometric embedding of P2 into F (Ψ := σ|{0}×P2 ). Then, for all e1 ∈ E1 and e2 ∈ P2 ,
∗ e1 , e2 K2 . SZ (z)e1 , e2 K2 = Ψ∗ PPF2 U (1 − zU )−1 D X
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
417
Therefore
∗
. SZ (z) = Ψ∗ PPF2 U (1 − zU )−1 |H D
X Choose the fundamental decomposition
⊕ P2 ⊕ U2∗ E + ) ⊕ U2∗ E − H = (D X
2
2
(4.2) (4.3)
and let · |H| be the corresponding quadratic norm. Observe that P2 , N , M are subspaces of H+ . Whence the restriction of ·, ·|H| to any of them equals the corresponding restriction of ·, ·H . In particular, we see that e2 |H| ≤ V e2 |H| for all e2 ∈ P2 . So V ≥ 1 when the operator norm is computed in terms of · |H| . The fundamental decomposition (4.3) yields the fundamental decomposition F = [(F H) ⊕ H+ ] ⊕ H− of F with norm · |F | . Since (F H) ⊕ N , (F H) ⊕ M ⊆ F + and U |(F H)⊕N is a unitary operator from (F H) ⊕ N onto(F H) ⊕ M, we get that, for all f ∈ F, F 2
U f 2|F | = V PDF f + U P(F H)⊕N f |F | F 2 = V PDF f 2|H| + U P(F H)⊕N f |F F 2 ≤ V 2 PDF f 2|H| + P(F H)⊕N f |F |
≤ V 2 f 2|F |. Therefore U = V when U is regarded as an operator on (F , · |F |). Set r := V PDH −1 . Then 1−zV PDH has a bounded inverse on (H, · |H|) for all |z| < r. Since P2 ⊆ H+ and U = V , we find out that, for all |z| < r, 1 − z(1 − PPH2 )V PDH on (H, · |H| ) and 1 − zU on (F , · |F | ) have bounded inverses. All those inverse operators can be computed by means of Neumann series. Then, when P2 is endowed with the quadratic norm · P2 = · |H| |P2 , it comes that z ∈ D −→ PPF2 U (1 − rzU )−1 h is an H 2 (P2 )-function for each h ∈ H. F Now, let ϑU (z) := PM U (1 − zPFF H U )−1 |N be the S(N , M)-function associated to U via the Arov-Grossmann functional model (Theorem 2.1). Taking into account that V n P2 ⊆ D and V n+1 P2 ⊥ P2 for all n ∈ N ∪ {0}, and by a straightforward computation we omit in the present discussion, we get that, for all |z| < r, PPF2 U (1 − zU )−1 |H H H −1 = PPH2 (V PDH + ϑU (z)PN )[1 − z(V PDH + ϑU (z)PN )] |H
= PPH2 V PDH [1 − z(1 − PPH2 )V PDH ]−1 + PPH2 1 + zV PDH [1 − z(1 − PPH2 )V PDH ]−1 (1 − PPH2 ) −1 H [1 − z(1 − PPH2 )V PDH ]−1 ϑU (z) × ϑU (z) 1 − zPN H × PN [1 − z(1 − PPH2 )V PDH ]−1 .
From this expression and (4.2) it follows that, for all |z| < r, SZ (z) = a(z) + b(z)ϑ(z)(1 − c(z)ϑ(z))−1 d(z),
(4.4)
418
Marcantognini and Morán
IEOT
where ϑ ≡ ϑU and
∗ ∈ L(E1 , P2 ), a(z) := Ψ∗ PPH2 V PDH [1 − z(1 − PPH2 )V PDH ]−1 D X ∗ H H H b(z):= Ψ PP2 1 + zV PD [1 − z(1 − PP2 )V PDH ]−1 (1 − PPH2 ) |M ∈ L(M, P2 ), H [1 − z(1 − PPH2 )V PDH ]−1 |M ∈ L(M, N ), c(z) := zPN H
∗ ∈ L(E1 , N ). [1 − z(1 − PPH2 )V PDH ]−1 D d(z) := PN
X
The Schur like formula (4.4) establishes the direct connection between S(N , M) and {SZ : Z ∈ HS(X)} while the map ϑ −→ SZ determined by (4.4) is a bijection between S(N , M) and HS(X) since the mappings Z ∈ HS(X) U ∈ U(V )
? ϑ ∈ S(N , M)
? {S Z (n)}
? SZ are all bijections (up to isomorphism as far as U ∈ U(V ) is concerned). Furthermore, we can realize the map ϑ → Z by a close formula as we show next. For each e1 ∈ E1 , z ∈ D → SZ (rz)e1 can be shown to be an H 2 (P2 )function when P2 is endowed with the quadratic norm · 2P2 = ·, ·K2 |P2 . Therefore, if Γ : H 2 (P2 ) → U2 P2 ⊕ U22 P2 ⊕ U23 P2 ⊕ · · · is the unitary operator given by ∞ ∞ ∞ n+1 n n 2 Γ z xn := U2 xn z xn ∈ H (P2 ) , x(z) = n=0
n=0
n=0
then ∞ n=0
−1 2 PUKn+1 Z| = Γ a(rz) + b(rz)ϑ(rz)(1 − c(rz)ϑ(rz)) d(rz) . E 1 P 2
2
K2 Consequently, PR Z|E1 : E1 → E2 ⊕ (R2 E2 ), 2 K2 Z|E1 PR 2
X
=
!
. Γ a(rz) + b(rz)ϑ(rz)(1 − c(rz)ϑ(rz))−1 d(rz)
K2 Since PR Z|E1 determines Z uniquely, the above representation gives a Schur 2 like formula for the map ϑ → Z. This completes the proof.
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
419
Remark 4.2. In the proof of Theorem 4.1 one can find the arguments to show that there exists a bijective correspondence between HS(X) and the set of all
If contractions Z ∈ L(R1 , R2 ) such that ZU1∗ |R1 = U2 Z and PER2 2 Z|E1 = X. the given contractions T1 and T2 are assumed to be coisometric contractions having Hilbert spaces for kernels, their corresponding minimal isometric dilations are unitary operators. Then, by assuming that X is contractive and considering
∗ U ∗ H1
X D N =D X
and
1
X ⊕ H2 : T2 h2 = 0 and D
X a + X
∗ h2 = 0 M = (a, h2 ) ∈ D
X , H1 ) a defect operator for X, no extra conditions on
X ∈ L(D with D {X, T1 , T2 } are required in Theorem 4.1 to grant that the set HS(X) is in bijective correspondence with S(N , M). The result reads: Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be coisometric contractions such that kernel (T1 ) and kernel (T2 ) are Hilbert spaces, and let U1 ∈ L(K1 ) and U2 ∈ L(K2 ) be their corresponding minimal isometric dilations. If X ∈ L(H1 , H2 ) is a contractive Hankel operator for T1 and T2 , then there exists a bijective correspondence between HS(X) and S(N , M) with N and M as above. Remark 4.3. Since HS(X) is in bijective correspondence with the Schur class S(N , M), it is clear that HS(X) has a single element if and only if either N = {0} or M = {0}. In particular: For fixed contractions T1 ∈ L(H1 ) and T2 ∈ L(H2 ) satisfying sup T1n < ∞ and sup T2n < ∞, any Hankel operator X for T1 K2
K1 |H1 , for some contraction operator and T2 such that X = (PR | )∗ XP R1 2 H2
∈ L(E1 , E2 ), has a unique Hankel symbol, say ZX , constrained to satisfy X that ZX |R1 is a contraction, if either PEK11 U1 |E1 is injective or PEK22 U2 |E2 is injective. Indeed, if PEK22 U2 |E2 is assumed to be injective then
∗ e2 = 0 = {0}. M := (a, e2 ) ∈ DX ⊕ E2 : PEK22 U2 e2 = 0 and DX a + X In a similar way one can show that if PEK11 U1 |E1 is injective then N = {0}. We last study the problem of establishing a necessary and sufficient condition for the set HS(X) to have a single element. We get the following criterion. Theorem 4.4. Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be Kre˘ın space contractions with minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ), respectively. Assume that sup T1n < ∞ and sup T2n < ∞. For j = 1, 2 consider the regular subspace Ej of Kj given by K
Ej := PRjj Hj
420
Marcantognini and Morán
IEOT
where ∞
Rj :=
Ujn Kj
n=0
is the regular subspace of Kj which reduces Uj such that Uj |Rj is unitary K and PRjj = limn→∞ Ujn Uj∗n . Let X ∈ L(H1 , H2 ) be a Hankel operator for
∈ L(E1 , E2 ) be a contraction operator such that X = T1 and T2 , and let X K2 ∗ K1 (PR2 |H2 ) XPR1 |H1 . On the Kre˘ın space E1 ⊕ E2 , with the standard inner product, consider the 2 × 2 block matrix operators K1 1 0 0 P U | T 1 := E1 1 E1 , T 2 := K2 0 PE2 U2 |E2 0 1 and
E :=
1
X
∗ X . 1
Then X has a unique Hankel symbol Z with Z|R1 contractive if and only if either (4.5) kernel (T 1 E) ⊆ kernel (E), or kernel (T 2 E) ⊆ kernel (E).
(4.6)
Proof. We apply Theorem 4.1 and use the notation introduced in its proof. In particular, if H is the Kre˘ın space built up in the proof of Theorem 4.1 and σ is the isometry from (E1 × E2 , [·, ·]) onto a dense subspace of H defined therein, then, for all e1 , e1 ∈ E1 and e2 , e2 ∈ E2 , σ(e1 , e2 ), σ(e1 , e2 )H = [(e1 , e2 ), (e1 , e2 )] = E(e1 , e2 ), (e1 , e2 )E1 ⊕E2 . (4.7) It then follows that a set A of E1 × E2 is such that H if and only if, σA = ∗ regarding A as a set of E1 ⊕ E2 , it holds that kernel (E|A ) ⊆ kernel (E). In particular, by the identifications T 1∗ (E1 ⊕ E2 ) = U1∗ E1 × E2 and T 2∗ (E1 ⊕ E2 ) = E1 × U2∗ E2 , we get that either σ(U1∗ E1 × E2 ) = H or σ(E1 × U2∗ E2 ) = H if and only if kernel (T j E) ⊆ kernel (E) for j = 1, 2 (which corresponds with (4.5) and (4.6) respectively.) The result is now obtained just by recalling that HS(X) has a single element if and only if either of the Hilbert spaces N and M given in the statement of Theorem 4.1 is {0} and that N = H σ(U1∗ E1 × E2 ) and M = H σ(E1 × U2∗ E2 ).
5. Appendix We herein include a sketch of the proof of Theorem 2.1.
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
421
∞ k (i) Given ϑ ∈ S(N , M), ϑ(z) = k=0 z ϑk , z ∈ D, let Uϑ be the linear operator defined in Fϑ := H ⊕ Eϑ by ⎤ ⎡ ⎡ ⎤ H h + φ(0) V PDH h + ϑ(0)PN h φ ⎥ ⎢ ∗ H ⎣ ⎦ S (φ + ϑPN h) Uϑ φ := ⎣ h ∈ H, ∈ Eϑ . ⎦ ψ ψ ∗ H S (ψ + ∆P h) N
Here Eϑ := H 2 (M) ⊕ ∆L2 (N ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (N )}⊥ , with 1
∆(ζ) := (1 − ϑ(ζ)∗ ϑ(ζ)) 2 , 2
|ζ| = 1,
2
and S is the shift on either H (M) or L (N ), depending on context. φ As Sϑ = ϑS|H 2 (N ) and S∆ = ∆S, given any h ∈ H and ∈ Eϑ , we ψ H H h) ∈ H 2 (M), S ∗ (ψ + ∆PN h) ∈ ∆L2 (N ) and that, for get that S ∗ (φ + ϑPN all u ∈ H 2 (N ), " ! # H S ∗ (φ + ϑPN h) ϑu H H H h, ϑSu+∆PN h, ∆Su = PN h, Su = 0. , = ϑPN H S ∗ (ψ + ∆PN h) ∆u Therefore, Uϑ ∈ L(Fϑ ).
φ ∈ Eϑ , It can be shown that Uϑ is unitary. In point of fact, if h ∈ H and ψ ⎡ ⎤ ⎡ ∗⎤ h h then ⎣ φ∗ ⎦ = Uϑ∗ ⎣ φ ⎦ is given by ψ∗ ψ H H ∗ h + PN (ϑ Sφ + ∆Sψ), h∗ = V ∗ PDH h + ϑ(0)∗ PM H H H h + Sφ − ϑ[ϑ(0)∗ PM h + PN ((ϑ∗ Sφ + ∆Sψ)], φ∗ = PM H H h + PN ((ϑ∗ Sφ + ∆Sψ). ψ ∗ = Sψ − ∆[ϑ(0)∗ PM
Also, it can be seen that Fϑ is the least Kre˘ın space containing Uϑn H for all n ∈ Z. From this and since Uϑ |D = V and Eϑ is a Hilbert space, it comes that Uϑ ∈ L(Fϑ ) is a minimal unitary Hilbert space extension of V . It remains to show that Fϑ Uϑ (1 − zPEFϑϑ Uϑ )−1 |N = ϑ(z) PM
for all z ∈ D. This follows from the relations ⎡ ⎤ 0 %k $ Fϑ PEϑ Uϑ h = ⎣ S ∗k ϑh ⎦ , h ∈ N , k ∈ N, S ∗k ∆h and
$ %k Fϑ Uϑ PEFϑϑ Uϑ h = ϑk h, PM
h ∈ N , k ∈ N ∪ {0}.
422
Marcantognini and Morán
IEOT
(ii) Let U ∈ L(F ) be a minimal unitary Hilbert space extension of V . Then U11 0 U= : [(F H) ⊕ N ] ⊕ D → [(F H) ⊕ M] ⊕ R 0 V with U11 a unitary operator from the Hilbert space (F H) ⊕ N onto the Hilbert space (F H) ⊕ M. Write E F : (F H) ⊕ N → (F H) ⊕ M. U11 = G H Then {F H, N , M; E, F, G, H} is a unitary colligation of operators with state space F H, input space N and output space M, all of them Hilbert spaces. It is known that its characteristic function z −→ H + zG(1 − zE)−1 F is an S(N , M)-function. Just note that H + zG(1 − zE)−1 F ≡ ϑU (z). (iii) For a given U ∈ L(F ) belonging to U(V ), set Ω(U ) = {z ∈ D : (1 − zU )−1 ∈ L(F )}. If z ∈ Ω(U ) and h ∈ H then U (1 − zU )−1 h = f ∈ F if and only if U h = f − zU f . Notice that U h = f − zU f if and only if H F V PDH h + U11 PN h = f − zV PDF f − zU11 (PN f + PFF H f ).
Whence U (1 − zU )−1 h = f if and only if F H F F F U11 PN h = PH f − zV PDF f − zPM U11 (PN f + PFF H f ) (5.1) V PDH h + PM
and H F F PFF H U11 PN h = PFF H f − zV PDF f − zPM U11 (PN f + PFF H f ).
(5.2)
From the arguments in the proof of (ii) it follows that, for all z ∈ D, (1−zPFF HU )−1 |(F H)⊕N = (1−zPFF HU11 )−1 |(F H)⊕N ∈ L((F H)⊕N ). H (z ∈ D). From (5.1) and (5.2) it can be Define TU (z) := V PDH + ϑU (z)PN seen that, for all z ∈ Ω(U ), (1 − zTU (z))−1 ∈ L(H) and F U (1 − zU )−1 |H = TU (z)(1 − zTU (z))−1 = (1 − zTU (z))−1 TU (z). PH
In particular, if U ∈ L(F ) and U ∈ L(F ) are two minimal unitary Hilbert space extensions of V such that ϑU (z) ≡ ϑU (z) then, for all z ∈ Ω(U ) ∩ Ω(U ),
F F PH U (1 − zU )−1 |H = PH U (1 − zU )−1 |H .
Therefore, U n h, HF = U n h, HF , h ∈ H, n ∈ N, and the operator defined on the linear span of {U n H}∞ n=−∞ and mapping U n h into U n h (h ∈ H, n ∈ Z) gives a weak isomorphism from F onto F .
Vol. 66 (2010)
Abstract Hankel Operators in Kre˘ın Spaces
423
Since F H is a Hilbert space, the weak isomorphism can be extended to an isometric isomorphism τ : F → F such that τ |H = 1 and τ U = U τ . As for the converse, let U ∈ L(F ), U ∈ L(F ) belong to U(V ) and assume that there exists an isometric isomorphism τ : F → F such that τ |H = 1 and τ U = U τ . Then τ PFF H U = PFF H U τ . So, for all x ∈ N , y ∈ M, and k ∈ N ∪ {0}, ( ) ' & $ %k F k F F F PM U PF H U x, y = PM U PF H U x, y . H
H
Hence,
F F U (1 − zPFF HU )−1 |N = PM U (1 − zPFF H U )−1 |N = ϑU (z) ϑU (z) = PM
for all z ∈ D.
References [1] T. Andˆ o, Linear Operators in Kre˘ın Spaces, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, 1979. [2] 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. [3] , Scattering matrices in the theory of unitary extensions of isometric operators, Math. Nachr., 157(1992), 105–123. [4] T. Azizov and I.S. Iokhvidov, Foundations of the Theory of Linear Operators in Spaces with Indefinite Metric, “Nauka”, Moscow, 1986; English transl. Linear Operators in Spaces with Indefinite Metric, Wiley, New York, 1989. [5] S. Bermudo, S.A.M. Marcantognini and M.D. Mor´ an, Operators of Hankel type, Czechoslovak Mathematical Journal, 56(2006), No. 4, 1147–1163. [6] J. Bognar, Indefinite Inner Product Spaces, Springer-Verlag, Berlin, 1974. [7] A. Dijksma, H. Langer and H.S.V. de Snoo, Generalized coresolvents of standard isometric operators and generalized resolvents of standard symmetric relations in Krein spaces, Operator Theory: Adv. Appl., Vol. 48(1990), Birkhuser, 261–274. [8] M.A. Dritschel and J. Rovnyak, Extension theorems for contraction operators on Kre˘ın spaces, Operator Theory: Adv. Appl., Vol. 47(1990), Birkhuser, 221– 305. , Operators on Indefinite Product Spaces, Lectures presented at the [9] meeting held at the Fields Institute for Research in Mathematical Sciences, Waterloo, Ontario, September 1994, edited by Peter Lancaster, Fields Institute Monographs 3, American Mathematical Society, Providence, RI, 1996. [10] B.W. McEnnis, Shifts on indefinite inner product spaces, Pacific J. Math., 81(1979), No. 1, 113–130. , Shifts on indefinite inner product spaces II, Pacific J. Math., [11] 100(1982), No. 1, 177–183. , Shifts on Krein spaces, Proceedings of Symposia in Pure Mathematics [12] 51(1990), Part 2, American Mathematical Society, Providence, RI, 201–211.
424
Marcantognini and Morán
IEOT
[13] I.S. Iokhvidov, M.G. Kre˘ın and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Akademie-Verlag, Berlin, 1982. [14] V. Pt´ ak, Factorization of Toeplitz and Hankel operators, Math. Bohem., 122(1997), No. 2, 131–140. [15] V. Pt´ ak and P. Vrbov´ a, Operators of Toeplitz and Hankel type, Acta Sci. Math. (Szeged), 52(1988), No. 1-2, 117–140. , Lifting intertwining relations, Integral Equations Operator Theory, [16] 11(1988), No. 1, 128–147. [17] B. Sz.-Nagy and C. Foia¸s, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London, 1970. S.A.M. Marcantognini Department of Mathematics Instituto Venezolano de Investigaciones Cient´ıficas P.O. Box 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: October 19, 2009. Revised: November 25, 2009.
Integr. Equ. Oper. Theory 66 (2010), 425–440 DOI 10.1007/s00020-010-1751-5 Published online February 16, 2010 © Birkhäuser/Springer Basel AG 2010
Integral Equations and Operator Theory
Completely Bounded and Ideal Norms of Multiplication Operators and Schur Multipliers T. Oikhberg Abstract. We estimate the completely bounded norms, the completely p-nuclear norms, and the completely p-summing norms of certain multiplication operators and Schur multipliers. Mathematics Subject Classification (2010). Primary: 46L07; Secondary: 47B10, 47B49, 47L20. Keywords. Operator space, Schatten space, Schur multiplier, multiplication operator, completely p-summing norm, completely p-nuclear norm.
1. Introduction The goal of this paper is to compute the c.b. norms, completely p-summing norms, and completely p-nuclear norms of multiplication operators and Schur multipliers. Throughout, we use standard operator space and Banach space terminology and results. The reader is referred to [5, 14, 17] for operator spaces, [12] for Banach spaces, and [4] for operator ideals. We use E ⊗ F to denote the minimal (or spatial) tensor product of operator spaces E and F . The space of n × n matrices (with its usual operator space structure) is referred to as Mn . The space of operators on 2 which belong to the Schatten p-class is denoted by S p , while Snp stands for its n × n version. The symbol S ∞ refers to the space of compact operators on 2 . We denote the norm of S p by · p , and write T p = ∞ if T ∈ / Sp. As common in the operator space literature, we denote the space of n × n matrices (that is, Sn∞ ) by Mn . For the sake of brevity, we often use Mn (E) instead of Mn ⊗ E (E is an operator space). The author wishes to thank the organizers of the Workshop on Operator Spaces and Quantum Groups at Fields Institute (December 2007), where part of this work was carried out.
426
Oikhberg
IEOT
The pairing between a matrix space and its conjugate arises from the parallel duality: formatrices (finite or infinite) T = (tij ) and S = (sij ), T, S = Tr T tS = i,j tij sij (tS is the transpose of S). p We first investigate the multiplication operators Mp,q → Sq : A,B : S 2 T → AT B, with A, B ∈ B( )\{0}. If there is no possibility of confusion regarding p and q, we use the notation MA,B . In Section 2, we estimate the c.b. norms of these operators. We prove that Mp,q A,B = A2r B2r , with 1/r = |1/p − 1/q| (Theorem 2.1). In Section 3, we deal with the completely s-summing and s-nuclear norms of multiplication operators (see below for the p,q o definition). Theorem 3.1 shows that πso (Mp,q A,B ) = νs (MA,B ) = Ar Br for certain 4-tuples (p, q, s, r) (see below for the definition and basic properties of πso (·) and νso (·)).
Next we turn our attention to Schur multipliers. Recall that, for a matrix φ = (φij ), the Schur multiplier with symbol φ (denoted by Sφ ) is the map taking an operator T = (Tij ) to (φij Tij ) (the latter matrix is sometimes denoted by φ T ). Note that, if A = diag (ai ) and B = diag (bj ), then MA,B is the Schur multiplier with the symbol (ai bj )∞ i,j=1 . In Section 4, we complement the results of [15] by computing the c.b. norms of certain Schur multipliers into 1 (N × N). In Section 5, we deal with completely p-summing norms of Schur multipliers. Theorem 5.1 gives a factorization result for completely p -summing Schur multipliers from B(2 ) to S p (1 ≤ p ≤ 2). In the particular case of p = 2, this is a complete description of completely p-summing Schur multipliers. Furthermore, for a Schur multiplier from B(2 ) to S 2 , we have π2o (Sφ ) = Sφ cb . Finally, we show that πpo (Sφ ) = φp when Sφ is considered as a map from S p into S p , where 1/p + 1/p = 1 (Proposition 5.5). In the commutative case, the operator norms of the diagonal maps from p to q are easy to compute. The summing norms (and other ideal norms) of such diagonal operators were computed by Garling and Carl [2, 6]. For the convolution operators Tµ : f → f ∗ µ, acting on L1 (G) or C(G) (G is a metrizable compact Abelian group), criteria for being nuclear or p-summing were given in [19] (that paper also covers convolutions with vector-valued measures). Further information on p-summing norms of convolutors from Lq to Lr can be found in [1]. In the non-commutative setting, certain estimates on ideal norms of formal identities between Schatten spaces were obtained in [3]. A few more ideal norms were estimated in [13]. Recall that the completely p-summing norm (sometimes also referred to as the operator p-summing norm) πpo (u) (1 ≤ p ≤ ∞) of an operator u : E → F (E and F are operator spaces) is defined as πpo (u) = IS p ⊗ u : S p ⊗ E → S p [F ]. This notion was introduced by G. Pisier in [16]. We refer the reader to that article for the necessary background, such as the definition and properties of the space S p [F ], the non-commutative Pietsch factorization, and a variety of other results. Here, we recall a few basic facts (having well-known classical counterparts).
Vol. 66 (2010)
Norms of Multiplication Operators
427
• The norm πpo (·) has is an ideal norm: πpo (v1 uv2 ) ≤ v1 cb πpo (u)v2 cb . Moreover, πpo (u) = supG πpo (u|G ), with the supremum taken over all finite dimensional subspaces of G of the domain of u. o (u) = ucb. • If q ≥ p, then πpo (u) ≥ πqo (u). In particular, πpo (u) ≥ π∞ p 2 p • For A, B ∈ S , πp (MA,B : B( ) → S ) ≤ A2p B2p (below, we see that actually we have equality). We also need a few easy facts concerning multiplication operators: • The composition of multiplication operators is again a multiplication operator: MA1 ,B1 MA2 ,B2 = MA1 A2 ,B2 B1 . • The adjoint of multiplication operators is again a multiplication operator: (MA,B )∗ = MtB,tA . One more ideal of operators will be investigated. Following Section 3.1.3 of [7], we say that an operator u : X → Y is completely p-nuclear (1 ≤ p ≤ ∞) if it has a factorization u = wMA,B v, with v : X → S ∞ , w : S p → Y , and MA,B : S ∞ → S p , with A, B ∈ S 2p . The completely p-nuclear norm νpo (u) is defined as the infimum of wcb A2p B2p vcb , running over all the factorizations as above. Denote the class of completely p-nuclear operators by Npo . The factorization results for completely p-summing operators show that, for any u, πpo (u) ≤ νpo (u). Furthermore, it was shown by M. Junge [7] that Πop (X, Y ) and Npo (Y, X) (1/p + 1/p ) are in trace duality, provided the spaces X and Y are finite dimensional. To finish this Introduction, recall that, by [16], Snp = Cnp ⊗h Rnp , where p Cn (respectively, Rnp ) is the column (row) subspace of Snp . Moreover, for any operator space X, Snp [X] = Cnp ⊗h X ⊗h Rnp . In the infinite dimensional case, we have S p = C p ⊗h Rp , and S p [X] = C p ⊗h X ⊗h Rp , where C p and Rp are the column and row subspaces of S p . It is known that Cn∞ = Cn , and Rn∞ = Rn (the usual column and row spaces). Moreover, Cnp∗ = Rnp = Cnp (1/p + 1/p = 1), and (Cnp0 , Cnp1 )θ = Cnp for 1/p = (1 − θ)/p0 + θ/p1 . These results remain true for C p and Rp .
2. Completely bounded norms Here we compute the c.b. norms of certain multiplication operators MA,B , with A, B ∈ B(2 )\{0}. Our main result is Theorem 2.1. For 1 ≤ p, q ≤ ∞, set 1/r = |1/p − 1/q|. Then MA,B CB(S p ,S q ) = A2r B2r . If either A or B does not belong to S r , then MA,B ∈ / CB(S p , S q ). The following lemma will be used extensively throughout this paper, in order to reduce our problems to finite dimensional ones. Denote by pn the projection onto the linear span of the first n elements of the canonical basis of 2 .
428
Oikhberg
IEOT
Lemma 2.2. Suppose A, B ∈ B(2 ), and E and F are either S r (1 ≤ r ≤ ∞) or B(2 ). Then MA,B CB(E,F ) = supn Mpn A,Bpn CB(E,F ). Moreover, for any 1 ≤ p < ∞, πpo (MA,B : E → F) = supn πpo (Mpn A,Bpn : E → F). A further technical lemma is needed. Below, Pn is the truncation operator T → pn T pn (in other words, Pn = Mpn ,pn ). p Lemma 2.3. Suppose 1 ≤ p < ∞, N ∈ N, and y ∈ SN [F ], where F is either q 2 p S (1 ≤ q ≤ ∞) or B( ). Then y = limn (ISN ⊗ Pn )(y).
Proof. The case of F = S q (with 1 ≤ q ≤ ∞) is easy. Indeed, the map z → (ISNp ⊗ Pn )(z) is contractive. Furthermore, we can approximate y by eleM p ments of the form i=1 ai ⊗ui , with ai ∈ SN and ui ∈ F. As limn Pn ui = ui for each i, we are done. The case of F = B(2 ) is slightly more complicated. Suppose, for the p sake of contradiction, that y ∈ SN [B(2 )] is such that y > 1, yet there 3 exists δ > 0 s.t. yn < (1 − δ) for any n (here, yn = (ISNp ⊗ Pn )(y)). Viewing Mn as the upper left corner of B(2 ), we can consider yn as an p [Mn ]. By Theorem 1.5 of [16], for each n there exist nonelement of SN 2p negative Cn , Dn ∈ SN and vn ∈ MN (Mn ), such that Cn = Dn = vn < 1 − δ, and yn = (Cn ⊗ I2n )vn (Dn ⊗ I2n ). By compactness, there ˜ exists n1 < n2 < . . . s.t. the sequences (Cnk ) and (Dnk ) converge to C, ˜ ˜ ˜ respectively D. Let C = σ + C and D = σ + D, where σ > 0 is so small that 2p . Let C and D lie in the unit ball of SN uk = (C −1 Cnk ⊗ I2n )vnk (Dnk D−1 ⊗ I2n ). k
k
Note that uk ≤ 1 − δ for each k, and ynk = (C ⊗ I2n )uk (D ⊗ I2n ), with k k uk MN (Mnk ) < 1. We can view (uk ) as a sequence in MN (B(2 )). Passing to a further subsequence if necessary, we can assume that (uk ) converges weak∗ to u in the unit ball of MN (B(2 )). Then weak∗ −lim ynk = (C ⊗I2 )u(D⊗I2 ) p [B(2 )]. On the other hand, weak∗ − lim ynk = y, lies in the unit ball of SN which is assumed to have norm greater than 1. This yields a contradiction. Proof of Lemma 2.2. We work with the completely p-summing norm, as the case of the c.b. norm is similar. We show that, if πpo (MA,B ) > 1, then p πpo (Mpn A,Bpn ) ≥ 1 for n sufficiently large. Find N ∈ N and x ∈ SN ⊗E s.t. x < 1, and (IMN ⊗ MA,B )xSNp [F ] = (I2N ⊗ A)x(I2N ⊗ B)SNp [F ] > 1. By Lemma 2.3, (I2N ⊗ A)x(I2N ⊗ B) = lim (ISNp ⊗ Pn )((I2N ⊗ A)x(I2N ⊗ B)) n
= lim (I2N ⊗ pn A)x(I2N ⊗ Bpn ), n
yielding the desired result.
The next lemma yields upper estimates for the c.b. norms of the multiplication operators.
Vol. 66 (2010)
Norms of Multiplication Operators
429
Lemma 2.4. For 1 ≤ p, q ≤ ∞ and 1/r = |1/p − 1/q|, MA,B CB(S p ,S q ) ≤ A2r B2r . The same estimate holds when S ∞ (when p or q equals ∞) is replaced by B(2 ). Proof. By Lemma 2.2, it suffices to show that MA,B CB(Snp ,Snq ) ≤ A2r B2r , for n × n matrices A and B. By [23], the natural identifications Φ : Sn2r → CB(Cnp , Cnq ) and Ψ : Sn2r → CB(Rnp , Rnq ) are isometries. In this notation, MA,B = Φ(A) ⊗ Ψ(B). Indeed, by polar decomposition we can assume that A and B are diagonal: write A = diag ((ai )ni=1 ), B = diag ((bi )ni=1 ). The matrix units in Snp are denoted by Eij (1 ≤ i, j ≤ n). Identify Cnp and Rnp with span[Ei1 : 1 ≤ i ≤ n] and span[E1j : 1 ≤ j ≤ n], respectively. The matrix unit Eij can be identified with Ei1 ⊗ E1j . For ξ = i ξi Ei1 , Aξ = i ai ξi Ei1 . The action of B can be described in a similar way. Thus MA,B Eij = Φ(A)Ei1 ⊗ Ψ(B)Ej1 . By the properties of tensor products, MA,B cb = Φ(A) ⊗ Ψ(B)CB(Cnp ⊗h Rpn ,Cnq ⊗h Rqn ) ≤ Φ(A)CB(Cnp ,Cnq ) Ψ(B)CB(Rpn ,Rqn ) ≤ A2r B2r ,
as desired. Lemma 2.5. For any A, B ∈ B(2 )\{0} MA,B CB(S 1 ,S ∞ ) = MA,B CB(S 1 ,B(2 )) = A2 B2 . Proof. We only need to consider the finite dimensional version: MA,B CB(Sn1 ,Sn∞ ) = A2 B2 .
By polar decomposition, we can assume that A = diag ((ai )ni=1 ) and B = diag ((bi )ni=1 ). Then MA,B Eij = ai bj Eij , where Eij are the matrix units. The canonical isometry between CB(E, F ) and E ∗ ⊗ F (for finite dimensional E and F ) allows us to identify the operator u = MA,B ∈ CB(Sn1 , Sn∞ ) with n u ˜ = i,j=1 ai bj Eij ⊗ Eij ∈ Mn ⊗ Mn . The space Mn ⊗ Mn can be identified with Mn2 , with Eij ⊗ Ek corresponding to Eik,j . Therefore, ucb
n n n n = ˜ u = ( ai Eii,1 )( bj E1,jj ) = ai Eii,1 · bj E1,jj i=1
=
n i=1
|ai |2
1/2
j=1 n
|bj |2
1/2
i=1
j=1
= A2 B2 .
j=1
Next we work with the “weighted transposition” operator. We denote by Θ the transposition operator A → tA on the space of n × n matrices. In other words, ΘEij = Eji , where (Eij )ni,j=1 are the matrix units. The same operator, acting from Snp to Snq , is denoted by Θp,q . Lemma 2.6. For 1 ≤ s ≤ 2, ΘCB(Sns ,Sns ) = 1. Together with Lemma 2.4, this lemma immediately implies
430
Oikhberg
IEOT
Corollary 2.7. Suppose 1 ≤ q ≤ ∞, 1 ≤ r ≤ 2, and 1/t = |1 − 1/r − 1/q|. Suppose, furthermore, that the sequences α = (αi )ni=1 and β = (βj )nj=1 satisfy α2t = β2t = 1. Then the operator T : S r → S q : Eji → αi βj Eij is completely contractive. Proof of Lemma 2.6. Clearly, Θcb ≥ ΘE11 = 1. To prove the converse inequality, note that, for s = 2, ΘCB(Sn2 ) = ΘB(Sn2 ) = 1, as the operator 1 Hilbert space Sn2 is 1-homogeneous. For s = 1, we identify Θ ∈ CB(Sn , Mn ) with the element ij Eji ⊗ Eij ∈ Mn (Mn ), which has norm 1. We complete the proof by using complex interpolation. Lemma 2.8. For every A, B ∈ B(2 )\{0}, MA,B CB(S ∞ ,S 1 ) = A2 B2 . Proof. As before, we can only consider the n × n case. The upper estimate for MA,B CB(S ∞ ,S 1 ) follows from Lemma 2.4. To prove the lower estimate, assume A = diag ((ai )ni=1 ) and B = diag ((bj )nj=1 ) with ai , bj non-negative, and A2 = B2 = 1. 1 Identify n the transposition map Θ : Sn → Mn : Eij → Eji with E ⊗ E ∈ M ⊗ M , which has norm 1. Thus, it suffices u ˜ = ji n n i,j=1 ij to show that ˜ v ≥ 1, where v˜ = (IMn ⊗ MA,B )˜ u=
n
Eij ⊗ aj bi Eji ∈ Mn ⊗ Sn1 .
i,j=1
But v˜ corresponds to v : Mn → Mn : Eji → aj bi Eij . To show that v is n u = completely contractive, note that (IMn ⊗ v)˜ i,j=1 Eij ⊗ aj bi Eij can 2 be identified with ( i bi Eii,1 )( j aj E1,jj ) ∈ Mn , the latter having norm one. Proof of Theorem 2.1. The case of p = q is trivial. Suppose p < q, and show that MA,B CB(S p ,S q ) = A2r B2r , where 1/r = 1/p − 1/q. The upper estimate for MA,B cb was established in Lemma 2.4. It suffices to obtain the lower estimate when A and B are positive n × n matrices. Let 1/r1 = 1−1/p and 1/r2 = 1/q −1/∞. Find A1 , B1 ∈ Sn2r1 and A2 , B2 ∈ Sn2r2 so that A1 2r1 = B1 2r1 = A2 2r2 = B2 2r2 = 1, A0 2 = A2r , and B0 2 = B2r , where A0 = A2 AA1 and B0 = B1 BB2 . Then MA0 ,B0 = MA2 ,B2 MA,B MA1 ,B1 , hence MA0 ,B0 CB(Sn1 ,Sn∞ ) ≤ MA1 ,B1 CB(Sn1 ,Snp ) MA,B CB(Snp ,Snq ) MA2 ,B2 CB(Snq ,Sn∞ ) . By Lemma 2.5, MA0 ,B0 CB(Sn1 ,Sn∞ ) = A0 2 B0 2 = A2r B2r . By Lemma 2.4, MA1 ,B1 and MA2 ,B2 are complete contractions. Thus, the centered inequality implies MA,B cb ≥ A2r B2r . The case of p > q is handled similarly, except that one uses Lemma 2.8 instead of Lemma 2.5.
Vol. 66 (2010)
Norms of Multiplication Operators
431
3. Completely p-summing norms Throughout this section, we use the notation w = w/(w − 1), for w ∈ [1, ∞] (in other words, w satisfies 1/w+1/w = 1). The main result of this section is Theorem 3.1. Consider MA,B as an operator from S p1 to S p2 . In each of the following situations, πpo (MA,B ) = νpo (MA,B ) = A2r B2r . 1. 2. 3. 4. 5.
p1 ≤ p ≤ p2 , 1/r = 1/p1 + 2/p − 1/p2 . p2 ≤ p ≤ p1 , 1/r = 1/p1 + 1/p2 . p2 ≤ min{2, p} ≤ max{2, p} ≤ p1 , 1/r = 1/p1 + 1/p2 . p1 ≤ p ≤ p2 ≤ 2, 1/r = 1/p2 + 2/p − 1/p1 . p1 = ∞, 1/r = 1/p + |1/p − 1/p2 |. In particular, if p1 = ∞ ≥ p ≥ p2 , then νpo (MA,B ) = πpo (MA,B ) = MA,B cb = A2p B2p .
Note that part (5) a strengthening of the Maurey Factorization Theorem for the class of multiplication operators (see Chapter 10 of [4] for the classical case, and [9, 10] for the recent non-commutative results). Observing that MI,I is the identity operator on Snq , we conclude: Corollary 3.2. For n ∈ N, q ∈ [2, ∞], and q/(q − 1) ≤ p ≤ q, πpo (ISnq ) = n2/p . More generally, for such p and q, πpo (ISnq 1 n2 ) = (n1 n2 )1/p . Remark 3.3. For p = 2,√this result follows from Chapter 6 of [16], where it was shown that π2o (IE ) = dim E for any finite dimensional operator space E. For p = 2, the completely p-summing norms of operators on OH, and some other homogeneous Hilbertian spaces, were recently computed by M. Junge, Q. Xu, and K.-L. Yew [11, 24]. In the classical case, the ideal norms (including the p-summing norms) of Iqn can be found in e.g. [2, 6]. Furthermore, by Lemma 5.2 of [3], π2 (id : Snp1 n2 → Snq 1 n2 ) =
√ max{1, min{n1 , n2 }1/q−1/2 } . n1 n2 max{1, min{n1 , n2 }1/p−1/2 }
By Lemma 2.2, it suffices to establish the finite dimensional case of Theorem 3.1. Accordingly, we restrict ourselves to the finite dimensional situation for the rest of this section. Furthermore, we assume that A = (diag (ai )ni=1 ) and B = (diag (bj )nj=1 ), with (ai ) and (bj ) positive. 1 ,p2 Lemma 3.4. In each of the following cases, νpo (MpA,B ) ≤ A2r B2r :
1.
1 1 1 1 1 1/p1 + 1/p2 , p ≥ p2 = + + − = 2/p + 1/p − 1/p , p ≤ p2 . r p1 p p p2 1 2
2. p1 ≤ 2 ≤ p2 ,
1 1 1 1 1 1/p1 + 1/p2 , p ≥ p2 = + + − = 2/p + 1/p1 − 1/p2 , p ≤ p2 . r p1 p p p2
432
Oikhberg
IEOT
1 ,p2 Lemma 3.5. In each of the following cases, πpo (MpA,B ) ≥ A2r B2r : 1. 1 1 1 1 1 1/p1 − 1/p2 + 2/p, p ≥ p1 = + − − = 1/p1 + 1/p2 , p ≤ p1 . r p2 p p p1
2.
1 1 1 1 1 p1 ≥ p 1/p1 + 1/p2 , = + − − = 2/p + 1/p2 − 1/p1 , p1 ≤ p. r p2 p p p1
3. p1 = ∞, 1/r = 1/p + |1/p − 1/p2 |. Proof of Lemma 3.4. (1) It suffices to consider the case of A2r = B2r = 1. Write A = A3 A2 A1 and B = B1 B2 B3 , where A1 2p1 = B1 2p1 = 1, A2 2p = B2 2p = 1, and A3 2q = B3 2q = 1, where 1/q = |1/p − 1/p2 |. We regard MA1 ,B1 , MA2 ,B2 , and MA3 ,B3 as acting from S p1 to S ∞ , from S ∞ to S p , and from S p to S p2 , respectively. Then νpo (MA,B ) ≤ MA1 ,B1 cb νpo (MA2 ,B2 )MA3 ,B3 cb ≤ 1.
p ,p
p1 ,p1 1 ,p2 1 2 = Θp2 ,p2 ◦ MtB, , Lemma 2.6, (2) The factorization MpA,B tA ◦ Θ and part (1) yield the result.
Proof of Lemma 3.5. By Lemma 2.2, it suffices to estimate πpo (MA,B ) from below when A and B are n × n matrices. By scaling and approximation, we can assume that A, B ∈ Mn , A2r = B2r = 1, A = diag ((ai )ni=1 ), and B = diag ((bj )nj=1 ), with ai , bj positive. (1) Let 1/s = 1/p2 + 1/p + |1/p1 − 1/p |, and note that 1/r + 1/s = 2. (1) (1) Find positive operators A1 = (diag (ai )ni=1 ) and B1 = (diag (bj )nj=1 ) s.t. A1 A1 = A1 2s = A2r = 1 = BB1 1 = B1 2s = B2r . (1)
(1)
Then MA1 B1 MAB Eij = ai ai bj bj Eij for any matrix unit Eij , hence the trace of MA1 B1 MAB (acting on Snp1 ) equals (1) (1) ai ai bj bj = A1 A1 BB1 1 = 1. i,j
By the trace duality, 1 = Tr (MA1 B1 MAB ) ≤ πpo (MAB )νpo (MA1 B1 ). By Lemma 3.4, νpo (MA1 B1 ) ≤ A1 2s B1 2s = 1, which yields the desired estimate for πpo (MAB ). (2) Suppose, for the sake of contradiction, that πpo (MA,B ) < 1. Let 1/s = |1/p2 − 1/p|, 1/t = |1/p − 1/p1 |, and q = min{p, p2 }. Find sequences (1) (1) (2) (2) a(1) = (ai )ni=1 , b(1) = (bj )nj=1 , a(2) = (ai )ni=1 , and b(2) = (bj )nj=1 in such a way that (1)
(2)
(1) (2)
(ai )2t = (ai )2s = (ai ai ai )p2 (1)
(2)
(1)
(2)
= (bj )2t = (bj )2s = (bj bj bj )q = 1
Vol. 66 (2010)
Norms of Multiplication Operators
433
(this is possible, since (ai )2r = (bj )2r = 1, and 1/r + 1/s + 1/t = 2/q). (1) (1) Consider u = ij ai bj Eji ⊗ Eij ∈ Snp ⊗ Snp1 . By Corollary 2.7, u ≤ 1. Let (1) (1) v = (ISnp ⊗ MA,B )(u) = ai ai bj bj Eji ⊗ Eij . ij
πpo (MA,B )
If < 1, then vSnp [Snp2 ] < 1. We achieve the desired contradiction by showing that vSnp [Snp2 ] ≥ 1. Suppose first p ≥ p2 . By [22], vSnp [Snp2 ] =
sup C2s ,D2s ≤1
(MC,D ⊗ ISnp2 )vSnp2 [Snp2 ] .
(2)
(3.1)
(2)
Taking C = (diag (bj )) and D = (diag (ai )), we have: vSnp [Snp2 ] ≥ =
n i,j=1
n
i,j=1
(1) (2)
(1) (2)
Eji ⊗ ai ai ai bj bj bj Eij Snp2 [Snp2 ] |ai ai ai bj bj bj |p2 (1) (2)
(1) (2)
(1) (2)
1/p2
(1) (2)
= (ai ai ai )p2 (bj bj bj )p2 = 1, which is the estimate we need. For p ≤ p2 , the proof proceeds along similar lines, with one exception: instead of (3.1), we use the inequality vSnp [Snp2 ] ≥ (I2n ⊗ C)v(I2n ⊗ D)Snp [Snp ] whenever C and D are in the unit ball of Sn2s (this follows from the fact that MC,D : Snp2 → Snp is completely contractive, and Corollary 1.2 of [16]). In particular, vSnp [Snp2 ] ≥ (I2n ⊗ A2 )v(I2n ⊗ B2 )Snp [Snp ] (1) (2) (1) (2) = ai ai ai bj bj bj Eji ⊗ Eij Snp [Snp ] (1) (2)
(1) (2)
= (ai ai ai )p (bj bj bj )p = 1. (3) In the case of p2 ≤ p, the inequality πpo (T ) ≥ T cb (valid for any operator T ), and the equality MA,B CB(S ∞ ,S p2 ) = A2p2 B2p2 (Theorem 2.1), yield the proof. For p2 ≥ p, Remark 5.10 of [16] yields a factorization MA,B = wMA1 ,B1 via Snp , with A1 2p = B2p = 1, and wcb = πpo (MA,B ). As the operators A and B are invertible, so are A1 and B1 , and o therefore, w = MA,B M−1 A1 ,B1 = MAA−1 ,B −1 B . Then wcb = πp (MA,B ) = 1
1
−1 AA−1 1 2s B1 B2s , with 1/s = 1/p−1/p2 . As 1/r = 2/p−1/p2 = 1/s+1/p, ∞,p2 −1 o A2r B2r ≤ AA−1 1 2s A1 2p B1 2p B1 B2s ≤ πp (MA,B ).
434
Oikhberg
IEOT
Proof of Theorem 3.1. For i, j ∈ {0, 1}, case (2i+j+1) of our theorem follows from Lemma 3.4(i) and Lemma 3.5(j). Case (5) follows from Lemma 3.4(1), Lemma 3.5(3), and Theorem 2.1.
4. Toeplitz Schur multipliers into 1 Here, we consider the c.b. norms of Toeplitz Schur multipliers from a matrix space into 1 (N × N). For a sequence m = (ms )s∈Z , we define the Toeplitz Schur multiplier Tm : (aij )i,j∈N → mj−i aij . The result below (essentially contained in [15]) describes the c.b. norm of Tm as acting from S 1 to 1 (N×N). Proposition 4.1. In the above notation, Tm cb = Tm = m1 . Proof. It is shown in [15] that Tm = m1 . It remains to show that Tm cb ≤ m1 . By an extreme point argument, it suffices to consider the case of ms = 1 for s = p, ms = 0 otherwise. For such an m, we have to show that IS 1 ⊗ Tm : S 1 [S 1 ] → S 1 [1 (N × N) ≤ 1. Note that S 1 [S 1 ] is isometric to S 1 (N × N). The extreme points of the closed unit ball of the latter space are of the form x ⊗ y, where x and y are unit vectors in 2 (N × N). Write 2 x = (xj ) and y = (yk ), where xj , yk ∈ , and j xj 2 = k yk 2 = 1. Then, xj ⊗ yk , j − k = p IS 1 ⊗ Tm (x ⊗ y) jk = 0, otherwise. If p ≥ 0, we have: ∞ xj yk = xk+p yk Tm (x ⊗ y) = ≤
j−k=p ∞
xk+p 2
k=1 ∞
1/2
k=1
yk 2
1/2
≤ 1.
k=1
The case of p ≤ −1 is handled similarly.
Next we consider a Schur multiplier from the upper triangular trace class matrices to 1 . More precisely, we consider the subspace HS 1 of S 1 , consisting of all trace class matrices (aij ) s.t. aij = 0 for i > j. A sequence (ms )s≥0 defines a multiplier Tm : HS 1 → 1 (N × N) : (aij ) → (mj−a aij ). Following [15], we define r(q+1) ∞ 2 1/2 ρ(m) = |m0 |2 + |m1 |2 + sup |ms |2 . r≥1 q=1
s=rq+1
Proposition 4.2. In the above notation, there exists a constant c such that ρ(m) cρ(m) ≥ Tm cb ≥ Tm ≥ √ . 3
√ Sketch of proof. By Theorem 3.3 of [15], cρ(m) ≥ Tm ≥ ρ(m)/ 3. In fact, that proof can be easily modified to yield cρ(m) ≥ Tm cb .
Vol. 66 (2010)
Norms of Multiplication Operators
435
Finally, we consider the Schur multipliers from B(2 ) (or S ∞ ) to (N × N). The space of bounded (completely bounded) Schur multipliers between matrix spaces E and F shall be denoted by B S (E, F ) (resp. CB S(E, F )). The space 1 (2 ) is viewed as the space of matrices a = (aij ), equipped with the norm a1 (2 ) = i ( j |aij |2 )1/2 . The space t1 (2 ) is equipped with the norm at1 (2 ) = ta1 (2 ) . Finally, we define the space 1
X = 1 (2 ) + t1 (2 ). That is, aX = inf b1 (2 ) + ct1 (2 ) . a=b+c
In this notation, we have: Proposition 4.3. The following spaces are isomorphic (via the natural identity): (1) X , (2) B S (B(2 ), 1 (N × N)), (3) CB S (B(2 ), 1 (N × N)), (4) B S (S ∞ , 1 (N × N)), (5) CB S (S ∞ , 1 (N × N)). Proof. It is easy to see that the spaces (2) and (4) are isometric, as are (3) and (5). It is shown in [15] that (1) and (3) are isomorphic. Moreover, id : CB S (B(2 ), 1 (N × N)) → B S (B(2 ), 1 (N × N)) is a contraction. By duality, it suffices to show that id : X → CB S (c0 (N×N), S 1 ) is a contraction. By symmetry, we need to prove that id : 1 (2 ) → CB S (c0 (N × N), S 1 ) is a contraction. By an extreme point argument, it suffices to show that Sa CB S (c0 (N×N),S 1 ) ≤ 1 whenever the matrix a = (aij ) is such that aij = 0 for i > 1, and j |a1j |2 = 1. This, in turn, is equivalent to proving that, for any family (xij ) of contractive n × n matrices, j a1j E1j ⊗ x1j S 1 ⊗Mn ≤ 1. The matrix units E1jspan a copy of C (the column space) in S 1 . Thus, we need to show that j a1j Ej1 ⊗ x1j S ∞ ⊗Mn ≤ 1. However, a1j Ej1 ⊗ x1j 2S ∞ ⊗Mn = |a1j |2 x∗1j x1j ≤ |a1j |2 x∗1j x1j ≤ 1, j
j
j
as desired.
5. Schur multipliers into Schatten spaces In this section we consider Schur multipliers from S p (or B(2 )) to S q . Theorem 5.1. Consider 1 ≤ p ≤ 2, a matrix φ = (φij ), and the following statements: 1. We have πpo (Sφ ) ≤ 1, where Sφ is viewed as acting from B(2 ) (or S ∞ ) to S p . 2. There exist sequences a = (ai )i∈N and b = (bj )j∈N in the unit ball of 2p , such that |φij | ≤ |ai bj | for any i, j ∈ N.
436
Oikhberg
IEOT
3. There exists a matrix ψ, the sequences a = (ai )i∈N and b = (bj )j∈N in the unit ball of 4 , and the sequences a = (ai )i∈N and b = (bj )j∈N in the unit ball of 2r (1/r = 1/p − 1/2), such that |ψij | ≤ |ai bj |, and Sφ = Mdiag (ai ),diag (bi ) Sψ . Here, π2o (Sψ ) ≤ 1 (Sψ is viewed as a map from B(2 ) to S 2 ), and furthermore, Mdiag (ai ),diag (bi ) CB(S 2 ,S p ) ≤ 1.
Then (1) ⇒ (2) ⇔ (3). For p = 2, (2) ⇔ (1).
By Theorem 5.5 and Remark 5.6, the implication (2) ⇒ (1) fails for p = 1. We do not know whether it is true for p ∈ (1, 2). The next result shows that, for p = 2, the c.b. norms and the completely 2-summing norms of Schur multipliers from B(2 ) to S 2 are equivalent. Theorem 5.2. There is a constant C such that π2o (Sφ ) ≤ CSφ cb for any Schur multiplier Sφ from B(2 ) (or S ∞ ) to S 2 . Remark 5.3. The classical Grothendieck Theorem tells us that any bounded operator from C(K) to a Hilbert space is 2-summing. This is no longer true in the non-commutative case. Indeed, by Corollary 5 of of [8], for every N there exists a projection P from B(2 ) to OHN s.t. P cb ≤ c N/(1 + log √N ). On o o the other hand, by Theorem 6.13 of [16], π2 (P ) ≥ π2 (idOHN ) = N . See [9, 10] for some positive results in this direction. Considering multiplication operators from L∞ (µ) to Lp (µ), instead of Schur multipliers, we complement the known results by computing the completely p-summing norm precisely. Theorem 5.4. Suppose µ is a σ-finite measure, and φ is a µ-measurable function. Denote by Mφ the operator of