Integral Equations and Operator Theory - Volume 68

Integr. Equ. Oper. Theory 68 (2010), 1–21 DOI 10.1007/s00020-010-1798-3 Published online June 8, 2010 c Springer Basel A...

Author: C. Tretter (Chief Editor)

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Integr. Equ. Oper. Theory 68 (2010), 1–21 DOI 10.1007/s00020-010-1798-3 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Essential Spectra of a 3 × 3 Operator Matrix and an Application to Three-Group Transport Equations Afif Ben Amar, Aref Jeribi and Bilel Krichen Abstract. In this paper we study spectral properties of a 3 × 3 block operator matrix with unbounded entries and with domain consisting of vectors which satisfy certain relations between their components. It is shown that, under certain conditions, this block operator matrix defines a closed operator, and the essential spectra of this operator are determined. These results are applied to a three-group transport equation. Mathematics Subject Classification (2010). Primary 39B42; Secondary 47A55, 47A53, 47A10. Keywords. Operator matrix, closability, essential spectra, transport operator.

1. Introduction In this work we are concerned with by a 3 × 3 block operator matrix ⎛ A ⎝D G

the essential spectra of operators deﬁned B E H

⎞ C F⎠ L

(1.1)

where the entries of the matrix are in general unbounded operators. The operator (1.1) is deﬁned on (D(A) ∩ D(D) ∩ D(G)) × (D(B) ∩ D(E) ∩ D(H)) × (D(C) ∩ D(F ) ∩ D(L)). Observe that this operator need not be closed. We need some standard notation from Fredholm theory. Let X and Y be two Banach spaces. By an operator T from X into Y , we mean a linear operator with domain D(T ) ⊂ X and range R(T ) ⊂ Y . By C(X, Y ) we denote the set of all closed, densely deﬁned linear operators from X into Y , by L(X, Y ) the Banach space of all bounded linear operators from X into Y . If T ∈ C(X, Y ) then ρ(T ) denotes the resolvent set of T , α(T ) the dimension of the kernel N (T ) and β(T ) the codimension of R(T ) in Y . The classes of

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Fredholm, upper semi-Fredholm and lower semi-Fredholm operators from X into Y are Φ(X, Y ) := {T ∈ C(X, Y ) : α(T ) < ∞, R(T ) closed in Y, β(T ) < ∞}, Φ+ (X, Y ) := {T ∈ C(X, Y ) : α(T ) < ∞, R(T ) closed in Y } and Φ− (X, Y ) := {T ∈ C(X, Y ) : β(T ) < ∞, R(T ) closed in Y }, respectively. If X = Y , the sets L(X, Y ), C(X, Y ), Φ(X, Y ), Φ+ (X, Y ) and Φ− (X, Y ) are replaced, by L(X), C(X), Φ(X), Φ+ (X) and Φ− (X), respectively. For an operator T ∈ Φ+ (X) or Φ− (X), its index is i(T ) := α(T ) − β(T ). An operator F ∈ L(X, Y ) is called a Fredholm perturbation, upper semi-Fredholm perturbation or lower semi-Fredholm perturbation, if T + F ∈ Φ(X, Y ), T + F ∈ Φ+ (X, Y ) or T + F ∈ Φ− (X, Y ) whenever T ∈ Φ(X, Y ), T ∈ Φ+ (X, Y ) or T ∈ Φ− (X, Y ), respectively. The sets of Fredholm, upper semi-Fredholm and lower semi-Fredholm perturbations are denoted by F(X, Y ), F+ (X, Y ) and F− (X, Y ), respectively. The intersections, Φ(X, Y )∩ L(X, Y ), Φ+ (X, Y )∩L(X, Y ), Φ− (X, Y )∩L(X, Y ), are denoted by Φb (X, Y ), Φb+ (X, Y ), Φb− (X, Y ), respectively. If we replace Φ(X, Y ), Φ+ (X, Y ) and Φ− (X, Y ) by the sets Φb (X, Y ), Φb+ (X, Y ) and Φb− (X, Y ), we obtain the sets b b (X, Y ) and F− (X, Y ). These classes of operators were introF b (X, Y ), F+ duced and investigated by Gohberg et al. [10]. Recently, it was shown in [4], that F(X, Y ), F+ (X, Y ) and F− (X, Y ) are closed subsets of L(X, Y ). An operator T ∈ L(X, Y ) is said to be weakly compact if T (M ) is relatively weakly compact in Y for every bounded subset M ⊂ X. The family of weakly compact operators from X into Y is denoted by W(X, Y ). If X = Y , this family of operators, denoted by W(X) := W(X, X), is a closed two-sided ideal of L(X) containing that of compact operators on X (see [9,11]). Note that if X is a Banach space with the Dunford–Pettis property (see [8]), then W(X) ⊂ F+ (X) ∩ F− (X). where F− (X) := F− (X, X) and F+ (X) := F+ (X, X). As an example, any L1 -space has the Dunford–Pettis property. If X is a Banach space and T ∈ C(X), various notions of essential spectra have been deﬁned in the literature. In this work, we are concerned with the following essential spectra: σew (T ) σes (T ) σeap (T ) σeδ (T )

:= {λ ∈ C : λ − T ∈ / Φ(X)}, := C\ρes (T ), := C\ρeap (T ), := C\ρeδ (T ),

where ρes (T ) := {λ ∈ C : λ − T ∈ Φ(X), i(λ − T ) = 0}, ρeap (T ) := {λ ∈ C : λ − T ∈ Φ+ (X), i(λ − T ) ≤ 0} ,

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and ρeδ (T ) := {λ ∈ C : λ − T ∈ Φ− (X), i(λ − T ) ≥ 0} . We call σew , σes , σeap and σeδ , the Wolf, Schechter, Rakoˇcevi´c and Schmoeger essential spectrum, respectively (see for instance [13,14,24–28, 33]). For an operator T ∈ C(X), it holds σew (T ) ⊂ σes (T ), σes (T ) = σeap (T ) ∪ σeδ (T ). During the last years, e.g. the papers [2,29] were devoted to the study of the Wolf essential spectrum of operators deﬁned by a block operator matrix A B A0 := C D that acts on the product X × Y of Banach spaces. An account of the research and a wide panorama of methods to investigate the spectrum of block operator matrices are presented by Tretter [30–32]. In general, the operators occurring in A0 are unbounded and A0 need not be a closed nor a closable operator, even if its entries are closed. However, under some conditions A0 is closable and its closure A can be determined. In the theory of unbounded block operator matrices, the Frobenius– Schur factorization is a basic tool to study the spectrum and various spectral properties. This was ﬁrst recognized by Nagel [22,23] and, independently and under slightly different assumptions, later in [2]. In fact, Atkinson et al. [2] are concerned with the Wolf essential spectrum, and they consider the situation where the domains satisfy the conditions D(A) ⊂ D(C) and D(B) ⊂ D(D). Moreover, the compactness of the operators (λ−A)−1 (see [2]) or C(λ−A)−1 and ((λ − A)−1 B)∗ (see [29]) for some (and hence for all) λ ∈ ρ(A) was assumed, whereas in [6], it was only assumed that (λ − A)−1 for λ ∈ ρ(A) belongs to a non-zero two-sided closed ideal I(X) ⊂ F(X) of L(X). In [20], Moalla, Damak and Jeribi extended these results to a large class of operators, described their essential spectra, and applied these results to describe the essential spectra of two-group transport operators with general boundary conditions in Lp -spaces. In [16], Jeribi, Moalla and Walha treated a 3 × 3 block operator matrix (1.1) on a Banach space. It was shown that under certain conditions, this block operator matrix deﬁnes a closable operator and its essential spectra are determined. However, to determine the essential spectra of the closure of (1.1), they have to know the essential spectrum of the entry A. In [3], B´ atkai, Binding, Dijksma, Hryniv and Langer consider a 2 × 2 block operator matrix and describe its essential spectrum under the assumption that D(A) ⊂ D(C), that the intersection of the domains of the operators B and D is sufﬁciently large, and that the domain of the operator matrix is deﬁned by an additional relation of the form ΓX x = ΓY y between the two components of its elements. Moreover, they suppose that the operator C(A1 − λ)−1 is compact for some (and hence for all) λ ∈ ρ(A1 ), where A1 := A|D(A)∩N (ΓX ) . However, in classical transport theory in L1 -spaces, this operator is only weakly compact (see Sect. 4). Recently in [5], Charﬁ and

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Jeribi extended the results of [3]; they are concerned with the investigation of the Rakoˇcevi´c essential spectrum σeap and the Schmoeger essential spectrum σeδ of A. In the present paper we extend these results to 3 × 3 block operator matrices (1.1), where the domain is deﬁned by additional relations of the form ΓX x = ΓY y = ΓZ z between the three components of its elements. We focus on the investigation of the closability and the description of the essential spectra. Compared with the papers [6,16,20], we can determine the essential spectra of the closure of (1.1) without knowing the essential spectra of the operator A but only that of one of its restrictions A1 , and we give an application to transport theory which is more general than the one considered in [16]. In fact, in the Banach space X1 × X1 × X1 where X1 := L1 ([−a, a] × [−1, 1]; dxdξ), a > 0, we consider an operator that describes the neutron transport in a plane-parallel domain with width 2a, or the transfer of unpolarized light in a plane-parallel atmosphere of optical thickness 2a (see Sect. 4). An outline of the paper is as follows. In Sect. 2 we describe the closure of the operator in (1.1) under certain assumptions on its entries, in Sect. 3 we study some essential spectra of this closure, and in Sect. 4, as an application, we describe the essential spectra of a three-group transport operator.

2. Closability and Closure of the Block Operator Matrix (1.1) Let X, Y , Z and W be Banach spaces. We consider the block operator matrix (1.1) in the space X × Y × Z, that is the linear operator A acts in X, E in Y and L in Z, B from Y to X, etc. Further, we suppose that operators ΓX , ΓY , ΓZ are given, acting from X, Y, Z, respectively, into W . Throughout this and the next section, we will consider the following assumptions. (H1) The operator A is densely deﬁned and closable. Then D(A), equipped with the graph norm x A = x + Ax can be completed to a Banach space XA which coincides with D(A), the domain of the closure of A in X. (H2) D(A) ⊂ D(ΓX ) ⊂ XA and ΓX : XA −→ W is a bounded mapping. Denote by ΓX the extension by continuity which is a bounded operator from XA into W . (H3) D(A) ∩ N (ΓX ) is dense in X and the resolvent set of the restriction A1 := A|D(A)∩N (ΓX ) is not empty: ρ(A1 ) = ∅. Remark 2.1. It follows from (H1), (H3) that ΓX (D(A1 )) = {0} and that the operator A1 is closed. Therefore D(A1 ) is a closed subset of XA . (H4)

The operator B is densely deﬁned and for some (and hence for all) μ ∈ ρ(A1 ) the operator (A1 − μ)−1 B is bounded. In fact, for μ, λ ∈ ρ(A1 ), we have by the resolvent identity (A1 − λ)−1 B − (A1 − μ)−1 B = (λ − μ)(A1 − λ)−1 (A1 − μ)−1 B.

(H5) (H6)

D(A) ⊂ D(D) ⊂ XA and D is a closable operator from XA into Y . D(A) ⊂ D(G) ⊂ XA and G is a closable operator from XA into Z.

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The closed graph theorem and the assumptions (H5), (H6) imply that for λ ∈ ρ(A1 ) the operators F1 (λ) := D(A1 − λ)−1 and F2 (λ) := G(A1 − λ)−1 are bounded from X into Y and X into Z, respectively. Under the assumptions (H1)–(H3), B´ atkai et al. [3], have proved the decomposition D(A) = D(A1 ) ⊕ N (A − μ) for every μ ∈ ρ(A1 ). It is easy to see that the restriction of ΓX to N (A − μ) is injective. Denote the inverse of ΓX |N (A−μ) by Kμ := (ΓX |N (A−μ) )−1 . By Remark 2.1, we can write Kμ : ΓX (D(A)) −→ N (A − μ) ⊂ D(A). For μ ∈ ρ(A1 ), and if assumptions (H1)–(H3) are satisﬁed, then (A − μ)x = (A1 − μ)(I − Kμ ΓX )x. (H7)

For some μ ∈ ρ(A1 ), Kμ is a bounded operator from ΓX (D(A)) into X; its extension by continuity to ΓX (D(A)) is denoted by K μ .

Since for x ∈ N (A − μ), x A = (1 + μ) x , the operator K μ : ΓX (D(A)) −→ XA is bounded and for z ∈ ΓX (D(A)), we have AK μ z = μK μ z, ΓX K μ z = z. Lemma 2.2. [3] For every λ, μ ∈ ρ(A1 ) and under the assumptions (H1)– (H3), we have Kμ − Kλ = (μ − λ)(A1 − μ)−1 Kλ . Since (A1 − λ) and (A1 − μ)−1 are boundedly invertible, Kμ is closable if and only if Kλ is closable, in which case we have K μ − K λ = (μ − λ)(A1 − μ)−1 K λ , hence K μ = (A1 − λ)(A1 − μ)−1 K λ . In the following, denote S(μ) := E + DKμ ΓY − D(A1 − μ)−1 B. The operator S(μ) is deﬁned on the domain: Y1 = {y ∈ D(B) ∩ D(E) : ΓY y ∈ ΓX (D(A))}.

(2.1)

For μ ∈ ρ(A1 ), denote the restriction of S(μ) to the set Y1 ∩ N (ΓY ) by S1 (μ). Lemma 2.3. For every λ, μ ∈ ρ(A1 ) we have S1 (μ) − S1 (λ) = −(μ − λ)D(A1 − μ)−1 (A1 − λ)−1 B.

(2.2)

Proof. Let λ, μ ∈ ρ(A1 ) S(μ) − S(λ) = [E + DKμ ΓY − F1 (μ)B] − [E + DKλ ΓY − F1 (λ)B] = (μ − λ)D(A1 − μ)−1 [Kμ ΓY − (A1 − λ)−1 B] = (μ − λ)F1 (μ)[Kμ ΓY − (A1 − λ)−1 B]. For y ∈ D(S1 (μ)), we have ΓY y = 0 and the relation (2.2) is veriﬁed. (H8)

For some μ ∈ ρ(A1 ), the operator S1 (μ) is closed.

Remark 2.4. By assumptions (H4) and (H5), the operator F1 (μ)(A1 − λ)−1 B is bounded on its domain, so if S1 (μ) is closed for some μ ∈ ρ(A1 ) then it is closed for all such μ.

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It was shown in [16, Remark 3.1] that if A1 and E generate C0 semi-groups, and B and D are bounded, then there exists μ ∈ C such that μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)). For μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), the set Y1 can be decomposed as follows: Y1 = D(S1 (μ)) ⊕ N (S(μ) − μ). Following [3], denote the inverse of ΓY |N (S(μ)−μ) by Jμ := (ΓY |N (S(μ)−μ) )−1 , Jμ : ΓY (Y1 ) −→ N (S(μ) − μ) ⊂ Y1 . Assume that for some μ ∈ ρ(A1 ), Jμ is bounded from ΓY (Y1 ) into Y and its extension by continuity to ΓY (Y1 ) is denoted by J μ . D(B) ∩ D(E) ⊂ D(ΓY ), D(B) ∩ D(H) ⊂ D(ΓY ), the set Y1 is dense in Y and the restriction of ΓY to Y1 is bounded as an operator from Y 0 into W . We denote the extension by continuity of ΓY |Y1 to Y by ΓY . (H10) L is densely deﬁned and closed with non-empty resolvent set, i.e., ρ(L) = ∅. (H11) For some (and hence for all) μ ∈ ρ(A1 ), the operator G2 (μ) := (A1 − μ)−1 C is bounded. (H12) D(C) ∩ D(F ) ∩ D(L) ⊂ D(ΓZ ), the set

(H9)

Z1 := {z ∈ D(C) ∩ D(F ) ∩ D(L) : ΓZ z ∈ ΓY (Y1 )}

(H13)

is dense in Z and the restriction of ΓZ to Z1 is bounded as an operator from Z into W . Denote the extension by continuity of ΓZ |Z1 to 0 Z by ΓZ . For some (and hence for all) μ ∈ ρ(A1 ), the operator F − D(A1 − μ)−1 C is closable and its closure F − D(A1 − μ)−1 C is bounded.

In the following we use these assumptions to show the closability of the operator in (1.1) and to describe the closure. The main idea is, as in the 2 × 2 case, a factorization of the 3 × 3 matrix with a diagonal matrix of Schur complements in the middle and invertible factors to the right and to the left (see for example [34]). We start with some lemmas that will be needed in the sequel. Lemma 2.5. For some μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)) and under the assumptions (H7), (H9), we have (S(μ) − μ) = (S1 (μ) − μ)(I − Jμ ΓY ); I − Jμ ΓY is the projection from Y1 on D(S1 (μ)) parallel to N (S(μ) − μ). Proof. For y ∈ Y1 we can write y = (I − Jμ ΓY )y + Jμ ΓY y. Observe that y1 = (I − Jμ ΓY )y ∈ D(S1 (μ)) and y2 = Jμ ΓY y ∈ N (S(μ) − μ), then (S(μ) − μ)y = (S1 (μ) − μ)y1 = (S1 (μ) − μ)(y − y2 ) = (S1 (μ) − μ)(I − Jμ ΓY )y.

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In the remainder of this section, assume that (H1)–(H13) are satisﬁed. We consider the Banach space X × Y × Z and deﬁne the operator L0 as follows: ⎧⎛ ⎞ ⎫ x ∈ D(A) ⎨ x ⎬ D(L0 ) = ⎝ y ⎠ : y ∈ D(B) ∩ D(E) , ΓX x = ΓY y = ΓZ z , ⎩ ⎭ z z ∈ D(C) ∩ D(F ) ∩ D(L) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x Ax + By + Cz x L0 ⎝ y ⎠:=⎝ Dx + Ey + F z ⎠, ⎝ y ⎠ ∈ D(L0 ). z Gx + Hy + Lz z One way to solve a system of linear equations is by row reduction Gauss elimination that transforms the block matrix into triangular form. By analogy with the case of a 2 × 2 operator matrix (see [2,31]), we introduce the following operators: G1 (μ) G3 (μ) Θ(μ) F3 (μ) S2 (μ)

:= := := := :=

−Kμ ΓY + (A1 − μ)−1 B, −Jμ ΓZ + (S1 (μ) − μ)−1 (F − D(A1 − μ)−1 C), H + GKμ ΓY − G(A1 − μ)−1 B, Θ(μ)(S1 (μ) − μ)−1 , L − F2 (μ)C + Θ(μ)(Jμ ΓZ − (S1 (μ) − μ)−1 (F − F1 (μ)C)).

Our aim is to describe the closure L of L0 . We start with the following Frobenius–Schur type factorization of L0 . Denote by Tμ the operator deﬁned for every μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)) by ⎞ ⎛ ⎞ ⎛ I G1 (μ) G2 (μ) I 0 0 I 0 ⎠ Δ(μ) ⎝ 0 I G3 (μ) ⎠, Tμ := ⎝ F1 (μ) F2 (μ) F3 (μ) I 0 0 I where

⎛

A1 − μ Δ(μ) = ⎝ 0 0

0 S1 (μ) − μ 0

⎞ 0 ⎠. 0 S2 (μ) − μ

Clearly, ⎧⎛ ⎞ ⎛ I ⎨ x D(Tμ ) = ⎝ y ⎠ = ⎝ 0 ⎩ z 0

−G1 (μ) I 0

⎫ ⎞⎛ ⎞ x x ∈ D(A1 ) ⎬ Ω(μ) −G3 (μ) ⎠ ⎝ y ⎠ , y ∈ Y1 ∩ N (ΓY ) ⎭ I z z ∈ Y2

where Y2 := {z ∈ D(C) ∩ D(F ) ∩ D(L) : ΓZ z ∈ ΓY (Y1 )} and Ω(μ) = G1 (μ)G3 (μ) − G2 (μ). Lemma 2.6. If μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), then L0 − μ = Tμ .

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⎛ ⎞ x Proof. It is easy to check that, for any ⎝ y ⎠ ∈ D(L0 ), z ⎛ ⎞ ⎛ ⎞ x x Tμ ⎝ y ⎠ = (L0 − μ) ⎝ y ⎠. z z It follows that Tμ is an extension of the operator L0 − μ, i.e., L0 − μ ⊂ Tμ . ⎛ ⎞ x It remains to show that D(Tμ ) ⊂ D(L0 ). Let ⎝ y ⎠ ∈ D(Tμ ). Then z x = x − G1 (μ)y + [G1 (μ)G3 (μ) − G2 (μ)] z, y = y − G3 (μ)z, z = z. Observe that z ∈ Y2 ⊂ D(C)∩D(F )∩D(L), y = y−G3 (μ)z ∈ N (S(μ)−μ) ⊂ Y1 , Y1 ⊂ D(B)∩D(E) and x = x−G1 (μ)y +(G1 (μ)G3 (μ)−G2 (μ))z ∈ D(A). It is easy to verify the boundary conditions ΓX x = ΓY y = ΓZ z, therefore L0 − μ = Tμ . Lemma 2.7. If μ ∈ ρ(A1 )∩ρ(S1 (μ)), λ ∈ ρ(A1 )∩ρ(S1 (λ)) and under assumptions (H1)–(H3), (H7), (H9), we have Jμ = (S1 (μ) − μ)−1 (S1 (λ) − λ)Jλ . Proof. Recall that Jμ = (ΓY |N (S(μ)−μ) )−1 : W ⊃ ΓY (Y1 ) −→ N (S(μ) − μ) ⊂ Y1 . Let w ∈ ΓY (Y1 ), set y = Jμ w and y = Jλ w, then S(μ)y = μy, ΓY y = w, S(λ)y = λy and ΓY y = w. Note that (y − y ) ∈ N (ΓY ) ∩ Y1 = D(S1 (μ)). Now we observe the action of (S1 (μ) − μ) on y − y : (S1 (μ) − μ)(y − y ) = −(S1 (μ) − μ)y = −S1 (μ)y + μy . By Lemma 2.3, we can write (S1 (μ) − μ)(y − y ) = − S1 (λ) − (μ − λ)F1 (μ)(A1 − λ)−1 B y + μy = (μ − λ)y +(μ − λ)D(A1 − μ)−1 (A1 −λ)−1 By , y − y = (S1 (μ)−μ)−1 × (μ − λ)I + (μ − λ)D(A1 − μ)−1 (A1 − λ)−1 B y = (S1 (μ) − μ)−1 [(μ − λ)I − S1 (μ) + S1 (λ)] y = (S1 (μ) − μ)−1 (S1 (λ) − λ) − I y . It follows that Jμ − Jλ = [(S1 (μ) − μ)−1 (S1 (λ) − λ) − I]Jλ .

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Since for μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)) and λ ∈ ρ(A1 ) ∩ ρ(S1 (λ)), the operator (S1 (μ) − μ)−1 (S1 (λ) − λ) is boundedly invertible, Jμ is closable if and only if Jλ is closable. Moreover, J μ = (S1 (μ) − μ)−1 (S1 (λ) − λ)J λ and we obtain the relation J μ − J λ = (S1 (μ) − μ)−1 (S1 (λ) − λ) − I J λ . Lemma 2.8. If the operator Θ(μ) = H + GKμ ΓY − G(A1 − μ)−1 B is closable for some μ ∈ ρ(A1 ), then it is closable for all such μ. Proof. Let μ, λ ∈ ρ(A1 ).

Θ(μ) − Θ(λ) = G(Kμ − Kλ ) − G (A1 − μ)−1 − (A1 − λ)−1 B = (μ − λ)G(A1 − μ)−1 Kλ ΓY − (A1 − λ)−1 B .

Here ΓY is bounded on Y1 by assumption (H9). From (H7), (H4) and (H6) it follows that the operators Kλ , (A1 − λ)−1 B and G(A1 − μ)−1 , respectively, are bounded. From Lemma 2.8, it is easy to check that 0

Θ(μ) − Θ(λ) = (μ − λ)G(A1 − μ)−1 [K λ ΓY − (A1 − λ)−1 B]. Using assumption (H13) we see that, for any μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), 0

J μ ΓZ − (S1 (μ) − μ)−1 (F − D(A1 − μ)−1 C) is bounded as an operator from Z into W , therefore 0

Θ(μ)(J μ ΓZ − (S1 (μ) − μ)−1 (F − D(A1 − μ)−1 C)) is bounded everywhere. Now, by assumption (H11), the operator (A1 − μ)−1 C is bounded, since G is closable, also G(A1 − μ)−1 C is bounded. By assumption (H10), L is densely deﬁned and closed. Hence S2 (μ) is closable. In fact, the next lemma shows that the closedness of S2 (μ) does not depend on the choice of μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)). Lemma 2.9. If for some μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)) the operator S2 (μ) is closable, then it is closable for all such μ. Proof. Let μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)) and λ ∈ ρ(A1 ) ∩ ρ(S1 (λ)). Using Lemma 2.7 and the resolvent identity we ﬁnd S2 (μ) − S2 (λ) = −(μ − λ)[F2 (μ)G2 (λ) − F3 (μ)F1 (λ)G2 (μ)] +(F3 (λ) − F3 (μ))(S1 (λ) − λ)G3 (λ). Since the operators Fi , i = 1, 2, 3 are bounded everywhere and the operator G2 (λ) is bounded on its domain, and, on the other hand, S1 (λ) − λ is closed and by assumptions (H13) and (H12), the operator G3 (λ) is bounded on its domain, the closedness of the operator S2 (μ) does not depend on the choice of μ.

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Denote the closure of S2 (μ) by S 2 (μ). Then we have S 2 (μ) = S 2 (λ) + (μ − λ) F3 (μ)F1 (λ)G2 (μ) − F2 (μ)G2 (λ) 0 + (F3 (μ)−F3 (λ))(S1 (λ)−λ) J λ ΓZ − (S1 (λ) − λ)−1 (F − F1 (λ)C) (2.3) In the following we consider the operators 1 (μ) := −K μ Γ0 + (A1 − μ)−1 B, G Y G2 (μ) := (A1 − λ)−1 C, 3 (μ) := −J μ Γ0 + (S1 (μ) − μ)−1 (F − D(A1 − μ)−1 C). G Z

(2.4) (2.5) (2.6)

Now we can formulate the main result of this section. Theorem 2.10. Under assumptions (H1)–(H13), the operator L0 is closable if and only if S2 (μ) is closable for some μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)). In this case the closure L of L0 is given by ⎛ ⎞ 0 0 A1 − μ ⎠ Gr (μ), 0 S1 (μ) − μ L = μI + Gl (μ) ⎝ 0 0 0 S 2 (μ) − μ where

⎛

I Gl (μ) := ⎝ F1 (μ) F2 (μ)

0 I F3 (μ)

⎞ ⎛ 0 I 0 ⎠ and Gr (μ) := ⎝ 0 I 0

1 (μ) G I 0

⎞ 2 (μ) G 3 (μ) ⎠ G I

or, spelled out, ⎧⎛ ⎫ ⎞⎛ ⎞ 1 (μ) x x ∈ D(A1 ) Ω(μ) ⎨ I −G ⎬ ⎠ ⎝ ⎠ ∩ N (Γ(Y )) y y ∈ Y , D(L) = ⎝ 0 I −G3 (μ) 1 ⎩ ⎭ z ∈ Y2 z 0 0 I ⎞ ⎛ ⎛ ⎞ 1 (μ)y + μΩ(μ)z 1 (μ)y + Ω(μ)z A1 x − μG x−G ⎠ = ⎝ Dx + S1 (μ)y − μG 3 (μ)z ⎠ . 3 (μ)z L⎝ y−G z Gx + Θ(μ)y + S 2 (μ)z where 3 (μ) − G 2 (μ). 1 (μ)G Ω(μ) := G Proof. Let μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)). Under the assumptions of the theorem, the external operators Gl (μ) and Gr (μ) are boundedly invertible, hence L0 − μ is closable if and only if S2 (μ) is closable. ⎞⎛ ⎞ ⎛ ⎛ ⎞ 1 (μ) x Ω(μ) I −G x 3 (μ) ⎠ ⎝ y ⎠ ∈ D(L), we Finally, for ⎝ y ⎠ = ⎝ 0 I −G z z 0 0 I obtain

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⎞ ⎛ ⎞ 1 (μ)y + μΩ(μ)z (A1 − μ)x + μx − μG x 3 (μ)z ⎠ L ⎝ y ⎠ = ⎝ F1 (μ)(A1 − μ)x + (S1 (μ) − μ)y + μy − μG z F2 (μ)(A1 − μ)x + F3 (μ)(S1 (μ) − μ)y + S 2 (μ)z ⎛ ⎞ 1 (μ)y + μΩ(μ)z A1 x − μG 3 (μ)z ⎠. = ⎝ Dx + S1 (μ)y − μG Gx + Θ(μ)y + S 2 (μ)z ⎛

3. Wolf, Schechter, Rakoˇcevi´c and Schmoeger Essential Spectra of L Having obtained the closure L of the operator L0 , in this section we discuss its essential spectra. As a ﬁrst step we prove the following stability lemma. Lemma 3.1. Let μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)). If the sets Φb (Y, X), Φb (Z, X) and Φb (Z, Y ) are not empty, and if F1 (μ) ∈ F b (X, Y ), F2 (μ) ∈ F b (X, Z) and F3 (μ) ∈ F b (Y, Z), then σew (S1 (μ)), σes (S1 (μ)), σew (S 2 (μ)) and σes (S 2 (μ)) do not depend on the choice of μ. Proof. Using (2.2), assumption (H4), [4, Theorem 3.1] and [10, Theorem 3.2 (ii)], we infer that σew (S1 (μ)) = σew (S1 (λ)). Hence σew (S1 (μ)) does not depend on μ. Clearly, [F3 (μ)F1 (μ)−F2 (μ)](A1 − λ)−1 C ∈ F b (Z) and (F3 (μ)− 0 F3 (λ))(S1 (λ) − λ)[J λ ΓZ − (S1 (λ) − λ)−1 (F − D(A1 − λ)−1 C)] ∈ F b (Z), so in the same way we can deduce from (2.3) and [4, Theorem 3.1] that σew (S 2 (μ)) = σew (S 2 (λ)). The same reasoning yields the claim for σes (.). In the sequel, we denote for μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)) by M (μ) the following operator: ⎞ ⎛ 2 (μ) 1 (μ) G 0 G ⎜ 1 (μ) 2 (μ) + G 3 (μ) ⎟ M (μ) := ⎝ F1 (μ) F1 (μ)G F1 (μ)G ⎠. F2 (μ) F2 (μ)G1 (μ) + F3 (μ) F2 (μ)G2 (μ) + F3 (μ)G3 (μ) Theorem 3.2. Suppose that the assumptions (H1)–(H13) are satisfied. If for some μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), we have F1 (μ) ∈ F b (X, Y ), F2 (μ) ∈ F b (X, Z), F3 (μ) ∈ F b (Y, Z) and M (μ) ∈ F(X × Y × Z), then σew (L) = σew (A1 ) ∪ σew (S1 (μ)) ∪ σew (S 2 (μ)) and σes (L) ⊆ σes (A1 ) ∪ σes (S1 (μ)) ∪ σes (S 2 (μ)). Moreover, if the sets C\σew (A1 ), C\σew (S1 (μ)) are connected and ρ(S1 (μ)) = ∅, then σes (L) = σes (A1 ) ∪ σes (S1 (μ)) ∪ σes (S 2 (μ)).

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Proof. Fix μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)). Then for λ ∈ C we have L − λI = (L − μI) + (μ − λ)I ⎛ 0 A1 − λ S1 (μ) − λ = Gl (μ) ⎝ 0 0 0

⎞ 0 ⎠ Gr (μ) − (μ − λ)M (μ). 0 S 2 (μ) − λ

Since M (μ) is a Fredholm perturbation, Gl (μ) and Gr (μ) are boundedly invertible. Then L − λI is Fredholm if and only if the diagonal matrix is Fredholm. It follows that σew (L) = σew (A1 )∪σew (S1 (μ))∪σew (S 2 (μ)). Moreover, i(L − λI) = i(A1 − λI) + i(S1 (μ) − λI) + i(S 2 (μ) − λI).

(3.1)

If i(L − λI) = 0, then one of the terms in (3.1) is non-zero, hence σes (L) ⊆ σes (A1 ) ∪ σes (S1 (μ)) ∪ σes (S 2 (μ)). By assumption (H3), we have ρ(A1 ) = ∅. Since the set C\σew (A1 ) is connected, by [1, Theorem 2.1], σew (A1 ) = σes (A1 ). Using the same argument as in Lemma 3.1, it follows that σew (S1 (μ)) = σes (S1 (μ)) and i(S1 (μ) − λI) = 0 for each λ ∈ C\σew (S1 (μ)). If λ ∈ C\σes (L), then λ ∈ C\σew (A1 ), λ ∈ C\σew (S1 (μ)) and λ ∈ C\σew (S 2 (μ)). Further, i(L − λI) = i(S 2 (μ) − λI), hence λ ∈ C\σes (S 2 (μ)) and, ﬁnally, σes (L) = σes (A1 )∪σes (S1 (μ))∪σes (S 2 (μ)). In the next result, we discuss the Rakoˇcevi´c and the Schmoeger essential spectrum. First we prove again a stability lemma. Denote by ρb the union of the resolvent set ρ and the discrete spectrum σd , i.e., the set of all points which are regular or isolated eigenvalues with a ﬁnite-dimensional Riesz projection. b b Lemma 3.3. (i) If F1 (μ) ∈ F+ (X, Y ), F2 (μ) ∈ F+ (X, Z) and F3 (μ) ∈ b F+ (Y, Z) for some μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), then σeap (S 2 (μ)) does not depend on the choice of μ.

(ii)

b b b If F1 (μ) ∈ F− (X, Y ), F2 (μ) ∈ F− (X, Z) and F3 (μ) ∈ F− (Y, Z) for some μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), then σeδ (S 2 (μ)) does not depend on the choice of μ.

Proof. (i) Using [4, Theorem 2.1], we deduce that the difference S 2 (μ)−S 2 (λ) b (Z, Z). Now from Theorem (3.1)(i) and Remark 3.3 in [15] in (2.3) is in F+ we conclude that σeap (S 2 (μ)) does not depend on the choice of μ. (ii) This assertion can be proved in a similar way as (i). Theorem 3.4. Assume that (H1)–(H13) are satisfied. b (X, Y ), F2 (μ) ∈ (i) If for some μ ∈ ρb (A1 ) ∩ ρ(S1 (μ)), we have F1 (μ) ∈ F+ b b F+ (X, Z) and F3 (μ) ∈ F+ (Y, Z), then

σeap (L) ∩ ρb (A1 ) ∩ ρ(S1 (μ)) = σeap (S 2 (μ)) ∩ ρb (A1 ) ∩ ρ(S1 (μ)).

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b (X, Y ), F2 (μ) ∈ If for some μ ∈ ρb (A1 ) ∩ ρ(S1 (μ)), we have F1 (μ) ∈ F− b b F− (X, Z) and F3 (μ) ∈ F− (Y, Z), then

σeδ (L) ∩ ρb (A1 ) ∩ ρ(S1 (μ)) = σeδ (S 2 (μ)) ∩ ρb (A1 ) ∩ ρ(S1 (μ)). Proof. We have: ρb (A1 ) ∩ ρ(S1 (μ)) = [ρ(A1 ) ∪ σd (A1 )] ∩ ρ(S1 (μ)) = [ρ(A1 ) ∩ ρ(S1 (μ))] ∪ [σd (A1 ) ∩ ρ(S1 (μ))] . First case: If μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), it is clear that the external factors Gl (μ) and Gr (μ) are bounded and have bounded inverses. Therefore it follows from [27, Theorem 6.4] that L − μI is an upper semi-Fredholm operator if and only if S 2 (μ) − μ is. Furthermore, the use of [21, Theorem 12] and [5, Remark 2.1] allows us conclude that i(L − μI) = i(S 2 (μ) − μ), hence σeap (L) = σeap (S 2 (μ)). Now, by Lemma 3.3, we deduce that σeap (L) = σeap (S 2 (λ)). Second case: If μ ∈ σd (A1 ) ∩ ρ(S1 (μ)), then there exists an ε > 0 such that for the disk D(μ, 2ε) we have D(μ, 2ε)\{μ} ⊂ ρ(A1 ) ∩ ρ(S1 (μ)). 1 := A1 +εPμ , where Pμ is the ﬁnite rank Riesz projection of A1 corDenote A responding to μ. We can easily check that D(μ, ε)\{μ} ⊂ ρ(A1 ) ∩ ρ(S1 (μ)) ∩ 1 ). ρ(A 1 ). Until further notice we ﬁx λ ∈ ρ(A1 ) ∩ ρ(S1 (μ)) ∩ ρ(A ⎛ ⎞ ⎛ B C Pμ 0 A ⎝ 0 L0 := D E F ⎠ = L0 + ε ⎝ 0 0 0 G H L 0 , we obtain For the closure L of L ⎛

Pμ L = L + ε ⎝ 0 0

0 0 0

Denote ⎞ 0 0⎠. 0

⎞ 0 0⎠. 0

= σeap (L) and Clearly, L is a ﬁnite rank perturbation of L, therefore σeap (L) i(L − λ) = i(L − λ). Next, we apply the obtained result of the ﬁrst part of the proof to the Denote operator L. 1 −λ)−1 C + Θ(λ)[ 1 − λ)−1 C)]. S2 (λ) := L − G(A Jλ ΓZ − (S1 (λ)−λ)−1 (F −D(A λ , Jλ , Θ(λ) and S1 (λ) are the operators deﬁned as Kλ and Jλ with Here K Hence A replaced by A. λ w = x ⇐⇒ x ∈ N (A − λ), ΓX x = w K and Jλ w = y ⇐⇒ y ∈ N (S(λ) − λ),

ΓY y = w.

If S2 (λ) is closable, then its perturbation S2 (λ) is also closable. Denote by S2 (λ) its closure. We claim the following:

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If the assumptions (H1)–(H13) are satisﬁed and S2 (λ) is closable, then the operator S2 (λ) − S2 (λ) is of ﬁnite rank and σeap (S2 (λ)) = σeap (S2 (λ)). Indeed, observe ﬁrst that 1 −λ)−1 C]−[F − D(A1 −λ)−1 C] = −εD(A 1 − λ)−1 Pλ (A1 − λ)−1 C. [F −D(A The assumptions (H5), (H11), and [21, Theorem 12] imply that this difference is of ﬁnite rank. Moreover, λ ΓY − D(A 1 − λ)−1 B) S1 (λ) − S1 (λ) = (DKλ ΓY − D(A1 − λ)−1 B) − (DK λ )ΓY − D((A1 − λ)−1 − (A 1 − λ)−1 )B. = D(Kλ − K λ , therefore λ = εPλ (A1 − λ)−1 K Evidently, Kλ − K λ ΓY + (A 1 − λ)−1 B]. S1 (λ) − S1 (λ) = εD(A1 − λ)−1 Pλ [K It follows that (S1 (λ)−λ)−1 −(S1 (λ) − λ)−1 = (S1 (λ) − λ)−1 [S1 (λ) − S1 (λ)](S1 (λ) − λ)−1 . Applying [21, Theorem 12], (H4) and (H5), we deduce that this difference is of ﬁnite rank. On the other hand, note that λ ΓY − εG(A1 − λ)−1 Pλ (A 1 − λ)−1 B Ψ(λ) := Θ(λ) − Θ(λ) = −εF2 (λ)Pλ K λ ΓY + (A 1 − λ)−1 B]. = −εF2 (λ)Pλ [K By the same argument, (H6) and [21, Theorem 12] imply that the operator Ψ(λ) is of ﬁnite rank. The operator 1 − λ)−1 C) Υ(λ) := Jλ ΓZ − (S1 (λ) − λ)−1 (F − D(A is bounded on its domain (here Υ(λ) = −G3 (λ), see Sect. 2), hence Υ(λ) Θ(λ) − Θ(λ)Υ(λ) = [Θ(λ) + Ψ(λ)]Υ(λ) − Θ(λ)Υ(λ) = Θ(λ)[Υ(λ) − Υ(λ)] + Ψ(λ)Υ(λ). By assumption, S2 (λ) is closable in Z, so its perturbation S2 (λ) is closable, and we conclude that S2 (λ) − S 2 (λ) is of ﬁnite rank. Therefore σeap (S2 (λ)) = σeap (S 2 (λ)). Now, using Lemma 3.3, we deduce that σeap (S2 (λ)) is independent of 1 ) ∩ ρ(S1 (μ)), we ρ(A1 ). Applying the ﬁrst part of this proof for μ ∈ ρ(A ﬁnd σeap (L) = σeap (S2 (λ)), and ﬁnally σeap (L) = σeap (L) = σeap (S2 (μ)) = σeap (S2 (λ)) = σeap (S 2 (λ)). The proof of (ii) is similar. Theorem 3.5. Suppose that the assumptions (H1)–(H13) are satisfied. (i)

b If for some μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), we have F1 (μ) ∈ F+ (X, Y ), F2 (μ) ∈ b b (X, Z), F3 (μ) ∈ F+ (Y, Z) and M (μ) ∈ F+ (X, Y, Z), then F+

σeap (L) ⊆ σeap (A1 ) ∪ σeap (S1 (μ)) ∪ σeap (S 2 (μ)).

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Moreover, if the sets C\σew (A1 ), C\σew (S1 (μ)), C\σew (S 2 (μ)) and C\σew (L) are connected and ρ(S1 (μ)), ρ(S 2 (μ)) and ρ(L) are not empty, then σeap (L) = σeap (A1 ) ∪ σeap (S1 (μ)) ∪ σeap (S 2 (μ)). (ii)

b If for some μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)), we have F1 (μ) ∈ F− (X, Y ), F2 (μ) ∈ b b F− (X, Z), F3 (μ) ∈ F− (Y, Z) and M (μ) ∈ F− (X, Y, Z), then

σeδ (L) ⊆ σeδ (A1 ) ∪ σeδ (S1 (μ)) ∪ σeδ (S 2 (μ)). Moreover, if the sets C\σew (A1 ), C\σew (S1 (μ)), C\σew (S 2 (μ)), and C\σew (L) are connected and ρ(S1 (μ)), ρ(S 2 (μ)) and ρ(L) are not empty, then σeδ (L) = σeδ (A1 ) ∪ σeδ (S1 (μ)) ∪ σeδ (S 2 (μ)). Proof. (i) Fix μ ∈ ρ(A1 ) ∩ ρ(S1 (μ)). As in the proofs of Theorems 3.2 and 3.4, we ﬁnd σeap (L) ⊆ σeap (A1 ) ∪ σeap (S1 (μ)) ∪ σeap (S 2 (μ)). Since C\σew (A1 ), C\σew (S1 (μ)), C\σew (S 2 (μ)) and C\σew (L) are connected and ρ(S1 (μ)), ρ(S 2 (μ)) and ρ(L) are not empty, the result follows from [5, Proposition 2.3] together with [20, Theorem 3.2]. The proof of (ii) is similar.

4. Application to a Three-Group Transport Equation In this section we apply Theorems 3.2 and 3.5 to a three-group transport operator in an L1 -space. Let a > 0 and X1 := L1 ([−a, a] × [−1, 1]; dxdξ) ,

X := Y := Z := X1 .

We consider the operator matrix L = T + K, where

⎛

⎜ ⎜ ⎜ Tψ = ⎜ ⎜ ⎝ ⎛

−ξ

T1 =: ⎝ 0 0 and

⎞

∂ψ1 − σ1 (ξ)ψ1 ∂x −ξ

0 0 0 T2 0

0

⎞⎛ ⎞ 0 ψ1 0 ⎠ ⎝ ψ2 ⎠ TQ ψ3 ⎛

0 K = ⎝ K21 K31

0

∂ψ2 − σ2 (ξ)ψ2 ∂x 0

K12 K22 K32

⎞ K13 0 ⎠ K33

0 ∂ψ3 − σ3 (ξ)ψ3 −ξ ∂x

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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where Kij , i, j = 1, 2, 3, (i, j) = (1, 1), (2, 3), are bounded linear operators in X1 , deﬁned by 1 Kij u(x, ξ) :=

κij (x, ξ, ξ )u(x, ξ )dξ ,

u ∈ X1 ;

(4.1)

−1

the kernels κij : [−a, a]×[−1, 1]×[−1, 1] −→ R are assumed to be measurable. The operators Ti , i = 1, 2, are the so-called streaming operators in X1 , deﬁned by ∂ϕ (x, ξ) − σi (ξ)ϕ(x, ξ), ϕ ∈ W1 , ∂x , and TQ where W1 is the partial Sobolev space W1 = ϕ ∈ X1 : ξ ∂ϕ ∈ X 1 ∂x Ti ϕ(x, ξ) = −ξ

is deﬁned on D(TQ ) = {ϕ ∈ W1 : ϕi = Qϕo } by ∂ϕ (x, ξ) − σ3 (ξ)ϕ(x, ξ); ∂x here σj ∈ L∞ [−1, 1], j = 1, 2, 3, and Q is a linear (boundary) operator. We consider the boundary spaces TQ ϕ(x, ξ) = −ξ

o o × X2,1 X1o := L1 ({−a} × [−1, 0]; |ξ|dξ) × L1 ({a} × [0, 1]; |ξ|dξ) =: X1,1

and i i × X2,1 . X1i := L1 ({−a} × [0, 1]; |ξ|dξ) × L1 ({a} × [−1, 0]; |ξ|dξ) =: X1,1

It is well known that any function ϕ ∈ W1 has traces on the spacial boundary sets {−a} × (−1, 0) and {a} × (1, 0) in X1o and X1i , respectively (see [7]). They are denoted by ϕo and ϕi , and represent the outgoing and the incoming ﬂuxes, respectively (“o” `for outgoing and “i” `for incoming). The function ϕ(x, ξ) represents the number density of gas particles with position x and the cosine of direction of propagation ξ; that is, ξ is the cosine of the angle between the velocity vector and the x-direction of the particles. The functions σj , j = 1, 2, 3, which are supposed to be measurable, are called collision frequencies. Let λ∗j ∈ R be deﬁned by λ∗j := lim inf |ξ|→0 σj (ξ), j = 1, 2, 3. We deﬁne the operator L on the domain ⎫ ⎧⎛ ⎞ ⎬ ⎨ ψ1 D(L) = ⎝ ψ2 ⎠ : ψ1 ∈ W1 , ψ2 ∈ D(T2 ), ψ3 ∈ D(TQ ), ψ1i = ψ2i = ψ3i . ⎭ ⎩ ψ3 and introduce the boundary operators ΓX , ΓY and ΓZ : ΓX : X1 → X1i , ΓX ψ1 = ψ1i ,

ΓY : X1 → X1i , ΓY ψ2 = ψ2i

and ΓZ : X1 → X1i , ΓZ ψ3 = Qψ3o . Let A1 be the operator deﬁned by D(A1 ) := {ψ1 ∈ W1 , ψ1i = 0} and A1 := T1 . Solving the equation (λ − T1 )ψ1 = ϕ1 for ψ1 ∈ D(A1 ), it follows that for λ

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such that Reλ > −λ∗1 we have λ ∈ ρ(A1 ). For these values of λ the operator Kλ is chosen as follows: u ∈ X1i , Kλ u = ϕ1 :⇐⇒ (T1 − λ)ϕ1 = 0, ϕ1 ∈ W1 , ϕi1 = u. It is easy to check that Kλ is bounded and Kλ ≤ [(Reλ + λ∗1 )]−1 . The domain Y1 from (2.1) is given by Y1 = {ψ2 ∈ W1 : ψ2i ∈ ΓX (W1 )}. The operator Jλ is deﬁned on the domain D(Jλ ) := {ψ2i : ψ2 ∈ Y1 } and u ∈ X1i , Jλ u = ϕ2 :⇐⇒ ϕ2 ∈ Y1 , ΓY ϕ2 = u, (S(λ) − λ)ϕ2 = 0. The equation (S(λ) − λ)ϕ2 = 0 leads to (T2 + K22 − λ)ϕ2 + K21 Kλ ϕi2 − K21 (T1 − λ)−1 K12 ϕ2 = 0, hence

(T2 + K22 − λ) − K21 (T1 − λ)−1 K12 ϕ2 = −K21 Kλ u.

Denote by rσ the spectral radius. For λ ∈ ρ(T1 ) ∩ ρ(T2 ) such that rσ ((T2 − λ)−1 K22 ) < 1, we have λ ∈ ρ(T1 ) ∩ ρ(T2 ) ∩ ρ(T2 + K22 ). Moreover, if rσ (T2 + K22 − λ)−1 K21 (T1 − λ)−1 K12 < 1, then the operator Jλ is given by n Jλ = − (T2 + K22 − λ)−1 K21 (T1 − λ)−1 K12 (T2 + K22 − λ)−1 K21 Kλ n≥0

and hence bounded. Now the operator S1 (λ), deﬁned on D(S1 (λ)) = {ϕ2 ∈ W1 : ϕi2 = 0}, is given by S1 (λ) = (T2 + K22 ) − K21 (T1 − λ)−1 K12 . To prove that the operator M (λ) as deﬁned in Sect. 3 is weakly compact on i (λ), i = 1, 2, 3 (see X1 × X1 × X1 we prove that the operators Fi (λ) and G Sect. 2 and (2.4)–(2.6)) are weakly compact. Notice that the collision operators Kij , i, j = 1, 2, 3, (i, j) = (1, 1), (2, 3), deﬁned in (4.1), act only on the velocity v, so x ∈ [−a, a] may be seen simply as a parameter. Thus, we consider Kij as a function Kij : x ∈ [−a, a] −→ Kij (x) ∈ L (L1 ([−1, 1]; dξ)) . Definition 4.1. [19] A collision operator K of the form (4.1) is said to be regular if the set {κ(x, ·, ξ ), (x, ξ ) ∈ (−a, a) × (−1, 1)} is a relatively weakly compact subset of the space L1 ((−1, 1), dξ). Lemma 4.2. [17] Let λ ∈ ρ(A1 ). (i) If the operators K21 , K31 are non-negative and their kernels κ31 (x,ξ,ξ ) |ξ | −λ∗1 , the

(ii)

κ21 (x,ξ,ξ ) , |ξ |

define regular operators, then for any λ ∈ C satisfying Reλ > operators F1 (λ) = K21 (A1 − λ)−1 , F2 (λ) = K31 (A1 − λ)−1 , respectively, are weakly compact on X1 . If K13 , K12 are non-negative regular operators, then for any λ ∈ C with Reλ > −λ∗1 the operators (A1 − λ)−1 K13 , (A1 − λ)−1 K12 , respectively, are weakly compact in X1 .

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Theorem 4.3. If Q is a weakly compact positive operator, K12 , K21 , K13 , ) and K31 , K33 are non-negative regular operators and, in addition, κ21 (x,ξ,ξ |ξ | κ31 (x,ξ,ξ ) |ξ |

define regular operators on X1 , then

σew (L) = σes (L) = σeap (L) = σeδ (L) = {λ ∈ C : Reλ ≤ −min(λ∗1 , λ∗2 , λ∗3 )} . Proof. It was shown in [18,20] that σeap (A1 ) = σeδ (A1 ) = σew (A1 ) = σes (A1 ) = {λ ∈ C : Reλ ≤ −λ∗1 }. If λ ∈ ρ(T1 ), the operator S1 (λ) is given by S1 (λ) = (T2 +K22 )−F1 (λ)K12 . By Lemma 4.2, S1 (λ)−T2 −K22 is weakly compact on X1 . Then [4, Theorem 3.1] and [15, Remark 3.3] allow us to conclude that σew (S1 (λ)) = σew (T2 + K22 ) and σeap (S1 (λ)) = σeap (T2 + K22 ). On the other hand, for λ ∈ ρ(T2 ) such that rσ ((T2 − λ)−1 K22 ) < 1, hence λ ∈ ρ(T2 ) ∩ ρ(T2 + K22 ), we have n (λ − T2 )−1 K22 (λ − T2 )−1 . (λ − T2 − K22 )−1 − (λ − T2 )−1 = n≥1

Since K22 is a non-negative regular operator, it follows from [12, Lemma 3.1] that the operator (λ − T2 − K22 )−1 − (λ − T2 )−1 is weakly compact on X1 . Then from [12, Theorem 2] we obtain σew (T2 + K22 ) = σew (T2 ) = {λ ∈ C : Reλ ≤ −λ∗2 },

(4.2)

and [15, Remark 3.3] implies σeap (T2 + K22 ) = σeap (T2 ) = {λ ∈ C : Reλ ≤ −λ∗2 }.

(4.3)

Now the operator Θ(λ) in Sect. 2 can be written as follows: Θ(λ) = K32 + K31 Kλ ΓY − K31 (T1 − λ)−1 K22 . Obviously, Θ(λ) is bounded on X1 . Since Q is weakly compact, so is ΓZ . With Lemma 4.2 we conclude that also the operator Jλ ΓZ + (S1 (λ) − λ)−1 F1 (λ)K13 is weakly compact in X1 . On the other hand, for λ ∈ ρ(A1 ) ∩ ρ(S1 (λ)) the operator S2 (λ) is given by S2 (λ) = (TQ + K33 ) − F2 (λ)K13 + Θ(λ)(Jλ ΓZ + (S1 (λ) − λ)−1 F1 (λ)K13 ). It follows from Lemma 4.2 and the fact that W(X1 ) is a two-sided ideal of L(X1 ), that the operator S2 (λ) − TQ − K33 is weakly compact in X1 . Hence σew (TQ + K33 ) = σew (S2 (λ)),

(4.4)

σeap (TQ + K33 ) = σeap (S2 (λ)),

(4.5)

and

In the same way as above, it follows that σew (TQ + K33 ) = σew (TQ ) = {λ ∈ C : Reλ ≤ −λ∗3 },

(4.6)

σeap (TQ + K33 ) = σeap (TQ ) = {λ ∈ C : Reλ ≤ −λ∗3 }.

(4.7)

and

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Applying Theorems 3.2 and 3.5 and Eqs. (4.2)–(4.7) we get σew (L) = σeap (L) = {λ ∈ C : Reλ ≤ −min(λ∗1 , λ∗2 , λ∗3 )}. The same reasoning implies the corresponding result for the essential spectrum σes (L) and σeδ (L). Remark 4.4. If the domain of the streaming operator T2 is chosen with a boundary condition, then for the collision operator K23 we have K23 = 0. Acknowledgements The authors would like thank Professor Heinz Langer who has made some valuable comments and suggestions which have improved the manuscript greatly.

References [1] Abdmouleh, F., Jeribi, A.: Gustafson, Weidmann, Kato, Wolf, Schechter, Browder, Rakoˇcevi´c and Schmoeger Essential spectra of the sum of two bounded operators and application to transport operator. Math. Nachr. (2010, in press) [2] Atkinson, F.V., Langer, H., Mennicken, R., Shkalikov, A.A.: The essential spectrum of some matrix operators. Math. Nachr. 167, 5–20 (1994) [3] B´ atkai, A., Binding, P., Dijksma, A., Hryniv, R., Langer, H.: Spectral problems for operator matrices. Math. Nachr. 278(12–13), 1408–1429 (2005) [4] Ben Amar, A., Jeribi, A., Mnif, M.: Some results on Fredholm and semiFredholm perturbations and applications. Preprint (2008) [5] Charfi, S., Jeribi, A.: On a characterization of the essential spectra of some matrix operators and applications to two-group transport operators. Math. Z. 262(4), 775–794 (2009) [6] Damak, M., Jeribi, A.: On the essential spectra of some matrix operators and applications. Electron. J. Differ. Equ. 11, 1–16 (2007) [7] Dautray, R., Lions, J.L.: Analyse Math´ematique et Calcul Num´erique, vol. 9. Masson, Paris (1988) [8] Dunford, N., Pettis, B.J.: Linear operations on summable functions. Trans. Am. Math. Soc. 47, 323–392 (1940) [9] Dunford, N., Schwartz, J.T.: Linear operators, Part I. General Theory. Interscience, New York (1958) [10] Gohberg, I.C., Markus, A.S., Fel’dman, I.A.: Normally solvable operators and ideals associated with them. Am. Math. Soc. Transl. Ser. 2 61, 63–84 (1967) [11] Goldberg, S.: Unbounded Linear Operators. Theory and Applications. McGraw-Hill, New York (1966) [12] Jeribi, A.: Quelques remarques sur les op´erateurs de Fredholm et application a l’´equation de transport. C. R. Acad. Sci. Paris S´er. I, Math. 325(1), 43–48 ` (1997) [13] Jeribi, A.: A characterization of the Schechter essential spectrum on Banach spaces and applications. J. Math. Anal. Appl. 271(2), 343–358 (2002)

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[14] Jeribi, A., Mnif, M.: Fredholm operators, essential spectra and application to transport equation. Acta Appl. Math. 89(1–3), 155–176 (2006) [15] Jeribi, A., Moalla, N.: A characterization of some subsets of Schechter’s essential spectrum and Singular transport equation. J. Math. Anal. Appl. 358(2), 434–444 (2009) [16] Jeribi, A., Moalla, N., Walha, I.: Spectra of some block operator matrices and application to transport operators. J. Math. Anal. Appl. 351(1), 315–325 (2009) [17] Jeribi, A., Walha, I.: Gustafson, Weidmann, Kato, Schechter and Browder Essential spectra of some matrix operator and application to two-group transport equation. Math. Nachr. (2010, in press) [18] Latrach, K., Jeribi, A.: Some results on Fredholm operators, essential spectra, and application. J. Math. Anal. Appl. 225(2), 461–485 (1998) [19] Lods, B.: On linear kinetic equations involving unbounded cross-sections. Math. Methods Appl. Sci. 27(9), 1049–1075 (2004) [20] Moalla, N., Damak, M., Jeribi, A.: Essential spectra of some matrix operators and application to two-group transport operators with general boundary conditions. J. Math. Anal. Appl. 323(2), 1071–1090 (2006) [21] M¨ uller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory: Advances and Applications, vol. 139. Birkh¨ auser Basel and Boston (2003) [22] Nagel, R.: Towards a “matrix theory” for unbounded operator matrices. Math. Z. 201(1), 57–68 (1989) [23] Nagel, R.: The spectrum of unbounded operator matrices with non-diagonal domain. J. Funct. Anal. 89(2), 291–302 (1990) [24] Rakoˇcevi´c, V.: On one subset of M. Schechter’s essential spectrum. Mat. Vesnik. 5(18)(33), 389–391 (1981) [25] Rakoˇcevi´c, V.: Approximate point spectrum and commuting compact perturbations. Glasg. Math. J. 28(2), 193–198 (1986) [26] Schechter, M.: On the essential spectrum of an arbitrary operator, I.. J. Math. Anal. Appl. 13, 205–215 (1966) [27] Schechter, M.: Principles of Functional Analysis. Academic Press, New York (1971) [28] Schmoeger, C.: The spectral mapping theorem for the essential approximate point spectrum. Colloq. Math. 74(2), 167–176 (1997) [29] Shkalikov, A.A.: On the essential spectrum of some matrix operators. Math. Notes 58(5–6), 1359–1362 (1995) [30] Tretter, C.: Spectral issues for block operator matrices. In: Differential Equations and Mathematical Physics (Birmingham AL, 1999). AMS/IP Stud. Adv. Math., vol. 16, pp. 407–423. Amer. Math. Soc., Providence (2000) [31] Tretter, C.: Spectral Theory of Block Operator Matrices and Applications. Impe. Coll. Press, London (2008) [32] Tretter, C.: Spectral inclusion for unbounded block operator matrices. J. Funct. Anal. 11, 3806–3829 (2009) [33] Wolf, F.: On invariance of the essential spectrum under a change of the boundary conditions of partial differential operators. Indag. Math. 21, 142–147 (1959) [34] Zhang, F. (ed.): The Schur Complement and its Applications. Numerical Methods and Algorithms, vol. 4. Springer, New York (2005)

Vol. 68 (2010)

Essential Spectra of a 3 × 3 Operator Matrix

Afif Ben Amar, Aref Jeribi (B) and Bilel Krichen D´epartement de Math´ematiques Facult´e des Sciences de Sfax Universit´e de Sfax Route de Soukra Km 3.5 B.P. 1171, 3000 Sfax Tunisie e-mail: [email protected] Afif Ben Amar e-mail: [email protected] Received: September 6, 2009. Revised: March 12, 2010.

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Integr. Equ. Oper. Theory 68 (2010), 23–60 DOI 10.1007/s00020-010-1783-x Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Solutions to Operator Equations on Hilbert C ∗-Modules II Xiaochun Fang and Jing Yu Abstract. In this paper, we study the solvability of the operator equations A∗ X + X ∗ A = C and A∗ XB + B ∗ X ∗ A = C for general adjointable operators on Hilbert C ∗ -modules whose ranges may not be closed. Based on these results we discuss the solution to the operator equation AXB = C, and obtain some necessary and sufﬁcient conditions for the existence of a real positive solution, of a solution X with B ∗ (X ∗ + X)B ≥ 0, and of a solution X with B ∗ XB ≥ 0. Furthermore in the special case that R(B) ⊆ R(A∗ ) we obtain a necessary and sufﬁcient condition for the existence of a positive solution to the equation AXB = C. The above results generalize some recent results concerning the equations for operators with closed ranges. Mathematics Subject Classification (2000). Primary 46L08; Secondary 47A05. Keywords. Hilbert C ∗ -module, solution, positive solution, real positive solution.

1. Introduction and Preliminary Much progress has been made on the study of the solvability of equations for ﬁnite matrices. As a generalization to inﬁnite case, equations for Hilbert space operators and even for adjointable Hilbert C ∗ -module operators have attracted more and more attention (see [1–9,12,10,11,13,17–20,22,23,25,26, 28–33]). The equation A∗ X + X ∗ A = C was studied for matrices by Braden [1], and for the Hilbert space operators by Djordjevi´c [13]. Yuan [32] studied the solvability of the operator equation A∗ XB + B ∗ X ∗ A = C for ﬁnite matrices under the condition that R(B ∗ ) is contained in R(A∗ ), where R(·) denotes the operator range, and then Xu et al. [30] generalized the result to Hilbert C ∗ -module for the operators with closed ranges under the same condition: The research reported in this article was supported by the National Natural Science Foundation of China (10771161).

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Let A ∈ LA (E, F ) and B ∈ LA (E, G) have closed ranges such that R(B ∗ ) ⊆ R(A∗ ). Let C ∈ LA (E) and S = NB A∗ such that R(S) is also closed. Then the equation A∗ XB + B ∗ X ∗ A = C,

X ∈ LA (G, F )

has a solution if and only if C = C ∗ and Re((A∗ A∗ + + SS + )CB ∗ B ∗ + ) = 2C, where (·)+ denotes the Moore–Penrose inverse and Re(X) = X + X ∗ . To use the generalized inverse as in the study of the equations for matrices, Yuan and Xu have to restrict their attention to the bounded (adjointable) linear operators with closed ranges (both in the case of Hilbert space and of Hilbert C ∗ -module). The real positive solutions to the equation AXB = C were studied by Cvekovi´c-Ili´c [8], and Wang and Yang [28] for ﬁnite matrices; the positive solutions to the same equation were studied by Khatri and Mitra [23], and Zhang [33] for ﬁnite matrices. In the special case that B = I, the real positive solutions to the equation AX = C were studied by Groß [18]; the study of positive solutions to the equation AX = C were found in [23] for matrices, and in [12,29] for Hilbert space operators and Hilbert C ∗ -module operators respectively. Xu et al.[30] proposed some new equivalent conditions for the existence of the real positive solution to the equation AXB = C for the adjointable Hilbert C ∗ -module operators with closed ranges, which was stated as follows: Let A ∈ LA (E, F ) and B ∈ LA (G, E) have closed ranges and C ∈ LA (G, F ). Suppose that T = NA B also has closed range. Let PR(A∗ )∩R(B) be the projection from E onto R(A∗ ) ∩ R(B). Then the following statements are equivalent: (i) AXB = C has a solution X ∈ LA (E) such that X + X ∗ ≥ 0; (ii) AXB = C has a solution X ∈ LA (E) such that B ∗ Re(X)B ≥ 0; (iii) AA+ CB + B = C, NT Re(B ∗ A+ C)NT ≥ 0; (iv) AA+ CB + B = C, PR(A∗ )∩R(B) Re(A+ CB + )PR(A∗ )∩R(B) ≥ 0. Based on this result he also gave some necessary and sufﬁcient conditions for the existence of a positive solution in the case that the range of B is contained in that of A∗ . The main method is still by use of the Moore–Penrose inverse. It is known that in the case of (inﬁnite dimensional) Hilbert space and Hilbert C ∗ -module, closed range is a very strong condition which general bounded (adjointable) linear operators may not satisfy. In fact an operator with closed range is also called a generalize Fredholm operator. However, both the way of proof and the constructed solutions in [1,13,32,30] depend on the existence of the Moore–Penrose inverse (i.e., generalized inverse) which is equivalent to closed range. The purpose of this paper is to provide a new approach to the study of the equations A∗ XB + B ∗ X ∗ A = C and AXB = C for more general adjointable Hilbert module operators than those with closed ranges. The paper is organized as follows. In Sect. 2, we will generalize the main results in [1] and [13] on the solvability of A∗ X + X ∗ A = C for Hilbert C ∗ -module

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operators (see Theorem 2.4, obtain a new result concerning the solvability of the equation A∗ XB + B ∗ X ∗ A = C (see Theorem 2.6) and then generalize the results in [32] and [30] (see Corollary 2.8). In Sect. 3, we will study the general and real positive solutions to the equation AXB = C and generalize the results concerning the existence of the real positive solution in [30] (see Theorem 3.4). Moreover some results concerning the solution X with B ∗ (X ∗ + X)B ≥ 0 are obtained (see Theorem 3.7). In Sect. 4, we will give some necessary and sufﬁcient conditions for the existence of solution X to the equation AXB = C with B ∗ XB ≥ 0. In particular, in the special case that R(B) ⊆ R(A∗ ) we obtain a necessary and sufﬁcient condition for the existence of a positive solution to the equation AXB = C (see Theorem 4.12) which generalize the results in [30]. First of all, we recall some knowledge about Hilbert C ∗ -modules. Throughout this paper, A is a C ∗ -algebra. An inner-product A-module is a linear space E which is a right A-module, together with a map (x, y) → x, y : E × E → A such that for any x, y, z ∈ E, α, β ∈ C and a ∈ A, the following conditions hold: (i) x, αy + βz = αx, y + βx, z; (ii) x, ya = x, ya; (iii) x, y = y, x∗ ; (iv) x, x ≥ 0, and x, x = 0 if and only if x = 0. An inner-product A-module E which is complete with respect to the 1 induced norm x = x, x 2 is called a (right) Hilbert A-module. Suppose that E, F are two Hilbert A-modules, let LA (E, F ) be the set of all maps T : E → F for which there is a map T ∗ : F → E such that T x, y = x, T ∗ y,

for each x ∈ E and y ∈ F.

It is known that any element T of LA (E, F ) must be a bounded linear operator, which is also A-linear in the sense that T (xa) = T (x)a for x ∈ E and a ∈ A. For any T ∈ LA (E, F ), the range, the null space of T are denoted by R(T ) and N (T ), respectively. We call LA (E, F ) the set of adjointable operators from E to F . We denote by BA (E, F ) the set of all bounded A-linear maps, and therefore LA (E, F ) ⊆ BA (E, F ). In case E = F , LA (E), to which we abbreviate LA (E, F ), is a C ∗ -algebra. Then for A ∈ LA (E), A is Hermitian (selfadjointable) if and only if Ax, y = x, Ay for any x, y ∈ E, and positive 1 if and only if Ax, x ≥ 0 for any x ∈ E, in which case, we denote by A 2 the unique positive element B such that B 2 = A in the C ∗ -algebra LA (E) 1 and then R(A) = R(A 2 ). Let LA (E)sa , LA (E)+ be the sets of Hermitian and positive elements of LA (E) respectively. For any A, B ∈ LA (E)sa , we say A ≥ B if (A − B)x, x ≥ 0 for any x ∈ E. For A+ , the set of positive elements of the C ∗ -algebra A, is a positive cone, we could easily verify that `ıs a partial order on LA (E). For an operator T ∈ LA (E), set Re(T ) = T + T ∗ , and T is called real positive if Re(T ) ≥ 0. We say that a closed submodule E1 of E is topologically complemented if there is a closed submodule E2 of E such that E1 + E2 = E and E1 ∩ E2 = 0, 2 , called the direct sum of E1 and brieﬂy denote the sum by E = E1 ⊕E

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and E2 . If moreover E2 = E1⊥ , where E1⊥ = {x ∈ E : x, y = 0 for all y ∈ E1 }, we say E1 is orthogonally complemented and brieﬂy denote the sum by E = E1 ⊕ E2 , called the orthogonal sum of E1 and E2 . In this case, E1 = E1⊥⊥ and there exists unique orthogonal projection (i.e., idempotent and selfadjointable operator in LA (E)) onto E1 . For two submodules E1 and F1 of E, if E1 ⊆ F1 , then E1 ⊥ ⊇ F1 ⊥ . Let T ∈ LA (E, F ), then (1) N (T ) = R(T ∗ )⊥ and N (T )⊥ ⊇ R(T ∗ ); (2) if R(T ) is closed, then so is R(T ∗ ), and in this case both R(T ) and R(T ∗ ) are orthogonally complemented and R(T )⊥ = N (T ∗ ), R(T ∗ )⊥ = N (T ) (see [24, Theorem 3.2] ). The reader may refer to [14–16,21,24,27] for details. Any element T − of {X ∈ LA (F, E) : T XT = T } is called the inner inverse of T and R(T T − ) = R(T ). R(T ) is closed if and only if T has a inner inverse. The Moore–Penrose inverse T + of T is the unique inner inverse of T which satisﬁes T +T T + = T +,

T T + = (T T + )∗ ,

T + T = (T + T )∗ .

In this case, (T + )∗ = (T ∗ )+ , R(T + ) = R(T ∗ ) and T + |R(T )⊥ = 0. Thus T T + , T + T are the projection onto R(T ) and R(T ∗ ) respectively. For this we will refer to [31]. Throughout this paper, E, F, G and H are Hilbert A-modules. For an operator T ∈ LA (E, F ) if the closure of R(T ∗ ) is orthogonally complemented, ⊥

then R(T ∗ ) = N (T ) and there exists an orthogonal decomposition E = R(T ∗ ) ⊕ N (T ). Let PT ∗ denote the orthogonal projection of E onto R(T ∗ ) and NT denote the projection I − PT ∗ , then PT ∗ + NT = IE . In [17], the authors have generalized the famous Douglas theorem from Hilbert space case to Hilbert C ∗ -module case and studied the solutions, Hermitian solutions and positive solutions to the equation AX = C and the common solutions to the equation system AX = C, XB = D, which could be stated as follows: Theorem 1.1. Let A ∈ LA (E, F ) and C ∈ LA (G, F ) with R(A∗ ) orthogonally complemented. (i) AX = C has a solution X ∈ LA (G, E) if and only if R(C) ⊆ R(A), and if and only if CC ∗ ≤ λAA∗ for some λ > 0. In this case, there exists a unique solution X ∈ LA (G, E) satisfying R(X) ⊆ N (A)⊥ , which we call the reduced solution and is denoted by D. Concretely, D is defined as follows: D = PA∗ A−1 C, D |N (A) = 0; D∗ A∗ y = C ∗ y, for all y ∈ F. ∗

(ii) (iii)

If G = E, then AX = C has a Hermitian solution if and only if R(C) ⊆ R(A), CA∗ ∈ LA (E)sa . If G = E, then AX = C has a positive solution if and only if R(C) ⊆ R(A), CA∗ ∈ LA (E)+ .

Theorem 1.2. Let A ∈ LA (E, F ), C ∈ LA (G, F ), and B ∈ LA (H, G), D ∈ LA (H, E). Suppose R(A∗ ), R(B) are orthogonally complemented submodules

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in E and G respectively, then AX = C, XB = D have a common solution X ∈ LA (G, E) if and only if R(C) ⊆ R(B),

R(D∗ ) ⊆ R(B ∗ ),

AD = CB.

In this case, the general common solution is of the form X = D1 + NA D2 ∗ + NA V NB ∗ , where D1 , D2 are the reduced solutions of AX = C and B ∗ X = D∗ respectively, NA , NB are the orthogonal projection onto the null spaces of A and B respectively, and V ∈ LA (G, E) is arbitrary. In this paper, much of our study depends on the above two theorems.

2. Solutions to Equations A∗ X + X ∗ A = C and A∗ XB + B ∗ X ∗ A = C In this section, we will ﬁrstly study the solvability of the equation A∗ X + X ∗ A = C. When R(A) is closed (i.e., A is a matrix or an adjointable module operator with closed range), the equation A∗ X + X ∗ A = C

(2.1)

has a solution if and only if C = C ∗,

NA CNA = 0,

+

where NA = I − A A is the projection onto the null space of A, and A+ denotes the Moore–Penrose inverse of A. For this we may refer to [1,13,30]. In the following discussion, we consider the existence of the solution to this equation for more general adjointable module operators than those with closed ranges. Set NA for the (real) solution space to A∗ X + X ∗ A = 0. The following proposition describes the solution space NA under the condition that R(A) is orthogonally complemented closed submodule of F . Proposition 2.1. Let A ∈ LA (E, F ) and R(A) be orthogonally complemented in F , then for any V1 ∈ LA (E, F ) and V2 ∈ LA (F )sa , X = NA∗ V1 + iV2 A ∈ NA . In particular, if R(A) is closed, NA = {NA∗ V1 NA + iV2 A : V1 ∈ LA (E, F ), V2 ∈ LA (F )sa } = {NA∗ V1 + iV2 A : V1 ∈ LA (E, F ), V2 ∈ LA (F )sa }. Proof. Since R(A) is orthogonally complemented in F , there exists an orthogonal projection NA∗ onto the null space of A∗ . For arbitrary operators V1 ∈ LA (E, F ), V2 ∈ LA (F )sa , we have A∗ (NA∗ V1 + iV2 A) + (NA∗ V1 + iV2 A)∗ A = iA∗ V2 A − iA∗ V2 A = 0, then NA∗ V1 + iV2 A ∈ NA . If R(A) is closed, it is only needed to prove that for each operator X0 of NA there exist V1 ∈ LA (E, F ), V2 ∈ LA (F )sa such that X0 = NA∗ V1 + iV2 A.

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Since A∗ X0 + X0 ∗ A = 0, multiplying the equation above by PA∗ from both sides, we have A∗ X0 PA∗ + PA∗ X0 ∗ A = 0. Since R(A) is closed, so is R(A∗ ). Then R(iPA∗ X0 ∗ ) ⊆ R(A∗ ), and (iPA∗ X0 ∗ )A = iPA∗ X0 ∗ A = −iA∗ X0 PA∗ = A∗ (iPA∗ X0∗ )∗ , By Theorem 1.1 (ii) the equation A∗ X = iPA∗ X0 ∗ has a selfadjointable solution V2 ∈ LA (F ). Since A∗ (PA X0 NA ) = A∗ X0 NA = −X0∗ ANA = 0, we have PA X0 NA = 0. So X0 = X0 NA + X0 PA∗ = (NA∗ + PA )X0 NA + X0 PA∗ = NA∗ X0 NA + X0 PA∗ = NA∗ X0 NA + (−iA∗ V2 )∗ = NA∗ X0 NA + iV2 A. NAB

Set NAB for the (real) solution space to A∗ XB + B ∗ X ∗ A = 0, then = (NBA )∗ .

Corollary 2.2. Let A ∈ LA (E, F ), B ∈ LA (E, G) and R(A), R(B) be orthogonally complemented in F and G respectively. Set Σ ={(V1 , V2 , V3 , V4 ) ∈ LA (E, F ) × LA (F )sa × LA (E, F ) × LA (G)sa : R(V1 ∗ NA∗ − iA∗ V2 ) ⊆ R(B ∗ ) and R(V3 ∗ NB ∗ − iB ∗ V4 ) ⊆ R(A∗ )}, then the operator of the form X = X + X is a solution to the equation A∗ XB + B ∗ X ∗ A = 0, where X and X are the solutions in LA (G, F ) to the equations X B = NA∗ V1 NA + iV2 A and (X )∗ A = NB ∗ V3 NB + iV4 B respectively for any (V1 , V2 , V3 , V4 ) ∈ Σ. In particular, (i) if R(A) is closed, set Σ1 = {(V1 , V2 ) ∈ LA (E, F ) × LA (F )sa : R(NA V1 ∗ NA∗ − iA∗ V2 ) ⊆ R(B ∗ )}, then the solution space to A∗ XB + B ∗ X ∗ A = 0 is

NAB = {X ∈ LA (G, F ) : X B = NA∗ V1 NA + iV2 A, for some (V1 , V2 ) ∈ Σ1 }. (ii)

if R(B) is closed, set Σ2 = {(V3 , V4 ) ∈ LA (E, F ) × LA (G)sa : R(NB V3 ∗ NB ∗ − iB ∗ V4 ) ⊆ R(A∗ )}, then the solution space to A∗ XB + B ∗ X ∗ A = 0 is

NAB = (NBA )∗ = {X ∈ LA (G, F ) : (X )∗ A = NB ∗ V3 NB + iV4 B, for some (V3 , V4 ) ∈ Σ2 }. Proposition 2.3. Let A ∈ LA (E, F ) with R(A∗ ) orthogonally complemented in E, and C ∈ LA (E). Then the following statements are equivalent: (i) R(PA∗ CPA∗ ) ⊆ R(A∗ ), R(CNA ) ⊆ R(A∗ );

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R(PA∗ C) ⊆ R(A∗ ), R(CNA ) ⊆ R(A∗ ); R(PA∗ CPA∗ + CNA ) ⊆ R(A∗ ). If additionally C = C ∗ and R(A) ⊆ F is orthogonally complemented, then (i), (ii) and (iii) are equivalent to (iv) R(PA∗ C + CNA ) ⊆ R(A∗ ).

(ii) (iii)

Proof. Since R(A∗ ) is orthogonally complemented in E, there exist orthogonal projections NA , PA∗ onto the null space of A and the closure of the range space of A∗ , respectively. (i)⇔(ii): It is well known that R(PA∗ C) = R(PA∗ CPA∗ + PA∗ CNA ). So if R(CNA ) ⊆ R(A∗ ), we obtain that R(PA∗ C) ⊆ R(A∗ ) ⇔ R(PA∗ CPA∗ ) ⊆ R(A∗ ). (i)⇒(iii): It is obvious that R(PA∗ CPA∗ + CNA ) ⊆ R(PA∗ CPA∗ ) + R(CNA ) ⊆ R(A∗ ). (iii)⇒(i): Suppose R(PA∗ CPA∗ + CNA ) ⊆ R(A∗ ), for any x ∈ E, we have PA∗ CPA∗ x = (PA∗ CPA∗ + CNA )PA∗ x ∈ R(A∗ ), CNA x = (PA∗ CPA∗ + CNA )NA x ∈ R(A∗ ), i.e., R(PA∗ CPA∗ ) ⊆ R(A∗ ) and R(CNA ) ⊆ R(A∗ ). Thus we have proven the equivalence of (i)–(iii). From discussion in (iii)⇒(i), we could obtain R(PA∗ C + CNA ) = R(PA∗ CPA∗ + PA∗ CNA + CNA ) ⊆ R(A∗ ). In particular, if R(A) is also orthogonally complemented in F and R(PA∗ C + CNA ) ⊆ R(A∗ ), then applying Theorem 1.1 (i) we know the equation A∗ X + X ∗ A = C has a solution X ∈ LA (E, F ). Multiplying NA from the right on both sides of the equation A∗ X + X ∗ A = C, we obtain CNA = A∗ XNA . So R(CNA ) ⊆ R(A∗ ) and then R(PA∗ C) = R(PA∗ C + CNA − CNA ) ⊆ R(A∗ ). Thus we obtain the equivalence of (iv) and (ii) under the additional condition that R(A) is orthogonally complemented in F and C = C ∗ . Theorem 2.4. Let A ∈ LA (E, F ) with R(A∗ ) orthogonally complemented in E, and C ∈ LA (E). (i) (ii)

If A∗ X + X ∗ A = C has a solution X ∈ LA (E, F ), then C = C ∗ and R(CNA ) ⊆ R(A∗ ). Suppose that R(A) is orthogonal complemented submodules of F , and R(PA∗ CPA∗ ) ⊆ R(A∗ ). If C = C ∗ and R(CNA ) ⊆ R(A∗ ), then the Eq. (2.1) has a solution X ∈ LA (E, F ). In this case, X0 =

1 PA A∗ −1 (CNA + PA∗ C) 2

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is a solution to Eq. (2.1) such that X0∗ A = 12 (NA C + CPA∗ ) and X0∗ NA∗ = 0, and the (real) solution space to Eq. (2.1) is 1 PA A∗ −1 (CNA + PA∗ C) + NA , 2 where NA is the (real) solution space to A∗ X + X ∗ A = 0. In particular, if R(A) is closed, then the equation (2.1) has a solution X ∈ LA (E, F ) if and only if C = C ∗ and R(CNA ) ⊆ R(A∗ ). In this case, the solution space to the Eq. (2.1) is 1 ∗ −1 PA A (CNA +PA∗ C)+NA∗ V1 NA + iV2 A : V1 ∈ LA (E), V2 ∈ LA (E)sa . 2 Proof. (i) By the assumption that R(A∗ ) is orthogonally complemented in E, there exist projections NA , PA∗ onto N (A) and R(A∗ ) respectively. Since the equation A∗ X + X ∗ A = C has a solution X ∈ LA (E, F ), Multiplying NA from the right on both sides of the above equation, we have CNA = A∗ XNA + X ∗ ANA = A∗ XNA , (ii)

and hence R(CNA ) ⊆ R(A∗ ). From Proposition 2.3, we could see that A, C satisfy R(CNA + PA∗ C) ⊆ R(A∗ ). Since R(A) ⊆ F is orthogonally complemented, by Theorem 1.1 (i) we know that the equation A∗ X = 12 (CNA + PA∗ C) has the reduced solution, denoted by X0 , in LA (E, F ), which is deﬁned as follows: 1 PA A∗ −1 (CNA + PA∗ C); 2 1 X0 ∗ |N (A∗ ) = 0; X0 ∗ Ax = (NA C + CPA∗ )x, for all x ∈ E. 2 Directly calculating, we verify X0 is a solution to the Eq. (2.1): X0 =

1 1 (CNA + PA∗ C) + (NA C + CPA∗ ) = C. 2 2 Thus for any X ∈ NA , where NA is as in Proposition 2.1, X0 + X is a solution to the Eq. (2.1). Conversely, if X ∈ LA (E, F ) is a solution to A∗ X + X ∗ A = C, then A∗ X0 + X0 ∗ A =

A∗ (X − X0 ) + (X − X0 )∗ A = 0 and so X − X0 ∈ NA . Thus we obtain the solution space to the Eq. (2.1) is X0 + NA . Particularly, if R(A) is closed, then it is orthogonally complemented in F and R(A∗ ) is closed which makes R(PA∗ CPA∗ ) ⊆ R(A∗ ) hold automatically. Combining (i) we obtain that the Eq. (2.1) has a solution X ∈ LA (E, F ) if and only if C = C ∗ and R(CNA ) ⊆ R(A∗ ).

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In this case, by Proposition 2.1, we could obtain the solution space concretely as follows:

1 PA A∗−1 (CNA +PA∗ C)+NA∗ V1 NA +iV2 A : V1 ∈ LA (E), V2 ∈ LA (E)sa . 2

The proof is completed.

Corollary 2.5. Let A ∈ LA (E, F ) with R(A∗ ) orthogonally complemented in E, and C ∈ LA (E). (i) (ii)

If A∗ X − X ∗ A = C has a solution X ∈ LA (E, F ), then C = −C ∗ and R(CNA ) ⊆ R(A∗ ). Suppose that R(A) is orthogonal submodules of F and R(PA∗ CPA∗ ) ⊆ R(A∗ ). If C = −C ∗ and R(CNA ) ⊆ R(A∗ ), then the equation A∗ X − X ∗ A = C has a solution X ∈ LA (E, F ). In this case, the operator X0 =

1 PA A∗ −1 (CNA + PA∗ C) 2

is a solution such that X0∗ A = 12 (NA C + CPA∗ ) and X0∗ NA∗ = 0, and the (real) solution space to equation A∗ X − X ∗ A = C is 1 PA A∗ −1 (CNA + PA∗ C) + iNA , 2 where NA is the (real) solution space to A∗ X + X ∗ A = 0. In particular, if R(A) is closed, then A∗ X − X ∗ A = C has a solution X ∈ LA (E, F ) if and only if C = −C ∗ and R(CNA ) ⊆ R(A∗ ). In this case, the solution space is 1 ∗ −1 PA A (CNA +PA∗ C)+iNA∗ V1 NA −V2 A : V1 ∈ LA (E), V2 ∈ LA (E)sa . 2 Remark. In the special case that R(A) is closed, R(A∗ ) is closed, so the assumption R(PA∗ CPA∗ ) ⊆ R(A∗ ) is automatically satisﬁed and R(CNA ) ⊆ R(A∗ ) is equivalent to NA CNA = 0. So Theorem 2.4 implies the results about the solvability of AX ∗ + XA∗ = C in [1,13,30]. With this preparation we may study the existence of the solution to the equation A∗ XB + B ∗ X ∗ A = C. (2.2) Theorem 2.6. Let A ∈ LA (E, F ), B ∈ LA (E, G) and C ∈ LA (E). (i) Suppose R(A∗ ) (R(B ∗ ), respectively) is orthogonally complemented in E. If the equation A∗ XB + B ∗ X ∗ A = C has a solution X ∈ LA (G, F ), then C = C ∗, (ii)

R(CNA ) ⊆ R(A∗ )

(R(CNB ) ⊆ R(B ∗ ), respectively).

Suppose R(A), R(B) are orthogonally complemented in F and G respectively. If one of the following conditions holds, then the Eq. (2.2) has a solution X ∈ LA (G, F ):

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R(A∗ ) is orthogonally complemented in E and C = C ∗ , R(PA∗ CPA∗ ) ⊆ R(A∗ ), R(CNA ) ⊆ R(A∗ ), R(NA C + CPA∗ ) ⊆ R(B ∗ ),

(2)

R(B ∗ ) is orthogonally complemented in E and C = C ∗ , R(PB ∗ CPB ∗ ) ⊆ R(B ∗ ), R(CNB ) ⊆ R(B ∗ ), R(NB C + CPB ∗ ) ⊆ R(A∗ ).

Moreover, if R(A) or R(B) is closed, then the fourth condition can be replaced by R(NA C + CPA∗ ) ⊆ R(B ∗ ) in case (1), and by R(NB C + CPB ∗ ) ⊆ R(A∗ ) in case (2). In case (1), we obtain a special solution X in LA (G, F ) which is defined as follows: 1 ∗ −1 (CNA + PA∗ C)x, z = Bx, for all x ∈ E; 2 PA A Xz = 0, z ∈ N (B ∗ ). In case (2), we obtain a special solution X in LA (G, F ) whose adjointable operator is defined as follows: 1 ∗ −1 (CNB + PB ∗ C)x, z = Ax, for all x ∈ E; 2 PB B ∗ X z= 0, y ∈ N (A∗ ). Proof. (i) If A∗ XB + B ∗ X ∗ A = C has a solution X ∈ LA (G, F ), multiplying the both sides of the above equation from the right by NA and NB respectively, we obtain that A∗ XBNA = CNA ,

B ∗ X ∗ ANB = CNB ,

and so R(CNB ) ⊆ R(B ∗ ), R(CNA ) ⊆ R(A∗ ). (ii) It is sufﬁcient to prove the result in case (1) for the symmetry of A and B. By Theorem 2.4 (ii), we obtain a special solution X0 ∈ LA (E, F ) to the Eq. (2.1) which is deﬁned as follows: 1 X0 = PA A∗ −1 (CNA + PA∗ C) 2 and 1 ∗ 2 (NA C + CPA )x, z = Ax, for all x ∈ E; ∗ X0 z = 0, z ∈ N (A∗ ). For any y ∈ F , since F = R(A) ⊕ N (A∗ ), there exist xn ∈ E, y1 ∈ N (A ) such that {Axn } is convergent in F and y = limn Axn + y1 , then X0 ∗ y = limn X0 ∗ Axn . Thus ∗

R(X0 ∗ ) ⊆ R(X0 ∗ A) = R(NA C + CPA∗ ) ⊆ R(B ∗ ). In the case that R(A) is closed, there exists some x ∈ E such that Axn → Ax as n → ∞ and then X0 ∗ y = X0 ∗ Ax, i.e., R(X0 ∗ ) = R(X0 ∗ A). Since R(B) ⊆ G is orthogonally complemented, applying Theorem 1.1 (i), we know the equation B ∗ X = X0 ∗ has the reduced solution D in LA (F, G).

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Set X1 = D∗ , then X1 is a special solution to the Eq. (2.2) and it is deﬁned as follows: X0 x, z = Bx, for all x ∈ E; X1 z = 0, z ∈ N (B ∗ ). If R(B) is closed, then R(B ∗ ) is closed and obviously R(NA C + CPA∗ ) ⊆ R(B ∗ ) ⇔ R(NA C + CPA∗ ) ⊆ R(B ∗ ). If R(A) is closed, by the discussion above we have R(X0 ∗ ) = R(X0 ∗ A) = R(NA C + CPA∗ ). Thus, for the existence of X1 , it is sufﬁcient that R(NA C + CPA∗ ) ⊆ R(B ∗ ). Therefore, if R(A) or R(B) is closed, the forth condition R(NA C + CPA∗ ) ⊆ R(B ∗ ) can be replaced by R(NA C + CPA∗ ) ⊆ R(B ∗ ). Corollary 2.7. Let A ∈ LA (E, F ), B ∈ LA (E, G) with closed ranges, and let C ∈ LA (E). Suppose CNA = NB C, then the equation A∗ XB + B ∗ X ∗ A = C has a solution X ∈ LA (G, F ) if and only if C = C ∗,

R(CNB ) ⊆ R(B ∗ ),

R(CNA ) ⊆ R(A∗ ).

Proof. By Theorem 2.6 (i) it is only to prove the sufﬁciency. Assume that C = C ∗ , R(CNB ) ⊆ R(B ∗ ) and R(CNA ) ⊆ R(A∗ ). By the assumption that CNA = NB C, we have NA C = CNB ,

PA∗ C = CPB ∗ and CPA∗ = PB ∗ C,

and it follows that R(NA C + CPA∗ ) = R(CNB + PB ∗ C) ⊆ R(B ∗ ). For R(PA∗ CPA∗ ) ⊆ R(A∗ ), applying Theorem 2.6 (ii) (1), we know the equa tion A∗ XB + B ∗ X ∗ A = C has a solution X ∈ LA (G, F ). Corollary 2.8. Let A ∈ LA (E, F ), B ∈ LA (E, G) and C ∈ LA (E). Suppose R(A∗ ), R(A), R(B ∗ ) and R(B) are orthogonally complemented, R(PB ∗ C) ⊆ R(A∗ ) ∩ R(B ∗ ), and R(A) or R(B) is closed. Set S = NB A∗ and assume R(S) is orthogonally complemented in E. Then the following statements are equivalent: (i) A∗ XB + B ∗ X ∗ A = C has a solution X ∈ LA (G, F ) such that R(B ∗ X ∗ A) ⊆ R(A∗ ); (ii) C = C ∗ , R(C) ⊆ R(A∗ ), R(CNB ) ⊆ R(B ∗ ); (iii) C = C ∗ , R(C) ⊆ R(A∗ ), Re((PA∗ + PS )CPB ∗ ) = 2C. Proof. Since R(S) is orthogonally complemented in E, R(PS ) = R(NB A∗ ) ⊆ N (B) and then PS PB ∗ = 0. Therefore, PS A∗ = PS (NB + PB ∗ )A∗ = PS NB A∗ = NB A∗ . (i)⇒(iii): Suppose X ∈ LA (G, F ) is a solution to the Eq. (2.2) such that R(B ∗ X ∗ A) ⊆ R(A∗ ), obviously C = C ∗ , R(C) = R(A∗ XB + B ∗ X ∗ A) ⊆ R(A∗ ),

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and we have the following equations: Re((PA∗ + PS )CPB ∗ ) = (PA∗ +PS )(A∗ XB +B ∗ X ∗ A)PB ∗ +PB ∗ (A∗ XB +B ∗ X ∗ A)(PA∗ + PS ) = (A∗ XB + B ∗ X ∗ A)PB ∗ + NB A∗ XB + PB ∗ (A∗ XB + B ∗ X ∗ A) + B ∗ X ∗ ANB = A∗ XB + B ∗ X ∗ A + (PB ∗ + NB )A∗ XB + B ∗ X ∗ A(PB ∗ + NB ) = 2C. (iii)⇒(ii): If R(C) ⊆ R(A∗ ), by Theorem 1.1 (i) there exists an operator Y ∈ LA (E, F ) such that A∗ Y = C and hence PS C = PS A∗ Y = NB A∗ Y = NB C. Therefore, we have that 2C = Re((PA∗ + PS )CPB ∗ ) = (PA∗ + PS )CPB ∗ + PB ∗ C(PA∗ + PS ) = CPB ∗ + PS CPB ∗ + PB ∗ C + PB ∗ CPS = CPB ∗ + NB CPB ∗ + PB ∗ C + PB ∗ CNB = CPB ∗ + (CPB ∗ − PB ∗ CPB ∗ ) + PB ∗ C + PB ∗ CNB = 2CPB ∗ + 2PB ∗ CNB . Thus CNB = PB ∗ CNB and then R(CNB ) ⊆ R(PB ∗ C) ⊆ R(B ∗ ). (ii)⇒(i): Since R(C) ⊆ R(A∗ ), we have R(NB C + CPB ∗ ) = R(C − PB ∗ C + CPB ∗ ) ⊆ R(A∗ ) and R(PB ∗ CPB ∗ ) ⊆ R(B ∗ ). By Theorem 2.6 (ii) (2), we know that the equation (2.2) has a solution X in LA (G, F ). For R(C) ⊆ R(A∗ ), we obtain R(B ∗ X ∗ A) = R(C − A∗ XB) ⊆ R(A∗ ). Remark. In Theorem 2.1 both of [32] and [30], Yuan and Xu showed, in the case of matrix and of Hilbert C ∗ -module respectively, that for (adjointable) operators A and B with R(B ∗ ) ⊆ R(A∗ ), and R(A), R(B) and R(S) closed, the equation A∗ XB + B ∗ X ∗ A = C has a solution X ∈ LA (G, F ) if and only if C = C ∗,

Re((A∗ A∗ + + SS + )CB ∗ B ∗ + ) = 2C,

where (·)+ denotes the Moore–Penrose inverse. In fact, A∗ A∗ + is the orthogonal projection onto R(A∗ ). Moreover, ∗ ∗ under the condition that R(B ∗ ) ⊆ R(A∗ ), Re((A∗ A+ + SS + )CB ∗ B + ) = 2C implies R(C) ⊆ R(A∗ ). In fact, since A∗ A∗ + S = A∗ A∗ + NB A∗ = A∗ A∗ + A∗ − A∗ A∗ + PB ∗ A∗ = A∗ − PB ∗ A∗ = S, we have R(S) ⊆ R(A∗ ) and then R(C) ⊆ R(A∗ ). Therefore, in this case, the results in [30] and [32] actually are: the equation A∗ XB + B ∗ X ∗ A = C has a solution X ∈ LA (G, F ) such that R(B ∗ XA) ⊆ R(A∗ ) if and only if C = C ∗,

Re((A∗ A∗ + + SS + )CB ∗ B ∗ + ) = 2C,

R(C) ⊆ R(A∗ ).

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Moreover, if B has closed range (so has B ∗ ) and R(B ∗ ) ⊆ R(A∗ ), then R(PB ∗ C) ⊆ R(A∗ ) ∩ R(B ∗ ) holds automatically. Therefore, Corollary 2.8 surely implies Theorem 2.1 both of [32] and [30].

3. Solutions and Real Positive Solutions to the Equation AXB = C In this section, we discuss the existence of solutions to the equation AXB = C. In the case of matrix, AXB = C has a solution if and only if AA− CB − B = C, in which case, the general solution X to equation AXB = C is of the form X = A− CB − + V − A− AV BB − , where A− denotes a inner inverse of A and V is arbitrary. Xu [30] obtained the same result for adjointable Hilbert module operators with closed ranges based on the Moore–Penrose inverse. We will generalize this result to more general adjointable module operators without the assumption of closed range. Theorem 3.1. Let A ∈ LA (E, F ), B ∈ LA (G, H) and C ∈ LA (G, F ). (i)

If the equation AXB = C has a solution X ∈ LA (H, E), then R(C) ⊆ R(A) and R(C ∗ ) ⊆ R(B ∗ ).

(ii)

Suppose R(B) and R(A∗ ) are orthogonally complemented submodules of H and E respectively. If R(C) ⊆ R(A) and R(C ∗ ) ⊆ R(B ∗ ) (or, R(C) ⊆ R(A) and R(C ∗ ) ⊆ R(B ∗ )), then AXB = C has a unique solution D ∈ LA (H, E) such that R(D) ⊆ N (A)⊥ and

R(D∗ ) ⊆ N (B ∗ )⊥ ,

which is called the reduced solution, and the general solution to AXB = C is of the form X = D + NA V 1 + V 2 N B ∗ ,

where V1 , V2 ∈ LA (H, E).

Moreover if at least one of R(A), R(B) and R(C) is a closed submodule, then the condition R(C ∗ ) ⊆ R(B ∗ ) (orR(C) ⊆ R(A)) can be replaced by R(C ∗ ) ⊆ R(B ∗ ) (orR(C) ⊆ R(A)). Proof. (i) If the equation AXB = C has a solution X ∈ LA (H, E), it is easy to know that R(C) ⊆ R(A),

R(C ∗ ) ⊆ R(B ∗ ).

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Since R(C) ⊆ R(A) and R(A∗ ) ⊆ E is orthogonally complemented, by Theorem 1.1 (i) we know that the equation AX = C has the reduced solution D3 ∈ LA (G, E) satisfying D3 = PA∗ A−1 C; D3 ∗ |N (A) = 0, D3 ∗ A∗ y = C ∗ y, for all y ∈ F.

From the definition of D3 , we could see that R(D3 ∗ ) ⊆ R(C ∗ ) ⊆ R(B ∗ ), so applying Theorem 1.1 (i) again we know that the equation B ∗ X = D3 ∗ also has the reduced solution which we denote by D4 , then we have D4 = PB B ∗ −1 D3 ∗ ; D4 ∗ |N (B ∗ ) = 0, D4 ∗ Bz = D3 z, for all z ∈ G. Set D = D4 ∗ , we have ADB = AD4 ∗ B = AD3 = C, and D is a solution to AXB = C such that R(D) ⊆ N (A)⊥ ,

R(D∗ ) ⊆ N (B ∗ )⊥ .

Next we show the uniqueness of the solution X such that R(X) ⊆ N (A)⊥ and R(X ∗ ) ⊆ N (B ∗ )⊥ . Assume X is a solution satisfying the above two conditions, then A(X − D)B = 0. Since R(X − D) ⊆ N (A)⊥ , we have (X − D)B = 0 and then B ∗ (X ∗ − D∗ ) = 0. From R(X ∗ − D∗ ) ⊆ N (B ∗ )⊥ , it could be seen that X = D. Thus the solution satisfying R(X) ⊆ N (A)⊥ and R(X ∗ ) ⊆ N (B ∗ )⊥ is unique. Finally, we give the general solution form to the equation AXB = C. We note that AXB = 0 if and only if PA∗ XPB = 0. In fact, if AXB = 0, we have AXPB = 0 by the orthogonal decomposition H = R(B) ⊕ N (B ∗ ) and continuity of AX ∈ LA (H, F ). For any x ∈ E, s ∈ H, without loss of generality we set x = A∗ y + x1 , y ∈ F, x1 ∈ N (A), then PA∗ XPB s, x = XPB s, A∗ y = AXPB s, y = 0, and so PA∗ XPB = 0. Conversely, if PA∗ XPB = 0, then AXPB = A(PA∗ + NA )XPB = APA∗ XPB = 0. So for any z ∈ G, we have AXBz = AXPB Bz = 0. Therefore, AXB = 0. For any V1 , V2 ∈ LA (H, E), PA∗ (NA V1 + V2 NB ∗ )PB = 0, and hence A(NA V1 + V2 NB ∗ )B = 0. If AXB = 0, then PA∗ XPB = 0 and XPB = NA V1 for some V1 ∈ LA (H, E), so that X = NA V1 + XNB ∗ . From this we could obtain that AXB = 0 if and only if X = NA V1 + V2 NB ∗ for some V1 , V2 ∈ LA (H, E). Therefore, we could obtain the general solution X ∈ LA (H, E) has the form of X = D + NA V1 + V2 NB ∗ , where D ∈ LA (H, E) is the reduced solution, V1 , V2 ∈ LA (H, E) are arbitrary.

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In particular, if R(B) or R(C) is closed, obviously R(C ∗ ) ⊆ R(B ∗ ) ⇔ R(C ∗ ) ⊆ R(B ∗ ). In the case that R(A) is closed (so is R(A∗ )), it is easy to prove that R(D3 ∗ ) = R(C ∗ ), so it is sufﬁcient that R(C ∗ ) ⊆ R(B ∗ ) for the existence of the solution to the equation B ∗ X = C ∗ in the proof of (ii). Therefore, if at least one of R(A), R(B) and R(C) is a closed submodule, then the condition R(C ∗ ) ⊆ R(B ∗ ) can be replaced by R(C ∗ ) ⊆ R(B ∗ ). For the case R(C) ⊆ R(A) and R(C ∗ ) ⊆ R(B ∗ ), it is only needed to note that AXB = C has a solution if and only if B ∗ Y A∗ = C ∗ has a solution. Remark. (i) It is obvious that in the case that A, B have closed ranges, we have that AA− CB − B = C if and only if R(C) ⊆ R(A) and R(C ∗ ) ⊆ R(B ∗ ). Hence, Theorem 3.1 implies Lemma 2.4 in [30] on the solvability of the equation AXB = C for operators with closed ranges. (ii) From the proof of Theorem 3.1 (ii), we could see that the reduced solution D ∈ LA (H, E) is deﬁned as follows: PA∗ A−1 Cz, s = Bz, for all z ∈ G; Ds = 0, s ∈ N (B ∗ ). PB B ∗ −1 C ∗ y, x = A∗ y, for all y ∈ F ; D∗ x = 0, x ∈ N (A). where PA∗ A−1 C ∈ LA (G, E), PB B ∗ −1 C ∗ ∈ LA (F, H). (iii) Suppose X0 ∈ LA (H, E) is a solution to AXB = C. If R(B), R(A∗ ) are orthogonally complemented submodules of H and E respectively, it is easy to know that APA∗ X0 PB B = C, and so PA∗ X0 PB is a solution to AXB = C such that R(PA∗ X0 PB ) ⊆ N (A)⊥ and R(PB X0 ∗ PA∗ ) ⊆ N (B ∗ )⊥ . By the uniqueness of the reduced solution, we obtain D = PA ∗ X 0 P B . By use of Theorem 3.1, we will obtain a characterization of the solutions to the equation AX = C. Corollary 3.2. Let A ∈ LA (E, F ) with R(A∗ ) ⊆ E orthogonally complemented, and let C ∈ LA (G, F ). Suppose G1 is an orthogonally complemented closed submodule of G, then the equation AX = C has a solution X ∈ LA (G, E) such that R(X ∗ ) ⊆ G1 if and only if R(C) ⊆ R(A) and R(C ∗ ) ⊆ G1 . Proof. Let P : G → G1 be the orthogonal projection. If AX = C has a solution X ∈ LA (G, E) such that R(X ∗ ) ⊆ G1 , then R(C ∗ ) ⊆ R(X ∗ ) ⊆ R(P ), and so P C ∗ = C ∗ , i.e., CP = C. Therefore, AXP = CP = C, i.e., the equation AXP = C has a solution. So AX = C has a solution X ∈ LA (G, E) such that R(X ∗ ) ⊆ G1 if and only if AXP = C has a solution. By Theorem 3.1 (i) and (ii), we obtain that AX = C has a solution X ∈ LA (G, E) such that

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R(X ∗ ) ⊆ G1 if and only if R(C) ⊆ R(A) and R(C ∗ ) ⊆ G1 . ∗

For the special case B = A with R(A) and R(C) closed, the solvability of AXB = C has been discussed by many papers, e.g., Theorem 2.2 in [9] and Lemma 3.2 in [30]. As a corollary of Theorem 3.1, we could express the general solution by the reduced solution in place of the Moore–Penrose inverse. Corollary 3.3. Let A ∈ LA (E, F ) and C ∈ LA (F ) such that R(A) and R(A∗ ) are orthogonally complemented, and A or C has the closed range. (i) The equation AXA∗ = C has a solution if and only if R(C) ⊆ R(A) and R(C ∗ ) ⊆ R(A). In this case, the general solution is of the form X = D + NA V 1 + V 2 N A , (ii)

where V1 , V2 ∈ LA (E) and D is the reduced solution. The equation AXA∗ = C has a Hermitian solution if and only if C = C ∗ and R(C) ⊆ R(A). In this case, the general solution is of the form X = D + NA V1 + V1 ∗ NA + NA V2 NA ,

(iii)

where V1 ∈ LA (E), V2 ∈ LA (E)sa and D is the reduced solution. The equation AXA∗ = C has a positive solution if and only if C ≥ 0 and R(C) ⊆ R(A). In this case, the operator X = D + DV1 NA + NA V1 ∗ D + NA V1 ∗ DV1 NA + NA V2 NA is a positive solution for any V1 ∈ LA (E) and V2 ∈ LA (E)+ , where D is the reduced solution. In particular, if R(A) and R(C) are closed, the general positive solution is of the form above.

Proof. (i) From Theorem 3.1, we could easily obtain the equivalence and the general solution form to the equation AXA∗ = C. (ii) The necessity is obvious. Conversely, for C = C ∗ and R(C) ⊆ R(A), from Theorem 3.1 we know the equation has the reduced solution D in LA (E). By the definition of D given in the Remark (ii) following Theorem 3.1, we obtain that the reduced solution D satisﬁes that D = D∗ . Thus the sufﬁciency is proved. Next we will give the general form of the Hermitian solution. If AXA∗ = C has a Hermitian solution X ∈ LA (E), set X = D +NA V1 + V2 NA for V1 , V2 ∈ LA (E) by Theorem 3.1, then we have that NA (V1 − V2 ∗ ) − (V1 ∗ − V2 )NA = 0, and so i(V1 − V2 ∗ ) ∈ NNA . By Proposition 2.1, there exist U1 ∈ LA (E) and U2 ∈ LA (E)sa such that i(V1 − V2 ∗ ) = PA∗ U1 PA∗ + iU2 NA , and then V1 − V2 ∗ = −iPA∗ U1 PA∗ + U2 NA .

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Thus we obtain X = D + NA V1 + (V1 + iPA∗ U1 PA∗ − U2 NA )∗ NA = D + NA V1 + V1 ∗ NA − NA U2 NA = D + NA V1 + V1 ∗ NA + NA (−U2 )NA . Therefore, the general Hermitian solution is of the form X = D + NA V 1 + V 1 ∗ N A + N A V 2 N A , (iii)

where V1 ∈ LA (E) and V2 ∈ LA (E)sa . As we discussed in case (ii), we only need to show the reduced solution D ∈ LA (E) is positive. Since E = R(A∗ ) ⊕ N (A), R(D) ⊆ N (A)⊥ = R(A∗ ) and D|N (A) = D∗ |N (A) = 0, we have that for any x = A∗ y + x1 , where y ∈ F and x1 ∈ N (A), Dx, x = DA∗ y, A∗ y = ADA∗ y, y = Cy, y ≥ 0, so we obtain D ≥ 0 and the equivalence is proved. Obviously, for any V1 ∈ LA (E), V2 ∈ LA (E)+ , X = D + DV1 NA + NA V1 ∗ D + NA V1 ∗ DV1 NA + NA V2 NA is a solution to AXA∗ = C. Moreover, we know 1

1

1

1

X = (D 2 + D 2 V1 NA )∗ (D 2 + D 2 V1 NA ) + NA V2 NA ≥ 0. Particularly, if R(A) and R(C) are all closed, in the similar way as that of Theorem 2.2 in [9], we could obtain the general positive solution is of the form above. The real positive solutions to the equation AXB = C were studied by Cvekovi´c-Ili´c [8] and Wang and Yang [28] for ﬁnite matrices, and by Xu [30] for adjointable Hilbert module operators with closed ranges by use of the Moore–Penrose inverse. In our discussion of this question, we will extend our line of sight to more general operators. Theorem 3.4. Let A ∈ LA (E, F ), B ∈ LA (G, E) and C ∈ LA (G, F ) such that R(A∗ ) and R(B) are orthogonally complemented submodules of E. Suppose that AXB = C has the reduced solution D ∈ LA (E). Set T = NA B. Assume that R(T ) and R(T ∗ ) are orthogonally complemented in E and G respectively, and R(PT ∗ B ∗ Re(D)B) ⊆ R(T ∗ ). If R(PT ∗ B ∗ Re(D)BPT ∗ − 2B ∗ Re(D)BPT ∗ ) ⊆ R(B ∗ ), or R(T ) is closed, or R(B) is closed, then the following statements are equivalent: (i) AXB = C has a real positive solution X ∈ LA (E); (ii) AXB = C has a solution X ∈ LA (E) such that B ∗ Re(X)B ≥ 0; (iii) NT B ∗ Re(D)BNT ≥ 0; (iv) For all x ∈ R(A∗ ) ∩ R(B), Re(D)x, x ≥ 0.

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In this case, we have a real positive resolution X0 to AXB = C as follows: X 0 = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ , where W is a solution to T ∗ XB + B ∗ X ∗ T = NT B ∗ Re(D)BNT − B ∗ Re(D)B. Moreover, if R(B) is a closed submodule of E, then (i)–(iv) are equivalent to (v) PA∗ ,B Re(D)PA∗ ,B ≥ 0, where PA∗ ,B is the orthogonal projection of E onto R(A∗ ) ∩ R(B). Proof. (i)⇒(ii): It is easy to obtain. (ii)⇒(iii): Suppose X0 is a solution in LA (E) such that B ∗ Re(X0 )B ≥ 0. We know that D = PA∗ X0 PB . Therefore, for all z ∈ G, we have NT B ∗ Re(D)BNT z, z = B ∗ (PA∗ X0 PB + PB X0 ∗ PA∗ )BNT z, NT z = B ∗ PA∗ X0 PB BNT z, NT z +B ∗ PB X0 ∗ PA∗ BNT z, NT z. We claim that R(BNT ) = R(A∗ ) ∩ R(B). Since NA BNT = T NT = 0, it could be seen that R(BNT ) ⊆ R(A∗ ) ∩ R(B). For x ∈ R(A∗ ) ∩ R(B), there exists z ∈ G such that x = Bz. As 0 = NA x = NA Bz = T z, we get z ∈ N (T ) and so x = BNT z. This completes the proof of the claim. Therefore, we have NT B ∗ Re(D)BNT z, z = X0 BNT z, BNT z + X0 ∗ BNT z, BNT z = B ∗ X0 BNT z, NT z + B ∗ X0 ∗ BNT z, NT z = B ∗ (X0 + X0 ∗ )BNT z, NT z ≥ 0. and then NT B ∗ Re(D)BNT ≥ 0. (iii)⇒(i): Set Z = NT B ∗ Re(D)BNT and we know that Z is a positive solution, even the reduced solution to the equation NT XNT = Z. Set C = Z − B ∗ Re(D)B, we consider the equation

T ∗ XB + B ∗ X ∗ T = C . ∗

(3.1)

∗

By assumption, we know that R(T ) ⊆ R(B ). It is easy to see

P T ∗ C P T ∗ + C NT = −PT ∗ B ∗ Re(D)BPT ∗ + NT B ∗ Re(D)BNT − B ∗ Re(D)BNT = −PT ∗ B ∗ Re(D)BPT ∗ − PT ∗ B ∗ Re(D)BNT = −PT ∗ B ∗ Re(D)B.

So we have R(PT ∗ C PT ∗ +C NT ) ⊆ R(T ∗ ), for R(PT ∗ B ∗ Re(D)B) ⊆ R(T ∗ ). Moreover

NT C + C PT ∗ = NT B ∗ Re(D)BNT − NT B ∗ Re(D)B − B ∗ Re(D)BPT ∗ = −NT B ∗ Re(D)BPT ∗ − B ∗ Re(D)BPT ∗ = PT ∗ B ∗ Re(D)BPT ∗ − 2B ∗ Re(D)BPT ∗ . Since R(PT ∗ B ∗ Re(D)B) ⊆ R(T ∗ ) ⊆ R(B ∗ ), we have that

R(NT C + C PT ∗ ) = R(PT ∗ B ∗ Re(D)BPT ∗ − 2B ∗ Re(D)BPT ∗ ) ⊆ R(B ∗ ).

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If moreover R(PT ∗ B ∗ Re(D)BPT ∗ + 2B ∗ Re(D)BPT ∗ ) ⊆ R(B ∗ ), then we have that R(NT C + C PT ∗ ) ⊆ R(B ∗ ).

By Proposition 2.3 and Theorem 2.6 (ii) (1), replacing A by T and C by C , the Eq. (3.1) has a solution W in LA (E). Set X0 = D − D∗ NB ∗ + NA W PB − PB W ∗ NA NB ∗ , we have AX0 B = ADB = C. Moreover, X 0 + X 0 ∗ = P B D + D ∗ P B + P B NA W P B + P B W ∗ NA P B . For arbitrary x ∈ E, without loss of generality we set x = Bz + x1 , z ∈ G, x1 ∈ N (B ∗ ), then (X0 +X0 ∗ )x, x = (PB D+D∗ PB )x, x+(PB NA W PB + PB W ∗ NA PB )x, x = (D + D∗ )Bz, Bz + (NA W + W ∗ NA )Bz, Bz = B ∗ Re(D)Bz, z + (B ∗ NA W B + B ∗ W ∗ NA B)z, z = (B ∗ Re(D)B + T ∗ W B + B ∗ W ∗ T )z, z = Zz, z ≥ 0. Therefore, we obtain a real positive solution X0 to AXB = C. (iii)⇔(iv): we know that NT B ∗ Re(D)BNT ≥ 0 ⇔ Re(D)BNT z, BNT z ≥ 0, for all z ∈ G. Since R(BNT ) = R(A∗ ) ∩ R(B) shown in (ii)⇒(iii), we have NT B ∗ Re(D)BNT ≥ 0 ⇔ Re(D)x, x ≥ 0, for all x ∈ R(A∗ ) ∩ R(B). Consequently, we have proven the equivalence of (i)–(iv). If R(B) is closed, then BNT ∈ LA (G, E) has closed range R(A∗ )∩R(B). By Theorem 3.2 in [24], there exists an projection from E onto R(BNT ), which is denoted by PA∗ ,B . Therefore, NT B ∗ Re(D)BNT ≥ 0 ⇔ PA∗ ,B Re(D)PA∗ ,B ≥ 0. Remark. Theorem 3.3 in [30] showed that the above result holds under the conditions that A, B, C and T have closed ranges by use of the Moore–Penrose inverse. That is a special case of Theorem 3.4. In the special case that G = E and B = IE , we obtain the necessary and sufﬁcient conditions for the existence of the real positive solution to the equation AX = C. Corollary 3.5. Let A, C ∈ LA (E, F ) and R(A∗ ) ⊆ E be orthogonally complemented. Then AX = C has a real positive solution X ∈ LA (E) if and only if R(C) ⊆ R(A),

Re(CA∗ ) ≥ 0.

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In this case, the general real positive solution to AX = C is of the form X = D − NA D∗ + NA V, where D is the reduced solution and V ∈ LA (E) satisfies that NA V + V ∗ NA ≥ −PA∗ (D + D∗ )PA∗ . Proof. Since B = IE , T = NA , NT = PA∗ , by Theorem 1.1 and Theorem 3.4 we have that AX = C has a real positive solution if and only if R(C) ⊆ R(A),

PA∗ Re(D)PA∗ ≥ 0.

For any x ∈ E with x = A∗ y + x (y ∈ F, x ∈ N (A)) under the orthogonal decomposition E = R(A∗ ) ⊕ N (A), we have that PA∗ (D∗ + D)PA∗ x, x = A(D∗ + D)A∗ y, y = (AC ∗ + CA∗ )y, y. Therefore, PA∗ Re(D)PA∗ ≥ 0 if and only if Re(CA∗ ) ≥ 0. In this case, since NA D = 0 (and so PA∗ D = D), the Eq. (3.1) becomes NA X + X ∗ NA = PA∗ Re(D)PA∗ − Re(D) = −NA D∗ − DNA . Then X = −D∗ is a solution to (3.1), and set W = −D∗ , we have a real positive solution X0 to AX = C: X0 = D − D∗ NB ∗ − NA D∗ PB + PB DNA NB ∗ = D − NA D∗ . Therefore, we could give the general form of real positive solution X ∈ LA (E) as follows: X = D − NA D ∗ + NA V 1 , where V1 ∈ LA (E) satisﬁes that NA V1 + V1 ∗ NA ≥ −PA∗ (D + D∗ )PA∗ . Set SA,B,C for the set consisting of all solutions X to the equation AXB = C such that B ∗ (X ∗ + X)B ≥ 0. It is clear that SA,B,C is a (real) Banach space. Definition 3.6. Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E). Set T = NA B and

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Z = NT B ∗ Re(D)BNT , and assume that R(T ) and R(T ∗ ) are the orthogonally complemented submodules of E and G respectively. Set Y(A, B, C) = {Z + ZV1 PT ∗ + PT ∗ V1 ∗ Z + PT ∗ V1 ∗ ZV1 PT ∗ + PT ∗ V2 PT ∗ : V1 ∈ LA (G), V2 ∈ LA (G)+ }, 1 Σ(A, B, C)1 = {X1 ∈ LA (E) : X1 B = PT T ∗−1 PT ∗ (Y − B ∗ Re(D)B)(NT + I) 2 for some Y ∈ Y(A, B, C), and X1 NB ∗ = 0}, Σ(A, B, C)2 = {X2 ∈ LA (E) : X2 B = NT ∗ V3 NT + iV4 T, for some V4 ∈ LA (E)sa and V3 ∈ LA (G, E) with R(V3 ∗ NT ∗ ) ⊆ R(B ∗ )}, Σ(A, B, C) = {X1 + X2 : X1 ∈ Σ(A, B, C)1 and X2 ∈ Σ(A, B, C)2 }, S(A, B, C) = {D − D∗ NB ∗ + NA W PB − PB W ∗ NA NB ∗ + V NB ∗ : V ∈ LA (E), W ∈ Σ(A, B, C)}.

It is clear that S(A, B, C) is a subset of the solution space to the equation AXB = C. Theorem 3.7. Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E). Set T = NA B and Z = NT B ∗ Re(D)BNT , and assume that R(T ) is closed. Then we have the following statements: (i) Σ(A, B, C)2 is the solution space to the equation T ∗ XB + B ∗ X ∗ T = 0. (ii) For any Y ∈ Y(A, B, C), there exists an unique operator XY ∈ LA (E) such that XY B = 12 PT T ∗ −1 PT ∗ (Y − B ∗ Re(D)B)(NT + I) and XY NB ∗ = 0, and so XY ∈ Σ(A, B, C)1 and Σ(A, B, C)1 is not empty. Furthermore, XY is a special solution to the equation T ∗ XB + B ∗ X ∗ T = Y − B ∗ Re(D)B. (iii) If Z ≥ 0, then Y(A, B, C) is contained in the positive solution space to the equation NT XNT = Z, and S(A, B, C) ⊆ SA,B,C . If moreover R(Z) is closed too, then Y(A, B, C) is just the positive solution space to the equation NT XNT = Z, and S(A, B, C) = SA,B,C . Proof. (i) By Corollary 2.2 we know that the solution space to the equation T ∗ XB + B ∗ X ∗ T = 0 is NTB = {X2 ∈ LA (E) : X2 B = NT ∗ V3 NT z + iV4 T, for some V4 ∈ LA (E)sa and V3 ∈ LA (G, E) such that R(NT V3 ∗ NT ∗ − iT ∗ V4 ) ⊆ R(B ∗ )}. Since R(T ) is closed (so is R(T ∗ )), R(PT ∗ ) = R(T ∗ ) ⊆ R(B ∗ ). Since R(NT V3 ∗ NT ∗ − iT ∗ V4 ) = R(V3 ∗ NT ∗ − PT ∗ V3 ∗ NT ∗ − iT ∗ V4 ) and R(PT ∗ ) = R(T ∗ ) ⊆ R(B ∗ ), we have R(NT V3 ∗ NT ∗ − iT ∗ V4 ) ⊆ R(B ∗ ) ⇔ R(V3 ∗ NT ∗ ) ⊆ R(B ∗ ), and so NTB = Σ(A, B, C)2 .

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Recall that Z = NT B ∗ Re(D)BNT . By Corollary 3.3, we know that Y ∈ Y(A, B, C) is a Hermitian solution to NT XNT = Z. For any Y ∈ Y(A, B, C), we consider the equation

T ∗ XB + B ∗ X ∗ T = Y − B ∗ Re(D)B = C .

(3.2)

Since R(PT ∗ ) = R(T ∗ ) ⊆ R(B ∗ ), R(Z) = R((1 − PT ∗ )B ∗ Re(D) BNT ) ⊆ R(B ∗ ). Therefore, R(C ) = R(Y − B ∗ Re(D)B) ⊆ R(B ∗ ). For NT C NT = 0, we have C NT = PT ∗ C NT and then R(C NT ) ⊆ R(T ∗ ). Obviously,

R(NT C + C PT ∗ ) = R(C − PT ∗ C + C PT ∗ ) ⊆ R(B ∗ ). By Theorem 2.6 (ii) (1), we know that the Eq. (3.2) has a special solution XY ∈ LA (E) deﬁned as follows: XY |N (B ∗ ) = 0;

XY Bz =

for all z ∈ G.

1 PT T ∗ −1 (C NT + PT ∗ C )z, 2

Since

C NT + P T ∗ C = P T ∗ C NT + P T ∗ C

= PT ∗ (Y − B ∗ Re(D)B)(NT + I), we have 1 PT T ∗ −1 PT ∗ (Y − B ∗ Re(D)B)(NT + I)z, 2 for all z ∈ G,

XY Bz =

and so XY ∈ Σ(A, B, C)1 . (iii)

Assume that Z = NT B ∗ Re(D)BNT ≥ 0, it is well known that Z is the reduced solution to equation NT XNT = Z. By Corollary 3.3, Y ∈ Y(A, B, C) is a positive solution to NT XNT = Z, and if R(Z) is closed, Y(A, B, C) is just the positive solution space to the equation NT XNT = Z. For any W ∈ Σ(A, B, C), there is a Y ∈ Y(A, B, C) such that W = XY + X2 for some X2 ∈ Σ(A, B, C)2 . By (i) and (ii) we know that XY + Σ(A, B, C)2 is the solution space to the equation T ∗ XB + B ∗ X ∗ T = Y − B ∗ Re(D)B. Then W is a solution to it. Clearly for any V ∈ LA (E), X = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ + V N B ∗ is a solution to AXB = C. Moreover, since NB ∗ B = 0, B ∗ (X + X ∗ )B = B ∗ (PB D + D∗ PB + NA W PB + PB W ∗ NA )B = B ∗ DB + B ∗ D∗ B + B ∗ NA W B + B ∗ W ∗ NA B = B ∗ Re(D)B + T ∗ W B + B ∗ W ∗ T = Y ≥ 0. Thus S(A, B, C) ⊆ SA,B,C .

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If moreover R(Z) is closed, by Corollary 3.3 we could see that Y(A, B, C) is the positive solution space to the equation NT XNT = Z. Suppose that X is a solution to AXB = C such that B ∗ (X +X ∗ )B ≥ 0. It is well known that D = PA∗ XPB , and then Z = NT B ∗ (D + D∗ )BNT = NT B ∗ (PA∗ XPB + PB X ∗ PA∗ )BNT = NT B ∗ (X + X ∗ )BNT , where the third equation holds for R(BNT ) = R(A∗ ) ∩ R(B). So B ∗ (X + X ∗ )B is a positive solution to the equation NT XNT = Z and B ∗ (X + X ∗ )B ∈ Y(A, B, C). Set Y = B ∗ (X + X ∗ )B. Since T ∗ NA XB + B ∗ X ∗ NA T = B ∗ NA XB + B ∗ X ∗ NA B = B ∗ XB + B ∗ X ∗ B − B ∗ PA∗ XB − B ∗ X ∗ PA∗ B = B ∗ XB + B ∗ X ∗ B − B ∗ PA∗ XPB B − B ∗ PB X ∗ PA∗ B = B ∗ (X + X ∗ )B − B ∗ (D + D∗ )B = Y − B ∗ Re(D)B, we know that W = NA X is a solution to T ∗ XB + B ∗ X ∗ T = Y − B ∗ Re(D)B. By (i) and (ii) we know that XY + Σ(A, B, C)2 is the solution space to the equation above, so W ∈ XY + Σ(A, B, C)2 , and then W ∈ Σ(A, B). Set V = X + PB X ∗ , we have D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ + V N B ∗ = PA∗ XPB − PB X ∗ PA∗ NB ∗ + NA XPB − PB X ∗ NA NB ∗ + V NB ∗ = XPB − PB X ∗ NB ∗ + V NB ∗ = X − (X + PB X ∗ )NB ∗ + V NB ∗ = X, and then X ∈ S(A, B, C). Thus if moreover R(Z) is closed, SA,B,C = S(A, B, C).

4. Positive Solution to AXB = C In this section, we will study the existence of positive solution to AXB = C. It is well known that if the equation AXB = C has a positive solution X ∈ LA (E), then B ∗ XB ≥ 0 and B ∗ (X + X ∗ )B ≥ 0. At the beginning, we will study the existence of solution X to AXB = C such that B ∗ XB ≥ 0 for general operators on Hilbert C ∗ -module. Lemma 4.1. Let A ∈ LA (E, F ), B ∈ LA (G, E) and C ∈ LA (G, F ). Suppose that R(A∗ ), R(B) are orthogonally complemented closed submodules of E and AXB = C has the reduced solution D in LA (E). Set T = NA B and assume that R(T ∗ ) is orthogonally complemented in G. If X ∈ LA (E) is a solution to AXB = C, then NT B ∗ DBNT = NT B ∗ XBNT .

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As a consequence, NT B ∗ DBNT is selfadjoint if there is a solution X to AXB = C such that B ∗ XB is selfadjoint, and NT B ∗ DBNT ≥ 0 if there is a solution X to AXB = C with B ∗ XB ≥ 0. Proof. In the proof of Theorem 3.4 (ii)⇒(iii), it has been shown that R(BNT ) = R(A∗ ) ∩ R(B), and so PA∗ BNT = BNT and NT B ∗ PA∗ = NT B ∗ . Suppose that AXB = C has a solution X, then D = PA∗ XPB . So for all z ∈ G, we have NT B ∗ DBNT z, z = NT B ∗ PA∗ XPB BNT z, z = XBNT z, BNT z = NT B ∗ XBNT z, z = NT B ∗ XBNT z, z, i.e., NT B ∗ DBNT = NT B ∗ XBNT .

Definition 4.2. Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E). Set T = NA B and Z = NT B ∗ Re(D)BNT , and assume that R(T ) and R(T ∗ ) are the orthogonally complemented submodule of E and G respectively. Set SY(A, B, C) = {Z + ZV1 PT ∗ + PT ∗ V1 ∗ Z + PT ∗ V1 ∗ ZV1 PT ∗ + PT ∗ V2 PT ∗ : V1 ∈ LA (G) such that NT B ∗ DBPT ∗ = NT B ∗ DBNT V1 PT ∗ , V2 ∈ LA (G)+ }, 1 SΣ(A, B, C)1 = {X1 ∈ LA (E) : X1 B = PT T ∗−1 PT ∗ (Y −B ∗ Re(D)B)(NT +I) 2 for some Y ∈ SY(A, B, C), and X1 NB ∗ = 0}, 1 SΣ(A, B, C)2 = {X2 ∈ LA (E) : X2 B = PT T ∗−1 (PT ∗ B ∗ (D∗ − D)BPT ∗ ) 2 + NT ∗ (V3 NT + iV4 T ) for some V4 ∈ LA (E), V3 ∈ LA (G, E) with R(V3 ∗ NT ∗ ) ⊆ R(B ∗ )}, SΣ(A, B, C) = {X1 + X2 : X1 ∈ SΣ(A, B, C)1 , X2 ∈ SΣ(A, B, C)2 }, SS(A, B, C) = {D−D∗ NB ∗ +NA W PB −PB W ∗ NA NB ∗ +V NB ∗ : V ∈ LA (E), W ∈ SΣ(A, B, C)}.

It is clear that SS(A, B, C) is a subset of the solution space to the equation AXB = C. Since SY(A, B, C) ⊆ Y(A, B, C), SΣ(A, B, C)1 ⊆ Σ(A, B, C)1 . If moreover R(T ) is closed, then by Theorem 3.7 (i) SΣ(A, B, C)2 ⊆ Σ(A, B, C)2 . Therefore (1)

if R(T ) is closed, then SΣ(A, B, C) ⊆ Σ(A, B, C),

(2)

SS(A, B, C) ⊆ S(A, B, C);

if R(T ) is closed and Z ≥ 0, then SS(A, B, C) ⊆ S(A, B, C) ⊆ SA,B,C ;

(3)

if R(T ) and R(Z) are closed and Z ≥ 0, then SS(A, B, C) ⊆ S(A, B, C) = SA,B,C .

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Proposition 4.3. Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E), and T = NA B has the closed range. If X ∈ S(A, B, C) is a solution to AXB = C such that B ∗ XB is selfadjoint, then there exists V ∈ LA (G) such that NT B ∗ DB = NT B ∗ DBNT V , and X ∈ SS(A, B, C). As a consequence, R(NT B ∗ DB) ⊆ R(NT B ∗ DBNT ). Proof. Recall that Z = NT B ∗ Re(D)BNT . If X ∈ S(A, B, C), then let X = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ + V N B ∗ for some V ∈ LA (E) and W ∈ Σ(A, B, C). Suppose that W = XY + X2 with 1 PT T ∗ −1 PT ∗ (Y − B ∗ Re(D)B)(NT + I) and XY NB ∗ = 0, 2 X2 B = NT ∗ V3 NT + iV4 T, Y = Z + ZV1 PT ∗ + PT ∗ V1 ∗ Z + PT ∗ V1 ∗ ZV1 PT ∗ + PT ∗ V2 PT ∗ ,

XY B =

for some V1 ∈ LA (G), V2 ∈ LA (G)+ , V4 ∈ LA (E)sa and V3 ∈ LA (G, E) such that R(V3 ∗ NT ∗ ) ⊆ R(B ∗ ). Set C = Y − B ∗ Re(D)B. Since Y is selfadjoint, so is C . Clearly NT C NT = 0, and so PT ∗ C NT = C NT . Then we have B ∗ XB = B ∗ DB + B ∗ NA W B = B ∗ DB + T ∗ W B 1 = B ∗ DB + PT ∗ C (I + NT ) + iT ∗ V4 T 2 1 = B ∗ DB + (C NT + PT ∗ C ) + iT ∗ V4 T 2 1 1 1 = Y − Y + B ∗ DB + (C NT + PT ∗ C ) + iT ∗ V4 T 2 2 2 1 1 = Y +B ∗ DB + (C NT +PT ∗ C −C −B ∗ (D + D∗ )B) + iT ∗ V4 T 2 2 1 1 1 = Y + (B ∗ DB − B ∗ D∗ B) − (C PT ∗ − PT ∗ C ) + iT ∗ V4 T. 2 2 2 Set 1 1 1 C0 = B ∗ XB − Y − (B ∗ DB − B ∗ D∗ B) + (C PT ∗ − PT ∗ C ), 2 2 2

then the equation iT ∗ XT = C0 has a Hermitian solution V4 in LA (E), and so, by Corollary 3.3, C0 ∗ = −C0 ,

NT C0 = 0.

From the ﬁrst equation it could be seen that B ∗ XB = 12 Y , and then C0 =

1 1 (C PT ∗ − PT ∗ C ) − (B ∗ DB − B ∗ D∗ B). 2 2

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Thus we have 1 1 0 = NT C0 = NT ( (C PT ∗ − PT ∗ C ) − (B ∗ DB − B ∗ D∗ B)) 2 2 1 = (NT C PT ∗ + NT B ∗ D∗ B − NT B ∗ DB) 2 1 = NT (C + B ∗ D∗ B − B ∗ DB) 2 1 = NT (Y − 2B ∗ DB) 2 1 = (Z + ZV1 PT ∗ ) − NT B ∗ DB. 2 By Lemma 4.1 it is known that NT B ∗ DBNT is selfadjoint and Z = 2NT B ∗ DBNT , so 0 = NT B ∗ DBNT + NT B ∗ DBNT V1 PT ∗ − NT B ∗ DB

= NT B ∗ DBNT V1 PT ∗ − NT B ∗ DBPT ∗ . Hence, we obtain that NT B ∗ DB = NT B ∗ DB(NT + PT ∗ ) = NT B ∗ DBNT (I + V1 PT ∗ ). In the following work, we will show that X ∈ SS(A, B, C). It is only needed to prove that X2 ∈ SΣ(A, B, C)2 . By discussion above, we know that Z = 2NT B ∗ DBNT = Z ∗ , ZV1 PT ∗ = 2NT B ∗ DBPT ∗ , and V4 is a Hermitian solution to the equation i T ∗ V4 T = − (C PT ∗ − PT ∗ C − B ∗ DB + B ∗ D∗ B). 2 First of all we will characterize V4 concretely. By direct computation we have that

C PT ∗ − PT ∗ C − B ∗ DB + B ∗ D∗ B = (Y − B ∗ Re(D)B)PT ∗ − PT ∗ (Y − B ∗ Re(D)B) − B ∗ DB + B ∗ D∗ B = Y PT ∗ − PT ∗ Y − B ∗ Re(D)BPT ∗ + PT ∗ B ∗ Re(D)B − B ∗ DB + B ∗ D∗ B = ZV1 PT ∗ −PT ∗ V1 ∗ Z −B ∗ Re(D)BPT ∗ +PT ∗ B ∗ Re(D)B −B ∗ DB +B ∗ D∗ B = 2NT B ∗ DBPT ∗ − 2PT ∗ B ∗ D∗ BNT − B ∗ Re(D)BPT ∗ +PT ∗ B ∗ Re(D)B − B ∗ DB + B ∗ D∗ B = 2NT B ∗ DB − 2NT B ∗ DBNT − 2PT ∗ B ∗ D∗ BNT −B ∗ Re(D)BPT ∗ + PT ∗ B ∗ Re(D)B − B ∗ DB + B ∗ D∗ B = 2NT B ∗ DB − 2NT B ∗ D∗ BNT − 2PT ∗ B ∗ D∗ BNT −B ∗ Re(D)BPT ∗ + PT ∗ B ∗ Re(D)B − B ∗ DB + B ∗ D∗ B = 2NT B ∗ DB − 2B ∗ D∗ BNT − B ∗ DBPT ∗ + PT ∗ B ∗ D∗ B − NT B ∗ DB +B ∗ D∗ BNT = NT B ∗ DB − B ∗ D∗ BNT − B ∗ DBPT ∗ + PT ∗ B ∗ D∗ B = NT B ∗ DBPT ∗ +NT B ∗ DBNT −B ∗ D∗ BNT −B ∗ DBPT ∗ + PT ∗ B ∗ D∗ B = NT B ∗ DBPT ∗ − PT ∗ B ∗ D∗ BNT − B ∗ DBPT ∗ + PT ∗ B ∗ D∗ B = PT ∗ B ∗ D∗ BPT ∗ − PT ∗ B ∗ DBPT ∗ = PT ∗ B ∗ (D∗ − D)BPT ∗ .

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By Corollary 3.3, we know that V4 = DT + NT ∗ U1 + U1 ∗ NT ∗ + NT ∗ U2 NT ∗ , where U1 ∈ LA (E), U2 ∈ LA (E)sa , and DT is the reduced solution to T ∗ V4 T = − 2i (C PT ∗ − PT ∗ C − B ∗ DB + B ∗ D∗ B). Therefore, DT T = i ∗ −1 (PT ∗ B ∗ (D − D∗ )BPT ∗ ). Furthermore, we have that 2 PT T V4 T = DT T + NT ∗ U1 T i = PT T ∗ −1 (PT ∗ B ∗ (D − D∗ )BPT ∗ ) + NT ∗ U1 T. 2 By definition X2 ∈ Σ(A, B, C)2 satisﬁes that X2 B = NT ∗ V3 NT + iV4 T 1 = NT ∗ V3 NT + PT T ∗ −1 (PT ∗ B ∗ (D∗ − D)BPT ∗ ) + iNT ∗ U1 T 2 1 = NT ∗ (V3 NT + iU1 T ) + PT T ∗ −1 (PT ∗ B ∗ (D∗ − D)BPT ∗ ). 2 It follows that X2 ∈ SΣ(A, B, C)2 ⊆ Σ(A, B, C)2 .

Proposition 4.4. Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E), and T = NA B has the closed range. If NT B ∗ DBNT is selfadjoint, and there exists V ∈ LA (G) with ∗ NT B DB = NT B ∗ DBNT V , then AXB = C has a solution X ∈ S(A, B, C) such that B ∗ XB is selfadjoint, and 2B ∗ XB ∈ SY(A, B, C) ⊆ Y(A, B, C). Proof. Recall that Z = NT B ∗ Re(D)BNT , then Z = 2NT B ∗ DBNT is selfadjoint. Let V1 be the element in LA (E) such that NT B ∗ DBNT V1 = NT B ∗ DB. Set Y = Z + ZV1 PT ∗ + PT ∗ V1∗ Z + PT ∗ V1∗ ZV1 PT ∗ ,

C = Y − B ∗ Re(D)B, then Y is selfadjoint and in SY(A, B, C), and C

∗

= C . It follows that

NT (C PT ∗ − PT ∗ C − B ∗ DB + B ∗ D∗ B)

= NT C PT ∗ − NT B ∗ DB + NT B ∗ D∗ B = ZV1 PT ∗ − NT B ∗ Re(D)BPT ∗ − NT B ∗ DB + NT B ∗ D∗ B = ZV1 PT ∗ − NT B ∗ DB(PT ∗ + I) + NT B ∗ D∗ BNT = 2(NT B ∗ DBNT V1 PT ∗ − NT B ∗ DBPT ∗ ) = 0. Therefore, applying Corollary 3.3 (ii) we know that the equation i T ∗ XT = − (C PT ∗ − PT ∗ C − B ∗ DB + B ∗ D∗ B) 2 has a Hermitian solution V4 ∈ LA (E). Set X1 = XY , which is deﬁned by Y as in Theorem 3.7 (ii), then 1 XY B = PT T ∗ −1 PT ∗ C (NT + I). 2

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Since R(T ∗ V4∗ ) ⊆ R(T ∗ ) ⊆ R(B ∗ ), there exists X2 ∈ LA (E) with X2 B = iV4 T , and so X2 ∈ Σ(A, B, C)2 . Set W = X1 + X2 and X = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ , then W ∈ Σ(A, B, C) and X ∈ S(A, B, C). Moreover, by the same discussion as in the proof of Proposition 4.3, we have B ∗ XB = B ∗ DB + B ∗ NA W B = B ∗ DB + T ∗ W B 1 = B ∗ DB + PT ∗ C (I + NT ) + iT ∗ V4 T 2 1 ∗ = B DB + (C NT + PT ∗ C ) + iT ∗ V4 T 2 1 1 1 = Y − Y + B ∗ DB + (C NT + PT ∗ C ) + iT ∗ V4 T 2 2 2 1 1 ∗ = Y + B DB + (C NT + PT ∗ C − C − B ∗ (D + D∗ )B) + iT ∗ V4 T 2 2 1 1 1 1 ∗ = Y + (B DB − B ∗ D∗ B) − (C PT ∗ − PT ∗ C ) + iT ∗ V4 T = Y. 2 2 2 2

Therefore, B ∗ XB is selfadjoint. Since Y SY(A, B, C).

∈ SY(A, B, C), 2B ∗ XB ∈

Combining Propositions 4.3 and 4.4, we have the following result. Theorem 4.5. Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E), and T = NA B has the closed range. Then the following statements are equivalent: (i) AXB = C has a solution X ∈ S(A, B, C) such that B ∗ XB is selfadjoint; (ii) NT B ∗ DBNT is selfadjoint and there exists V ∈ LA (G) with NT B ∗ DB = NT B ∗ DBNT V . Moreover, if NT B ∗ DBNT is selfadjoint, then SS(A, B, C) = {X ∈ S(A, B, C) : B ∗ XB is selfadjoint}, (iii)

and in this case (i) and (ii) are equivalent to AXB = C has a solution X ∈ SS(A, B, C).

Proof. It is obvious that (i)⇔(ii) and {X ∈ S(A, B, C) : B ∗ XB is selfadjoint} ⊆ SS(A, B, C) by Lemma 4.1, Propositions 4.3 and 4.4. For any X ∈ SS(A, B, C), set X = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ + V N B ∗ for some V ∈ LA (E), W ∈ SΣ(A, B, C). Set W = XY + X2 for some Y ∈ SY(A, B, C) and X2 ∈ SΣ(A, B, C)2 . Since NT B ∗ DBNT is selfadjoint, Z = 2NT B ∗ DBNT is selfadjoint. Since Y ∈ SY(A, B, C), we may set Y = Z + ZV1 PT ∗ + PT ∗ V1∗ Z + PT ∗ V1∗ ZV1 PT ∗ + PT ∗ V2 PT ∗ ,

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for some V2 ∈ LA (E)+ and V1 ∈ LA (E) with NT B ∗ DBNT V1 PT ∗ = NT B ∗ DBPT ∗ , and so ZV1 PT ∗ = 2NT B ∗ DBNT PT ∗ . Set C = Y −B ∗ Re(D)B, then C = C ∗ . By the same computation as in the proof of Proposition 4.3, we have

C PT ∗ − PT ∗ C − B ∗ DB + B ∗ D∗ B = PT ∗ B ∗ (D∗ − D)BPT ∗ . So we have B ∗ XB = B ∗ DB + B ∗ NA W B = B ∗ DB + T ∗ W B = B ∗ DB + T ∗ XY B + T ∗ X2 B 1 1 = B ∗ DB + PT ∗ C (I + NT ) + (PT ∗ B ∗ (D∗ − D)BPT ∗ ) 2 2 1 1 = B ∗ DB + PT ∗ C (I +NT )+ (C PT ∗ −PT ∗ C −B ∗ DB + B ∗ D∗ B) 2 2 1 1 1 = B ∗ Re(D)B + PT ∗ C NT + C PT ∗ 2 2 2 1 1 1 1 = B ∗ Re(D)B + PT ∗ C − PT ∗ C PT ∗ + C PT ∗ . 2 2 2 2 Therefore, B ∗ XB is selfadjoint and thus SS(A, B, C) = {X ∈ S(A, B, C) : B ∗ XB is selfadjoint}. Corollary 4.6. Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E), and T = NA B has the closed range. Then the following statements are equivalent: (i) AXB = C has a solution X ∈ S(A, B, C) such that B ∗ XB ≥ 0; (ii) NT B ∗ DBNT ≥ 0 and there exists V ∈ LA (G) such that NT B ∗ DB = NT B ∗ DBNT V . Furthermore, if R(NT B ∗ DBNT ) is orthogonally complemented in G, then (i) and (ii) are equivalent to (iii) NT B ∗ DBNT ≥ 0, R(NT B ∗ DB) ⊆ R(NT B ∗ DBNT ); and if moreover R(NT B ∗ DBNT ) is closed, then (i), (ii) and (iii) are equivalent to (iv) AXB = C has a solution X ∈ LA (E) such that B ∗ XB ≥ 0. Proof. (i)⇒(ii) It is obvious by Lemma 4.1 and Proposition 4.3. (ii)⇒(i) By Proposition 4.4, we know there exists a solution X ∈ S(A, B, C) with 2B ∗ XB ∈ Y(A, B, C). Recall Z = NT B ∗ Re(D)BNT . For NT B ∗ DBNT ≥ 0, we know that Z ≥ 0 and then each Y ∈ Y(A, B, C) is positive by Theorem 3.7 (iii). So B ∗ XB ≥ 0. In the case that R(NT B ∗ DBNT ) is orthogonally complemented in G, the equivalence of (ii) and (iii) is from Theorem 1.1. Now suppose that R(NT B ∗ DBNT ) is closed. If NT B ∗ DBNT ≥ 0, then Z = 2NT B ∗ DBNT ≥ 0 has the closed range. By Theorem 3.4 (iii) we know that the solution X to AXB = C such that B ∗ XB ≥ 0 is in SA,B,C = S(A, B, C). So (i) is equivalent to (iv).

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Remark. Theorem 4.2 in [30] showed that for A, B with closed ranges, if R(T ) and R(NT B ∗ DBNT ) are closed, then (iii) ⇔ (iv). Corollary 4.6 proves that it is right for general operators A, B, C with R(A∗ ) and R(B) orthogonally complemented. So Corollary 4.6 implies Theorem 4.2 in [30]. Proposition 4.7. Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E), T = NA B has the closed range, and NT B ∗ DBNT is selfadjoint. Let X ∈ SS(A, B, C) with X = D − D∗ NB ∗ + NA W PB − PB W ∗ NA NB ∗ + V NB ∗ for some V ∈ LA (E) and W ∈ SΣ(A, B, C). Then X = X ∗ if and only if there exist V1 ∈ LA (E) and V2 ∈ LA (E)sa such that 1 V = − (PB (D + NA W PB ) − (D∗ + PB W ∗ NA )(I + 3NB ∗ )) 2 − iPB V1 ∗ PB − NB ∗ V2 , In which case, X = D +D∗ NB ∗ +NA W PB +PB W ∗ NA NB ∗ +NB∗ (−V2 )NB ∗ . As a consequence, SS(A, B, C)sa = {D + D∗ NB ∗ + NA W PB + PB W ∗ NA NB ∗ + NB ∗ V NB ∗ : V ∈ LA (E)sa , W ∈ SΣ(A, B, C)}. Proof. Since X = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ + V N B ∗ , then X ∗ − X = (D∗ − NB ∗ D + PB W ∗ NA − NB ∗ NA W PB + NB ∗ V ∗ ) −(D − D∗ NB ∗ + NA W PB − PB W ∗ NA NB ∗ + V NB ∗ ) = (D∗ + PB W ∗ NA )(I + NB ∗ ) − (I + NB ∗ )(D + NA W PB ) + (NB ∗ V ∗ − V NB ∗ ). Therefore, X = X ∗ if and only if V is a solution to the equation NB ∗ X ∗ − XNB ∗ = (I + NB ∗ )(D + NA W PB ) − (D∗ + PB W ∗ NA )(I + NB ∗ ). Set C1 = (I +NB ∗ )(D +NA W PB )−(D∗ +PB W ∗ NA )(I +NB ∗ ). Firstly, we show the existence of the solution to the equation NB ∗ X ∗ − XNB ∗ = C1 . Since C1 = −C1 ∗ , from Corollary 2.5 we know that there exists a solution in LA (E) to NB ∗ X ∗ − XNB ∗ = C1 if and only if PB C1 PB = 0. Since D is the reduced solution to AXB = C, PA∗ DPB = D. Directly computing, we have PB C1 PB = PB ((I + NB ∗ )(D + NA W PB ) − (D∗ + PB W ∗ NA )(I + NB ∗ ))PB = PB (D + NA W PB ) − (D∗ + PB W ∗ NA )PB = PB (D + NA W − D∗ − W ∗ NA )PB . Therefore, the equation NB ∗ X ∗ − XNB ∗ = C1 has a solution if and only if PB (D + NA W − D∗ − W ∗ NA )PB = 0, which is equivalent to B ∗ (D + NA W − D∗ − W ∗ NA )B = 0.

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In fact, since NT B ∗ DBNT is selfadjoint and X ∈ SS(A, B, C), B XB = B ∗ DB + T ∗ W B is selfadjoint by Theorem 4.5. Therefore, ∗

B ∗ (D + NA W − D∗ − W ∗ NA )B = B ∗ DB + T ∗ W B − B ∗ D∗ B − BW ∗ T = B ∗ XB − B ∗ X ∗ B = 0. Thus the equation NB ∗ X ∗ − XNB ∗ = C1 has a solution in LA (E). By Corollary 2.5, V is a solution, which is equivalent to X = X ∗ , if and only if there exist V1 ∈ LA (E) and V2 ∈ LA (E)sa such that 1 V = − (PB C1 + C1 NB ∗ ) − iPB V1 ∗ PB − NB ∗ V2 2 1 = − (PB (D + NA W PB ) − (D∗ + PB W ∗ NA )(I + NB ∗ ) 2 − 2(D∗ + PB W ∗ NA )NB ∗ ) − iPB V1 ∗ PB − NB ∗ V2 1 = − (PB (D + NA W PB ) − (D∗ + PB W ∗ NA )(I + 3NB ∗ )) 2 − iPB V1 ∗ PB − NB ∗ V2 . Therefore, X = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ + V N B ∗ = D − D∗ NB ∗ + NA W PB − PB W ∗ NA NB ∗ + 2(D∗ + PB W ∗ NA )NB ∗ − N B ∗ V 2 NB ∗ = D + D∗ NB ∗ + NA W PB + PB W ∗ NA NB ∗ + NB∗ (−V2 )NB ∗ . Thus SS(A, B, C)sa ⊆ {D + D∗ NB ∗ + NA W PB + PB W ∗ NA NB ∗ + NB ∗ V NB ∗ : V ∈ LA (E)sa , W ∈ SΣ(A, B, C)}. Conversely, for any V ∈ LA (E)sa and W ∈ SΣ(A, B, C), we have D + D ∗ NB ∗ + N A W P B + P B W ∗ NA NB ∗ + N B ∗ V N B ∗ = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ +(2PB W ∗ NA − 2D∗ + NB ∗ V )NB ∗ ∈ SS(A, B, C), and so B ∗ (D + D∗ NB ∗ + NA W PB + PB W ∗ NA NB ∗ + NB ∗ V NB ∗ )B = B ∗ (D + NA W )B is selfadjoint by Theorem 4.5, which is equivalent to PB (D + NA W )PB is selfadjoint. Then the last statement of the proposition is from that D + D∗ NB ∗ + NA W PB + PB W ∗ NA NB ∗ + NB∗ V NB ∗ = PB (D + NA W )PB + PB (D∗ + W ∗ NA )NB ∗ + NB ∗ (D + NA W )PB + NB∗ V NB ∗ is selfadjoint.

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Lemma 4.8. Let A ∈ LA (E)+ and S ⊆ E be an orthogonally complemented 12 under the orthogonal decomposition closed submodule. Set A = AA1211∗ A A22

E = S ⊕ S ⊥. (i) A12 A12 ∗ ≤ λA11 for some positive number λ > 0. (ii) If R(A11 ) is an orthogonally complemented closed submodule of E, then 1 1 R(A12 ) ⊆ R(A11 2 ), i.e., the equation A11 2 X = A12 has the reduced solution. S12 under orthogonal decomposition Proof. Set A = S ∗ S and S = SS11 21 S22 ⊥ E = S ⊕ S . Then we have

S11 S12 S11 ∗ S12 ∗ A= S21 ∗ S22 ∗ S21 S22

=

S11 ∗ S11 + S21 ∗ S21 S12 ∗ S11 + S22 ∗ S21

S11 ∗ S12 + S21 ∗ S22 S12 ∗ S12 + S22 ∗ S22

.

It follows that A12 A∗12 = (S11 ∗ S12 + S21 ∗ S22 )(S11 ∗ S12 + S21 ∗ S22 )∗ = S11 ∗ S12 S12 ∗ S11 +S21 ∗ S22 S22 ∗ S21 + (S11 ∗ S12 S22 ∗ S21 + S21 ∗ S22 S12 ∗ S11 ) ≤ S11 ∗ S12 S12 ∗ S11 + S21 ∗ S22 S22 ∗ S21 + S11 ∗ S12 S12 ∗ S11 +S21 ∗ S22 S22 ∗ S21 = 2(S11 ∗ S12 S12 ∗ S11 + S21 ∗ S22 S22 ∗ S21 )

≤ 2( S12 2 S11 ∗ S11 + S22 2 S21 ∗ S21 ). Set λ = max{2 S12 2 , 2 S22 2 }, then we have A12 A∗12 ≤ λA11 , and applying Theorem 1.1 (i) we have completed the proof of (ii).

Proposition 4.9. Let A ∈ LA (E)sa ,and S ⊆E be an orthogonally comple12 under the orthogonal decommented closed submodule. Set A = AA1211∗ A A22 1

position E = S ⊕ S ⊥ and suppose that A11 ≥ 0 and A11 2 X = A12 has the ∗ D12 ≥ reduced solution D12 ∈ LA (S ⊥ , S). Then A ≥ 0 if and only if A22 −D12 0. 1

Proof. Since A11 ≥ 0 and A11 2 X = A12 has the reduced solution D12 , 1

R(D12 ) ⊆ R(A11 2 ). For any x ∈ E, we let x = y + z, y ∈ S, z ∈ S ⊥ , then it follows that Ax, x = A11 y, y + A12 z, y + A12 ∗ y, z + A22 z, z 1

1

∗ A11 2 y, z + A22 z, z = A11 y, y + A11 2 D12 z, y + D12 1

1

= A11 y, y + D12 z, A11 2 y + A11 2 y, D12 z + A22 z, z 1

1

∗ D12 )z, z. = A11 2 y + D12 z, A11 2 y + D12 z + (A22 − D12 ∗ Hence, if A22 − D12 D12 ≥ 0, then A ≥ 0.

1

Conversely for any z ∈ S ⊥ , since R(D12 ) ⊆ R(A11 2 ), there exists yn ∈ S 1 such that A11 2 yn + D12 z < 1/n. Set xn = yn + z, then by the above

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∗ D12 )z, z. Therefore if A ≥ 0, equation we have Axn , xn → (A22 − D12 ∗ (A22 − D12 D12 )z, z ≥ 0. By the Arbitrariness of z, we obtain that if A ≥ 0, ∗ D12 ≥ 0. then A22 − D12

Proposition 4.10. Let A ∈ LA (E)sa, and S ⊆E be an orthogonally comple12 under the orthogonal decommented closed submodule. Set A = AA1211∗ A A22 position E = S ⊕ S ⊥ and suppose that R(A11 ) ⊆ S is orthogonally complemented. Then A ≥ 0 if and only if A11 ≥ 0, A12 A12 ∗ ≤ λA11 for some positive number λ, and 1

1

A22 − (PA11 A11 − 2 A12 )∗ PA11 A11 − 2 A12 ≥ 0. Proof. It is easy to know that if A ≥ 0, then A11 ≥ 0 and A22 ≥ 0. If there exists a positive number λ > 0 such that A12 A12 ∗ ≤ λA11 , then by Theorem 1 1.1 (i) we know that the equation A11 2 X = A12 has the reduced solution 1 D12 ∈ LA (S ⊥ , S) and D12 = PA11 A11 − 2 A12 . Then by Lemma 4.8 and Proposition 4.9, we obtain A ≥ 0 if and only if A11 ≥ 0, A12 A12 ∗ ≤ λA11 for some positive number λ and 1

1

A22 − (PA11 A11 − 2 A12 )∗ PA11 A11 − 2 A12 ≥ 0. Based on Propositions 4.9 and 4.10, we will characterize the positive solution to AXB = C and obtain some sufﬁcient and necessary conditions for the existence of the positive solution in the case that R(B) ⊆ R(A∗ ). Let A ∈ LA (E, F ), B ∈ LA (G, E), and let C ∈ LA (G, F ). Suppose that R(A∗ ) and R(B) are orthogonally complemented submodules of E, AXB = C has the reduced solution D ∈ LA (E), and T = NA B has the closed range. Set Z = NT B ∗ Re(D)BNT . Let X be an element in SS(A, B, C)sa . By Lemma 4.1 NT B ∗ DBNT is selfadjoint. By Proposition 4.7, X is of the form D + D ∗ NB ∗ + N A W P B + P B W ∗ NA NB ∗ + N B ∗ V N B ∗ for V ∈ LA (E)sa and W ∈ SΣ(A, B, C). Then B ∗ XB = B ∗ (D + NA W )B. 11 X12 under the orthogonal decomposition E = R(B) + Set X = X X21 X22 ∗ N (B ). With Xij viewed as elements in LA (E), we have X = X11 + X12 + X12 ∗ + X22 with X11 = PB (D + NA W )PB ,

X12 = PB (D∗ + W ∗ NA )NB ∗ ,

X22 = NB ∗ V NB ∗ .

Moreover we assume Z ≥ 0, which is equivalent to NT B ∗ DBNT ≥ 0 for NT B DBNT is selfadjoint. By the definitions of S(A, B, C) and SS(A, B, C) and Theorem 3.7 (i, iii), it is easy to see that SS(A, B, C) ⊆ S(A, B, C) ⊆ SA,B,C . In this case, we have B ∗ XB = 12 B ∗ (X + X ∗ )B ≥ 0. Then B ∗ (D + NA W )B = B ∗ XB ≥ 0, and so we have that ∗

PB (D + NA W )PB ≥ 0, i.e., X11 ≥ 0.

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Proposition 4.11. Let A ∈ LA (E, F ), B ∈ LA (G, E) and C ∈ LA (G, F ). Suppose that R(A∗ ), R(B) are orthogonally complemented closed submodules of E, and AXB = C has the reduced solution D in LA (E) and a solution X in SS(A, B, C)sa of the form D + D ∗ NB ∗ + N A W P B + P B W ∗ NA NB ∗ + N B ∗ V N B ∗ ,

11 X12 for V ∈ LA (E)sa and W ∈ SΣ(A, B, C). Set T = NA B and X = X X21 X22 under the orthogonal decomposition E = R(B) + N (B ∗ ). Suppose that T has the closed range, and NT B ∗ DBNT ≥ 0. (i) If X ≥ 0, then there exists a positive number λ > 0 such that B ∗ (D∗ + W ∗ NA )NB ∗ (D + NA W )B ≤ λ(B ∗ DB + T ∗ W B). (ii)

If R(X11 ) is orthogonally complemented in E, then X ≥ 0 if and only if there exists a positive number λ > 0 such that B ∗ (D∗ + W ∗ NA )NB ∗ (D + NA W )B ≤ λ(B ∗ DB + T ∗ W B), and 1

∗

1

X22 ≥ (PX11 X11 − 2 X12 ) PX11 X11 − 2 X12 . Proof. Firstly by the discussion above X11 ≥ 0. For any λ > 0, it is easy to see that B ∗ (D∗ + W ∗ NA )NB ∗ (D + NA W )B ≤ λ(B ∗ DB + T ∗ W B) if and only if PB (D∗ + W ∗ NA )NB ∗ (D + NA W )PB ≤ λPB (D + NA W )PB , i.e., X12 X12 ∗ ≤ λX11 . Then the proof can be completed by Proposition 4.10.

Theorem 4.12. Let A ∈ LA (E, F ), B ∈ LA (G, E), C ∈ LA (G, F ) such that R(B) ⊆ R(A∗ ), and R(A∗ ), R(B) be orthogonally complemented submodules of E. Suppose that AXB = C has the reduced solution D ∈ LA (E). (i) If AXB = C has a positive solution X ∈ LA (E), then B ∗ DB ≥ 0 and there exists a positive number λ such that B ∗ D∗ NB ∗ DB ≤ λB ∗ DB. (ii)

Suppose R(PB DPB ) is orthogonally complemented in E. If B ∗ DB ≥ 0 and B ∗ D∗ NB ∗ DB ≤ λB ∗ DB for some λ > 0, then AXB = C has a positive solution X ∈ SS(A, B, C)sa .

Proof. Since R(B) ⊆ R(A∗ ), we have N (A) ⊆ N (B ∗ ), and so that PA ∗ PB = PB ,

NA NB ∗ = NA ,

NA PB = 0.

Moreover T = NA B = 0, and so that NT = IG and Z = B ∗ (D + D∗ )B. Thus Y(A, B, C) = SY(A, B, C) = {Z},

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and then Σ(A, B, C)1 = SΣ(A, B, C)1 = {0},

Σ(A, B)2 = SΣ(A, B, C)2 = LA (E).

So Σ(A, B, C) = SΣ(A, B, C) = LA (E). Therefore we have S(A, B, C) = SS(A, B, C) = {D−D∗ NB ∗ +NA W PB −PB W ∗ NA +V NB ∗ : V, W ∈ LA (E)}. (i)

Suppose that X ∈ LA (E) is a positive solution to the equation AXB = C, then D = PA∗ XPB . It follows that X = XPB − PB XNB ∗ + (X + PB X)NB ∗ = PA∗ XPB + NA XPB − (PB XPA∗ NB ∗ + PB XNA NB ∗ ) + (X + PB X)NB ∗ = D − D∗ NB ∗ + NA XPB − PB XNA NB ∗ + (X + PB X)NB ∗ .

Set W = NA X and V = X + PB X, then

X = D − D ∗ NB ∗ + N A W P B − P B W ∗ NA NB ∗ + V N B ∗ . Consequently, X ∈ SS(A, B, C)sa and NT B ∗ DBNT = B ∗ DB = B ∗ XB ≥ 0. From Proposition 4.11 (i), for T = 0, there is λ > 0 such that B ∗ (D∗ + W ∗ NA )NB ∗ (D + NA W )B ≤ λ(B ∗ DB + T ∗ W B) = λB ∗ DB. Since NA NB ∗ = NA and D = PA∗ DPB , NA NB ∗ D = NA PA∗ D = 0. Then B ∗ (D∗ + W ∗ NA )NB ∗ (D + NA W )B = B ∗ D∗ NB ∗ DB + B ∗ W ∗ NA W B. Therefore B ∗ D∗ NB ∗ DB ≤ λB ∗ DB. (ii)

Take W ∈ LA (E) satisfying B ∗ W ∗ NA W B ≤ λB ∗ DB − B ∗ D∗ NB ∗ DB, for example take W = 0. Since NA NB ∗ = NA and D = PA∗ DPB , we have B ∗ (D∗ + W ∗ NA )NB ∗ (D + NA W )B = B ∗ D∗ NB ∗ DB + B ∗ W ∗ NA W B ≤ λB ∗ DB, i.e. PB (D∗ + W ∗ NA )NB ∗ (D + NA W )PB ≤ λPB DPB . Since R(PB DPB ) is orthogonally complemented in E and PB DPB ≥ 0 which is equivalent to B ∗ DB ≥ 0, applying Theorem 1.1 (i), the equation 1

(PB DPB ) 2 X = PB (D∗ + W ∗ NA )NB ∗ = D∗ NB ∗ + PB W ∗ NA

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= PP DP (PB DPB )− 12 (D∗ NB ∗ +PB W ∗ NA ). has the reduced solution D B B = DN B ∗ , we may chose V ∈ LA (E)sa such that Since D By Proposition 4.7 ∗ D. NB ∗ V N B ∗ ≥ D X = D + D ∗ NB ∗ + N A W P B + P B W ∗ NA + N B ∗ V N B ∗ is a solution to AXB = C in SS(A, B, C)sa . Since T = 0, NT B ∗ DBNT = B ∗ DB ≥ 0. By Proposition 4.11, we could see X ≥ 0. Remark. In the special case that R(A) and R(B) are closed, Theorem 5.6 in [30] showed that if R(B ∗ DB) is closed, then AXB = C has a positive solution if and only if B ∗ DB ≥ 0,

R(B ∗ D∗ ) ⊆ R(B ∗ D∗ B).

We know that if R(B) is closed (so is R(B ∗ )), then R(PB DB) = R(PB DPB ). Since B ∗ : R(B) → R(B ∗ ) is invertible, R(B ∗ DB) = R(B ∗ PB DB) is closed if and only if R(PB DB) is closed, i.e., R(B ∗ DB) is closed if and only if R(PB DPB ) is closed. Thus if B ∗ DB ≥ 0, by Theorem 1.1 (i), we have 1

B ∗ D∗ NB ∗ DB ≤ λB ∗ DB (for some λ > 0) ⇔ R(B ∗ D∗ NB ∗ ) ⊆ R((B ∗ DB) 2 ). If R(PB DPB ) is closed (so is R(B ∗ DB) by discussion above), then 1

R((B ∗ DB) 2 = R(B ∗ DB) = R(B ∗ D∗ B). Since R(B ∗ D∗ PB ) ⊆ R(B ∗ D∗ B) = R(B ∗ D∗ B), we obtain that R(B ∗ D∗ ) ⊆ R(B ∗ D∗ B) ⇔ R(B ∗ D∗ NB ∗ ) ⊆ R(B ∗ D∗ B). Thus B ∗ D∗ NB ∗ DB ≤ λB ∗ DB (for some λ > 0) ⇔ R(B ∗ D∗ ) ⊆ R(B ∗ D∗ B). Therefore, Theorem 4.12 generalizes Theorem 5.6 in [30].

References [1] Braden, H.: The equations AT X ± X T A = B. SIAM J. Matrix Anal. Appl. 20, 295–302 (1998) [2] Choi, M.D., Chandler, D.: The spectral mapping theorem for joint approximate point spectrum. Bull. Am. Math. Soc. 80, 317–321 (1974) [3] Choi, M.D., Holbrook, J.A., Kribs, D.W., Zyczkowski, K.: Higher-rank numerical ranges of unitary and normal matrices. Oper. Matrices 1, 409–426 (2007) [4] Choi, M.D., Kribs, D.W., Zyczkowski, K.: Quantum error correcting codes from the compression formalism. Rep. Math. Phys. 58, 77–86 (2006) [5] Choi, M.D., Kribs, D.W., Zyczkowski, K.: Higher-rank numerical ranges and compression problems. Linear Algebra Appl. 418, 828–839 (2006) [6] Choi, M.D., Kribs, D.W.: Method to ﬁnd quantum noiseless subsystems. Phys. Rev. Lett. 96, 050501–050504 (2006) [7] Choi, M.D., Li, C.K.: The ultimate estimate of the upper norm bound for the summation of operators. J. Funct. Anal. 232, 455–476 (2006)

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[8] Cvetkovi´c-Ili´c, D.S.: Re-nnd solutions of the matrix equation AXB = C. J. Aust. Math. Soc. 84, 63–72 (2008) [9] Cvetkovi´c-Ili´c, D.S., Daji´c, A., Koliha, J.J.: Positive and real-positive solutions to the equation axa∗ = c in C ∗ -algebras. Linear Multilinear Algebra 55, 535– 543 (2007) [10] Cross, R.W.: On the perturbation of unbounded linear operators with topologically complemented ranges. J. Funct. Anal. 92, 468–473 (1990) [11] Crouzeix, M.: Numerical range and functional calculus in Hilbert space. J. Funct. Anal. 244, 668–690 (2007) [12] Daji´c, A., Koliha, J.J.: Positive solutions to the equations AX = C and XB = D for Hilbert space operators. J. Math. Anal. Appl. 333, 567–576 (2007) [13] Djordjevi´c, D.S.: Explicit solution of the operator equation A∗ X + X ∗ A = B. J. Comput. Appl. Math. 200, 701–704 (2007) [14] Fang, X.: The representation of topological groupoid. Acta Math. Sin. 39, 6–15 (1996) [15] Fang, X.: The induced representation of C*-groupoid dynamical system. Chin. Ann. Math. (B) 17, 103–114 (1996) [16] Fang, X.: The realization of multiplier Hilbert bimodule on bidule space and Tietze extension theorem. Chin. Ann. Math.(B) 21, 375–380 (2000) [17] Fang, X., Yu, J., Yao, H.: Solutions to operator equations on Hilbert C ∗ -Modules. Linear Algebra Appl. 431, 2142–2153 (2009) [18] Groß, J.: Explicit solutions to the matrix inverse problem AX = B. Linear Algebra Appl. 289, 131–134 (1999) [19] Giribet, J.I., Maestripieri, A., Per´ıa, F.M.: Shorting selfadjoint operators in Hilbert spaces. Linear Algebra Appl. 428, 1899–1911 (2008) [20] Hansen, A.C.: On the approximation of spectra of linear operators on Hilbert spaces. J. Funct. Anal. 254, 2092–2126 (2008) [21] Jensen, K.K., Thomsen, K.: Elements of KK-Theory. Birkhauser, Boston (1991) [22] Karaev, M.T.: Berezin symbol and invertibility of operators on the functional Hilbert spaces. J. Funct. Anal. 238, 181–192 (2006) [23] Khatri, C.G., Mitra, S.K.: Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. Math. 31, 579–585 (1976) [24] Lance, E.C.: Hilbert C ∗ -Modules: A Toolkit for Operator Algebraists. Cambridge University Press, Cambridge (1995) [25] Lauzon, M.M., Treil, S.: Common complements of two subspaces of a Hilbert space. J. Funct. Anal. 212, 500–512 (2004) [26] Li, C.K., Tsing, N.K.: On the kth matrix numerical range. Linear Multilinear Algebra 28, 229–239 (1991) [27] Wegge-Olsen, N.E.: K-Theory and C ∗ -Algebras: A Friendly Approach. Oxford University Press, Oxford (1993) [28] Wang, Q., Yang, C.: The Re-nonnegative definite solutions to the matrix equation AXB = C. Comment. Math. Univ. Carolinae 39, 7–13 (1998) [29] Xu, Q.: Common Hermitian and positive solutions to the adjointable operator equations AX = C, XB = D. Linear Algebra Appl. 429, 1–11 (2008) [30] Xu, Q., Sheng, L., Gu, Y.: The solutions to some operator equations. Linear Algebra Appl. 429, 1997–2024 (2008)

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[31] Xu, Q., Sheng, L.: Positive semi-definite matrices of adjointable operators on Hilbert C ∗ -modules. Linear Algebra Appl. 428, 992–1000 (2008) [32] Yuan, Y.: Solvability for a class of matrix equation and its applications. J. Nanjing Univ. (Math. Biquarterly) 18, 221–227 (2001) [33] Zhang, X.: Hermitian nonnegative-definite and positive-deﬁne solutions of the matrix equation AXB = C. Appl. Math. E-Notes 4, 40–47 (2004) Xiaochun Fang and Jing Yu Department of Mathematics Tongji University 200092 Shanghai China e-mail: [email protected]; [email protected] Received: September 12, 2009. Revised: October 5, 2009.

Integr. Equ. Oper. Theory 68 (2010), 61–74 DOI 10.1007/s00020-010-1813-8 Published online July 13, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

The Trace of Nuclear Operators on Lp(μ) for σ-Finite Borel Measures on Second Countable Spaces Julio Delgado Abstract. Let Ω be a second countable topological space and μ be a σ−ﬁnite measure on the Borel sets M. Let T be a nuclear operator on Lp (Ω, M, μ), 1 < p < ∞, in this work we establish a formula for the trace of T . A preliminary trace formula is established applying the general theory of traces on operator ideals introduced by Pietsch and a characterization of nuclear operators for σ−ﬁnite measures. We also use the Doob’s maximal theorem for martingales with the purpose of studying the kernel k(x, y) of T on the diagonal. Mathematics Subject Classification (2010). Primary 47B10; Secondary 47G10, 47B38, 60G46. Keywords. Integral operators, nuclear operators, trace formula, martingales.

1. Introduction Let H be a complex and separable Hilbert space endowed with an inner product denoted by by <, >, and let T : H → H be a linear compact operator. If we denote by T ∗ : H → H the adjoint of T , then the linear operator 1 (T ∗ T ) 2 : H → H is positive and compact. Let (ψn )n be an orthonormal basis 1 for H consisting of eigenvectors of (T ∗ T ) 2 , and let sn (T ) be the eigenvalue corresponding to the eigenvector ψn , n = 1, 2, . . . We call sn (T ), n = 1, 2, . . ., the singular values of T : H → H. If ∞

sn (T ) < ∞,

n=1

This work has been partially supported by Universidad del Valle, Vicerrectoria Inv. Grant#7756.

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then the linear operator T : H → H is said to be in the trace class S1 . It can be shown that S1 is a Banach space in which the norm S1 is given by ∞ sn (T ), T ∈ S1 . T S1 = n=1

Let T : H → H be an operator ∞ in S1 and let (φn )n be any orthonormal basis for H. Then, the series n=1 < T φn , φn > is absolutely convergent and the sum is independent of the choice of the orthonormal basis (φn )n . Thus, we can deﬁne the trace of any linear operator T : H → H in S1 by ∞ trace(T ) = < T φn , φn>, n=1

where {φn : n = 1, 2, . . .} is any orthonormal basis for H. If the singular values are square-summable T is called a Hilbert–Schmidt operator. It is clear that every trace class operator is a Hilbert–Schmidt operator. In general, if the sequence of singular values is p-summable is said that T belongs to the Schatten class Sp and it is well known that Sp is an ideal in L(H). We are interested in integral operators. In the case H = L2 (Ω, M, μ), T is a Hilbert–Schmidt operator if and only if T can be represented by a kernel k(x, y) in L2 (Ω × Ω, μ ⊗ μ) (cf. [11]). In the general setting of Banach spaces the concept of trace class operator can be generalized as follows. Let E and F be two Banach spaces, a linear operator T from E to F is called nuclear if there exist sequences (xn ) in E and (yn ) in F so that Ax = x, xn yn and xn E yn F < ∞. n

n

This definition agrees with the concept of trace class operator in the setting of Hilbert spaces (E = F = H). The set of nuclear operators from E into F forms the ideal of nuclear operators N(E, F ) endowed with the norm N (A) = inf xn E yn F : A = xn ⊗ yn . n

n

In order to ensure the existence of a good definition of the trace on the ideal of nuclear operators N(E) one is constraints to consider the Banach spaces E enjoying the approximation property (cf. [9,4]). In that case, if T : E → E is nuclear, the trace is deﬁned by ∞ trace(T ) = xn (yn ), ∞

n=1 xn

n=1

where T = ⊗ yn is a representation of T . It can be shown that this definition is independent of the representation. The trace can also be deﬁned in the general setting of quasi-Banach operator ideals introduced by Pietsch (cf. [9]). An historical exposition about the concept of the trace and many others aspects of Banach space theory and linear operators can be found in the excellent book by Pietsch [10]. Others good references on traces are [7,12]. Now we present specifically the problem which we are interested. Let Ω be a second countable topological space and μ a Borel and σ−ﬁnite measure.

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In Chris Brislawn [2], considered the class of nuclear operators or trace class on L2 (Ω, M, μ) and uses an averaging process with the purpose of deﬁning and analyze the kernel k(x, y) on the diagonal in Ω × Ω. The main tool to study k(x, y) on the diagonal is the Doob’s maximal theorem for martingales. ˜ x) the pointwise values of this averaging process, Brislawn Denoting by k(x, proved the following formula for a trace class operator T on L2 (μ) ˜ x)dμ(x). trace(T ) = k(x, (1.1) Ω

In [5] we have extended this formula for 1 < p < ∞ in the case of the Lebesgue measure on Rn , previously established for p = 2 in [1]. Our goal consists in obtaining the same result for second countable topological spaces and Borel measures. In the second section of this work we present some basic results about nuclear operators on Lp (μ) for σ−ﬁnite measures, the Chap. 4.2 of the book by Pietsch [9] plays an essential role. The general point of view of traces on operator ideals introduced by Pietsch is used in order to obtain a basic trace formula in the Lp setting. This formula will be the starting point in the next section. The third section is devoted to the study of the trace using the martingale maximal function in the context of second countable topological spaces and σ-ﬁnite and Borel measures. The main result is the extension of the formula (1.1) for a nuclear operator on Lp (Ω, M, μ), 1 < p < ∞ which is contained in Theorem 3.8.

2. Nuclear Operators on Lp (μ) and the Trace In this section we recall some basic facts about the concept of trace on Operator Ideals. In particular we consider the trace of nuclear operators on Lp (μ). We refer the reader to the Chap. 4.2 of [9] for the general theory of traces on operator ideals and the notation used in this section, for the theory of tensor products we refer the reader to [4]. Let E and F be Banach spaces, we denote by L(E, F ) the algebra of bounded linear operators from E into F , and by F(E, F ) the ideal of ﬁnite rank operators from E into F . If E = F we shall write F(E). An element T n of F(E, F ) can be written in the form j=1 xj ⊗ yj (not necessarily unique), where xj in E and yj in F , j = 1, . . . , n. An operator T in N(E, F ) can be ∞ ˜ π F (the completetion of the represented by an element n=1 xn ⊗ yn of E ⊗ tensor product E ⊗ F with respect to the π-norm). Let A = A(E) be an operator ideal, a trace on A is a function τ : A → C such that the following conditions hold: (T1 ) (T2 ) (T3 ) (T4 )

τ (a ⊗ x) = < x, a > for a ∈ E , and x ∈ E. τ (XT ) = τ (T X) for T ∈ A(E, F ) and X ∈ L(F, E). τ (S + T ) = τ (S) + τ (T ) for S, T ∈ A(E). τ (λT ) = λτ (T ) for T ∈ A(E), and λ ∈ C.

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One begins by deﬁning a trace on ﬁnite rank operators. If T ∈ F(E) we deﬁne a trace by trace(T ) :=

n

<xj , aj > .

(2.1)

j=1

One can prove that the sum is independent of the choice of the representation. Moreover, the definition above furnish the unique trace on the operator ideal F (cf. [9], Theorem 4.2, and the previous Lemma). A trace τ on a quasiBanach operator ideal A is said to be continuous if τ veriﬁes this property on all components A(E). Then there exists a constant c ≥ 1 such that |τ (T )| ≤ cT |A,

for all T ∈ A(E).

Now, suppose that A is a quasi-Banach operator ideal such that |trace(T )| ≤ cT |A,

for all T ∈ F(E),

(2.2)

where the constant c ≥ 1 is independent of the Banach space E. If the ﬁnite rank operators are dense in all components A(E), then the functional T → trace(T ) can be extended to a unique continuous trace on A (cf. [9], 4.2.5 Trace extension theorem). Due to the lackness of the approximation property for general Banach spaces, this extension process does not apply to the ideal of nuclear operators. One is then forced to restrict the ideal of nuclear operators to the class of Banach spaces with the approximation property. One can deﬁne then traceN (T ) :=

∞

<xj , yj >

(2.3)

n=1

as an extension of (2.1) and with the value of the sum independent of the representation T =

∞

xn ⊗ yn .

n=1

In particular we shall consider (Ω1 , M1 , μ1 ) and (Ω2 , M2 , μ2 ) two σ-ﬁnite measure spaces where 1 ≤ p1 , p2 < ∞ and q1 , q2 such that p1i + q1i = 1 (i = 1, 2). A ﬁnite rank operator T from Lp1 (μ1 ) into Lp2 (μ2 ) can be expressed as an integral operator ⎛ ⎞ n gj (x)hj (y)⎠ f (y)dμ1 (y), μ2 − a.e. x, (2.4) T f (x) = ⎝ Ω1

j=1

where gj and hj are functions in Lp2 (μ2 ) and Lq1 (μ1 ) for j = 1, . . . , n. If Ω = Ω1 = Ω2 , μ = μ1 = μ2 , p1 = p2 = p, the trace of T deﬁned by (2.1) is given by ⎛ ⎞ n gj (x)hj (x)⎠ dμ(x). (2.5) trace(T ) = ⎝ Ω

j=1

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n We observe that the H¨ older inequality implies that j=1 gj (x)hj (x) ∈ L1 (μ). In order to extend the formula (2.5) to nuclear operators we shall ﬁrst recall that the representation (2.4) can be extended to nuclear operators, every → Lp2 (μ2 ) can be written in the form of an nuclear operator T : Lp1 (μ1 ) ∞ integral operator with kernel n=1 gn ⊗ hn , where ∞gn and hn are sequences p2 q1 in L (μ2 ) and L (μ1 ) respectively such that n=1 gn Lp2 hn Lq1 < ∞. The discussion above is summarized by the following theorem (see [5] for a self-contained proof). Theorem 2.1. An operator T : Lp1 (μ1 ) → Lp2 (μ2 ) is nuclear if and only if there exist sequences (gn )n in Lp2 (μ2 ), and (hn )n in Lq1 (μ1 ) such that ∞ p1 p q n=1 gn L 2 hn L 1 < ∞, and for all f ∈ L T f (x) =

∞

gn (x)hn (y) f (y)dμ1 (y) , a.e. x.

n=1

Remark 2.2. (i) We recall that the ideal I(E, F ) of integral operators in the sense of Grothendieck contains the ideal N(E, F ), and I(E, F ) agrees whith N(E, F ) whenever that F is relexive or a separable dual. In particular if 1 < p2 < ∞ then I(Lp1 , Lp2 ) = N(Lp1 , Lp2 ). Notice that the above characterization includes the case p2 = 1. (ii) If both measures μ1 and μ2 are ﬁnite, and T is nuclear it is easy to see using Hlder inequality that k ∈ L1 (μ2 ⊗ μ1 ), this fact will be exploit in the next section. (iii) In the case of σ−ﬁnite measures the assertion k ∈ L1 (μ2 ⊗ μ1 ) is false. Take for example Ω1 = Ω2 = Rn , and μ1 , μ2 = λ, the Lebesgue measure, then using the fact that q2 > 1, we deﬁne k(x, y) = g(x)h(y), with g ∈ Lp2 (λ)\{0}, h ∈ Lq1 (λ)\L1 (λ). Then

|k(x, y)|dλ(x)dλ(y) =

Rn

(iv)

Rn

|g(x)|dλ

Rn

|h(y)|dλ = ∞.

Rn

If Ω = Ω1 = Ω2 , p1 = p2 and μ = μ1 = μ2 is a σ-ﬁnite measure, the kernel k in the above theorem is integrable on the diagonal, this is obtained integrating k(x, x) on each Ωj of a partition of Ω in μ-ﬁnite mesure sets, and applying the H¨ older inequality. Moreover, one has |k(x, x)|dμ(x) ≤ Ω

∞

gn Lp hn Lq .

(2.6)

n=1

Now, applying the characterization given by Theorem 2.1, the continuity of the trace (2.2) restricted to N(Lp (μ)) having into account that our Lp spaces posses the approximation property (cf. [8]), and the inequality (2.6), we can extend the formula (2.5) to the ideal N(Lp (μ)) of nuclear operators on Lp (μ) that will be useful to establish our main result in the next section.

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Corollary 2.3. Let (Ω, M, μ) be a σ-finite measure space μ. If T is a nuclear operator from Lp (μ) to Lp (μ) then the trace of T is given by tr T = k(x, x)dμ(x), (2.7) Ω

where k(x, y) is a kernel of T obtained from Theorem 2.1. The above formula shows how to calculate the trace using a special ker∞ nel k given in the form n=1 gn ⊗ hn , but a trace formula for an arbitrary kernel α(x, y) is not directly deducible from the corresponding for k using (2.7). However this problem disappears in the case for example of the counting measure on Z due to the uniqueness of the kernel. Moreover, one disposes of the following mild sufﬁcient condition for nuclearity in terms of the kernel. Proposition 2.4. Let K : Z × Z → C be a function satisfying

p1 2 p2 |K(j, m)| < ∞. j∈Z

(2.8)

m∈Z

Then the relation K(j, m) = < T ej , em >, (m, j) ∈ Z × Z defines a nuclear operator T : Lp1 (Z) → Lp2 (Z). In particular if p1 = p2 we have k(n, n). tr T = n∈Z

Proof. We observe that T ej Lp2 ej Lq1 = T ej Lp2 j∈Z

j∈Z

=

j∈Z

(2.9)

p1

|K(m, j)|p2

2

< ∞.

(2.10)

m∈Z

This justiﬁes the following calculus for f ∈ Lp1 (Z): < f, ej > ej , f = j∈Z

Tf =

< f, ej > T ej .

j∈Z

last equality holds from the facts (< f, ej >)j ∈ L∞ and p1 p1 p j∈Z T ej L 2 < ∞. Then T is a bounded operator from L (Z) to L (Z). p1 Hence, applying (2.9) and (2.10), T is a nuclear operator from L (Z) to Lp2 (Z), with kernel given by K(j, m) = . The last conclusion in the proposition is now clear from Corollary 2.3. The

The inequality in the above proposition is not a neccesary condition for nuclearity. We distinguish two cases, 1 < p2 < ∞ or p2 = 1. In the ﬁrst case, we deﬁne a rang one operator with kernel K(j, m) = g(j)h(m), where

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g ∈ Lp2 (Z)\L1 (Z) and h ∈ Lq1 (Z)\{0}. Then the operator T deﬁned by K is nuclear. However,

p1

p1 2 2 p2 p2 |K(j, m)| = |g(j)| |h(m)| j∈Z

m∈Z

j∈Z

m∈Z

= gL1 h

Lq1

= ∞. The other case is similar, if p2 = 1 we choose g ∈ Lp2 (Z)\{0} and h ∈ L∞ (Z)\Lp2 (Z), having in account that p2 < ∞, and q1 = ∞. The sufﬁcient condition given by (2.8) is not longer valid in the case of σ-ﬁnite measure on Borel sets, Carleman [3] has constructed a periodic continuous function ϕ(x) (ϕ(x) = ϕ(x + 1)) so that its Fourier series coefﬁcients cn obey ∞

|cn |p = ∞,

n=−∞

for any p < 2. Then, considering the normal operator 1 ϕ(x − y)f (y)dy,

T f (x) = 0

acting on L2 ([0, 1]) one obtains that the sequence (cn )n forms a complete system of eigenvalues of this operator corresponding to the complete orthonormal system φn (x) = e2πnx , T φn = cn φn . Then the singular values of T are given by sn (T ) = |cn | and ∞

sn (T )p = ∞,

n=1

for p < 2. In particular the operator T is not nuclear. However, the continuous kernel K(x, y) = ϕ(x − y) satisﬁes an integral condition of the form (2.8) due to the boudedness of K. This explains the impossibility for obtaining a sufﬁcient condition of this type for nuclearity in the general case of σ-ﬁnite Borel measures. Now, coming back to the calculus of the trace, the way to intertwine the informations on two kernels α and k, where k(x, y) is the kernel given by Theorem 2.1, consist in consider an averaging process on the diagonal. This explains the role that the maximal function will play in our analysis.

3. The Martingale Maximal Function and the Trace on Lp (μ) In this section we shall introduce the martingale maximal function and some results from probability theory involving the averaging process above mentioned. For a more comprehensive accounts on these concepts, the book [6] is a good reference. Two classical references for the Maximal function of

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Hardy-Littlewood are [13,14]. This process enable us to obtain a more general formula for the trace valid for every kernel of a nuclear operator. We will exploit the good properties of the particular kernel k using the averaging process and the martingale maximal function. Let (Ω, M, μ) be a σ−ﬁnite measure space and let {Mn }n a sequence of sub-σ-algebras such that Mn . Mn ⊂ Mn+1 and M = n p

Let f be in L (μ), in order to deﬁne a conditional expectation we assume that μ is σ-ﬁnite on each Mn . In that case, E(f |Mn ) exists. We say that a sequence {fn }n of functions on Ω is a martingale if each fn is Mn -measurable and E(fn |Mj ) = fj ,

for j < n.

(3.1)

In order to obtain a generalization of the Hardy-Littlewood maximal function we consider the particular case of martingales generated by a single M-measurable function f . The maximal function in this setting is deﬁned by M f (x) = sup E(|f ||Mn )(x). n

(3.2)

The Lp boudedness of the martingale maximal operator is given by the following theorem. Theorem 3.1 (Doob’s Martingale Maximal Theorem). The martingale maximal operator, M is weak-type (1, 1) on L1 (μ) and bounded on Lp (μ) for 1 < p ≤ ∞. In order to study the kernel of a nuclear operator on the diagonal we will need the following generalization of the Lebesgue’s differentiation theorem: Theorem 3.2 (Doob’s Martingale Convergence Theorem). If f ∈ Lp (μ) and 1 ≤ p ≤ ∞ then the sequence E(f |Mn ) converges to f μ-almost everywhere. We shall now deﬁne the conditional expectation. Let Pi , Pj be two partitions of Ω, the notation Pi ≺ Pj means that Pi is a reﬁnement of Pj . For a sequence {Pn }n of partitions of Ω such that Pn+1 ≺ Pn we consider Mn = σ(Pn ), and suppose further that M = n Mn . Then for each x ∈ Ω and each n ∈ N there is a unique set Cn (x) ∈ Pn containing x. Let N = {x ∈ Ω : μ(Cn (x)) = 0, for some n}. If x ∈ N and m > n, then μ(Cn (x)) = 0 since Cm (x) ⊂ Cn (x). Having into account that {Cj (x) : x ∈ N and μ(Cj (x)) = 0} is a countable set we obtain μ(N ) = 0. The conditional expectation of f with respect to Mn is given by 1 E(f |Mn ) = f dμ, if x ∈ N c . (3.3) μ(Cn (x)) Cn (x)

In this way the formula 3.3 holds μ-almost everywhere. We can now deﬁne our averaging process which is a generalization of the Hardy-Littlewood process:

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for x ∈ N c .

f dμ,

Cn (x)

The operator An posses the following important properties of convergence: Lemma 3.3. Let n ∈ N, 1 ≤ p ≤ ∞, and x ∈ N c if fj → f in Lp (μ) then An fj (x) → An f (x) as j → ∞. Corollary 3.4. If fj converges to f in Lp (μ)-norm then j An fj (x) converges to An f (x) for each x ∈ N c . In order to apply our averaging process to the kernel k(x, y) we need to consider partitions on Ω × Ω. From a partition Pn of Ω we induce a partition Pn × Pn , for each point (x, y) ∈ Ω × Ω we have (x, y) ∈ Cn (x) × Cn (y) and Pn+1 × Pn+1 ≺ Pn × Pn . The set N c × N c is conull with respect to product measure since (N c × N c )c ⊂ N × N , and one can also see that the diagonal of Ω × Ω which we denoted by Ω×Ω is contained μ-almost everywhere in N c × N c , indeed one has {x : (x, x) ∈ N c × N c }c = {x : (x, x) ∈ / N c × N c} = {x : x ∈ / N c or x ∈ / N c} = N. (2) An

We denote by the averaging operators on Ω × Ω. Let f ∈ L1loc (μ ⊗ μ), c for each (x, y) ∈ N × N c we have 1 (2) An f (x, y) = f (s, t)dμ(t)dμ(s). (3.4) μ(Cn (x))μ(Cn (y)) Cn (x) Cn (y)

(2)

The operator An satisﬁes the following fundamental property of multiplicativity on the tensorial product Lp ⊗ Lq ( p1 + 1q = 1): Lemma 3.5. Let g ∈ Lp (μ) and h ∈ Lq (μ) with

1 p

+

1 q

= 1. Then

A(2) n (g ⊗ h)(x, y) = An g(x)An h(y), for all (x, y) ∈ N c × N c . Denoting by M (2) the maximal function on Ω×Ω, we have the following consequence: Lemma 3.6. Let g ∈ Lp (μ) and h ∈ Lq (μ) with M c

(2)

1 p

+

1 q

= 1. Then

(g ⊗ h)(x, y) ≤ M g(x)M h(y),

c

for all (x, y) ∈ N × N . We also dispose of the subadditivity as a direct consequence of the definition of the martingale maximal operator M (g + h)(x) ≤ M g(x) + M h(x); for all x ∈ N c .

g, h ∈ Lp (μ),

(3.5)

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Now, we shall consider our particular setting. If Ω is a second countable topological space, M is a σ-algebra of Borel sets of Ω, and μ is a σ-ﬁnite measure, we will construct a suitable sequence of partitions of Ω. Let A = {Un }n be a countable base of M. Let P0 be a countable partition of Ω consisting of sets of ﬁnite measure. We can deﬁne Pn inductively in the following way Pn = {Un , Unc } ∩ Pn−1 ,

n ∈ N.

(3.6)

The sequence {Pn }n is increasing and Mn = σ(Pn ) is an increasing sequence of sub-σ-algebras of M, and M = n Mn . For f ∈ Lp (μ) we deﬁne f˜(x) = lim An f (x). n→∞

(3.7)

Hence and by Theorem 3.2 f˜(x) = f (x) a.e. x.

(3.8)

c

We say that x ∈ N is a regular point for f if the limit (3.7) exists. The set Rf of regular points of f is conull in Ω. Let A be a countable base for the topology on Ω. The σ-algebras σ(Pn ) converge to the Borel σ-algebra. Since the averaging process for this martingale is based on the topology of Ω, we have the following theorem: Theorem 3.7. Let Ω be a second countable topological space and μ a σ-finite Borel measure. If f ∈ Lp (μ), 1 ≤ ∞, then each point of continuity x ∈ N c is a regular point of f and satisfies f˜(x) = f (x). In order to see that this theorem also holds in Ω × Ω we note that almost all of the diagonal lies in N c × N c . Then, if f (x, y) is continuous almost everywhere along the diagonal then A(2) n f (x, x) → f (x, x)

μ − a.e. x.

(3.9)

In general, without the continuity hypotheses, we only have almost everywhere existence with respect to the product measure (3.10) f˜(x, y) = lim A(2) n f (x, y). n→∞

We shall now consider a nuclear operator T from Lp (μ) to Lp (μ) and a representation ∞ gj (x)hj (y), (3.11) k(x, y) = j=1

as in Theorem 2.1 with gj ∈ L (μ) , hj ∈ Lq (μ) (j = 1, 2, . . .), p1 + 1q = 1 and gj p hj q < ∞. We recall that from Remark 2.2 (ii) we are authorized to say that k(x, y) is integrable on ﬁnite measure subsets of Ω × Ω, to see this, it is sufﬁcient to consider the restriction of T to a space of Lp functions deﬁned on a ﬁnite measure subset of Ω. We will say that x ∈ N c is a regular point of the expansion (3.11) if the following limits exists at x, for all j ˜ j (x). lim An hj (x) = h (3.12) lim An gj (x) = g˜j (x) and p

n→∞

n→∞

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Hence, the set R of regular points of (3.11) is R = (Rgj ∩ Rhj ). j∈N

From Theorem 3.2 we have that almost every point in Ω is a regular point. Now, considering gj and hj to be deﬁned pointwise by (3.12) for every x ∈ R . By Corollary 3.4 and Lemma 3.5 we have A(2) n k(x, y) =

∞

An gj (x)An hj (y)

(3.13)

j=1

˜ y) almost everywhere by at every point (x, y) ∈ N c × N c . We deﬁne k(x, ˜ y) = lim A(2) k(x, y). k(x, n n→∞

We are now ready to establish a trace formula for nuclear operators on Lp (μ), the L2 (μ) case corresponding to the Theorem 3.1 proved in [2]. In the next theorem and the corollary we consider a measure space (Ω, bor(Ω), μ), where Ω is a second countable topological space and μ is a σ-ﬁnite measure deﬁned on the Borel σ-algebra bor(Ω). Theorem 3.8 (Main Theorem). Let 1 < p < ∞ and T : Lp (μ) −→ Lp (μ) be a nuclear operator with kernel k(x, y) as in Theorem 2.1. Then M (2) k(x, x) ∈ ˜ x) = k(x, x) for almost every x and consequently L1 (μ), k(x, ˜ x)dμ(x). tr T = k(x, (3.14) Ω

Proof. We begin by seeing that M (2) k(x, x) ∈ L1 (μ). Applying the subadditivity, the submultiplicativity and the boundedness of the maximal operator older inequality we obtain on Lp (μ) and Lq (μ), and the H¨ ⎛ ⎞ ∞ ⎝ M (2) k(x, x)dμ(x) = M gj (x)M hj (x)⎠ dμ(x) Ω

j=1

Ω

=

∞

⎛ ⎝

j=1

≤ C

∞

⎞ M gj (x)M hj (x)dμ(x)⎠

Ω

gj Lp hj Lq

j=1

< ∞. This proves the ﬁrst assertion and the fact that the sum ∞ j=1

M gj (x)M hj (x),

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is ﬁnite μ-a.e.x. From Corollary 2.3 k(x, x) is ﬁnite μ-a.e. x. Now we consider the sums ∞ ∞ gj (x)hj (x), M gj (x)M hj (x). j=1

j=1

We choose a conull set of regular points Γ ⊂ R so that for all x ∈ Γ both of the above series are ﬁnite. For each point x in Γ and all j ∈ N we have lim An gj (x) = gj (x),

and

n→∞

lim An hj (x) = hj (x).

n→∞

Now, using the fact that |An gj (x)||An hj (x)| ≤ |M gj (x)||M hj (x)|, for x in Γ, j ∈ N and n ∈ N. Then the series ∞

An gj (x)An hj (x)

j=1

converges absolutely and uniformly with respect to n ∈ N. Now, by (3.13) we have for every n ∈ N that An k(x, x) =

∞

An gj (x)An hj (x).

j=1

Hence, letting n → ∞ we obtain for each x ∈ Γ that ⎛ ⎞ ∞ ˜ x) = lim An ⎝ k(x, gj (x)hj (x)⎠ n→∞

=

=

=

∞ j=1 ∞ j=1 ∞

j=1

lim An (gj (x)hj (x))

n→∞

lim An gj (x)An hj (x)

n→∞

˜ j (x) g˜j (x)h

j=1

=

∞

gj (x)hj (x) = k(x, x).

j=1

Applying the Corollary 2.3 we have ˜ x)dx. tr T = k(x, Ω

Corollary 3.9. Let T be a nuclear operator on Lp (μ), 1 < p < ∞. Let k as in Theorem 2.1 and suppose that α(x, y) is a measurable function defined

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almost everywhere on Ω × Ω so that α(x, y) = k(x, y) for a. e. (x, y). Then α is integrable on the diagonal and tr T = α ˜ (x, x)dμ(x). Ω

Consequently, if α is a continuous, then tr T = α(x, x)dμ(x). Ω

Proof. The ﬁrst formula because equality almost everywhere on Ω × Ω for α ˜ x) = α(x, x) for all x. Then α is integrable on the diagonal and k implies k(x, and the formula follows. The second is obtained immediately. Acknowledgements I would like to thank an anonymous refeere for the valuable comments on the results and presentation helping to improve this manuscript.

References [1] Brislawn, C.: Kernels of trace class operators. Proc. Am. Math. Soc. 104, 1181– 1190 (1988) [2] Brislawn, C.: Traceable integral kernels on countably generated measure spaces. Paciﬁc J. Math. 150(2), 229–240 (1991) [3] Carleman, T.: ber die Fourierkoefﬁzienten einer steingen Function. Acta Math. 41, 377–384 (1918) [4] Defant, A., Floret, K.: Tensor Norms and Opertor Ideals, North-Holland Math. Studies 176, Amsterdam (1993) [5] Delgado, J.: A trace formula for nuclear operators on Lp , to appear in pseudodifferential operators: complex analysis and partial differential equations. In: Schulze, B.-W., Wong, M.W. (eds.) Operator Theory, Advances and Applications, vol. 205, p. 181–193. Birh¨ auser, Basel (2009) [6] Doob, J.L.: Stochastic Processes. Wiley, New York (1953) [7] Gohberg, I., Goldberg, S., Krupnik, N.: Traces and Determinants of Linear Operators. Birkh¨ auser, Basel (2001) [8] Grothendieck, A.: Produits tensoriels topologiques et espaces nucl´eaires. Memoirs Am. Math. Soc., Providence, 16 (1955) [9] Pietsch, A.: Eigenvalues and s-numbers. Cambridge University Press, New York (1986) [10] Pietsch, A.: History of Banach Spaces and Linear Operators. Birkh¨ auser, Basel (2007) [11] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II Fourier Analysis Self-Adjointness. Academic Press, New York (1975) [12] Simon, B.: Trace ideals and their applications. Cambridge University Press, Cambridge (1979)

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[13] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton (1970) [14] Stein, E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993) Julio Delgado (B) Universidad del Valle Calle 13, 100-00, Cali, Colombia e-mail: [email protected] Received: September 17, 2009. Revised: May 28, 2010.

Integr. Equ. Oper. Theory 68 (2010), 75–99 DOI 10.1007/s00020-010-1789-4 Published online March 24, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Spectral Theory of Discontinuous Functions of Self-Adjoint Operators: Essential Spectrum Alexander Pushnitski Abstract. Let H0 and H be self-adjoint operators in a Hilbert space. In the scattering theory framework, we describe the essential spectrum of the difference ϕ(H) − ϕ(H0 ) for piecewise continuous functions ϕ. This description involves the scattering matrix for the pair H, H0 . Mathematics Subject Classification (2000). Primary 47A40; Secondary 35P25, 47B25, 47F05. Keywords. Scattering matrix, essential spectrum, spectral projections.

1. Introduction Let H0 and H be self-adjoint operators in a Hilbert space H and suppose that the difference V = H − H0 is a compact operator. If ϕ : R → R is a continuous function which tends to zero at inﬁnity then a well known simple argument shows that the difference ϕ(H) − ϕ(H0 ) is a compact operator. Moreover, there is a large family of results that assert that if the function ϕ is sufﬁciently “nice” and V belongs to some Schatten– von Neumann class of compact operators, then ϕ(H) − ϕ(H0 ) also belongs to this class. See [4,11] or the survey [5] for early results of this type; they were later made much more precise by Peller, see [13,14]. See also [1,2,12] for some recent progress in this area. In all of the above mentioned results, the function ϕ is assumed to be continuous. If ϕ has discontinuities on the essential spectrum of H0 , then the difference ϕ(H) − ϕ(H0 ) in general fails to be compact even if V is a rank one operator; see [10,11]. In this paper we study the essential spectrum of ϕ(H) − ϕ(H0 ) for piecewise continuous functions ϕ. Some initial results in this direction have been obtained in [18]; we begin by describing these results.

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For a Borel set Λ ⊂ R, we denote by E(Λ) (resp. E0 (Λ)) the spectral projection of H (resp. H0 ) corresponding to the set Λ. If Λ is an interval, say Λ = [a, b), we write E[a, b) instead of E([a, b)) in order to make our formulas more readable. In [18], under some assumptions typical for smooth scattering theory, it was proven that for compact V one has (1.1) σess (E(−∞, λ) − E0 (−∞, λ)) = − 12 S(λ) − I , 12 S(λ) − I , where S(λ) is the scattering matrix for the pair H0 , H. In this paper, we prove the following generalisation of (1.1) (see Sect. 2.2). Assume that for some λ ∈ R the derivatives d |V |1/2 E0 (−∞, λ)|V |1/2 , dλ

d |V |1/2 E(−∞, λ)|V |1/2 dλ

(1.2)

exist in the operator norm. Then we prove (see Theorem 2.1) that the limit π α(λ) = lim E0 (λ − ε, λ + ε)V E(λ − ε, λ + ε) (1.3) ε→+0 2ε exists and the identity σess (E(−∞, λ) − E0 (−∞, λ)) = [−α(λ), α(λ)]

(1.4)

holds true. If the standard assumptions of either trace class or smooth variant of scattering theory are fulﬁlled, we prove (see Sect. 2.4) that α(λ) = 1 2 S(λ) − I. Thus, (1.1) becomes a corollary of (1.4). Using (1.4), we obtain the following results: (i)

Applying (1.4) in the trace class framework, we prove (see Sect. 2.3 for the definition of the core of the absolutely continuous spectrum): Theorem. Let V be a trace class operator. Then for a.e. λ ∈ R the derivatives in (1.2) exist, the relation (1.4) holds true and for a.e. λ in the core of the absolutely continuous spectrum of H0 , the relation (1.1) holds true.

This is stated as Theorem 2.3 below. In Sect. 2.5 we describe the essential spectrum of the difference ϕ(H) − ϕ(H0 ) for piecewise continuous functions ϕ. (iii) In Sect. 2.6 we give a convenient criterion for E0 (−∞, λ), E(−∞, λ) to be a Fredholm pair of projections. (iv) In Sects. 2.7, 2.8, we give some applications to the Schr¨ odinger operator. (ii)

In the proof of (1.4) we use the technique of [18] with some minor improvements. In (ii) above, we follow the method of proof used by S. Power in his description [16] of the essential spectrum of Hankel operators with piecewise continuous symbols. Finally, we note that a description of the absolutely continuous spectrum of the difference E(−∞, λ) − E0 (−∞, λ)

(1.5)

is also available in terms of the spectrum of the scattering matrix; see [18,20].

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2. Main Results 2.1. The Definition of the Operator H Let H0 be a self-adjoint operator in a Hilbert space H. We would like to introduce a self-adjoint perturbation V and deﬁne the sum H = H0 + V . Informally speaking, we would like to deﬁne H0 + V as a quadratic form sum; however, since we do not assume H0 or V to be semi-bounded, the language of quadratic forms is not applicable here. The definition of H0 + V requires some care; we follow the approach which goes back at least to [9] and was developed in more detail in [28, Sections 1.9, 1.10]. We assume that V is factorised as V = G∗ JG, where G is an operator from H to an auxiliary Hilbert space K and J is an operator in K. We assume that J = J ∗ is bounded in K, Dom|H0 |1/2 ⊂ Dom G and G(|H0 | + I)−1/2 is compact.

(2.1)

We denote by (·, ·) and · the inner product and the norm in H and by (·, ·)K and ·K the inner product and the norm in K. In applications a factorisation V = G∗ JG with these properties often arises naturally from the structure of the problem. In any case, one can always take K = H, G = |V |1/2 and J = sign(V ). For z ∈ C\σ(H0 ), we denote R0 (z) = (H0 − zI)−1 . Formally, we deﬁne the operator T0 (z) (sandwiched resolvent) by setting T0 (z) = GR0 (z)G∗ ;

(2.2)

more precisely, this means

∗ T0 (z) = G(|H0 | + I)−1/2 (|H0 | + I)R0 (z) G(|H0 | + I)−1/2 .

By (2.1), the operator T0 (z) is compact. It can be shown (see [28, Sections 1.9,1.10]) that under the assumption (2.1) the operator I + T0 (z)J has a bounded inverse for all z ∈ C\R and that the operator valued function ∗

R(z) = R0 (z) − (GR0 (z)) J (I + T0 (z)J)

−1

GR0 (z),

z ∈ C\R,

(2.3)

is a resolvent of a self-adjoint operator; we denote this self-adjoint operator by H. Thus, formula (2.3), which is usually treated as a resolvent identity for H0 and H = H0 + V , is now accepted as the definition of H. If V is bounded, then the above deﬁned operator H coincides with the operator sum H0 + V . If H0 is semi-bounded from below, then (2.1) means that V is H0 -form compact and then H coincides with the quadratic form sum H0 + V (in the sense of the KLMN Theorem, see [22, Theorem X.17]). In general, we have (f0 , Hf ) = (H0 f0 , f ) + (JGf0 , Gf )K ,

∀f0 ∈ Dom H0 ,

∀f ∈ Dom H. (2.4)

Finally, it is not difﬁcult to check that by (2.1) and (2.3), the resolvent R(z) can be written as R(z) = (|H0 | + I)−1/2 B(z)(|H0 | + I)−1/2

(2.5)

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with a bounded operator B(z). In particular, this implies that the operator GR(z) is well deﬁned and compact for any z ∈ C\R.

(2.6)

2.2. Main Result Let us ﬁx a “reference point” ν ∈ R and for λ > ν denote ∗

F0 (λ) = GE0 [ν, λ) (GE0 [ν, λ)) ,

(2.7)

∗

F (λ) = GE[ν, λ) (GE[ν, λ)) .

Note that by (2.1) and (2.6), the operators GE0 [ν, λ) and GE[ν, λ) are well deﬁned and compact. For ν < λ1 < λ2 , we have F0 (λ2 ) − F0 (λ1 ) = GE0 [λ1 , λ2 ) (GE0 [λ1 , λ2 ))

∗

(2.8)

and a similar identity holds true for F (λ). In what follows, we discuss the derivatives d d F0 (λ), F (λ) = F (λ) (2.9) F0 (λ) = dλ dλ understood in the operator norm sense. By (2.8), it is clear that neither the existence nor the values of these derivatives depend on the choice of the reference point ν. In fact, if H0 is semi-bounded from below, then we can take ν = −∞. It is also clear that if these derivatives exist in the operator norm, then F0 (λ) ≥ 0 and F (λ) ≥ 0 in the quadratic form sense. Theorem 2.1. Assume (2.1) and suppose that for some λ > ν, the derivatives F0 (λ), F (λ) exist in the operator norm. Then the limit π def (GE0 (λ − ε, λ + ε))∗ JGE(λ − ε, λ + ε) α(λ) = lim (2.10) ε→+0 2ε exists and the identity σess (E(−∞, λ) − E0 (−∞, λ)) = [−α(λ), α(λ)]

(2.11)

holds true. One also has α(λ) = πF0 (λ)1/2 JF (λ)1/2 .

(2.12)

The proof is given in Sect. 3. Remark. 1. It is easy to see that σ(E(−∞, λ) − E0 (−∞, λ)) ⊂ [−1, 1]. Thus, Theorem 2.1 implies, in particular, that α(λ) ≤ 1. 2. If λ ∈ / σ(H0 ) ∪ σ(H), then F0 (λ) = F (λ) = 0, and we obtain that the difference of the spectral projections in (2.11) is compact. This is not difﬁcult to prove directly (see Remark 3.5). 3. If the operator V R0 (i) is bounded, then it is obvious that (2.10) can be rewritten as (1.3). In what follows we prove that under the standard assumptions of either trace class or smooth version of scattering theory, one has α(λ) = S(λ) − I/2,

(2.13)

where S(λ) is the scattering matrix for the pair H0 , H. Thus, the veriﬁcation of (1.1) splits into two parts: (2.11) and (2.13). The statement (2.11) is more general than (2.13). Indeed, in order to prove (2.13), one has to ensure that

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the scattering matrix S(λ) is well deﬁned; this requires some assumptions stronger than those of Theorem 2.1, see Sects. 2.3 and 2.4. 2.3. The Scattering Matrix Here, following [28], we recall the definition of the scattering matrix in abstract scattering theory. This requires some rather lengthy preliminaries. First we need to recall the definition of the core of the absolutely continuous (a.c.) (ac) spectrum of H0 . Let E0 (·) (resp. E (ac) (·)) be the a.c. part of the spectral measure of H0 (resp. H) and let σac (H0 ) be the a.c. spectrum of H0 deﬁned (ac) as usual as the minimal closed set such that E0 (R\σac (H0 )) = 0. The set σac (H0 ) is “too large” for general scattering theory considerations. Indeed, it is not difﬁcult to construct examples when σac (H0 ) con(ac) tains a closed set A of a positive Lebesgue measure such that E0 (A) = 0 (ac) (consider E0 being supported on the intervals (an − 2−n , an + 2−n ), where a1 , a2 , . . . is a dense sequence in R). Thus, it is convenient to use the notion ˆac (H0 ) and deﬁned as a of the core of the a.c. spectrum of H0 , denoted by σ Borel set such that: (ac)

(ac)

(i) σ ˆac (H0 ) is a Borel support of E0 , i.e. E0 (R\ˆ σac (H0 )) = 0; (ac) ˆac (H0 )\A has a (ii) if A is any other Borel support of E0 , then the set σ zero Lebesgue measure. The set σ ˆac (H0 ) is not unique but is deﬁned up to a set of a zero Lebesgue measure. Suppose that for some interval Δ ⊂ R, the (local) wave operators (ac)

W± = W± (H0 , H; Δ) = s-lim eitH e−itH0 E0 t→±∞

(Δ)

exist and Ran W+ (H0 , H; Δ) = Ran W− (H0 , H; Δ). Then the (local) scatter(ac) ing operator S = W+∗ W− is unitary in Ran E0 (Δ) and commutes with H0 . Consider the direct integral decomposition (ac) Ran E0 (Δ)

⊕ h(λ)dλ

=

(2.14)

σ ˆac (H0 )∩Δ

which diagonalises H0 . Since S commutes with H0 , the decomposition (2.14) represents S as the operator of multiplication by the operator valued function S(λ) : h(λ) → h(λ). The unitary operator S(λ) is called the scattering matrix. With this definition, S(λ) is deﬁned for a.e. λ ∈ σ ˆac (H0 ). In abstract scattering theory, it does not make sense to speak of S(λ) at an individual ˆac (H0 ) is deﬁned only up to addition point λ ∈ σ ˆac (H0 ), since even the set σ or subtraction of sets of zero Lebesgue measure. Also, in general there is no distinguished choice of the direct integral decomposition (2.14); any unitary transformation in the ﬁber spaces h(λ) yields another suitable decomposition. Thus, the scattering matrix is, in general, deﬁned only up to a unitary equivalence. The above discussion refers only to the “abstract” version of the mathematical scattering theory. In concrete problems, there is often a natural

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distinguished choice of the core σ ˆac (H0 ) and of the direct integral decomposition (2.14). This usually allows one to consider S(λ) as an operator deﬁned for all (rather than for a.e.) λ ∈ σ ˆac (H0 ). In what follows we set S(λ) = I

for λ ∈ R\ˆ σac (H0 );

(2.15)

thus, S(λ) is now deﬁned for a.e. λ ∈ R. This will make the statements below more succinct. 2.4. The Scattering Matrix and α(λ) Similarly to the definition (2.2) of T0 (z), let us formally deﬁne T (z) = GR(z)G∗ . More precisely, using (2.5), we set ∗ T (z) = G(|H0 | + I)−1/2 B(z) G(|H0 | + I)−1/2 . By (2.1), the operator T (z) is compact. From the resolvent identity (2.3) it follows that T (z) = T0 (z) − T0 (z)J(I + T0 (z)J)−1 T0 (z) = (I + T0 (z)J)−1 T0 (z). (2.16) First let us consider the framework of smooth perturbations. Suppose that for some bounded open interval Δ ⊂ R, T0 (z) and T (z) are uniformly continuous in the operator norm in the rectangle Re z ∈ Δ, Im z ∈ (0, 1). (2.17) Of course, from here it trivially follows that the limits T0 (λ + i0), T (λ + i0) exist in the operator norm for all λ ∈ Δ. Under the assumption (2.17) the operator G is locally H0 -smooth and H-smooth on Δ, and therefore the local wave operators W± (H0 , H; Δ) exist and are complete (see e.g. [28] for the details). The scattering matrix S(λ) is deﬁned for a.e. λ ∈ σ ˆac (H0 ) ∩ Δ. Theorem 2.2. Assume (2.1) and (2.17). Then for all λ ∈ Δ, the derivatives F0 (λ) and F (λ) exist in the operator norm and so (2.11) holds true. For a.e. λ ∈ Δ, the identities (2.13) and (1.1) hold true. The proof is given in Sect. 4.2. In [18], formula (1.1) was proven under the additional assumptions of the compactness of G (which is a stronger assumption than (2.1)) and the H¨ older continuity of F0 (λ) and F (λ). Next, consider the trace class scheme. Let S2 be the Hilbert–Schmidt class. Suppose that H = H0 + V , where V = V ∗ is a trace class operator. Then we can factorise V = GJG∗ with G = |V |1/2 ∈ S2 and J = sign(V ). It is well known that under these assumptions, the derivatives F0 (λ) and F (λ) exist in the operator norm for a.e. λ ∈ R (see e.g. [28, Section 6.1]). We have Theorem 2.3. Let H = H0 + V , where V is a trace class operator. Set G = |V |1/2 . Then for a.e. λ ∈ R, the derivatives F0 (λ), F (λ) exist and (2.11), (2.13) and (1.1) hold true. Alternatively, we have the following statement more suitable for applications to differential operators:

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Theorem 2.4. Let H0 be semi-bounded from below; assume that (2.1) holds true and also, for some m > 0, G(|H0 | + I)−m ∈ S2 .

(2.18)

Then the conclusion of Theorem 2.3 holds true. The proofs of Theorems 2.3 and 2.4 are given in Sect. 4.2. Remark. 1. The existence and completeness of the wave operators under the assumptions of Theorems 2.3 and 2.4 is well known; see e.g. [28, Section 4.5 and Section 6.4]. 2. According to our convention (2.15), we have S(λ) − I = 0

for λ ∈ R\ˆ σac (H0 ).

Thus, Theorems 2.2–2.4 in particular, mean that for a.e. λ ∈ R\ˆ σac (H0 ), the difference of the spectral projections (1.5) is compact. 2.5. Piecewise Continuous Functions ϕ Let us consider the essential spectrum of ϕ(H) − ϕ(H0 ) for piecewise continuous functions ϕ. It is natural to consider complex-valued functions ϕ; in this case ϕ(H) − ϕ(H0 ) is non-selfadjoint. For a bounded operator M , we denote by σess (M ) the compact set of all z ∈ C such that the operator M − zI is not Fredholm. The reader should be warned that there are several non-equivalent definitions of the essential spectrum of a non-selfadjoint operator in the literature; see e.g. [6, Sections 1.4 and 9.1] for a comprehensive discussion. However, as we shall see, the essential spectrum of ϕ(H) − ϕ(H0 ) has an empty interior and a connected complement in C, and so in our case most of these definitions coincide. A function ϕ : R → C is called piecewise continuous if for any λ ∈ R the limits ϕ(λ±0) = limε→ + 0 ϕ(λ±ε) exist. We denote by P C0 (R) (resp. C0 (R)) the set of all piecewise continuous (resp. continuous) functions ϕ : R → C such that lim|x|→∞ ϕ(x) = 0. For ϕ ∈ P C0 (R) we denote κλ (ϕ) = ϕ(λ + 0) − ϕ(λ − 0),

sing supp ϕ = {λ ∈ R | κλ (ϕ) = 0}.

It is easy to see that for any ε > 0, the set {λ ∈ R | |κλ (ϕ)| > ε} is ﬁnite. For z1 , z2 ∈ C, we denote by [z1 , z2 ] the closed interval of the straight line in C that joins z1 and z2 . Theorem 2.5. Assume (2.1) and let (2.17) hold true for some open bounded interval Δ ⊂ R. Let ϕ ∈ P C0 (R) be a function with sing supp ϕ ⊂ Δ. Then we have σess (ϕ(H) − ϕ(H0 )) = ∪λ∈Δ [−α(λ)κλ (ϕ), α(λ)κλ (ϕ)] , where α(λ) is defined by (2.10). In particular, if ϕ is real valued, then σess (ϕ(H) − ϕ(H0 )) = [−a, a], The proof is given in Sect. 5.

a = sup |α(λ)κλ (ϕ)|. λ∈Δ

(2.19)

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2.6. The Fredholm Property A pair of orthogonal projections P , Q in a Hilbert space is called Fredholm, if (see e.g. [3]) ±1∈ / σess (P − Q).

(2.20)

If P, Q is a Fredholm pair, one deﬁnes the index of P, Q by index(P, Q) = dim Ker(P − Q − I) − dim Ker(P − Q + I). In a forthcoming publication [19], we study the index of the pair E(−∞, λ),

E0 (−∞, λ).

(2.21)

In connection with this (and perhaps otherwise) it is interesting to know whether the pair (2.21) is Fredholm. Under the assumptions of Theorem 2.1, the question reduces to deciding whether α(λ) < 1 or α(λ) = 1. If (2.13) holds true, then, clearly, the pair (2.21) is Fredholm if and only if −1 is not an eigenvalue of the scattering matrix S(λ). Below we give a convenient criterion for this in terms of the operators T0 , T . For a bounded operator M , we denote Re M = (M +M ∗ )/2 and Im M = (M − M ∗ )/2i. If the limits T0 (λ + i0), T (λ + i0) exist, we denote A0 (λ) = Re T0 (λ + i0),

A(λ) = Re T (λ + i0).

(2.22)

Theorem 2.6. Assume (2.1). Suppose that for some λ ∈ R, the limits T0 (λ + i0), T (λ + i0) and the derivatives F0 (λ), F (λ) exist in the operator norm. Then the following statements are equivalent: (i) the pair (2.21) is Fredholm; (ii) Ker(I + A0 (λ)J) = {0}; (iii) Ker(I − A(λ)J) = {0}. The proof is given in Sect. 4. Theorem 2.6 can be applied to either smooth or trace class framework. In applications, one can often obtain some information about the spectrum of A0 (λ) or A(λ); for example, one can sometimes ensure that the norm of A0 (λ) is small. By Theorem 2.6, this can be used to ensure that the pair (2.21) is Fredholm. Remark. 1. Since dim Ker(I + XY ) = dim Ker(I + Y X) for any bounded operators X, Y , we can equivalently restate (ii), (iii) as (iia) Ker(I + JA0 (λ)) = {0}; (iiia) Ker(I − JA(λ)) = {0}. 2. If the operator J has a bounded inverse, we can equivalently restate (ii), (iii) in a more symmetric form as (iib) Ker(J −1 + A0 (λ)) = {0}; (iiib) Ker(J −1 − A(λ)) = {0}. 2.7. Schr¨ odinger Operator: Smooth Framework Let H0 = −Δ in H = L2 (Rd ), d ≥ 1, and let H = H0 + V , where V is the operator of multiplication by a function V : Rd → R which is assumed to satisfy |V (x)| ≤ C(1 + |x|)−ρ ,

ρ > 1.

(2.23)

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Let K = H, G = |V |1/2 , J = sign V . Under the assumption (2.23), the hypotheses (2.1) and (2.17) hold true with any Δ = (c1 , c2 ), 0 < c1 < c2 < ∞, see e.g. [24, Theorem XIII.33]. It is also easy to see that the derivatives F0 (λ), F (λ) exist in the operator norm for all λ > 0. Thus, for any λ > 0, the conclusions of Theorems 2.2 and 2.6 hold true. In [18], formula (1.1) was proven for H0 and H as above only for d ≤ 3. We also see that the conclusion of Theorem 2.5 holds true for any ϕ ∈ P C0 (R) which is continuous in an open neighbourhood of zero. In this example there is a well known natural choice of the core σ ˆac (H0 ) = (0, ∞) and of the direct integral decomposition (2.14) with h(λ) = L2 (Sd−1 ). Moreover, in this case the scattering matrix S(λ) : L2 (Sd−1 ) → L2 (Sd−1 ) is continuous in λ > 0. Thus, in this case the statement (1.1) holds true for all λ > 0. 2.8. Schr¨ odinger Operator: Trace Class Framework Let H0 = −Δ + U in L2 (Rd ), d ≥ 1, where U is the operator of multiplication by a real valued bounded function. Next, let H = H0 + V , where V is the operator of multiplication by a real valued function V ∈ L1 (Rd ) such that V is (−Δ)-form compact. Then V is also H0 -form compact and H = H0 + V is well deﬁned as a form sum. It is well known (see e.g. [26, Theorem B.9.1]) that under the above assumptions, (2.18) holds true with G = |V |1/2 for m > d/4. Thus, the conclusions of Theorem 2.4 hold true. The assumptions on H0 in this example can be considerably relaxed by allowing U to have local singularities, by including a background magnetic ﬁeld, etc. Note that in this example we have no information on the a.c. spectrum of H0 .

3. Proof of Theorem 2.1 We follow the method of [18] with some minor technical improvements. In order to simplify our notation, we assume λ = 0 and denote R+ = (0, ∞), R− = (−∞, 0). 3.1. The Proof of (2.12) Let us prove that if the derivatives F0 (0) and F (0) exist in the operator norm, then the limit (2.10) also exists and the identity (2.12) holds true. Let us start from the r.h.s. of (2.12). Denoting δε = (−ε, ε) and using the identities X2 = XX ∗ = X ∗ X, we get F0 (0)1/2 JF (0)1/2 2 = F0 (0)1/2 JF (0)JF0 (0)1/2 = lim

1 F0 (0)1/2 JGE(δε )(GE(δε ))∗ JF0 (0)1/2 2ε

= lim

1 (GE(δε ))∗ JF0 (0)JGE(δε ) 2ε

ε→+0

ε→+0

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= lim

1 (GE(δε ))∗ JGE0 (δε )(GE0 (δε ))∗ JGE(δε ) (2ε)2

= lim

1 (GE(δε ))∗ JGE0 (δε )2 , (2ε)2

ε→+0

ε→+0

as required. In the rest of this section, we prove that if the derivatives F0 (0) and F (0) exist in the operator norm, then the identity σess (E(R− ) − E0 (R− )) = [−α(0), α(0)]

(3.1)

holds true with α(0) = πF0 (0)1/2 JF (0)1/2 . 3.2. The Kernels of H0 and H Lemma 3.1. Assume (2.1) and suppose that the derivatives F0 (0) and F (0) exist in the operator norm. Then Ker H0 = Ker H. We note that this phenomenon is well known in scattering theory; see, e.g. [24, Theorem XIII.23]. Proof. 1. By our assumptions, GE0 ({0}) = 0 (otherwise F0 (0) cannot exist). Suppose ψ ∈ Ker H0 ; then Gψ = 0 and the resolvent identity (2.3) yields

2.

1 R(z)ψ = R0 (z)ψ = − ψ. z Thus, Hψ = 0. This proves that Ker H0 ⊂ Ker H. From (2.3) it is not difﬁcult to obtain the “usual” resolvent identity (see e.g. [28, Section 1.10]): R(z) = R0 (z) − (GR0 (z))∗ JGR(z).

(3.2)

Now let ψ ∈ Ker H. As above, GE({0}) = 0, and so from (3.2) one obtains 1 R0 (z)ψ = R(z)ψ = − ψ. z Thus, H0 ψ = 0 and so Ker H ⊂ Ker H0 . 3.3. Reduction to the Products of Spectral Projections Let us denote D = E(R− ) − E0 (R− ) and H+ = Ker(D − I),

H− = Ker(D + I),

H0 = (H+ ⊕ H− )⊥ .

It is well known (see e.g. [7] or [3]) that D|H0 is unitarily equivalent to (−D)|H0 .

(3.3)

Therefore, the spectral analysis of D reduces to the spectral analysis of D2 and to the evaluation of the dimensions of H+ and H− . Next, using Lemma 3.1, by a simple algebra we obtain the identity D2 = E0 (R− )E(R+ )E0 (R− ) + E0 (R+ )E(R− )E0 (R+ ),

(3.4)

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where the r.h.s. provides a block-diagonal decomposition of D2 with respect to the direct sum H = Ran E0 (R− ) ⊕ Ran E0 ({0}) ⊕ Ran E0 (R+ ). Thus, the spectral analysis of D2 reduces to the spectral analysis of the two terms in the r.h.s. of (3.4). In Sects. 3.4–3.7 we prove Lemma 3.2. Assume (2.1). Then the differences E0 (R+ )E(R− )E0 (R+ ) − E0 (0, 1)E(−1, 0)E0 (0, 1) E0 (R− )E(R+ )E0 (R− ) − E0 (−1, 0)E(0, 1)E0 (−1, 0)

(3.5) (3.6)

are compact operators. Theorem 3.3. Assume (2.1) and suppose that the derivatives F0 (0) and F (0) exist in the operator norm. Then σess (E0 (0, 1)E(−1, 0)E0 (0, 1)) = [0, α(0)2 ],

(3.7)

σess (E0 (−1, 0)E(0, 1)E0 (−1, 0)) = [0, α(0)2 ],

(3.8)

where α(0) is given by α(0) = πF0 (0)1/2 JF (0)1/2 . In particular, α(0) ≤ 1. With these two statements, it is easy to provide Proof of Theorem 2.1. Combining Lemma 3.2, Theorem 3.3 and Weyl’s theorem on the invariance of the essential spectrum under compact perturbations, we obtain σess (E0 (R− )E(R+ )E0 (R− )) = σess (E0 (R+ )E(R− )E0 (R+ )) = [0, α(0)2 ]. (3.9) By (3.4), it follows that σess (D2 ) = [0, α(0)2 ].

(3.10)

Suppose ﬁrst that α(0) = 1. Then from (3.10) and (3.3) we obtain σess (D) = [−1, 1], as required. Next, suppose α(0) < 1. Then from (3.10) it follows that the dimensions of H− and H+ are ﬁnite, and therefore σess (D) = σess (D|H0 ) and

σess (D2 ) = σess ((D|H0 )2 ).

Recalling (3.3), we obtain σess (D|H0 ) = [−α(0), α(0)], and (3.1) follows.

3.4. Proof of Lemma 3.2 Lemma 3.4. Assume (2.1). Let ϕ ∈ C(R) be a function such that the limits limx→±∞ ϕ(x) exist. Then the difference ϕ(H) − ϕ(H0 ) is compact. Proof. As is well known (and can easily be deduced from the compactness of R(z) − R0 (z) for Im z = 0), the operator ϕ(H) − ϕ(H0 ) is compact for any function ϕ ∈ C0 (R). Therefore, it sufﬁces to prove that ϕ(H) − ϕ(H0 ) is compact for at least one function ϕ ∈ C(R) such that limx→∞ ϕ(x) = limx→−∞ ϕ(x) and both limits exist. The latter fact is provided by

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[17, Theorem 7.3] where it is proven that if (2.1) holds true then the dif ference tan−1 (H) − tan−1 (H0 ) is compact. Remark 3.5. Let μ ∈ R\(σ(H0 ) ∪ σ(H)). Then E(−∞, μ) − E0 (−∞, μ) = ϕ(H) − ϕ(H0 )

(3.11)

for an appropriately chosen continuous function ϕ with ϕ(x) = 1 for x ∈ σ(H) ∪ σ(H0 ), x < μ and ϕ(x) = 0 for x ∈ σ(H) ∪ σ(H0 ), x > μ. It follows that the difference (3.11) is compact. Proof of Lemma 3.2. 1. Let ϕ1 ∈ C(R) be such that ϕ1 (x) = 1 for x ≤ −1 and ϕ1 (x) = 0 for x ≥ 0. Then E(−∞, −1)E0 (R+ ) = E(−∞, −1)(ϕ1 (H) − ϕ1 (H0 ))E0 (R+ ) 2.

and so by Lemma 3.4 the r.h.s. is compact. Let ϕ2 ∈ C(R) be such that ϕ2 (x) = 1 for x ≥ 1 and ϕ2 (x) = 0 for x ≤ 0. Then E0 (1, ∞)E(R− ) = E0 (1, ∞)(ϕ2 (H0 ) − ϕ2 (H))E(R− ),

3.

(3.12)

(3.13)

and so by Lemma 3.4 the r.h.s. is compact. From the compactness of the l.h.s. of (3.12) and (3.13), the compactness of the difference (3.5) follows by some simple algebra. Compactness of (3.6) is proven in the same way.

3.5. Hankel Operators In order to prove Theorem 3.3, we need some basic facts concerning operator valued Hankel integral operators. Suppose that for each t > 0, a bounded self-adjoint operator K(t) in K is given. Suppose that K(t) is continuous in t > 0 in the operator norm. Deﬁne a Hankel integral operator K in L2 (R+ , K) by ∞ ∞ (Kf, g)L2 (R+ ,K) = (K(t + s)f (t), g(s))K dt ds, (3.14) 0

0

when f, g ∈ L2 (R+ , K) are functions with compact support in R+ . The statement below is a straightforward generalisation of [8, Proposition 1.1] to the operator valued case. Proposition 3.6. (i) Suppose K(t) ≤ C/t for all t > 0. Then the operator K is bounded and K ≤ πC. (ii) Suppose K(t) is compact for all t and K(t) = o(1/t) as t → +0 and as t → +∞. Then K is compact. Proof. Since the Carleman operator on L2 (R+ ) with the kernel (t + s)−1 is bounded with the norm π, we have ∞ ∞ f (t)K g(s)K |(Kf, g)L2 (R+ ,K) | ≤ C dt ds t+s 0

0

≤ πCf L2 (R+ ,K) gL2 (R+ ,K) ,

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which proves (i). To prove (ii), we need to approximate K by compact operators. Let Kn (t) = K(t)χn (t), where χn is the characteristic function of the interval (1/n, n) and let Kn be the corresponding operator in L2 (R+ , K). By (i), we have K − Kn L2 (R+ ,K) → 0 as n → ∞. Thus, it remains to show that each Kn is compact. For each n, the Hankel type integral operator with the kernel χn (t + s)/(t + s) in L2 (R+ ) is compact (in fact, Hilbert-Schmidt). It follows that Kn is compact if K(t) is independent of t. Now the result follows from the fact that K(t) can be uniformly approximated by piecewise constant functions on the interval (1/n, n). Important model operators in our construction below are the Hankel integral operators in L2 (R+ , K) of the type (3.14) with K(t) given by 1 − e−t 1 − e−t F0 (0) and F (0). (3.15) t t For this reason, we need to discuss the integral Hankel operator Γ in L2 (R+ ) −t−s with the integral kernel Γ(t, s) = 1−et+s . One can show (see e.g. [18, Lemma 7]) that σ(Γ) = [0, π].

(3.16)

In fact, the spectrum of Γ is purely absolutely continuous, but we will not need this fact. Identifying L2 (R+ , K) with L2 (R+ ) ⊗ K, we denote the operators (3.15) by Γ ⊗ F0 (0) and Γ ⊗ F (0). 3.6. The Operators L and L0 The crucial point of our proof of Theorem 3.3 is the representation E(−1, 0)E0 (0, 1) = −LJL∗0

(3.17)

in terms of some auxiliary operators L0 and L which we proceed to deﬁne. These operators act from L2 (R+ , K) to H. On the dense set L2 (R+ , K) ∩ L1 (R+ , K) we deﬁne L0 , L by ∞ L0 f = e−tH0 (GE0 (0, 1))∗ f (t)dt, (3.18) 0

∞

Lf =

etH (GE(−1, 0))∗ f (t)dt.

(3.19)

0

Lemma 3.7. Assume (2.1) and suppose that the derivatives F0 (0), F (0) exist in the operator norm. Then: (i) The operators L0 and L defined by (3.18) and (3.19) extend to bounded operators from L2 (R+ , K) to H. (ii) The differences L∗0 L0 − Γ ⊗ F0 (0), (iii)

are compact operators. The identity (3.17) holds true.

L∗ L − Γ ⊗ F (0)

(3.20)

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Proof. (i) Let us prove that L0 is bounded; the boundedness of L is proven in the same way. For f ∈ L2 (R+ , K) ∩ L1 (R+ ; K) we have ∞ ∞

L0 f = 2

GE0 (0, 1)e−(t+s)H0 (GE0 (0, 1))∗ f (t), f (s) dt ds, K

0

0

and so the above expression is a quadratic form of the operator of the type (3.14) with the kernel K(t) = GE0 (0, 1)e−tH0 (GE0 (0, 1))∗ . By Proposition 3.6, it sufﬁces to prove the bound K(t)K ≤ C/t, t > 0. Let f ∈ H and ρ(λ) = (E0 (−∞, λ)f, f ). Integrating by parts, one obtains 1

−tλ

e

−t

1

dρ(λ) = e

0

1 dρ(λ) + t

0

−tμ

μ

dμ e 0

dρ(λ). 0

It follows that −tH0

e

1

−t

E0 (0, 1) = e E0 (0, 1) + t

e−tμ E0 (0, μ)dμ.

0

Using this expression, the relation (2.8) and the fact that GE0 ({0}) = 0, we get 1

−t

K(t) = e (F0 (1) − F0 (0)) + t

e−tμ (F0 (μ) − F0 (0))dμ.

(3.21)

0

By our assumption on the differentiability of F0 , we have F0 (μ) − F0 (0) ≤ C|μ| for |μ| ≤ 1. Using this, we obtain: 1

−t

K(t) ≤ e F0 (1) − F0 (0) + t

e−tμ F0 (μ) − F0 (0)dμ

0

≤ Ce−t + Ct

1

e−tμ μdμ = C(1 − e−t )/t ≤ C/t,

t > 0,

0

(ii)

as required. Let us consider the ﬁrst of the differences (3.20); the second one is considered in the same way. By the same reasoning as above, L∗0 L0 − Γ ⊗ F0 (0) is the operator of the type (3.14) with K(t) = GE0 (0, 1)e−tH0 (GE0 (0, 1))∗ − F0 (0)(1 − e−t )/t. By (2.1), F0 (λ) is compact for all λ. Since the derivative F0 (0) exists in the operator norm, the operator F0 (0) is also compact. Thus, K(t) is compact for all t > 0. By Proposition 3.6(ii), it sufﬁces to prove that K(t) = o(1/t) as t → 0 and t → ∞. For t → 0, the statement is

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obvious. Consider the limit t → ∞. By the same calculation as in part (i) of the proof (see (3.21)), we have 1

−t

K(t) = e (F0 (1) − F0 (0)) + t

e−tμ (F0 (μ) − F0 (0))dμ

0

−F0 (0)t

1

e−tμ μ dμ − F0 (0)e−t .

0

It follows that −t

K(t) ≤ e

F0 (1)−F0 (0)−F0 (0)+t

1

e−tμ F0 (μ)−F0 (0) − F0 (0)μdμ.

0

(3.22) By our assumption, F0 (μ) − F0 (0) − F0 (0)μ = o(μ) as μ → 0. (iii)

(3.23)

Using (3.22) and (3.23), it is easy to see that K(t) = o(1/t) as t → ∞. Let f, f0 ∈ H. Using (2.4), we obtain

d E0 (0, 1)e−tH0 f0 , E(−1, 0)etH f dt = E0 (0, 1)e−tH0 f0 , HE(−1, 0)etH f − H0 E0 (0, 1)e−tH0 f0 , E(−1, 0)etH f . = JGE0 (0, 1)e−tH0 f0 , GE(−1, 0)etH f K

Using this and the easily veriﬁable relations E0 (0, 1)e−tH0 f0 → 0,

E0 (−1, 0)etH f → 0

as t → ∞,

we get (JL∗0 f0 , L∗ f )L2 (R+ ,K)

∞ = 0 ∞

=

JGE0 (0, 1)e−tH0 f0 , GE(−1, 0)etH f

K

dt

d E0 (0, 1)e−tH0 f0 , E(−1, 0)etH f dt dt

0

= −(E0 (0, 1)f0 , E(−1, 0)f ), which proves (3.17). 3.7. Proof of Theorem 3.3 We will prove (3.7); the relation (3.8) is proven in the same manner. 1.

First we introduce some notation. For bounded self-adjoint operators M and N we shall write M ≈ N if M |(Ker M )⊥ is unitarily equivalent to N |(Ker N )⊥ .

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It is well known that M ∗ M ≈ M M ∗ for any bounded operator M ; below we use this fact. 2. Using Lemma 3.7 we get, for some compact operators X0 and X: E0 (0, 1)E(−1, 0)E0 (0, 1) = L0 JL∗ LJL∗0 = L0 (Γ ⊗ JF (0)J)L∗0 + X, L0 (Γ ⊗ JF (0)J)L∗0 = L0 (Γ1/2 ⊗ JF (0)1/2 )(Γ1/2 ⊗ F (0)1/2 J)L∗0 ≈ (Γ1/2 ⊗ F (0)1/2 J)L∗0 L0 (Γ1/2 ⊗ JF (0)1/2 ) = (Γ1/2 ⊗ F (0)1/2 J)(Γ ⊗ F0 (0))(Γ1/2 ⊗ JF (0)1/2 )+X0 = Γ2 ⊗ (F (0)1/2 JF0 (0)JF (0)1/2 ) + X0 . Thus, by Weyl’s theorem, we obtain σess (E0 (0, 1)E(−1, 0)E0 (0, 1)) = σess (Γ2 ⊗ Q), Q = F (0)1/2 JF0 (0)JF (0)1/2 . 3.

The operator Q above is compact, selfadjoint and Q ≥ 0. Let Q = ∞ n=1 λn (·, fn )fn be the spectral decomposition of Q, where λ1 ≥ λ2 ≥ · · · are the eigenvalues of Q. Then ∞

Γ2 ⊗ Q = λn Γ2 ⊗ (·, fn )fn n=1

is an orthogonal sum decomposition of Γ2 ⊗ Q, and therefore 2 σess (Γ2 ⊗ Q) = ∪∞ n=1 σess (λn Γ ⊗ (·, fn )fn ).

Taking into account (3.16) and recalling that λ1 = Q, we obtain 2 2 2 1/2 σess (Γ2 ⊗ Q) = ∪∞ JF0 (0)JF (0)1/2 ] n=1 [0, λn π ] = [0, π Q] = [0, π F (0)

= [0, π 2 F0 (0)1/2 JF (0)1/2 2 ] = [0, α(0)2 ],

as required.

4. Proofs of Theorems 2.2, 2.3, 2.4 and 2.6 4.1. Existence of F0 , F and T0 , T Here we recall various statements concerning the existence of the derivatives F0 (λ), F (λ) and the limits T0 (λ + i0), T (λ + i0) under the assumptions of Theorems 2.2, 2.3, 2.4. All of these statements are essentially well known. If the limits T0 (λ + i0), T (λ + i0) exist, we denote B0 (λ) = Im T0 (λ + i0),

B(λ) = Im T (λ + i0).

F0 (λ)

We ﬁrst note that if the derivatives T (λ + i0) exist at some point λ, then πF0 (λ) = B0 (λ),

and F (λ) and the limits T0 (λ+i0),

πF (λ) = B(λ).

(4.1)

Indeed, this follows from the spectral theorem and the following well known fact (see e.g. [25, Theorem 11.22]): if μ is a measure on R and the derivative d dλ μ(−∞, λ) exists, then d dμ(t) π μ(−∞, λ) = lim Im . ε→+0 dλ t − λ − iε R

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Lemma 4.1. Assume (2.1) and suppose that (2.17) holds true for some bounded open interval Δ ⊂ R. Then for all λ ∈ Δ the derivatives F0 (λ), F (λ) exist in the operator norm. Proof. From the obvious operator inequality 0 ≤ E0 ({λ}) ≤

ε2 , (H0 − λ)2 + ε2

ε > 0,

we get 0 ≤ GE0 ({λ})(GE0 ({λ}))∗ ≤ εIm T0 (λ + iε),

ε > 0.

This implies that GE0 ({λ}) = 0 for all λ ∈ Δ. Using this, Stone’s formula (see e.g. [21, Theorem VII.13]) yields 1 ((F0 (b) − F0 (a))f, f ) = lim ε→+0 π

b Im (T0 (λ + iε)f, f )dλ a

1 = π

b (B0 (λ)f, f )dλ a

for any interval [a, b] ⊂ Δ. From here and the continuity of B0 (λ) we get the statement concerning F0 (λ). The case of F (λ) is considered in the same way. Lemma 4.2. (i) Assume that G is a Hilbert–Schmidt operator. Then for a.e. λ ∈ R, the derivatives F0 (λ), F (λ) and the limits T0 (λ + i0), T (λ + i0) exist in the operator norm. (ii) Under the assumptions of Theorem 2.4, for a.e. λ ∈ R the derivatives F0 (λ), F (λ) and the limits T0 (λ + i0), T (λ + i0) exist in the operator norm. Proof. (i) is one of the key facts of the trace class scattering theory, see e.g. [28, Section 6.1]. (ii) First consider F0 and T0 . Let us apply a standard argument: let Δ1 = (−R, R), Δ2 = R\Δ1 and write Gj = GE0 (Δj ), j = 1, 2. Then G1 ∈ S2 . Thus, by part (i) of the lemma, the derivative d d F0 (λ) = G1 E0 (−∞, λ)G∗1 , λ ∈ Δ1 , dλ dλ exists in the operator norm. Let us consider T0 (z); we have T0 (z) = G1 R0 (z)G∗1 + G2 (G2 R0 (z))∗ .

(4.2)

By part (i) of the lemma, the ﬁrst term in the r.h.s. of (4.2) has a limit as z → λ + i0 for a.e. λ ∈ Δ1 . Since R0 (z)E0 (Δ2 ) is analytic in z ∈ C\Δ2 , the second term in the r.h.s. of (4.2) has a limit as z → λ + i0 for all λ ∈ Δ1 . It follows that T0 (z) has boundary values as z → λ + i0 for a.e. λ ∈ Δ1 . Since R in the definition of Δ1 can be taken arbitrary large, this gives the desired statement for a.e. λ ∈ R.

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Consider F and T . First, exactly as in the proof of [23, Theorem XI.30], using (2.1) and (2.18), one shows that G(|H| + I)−m ∈ S2 .

(4.3)

After this, the proof follows the same argument as above.

Lemma 4.3. Assume (2.1) and let Δ ⊂ R be a bounded interval. Suppose that for a.e. λ ∈ Δ, the derivative F0 (λ) exists in the operator norm. Then for a.e. λ ∈ Δ\ˆ σac (H0 ), one has F0 (λ) = 0. Proof. 1. Recall the following measure theoretic statement. Let μ be a ﬁnite Borel measure on R and let Z be a Borel support of μ, i.e. μ(R\Z) = 0. Then d μ(−∞, λ) = 0 for Lebesgue-a.e. λ ∈ R\Z. (4.4) dλ Indeed, let μ = μac + μs be the decomposition of μ into the a.c. and singular components with respect to the Lebesgue measure. Let 0 ≤ f ∈ L1 (R) be the Radon-Nikodym derivative of μac with respect to the Lebesgue measure. Then (see e.g. [25, Section 8.6]) d d μac (−∞, λ) = f (λ), μs (−∞, λ) = 0, dλ dλ The statement μ(R\Z) = 0 implies that f (λ)dλ = 0.

for Lebesgue-a.e. λ ∈ R.

R\Z

Thus, f (λ) = 0 for Lebesgue-a.e. λ ∈ R\Z. From here we get (4.4). Let Zs be a Borel support of the singular part of the spectral measure ˆ=σ ˆac (H0 )∪Zs is E0 . Since the Lebesgue measure of Zs is zero, the set σ ˆ is a Borel support again a core of the a.c. spectrum of H0 . Moreover, σ σ ) = 0. of E0 , i.e. E0 (R\ˆ 3. Let GΔ = GE0 (Δ); by (2.1), GΔ is a compact operator. Let {en }∞ n=1 be an orthonormal basis in K. Consider the complex valued measures

2.

μnm (Λ) = (E0 (Λ)G∗Δ en , G∗Δ em ) ,

n, m ∈ N,

Λ ⊂ Δ.

We have μnm (Δ\ˆ σ ) = 0. Representing each μnm as a linear combination of four non-negative measures and applying (4.4), we obtain d μnm (−∞, λ) = 0, λ ∈ (Δ\ˆ σ )\Λnm , n, m ∈ N, dλ where the Lebesgue measure of the set Λnm is zero. It follows that d μnm (−∞, λ) = 0, λ ∈ (Δ\ˆ σ )\Λ, n, m ∈ N, (4.5) dλ where Λ = ∪n,m Λnm and the Lebesgue measure of Λ is zero. 4. Let D ⊂ K be the dense set of all ﬁnite linear combinations of elements of the basis {en }∞ n=1 . It follows from (4.5) that

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d (E0 (−∞, λ)G∗Δ f, G∗Δ g) = 0, ∀f, g ∈ D, dλ and therefore F0 (λ) = 0 for a.e. λ ∈ Δ\ˆ σ.

93

a.e. λ ∈ Δ\ˆ σ,

4.2. Connection Between α(λ) and S(λ) First we establish a connection between α(λ) and some auxiliary unitary operator S(λ). The idea to use the operator S(λ) is due to Sobolev and Yafaev [27]. Lemma 4.4. Assume (2.1) and suppose that the derivatives F0 (λ) and F (λ) and the limits T0 (λ + i0), T (λ + i0) exist for some λ ∈ R. Then the operator S(λ) = I − 2iB0 (λ)1/2 (J − JT (λ + i0)J)B0 (λ)1/2

(4.6)

in K is unitary and 1 S(λ) − I = α(λ). 2 Proof. 1.

(4.7)

From (2.16) one easily obtains the identity I − T (z)J = (I + T0 (z)J)−1 ,

Im z > 0.

(4.8)

Since the limits T0 (λ + i0) and T (λ + i0) exist in the operator norm, we conclude that the operator I + T0 (λ + i0)J has a bounded inverse and I − T (λ + i0)J = (I + T0 (λ + i0)J)−1 .

(4.9)

In the same way, one obtains I − JT (λ + i0) = (I + JT0 (λ + i0))−1 .

(4.10)

Taking adjoints in (4.10) and subtracting from (4.9), after some simple algebra we get JB(λ)J = (J − JT (λ + i0)J)B0 (λ)(J − JT (λ + i0)∗ J). From here the unitarity of S(λ) follows by a direct calculation. 2. Using the unitarity of S(λ) and the identity (4.1), we obtain (S(λ) − I)∗ (S(λ) − I) = 2I − 2Re S(λ) = 4 Im (B0 (λ)1/2 JT (λ + i0)JB0 (λ)1/2 ) = 4 Im (B0 (λ)1/2 JB(λ)JB0 (λ)1/2 ) = 4π 2 Im (F0 (λ)1/2 JF (λ)JF0 (λ)1/2 ). From here, taking into account (2.12), we get 1 S(λ) − I2 = π 2 F0 (λ)1/2 JF (λ)JF0 (λ)1/2 4 = π 2 F0 (λ)1/2 JF (λ)1/2 2 = α(λ)2 , as required. The following Lemma is essentially contained in [28, Section 7.7].

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Lemma 4.5. (i) Under the assumptions (2.1), (2.17), the local wave operaˆac (H0 ) ∩ Δ tors W± (H0 , H; Δ) exist and are complete, and for a.e. λ ∈ σ we have S(λ) − I = S(λ) − I.

(4.11)

(ii) Under the assumptions of Theorems 2.3 or 2.4, the wave operators W± (H0 , H) exist and are complete, and for a.e. λ ∈ σ ˆac (H0 ), the relation (4.11) holds true. Proof. (i) For the existence and completeness of wave operators, we refer to [28, Section 4.5]. Next, for a.e. λ ∈ σ ˆac (H0 ) ∩ Δ, the scattering matrix can be represented as S(λ) = I − 2πiZ(λ)(J − JT (λ + i0)J)Z(λ)∗ ,

(4.12)

where Z(λ) : K → h(λ) is an operator such that πZ(λ)∗ Z(λ) = B0 (λ).

(4.13)

This is the well known stationary representation for the scattering matrix, see e.g. [28, Section 5.5(3)]. Let us use the polar decomposition of Z(λ), Z(λ) = U |Z(λ)|, where |Z(λ)| = Z(λ)∗ Z(λ) = B0 (λ)1/2 /π, and U is an isometry which maps Ran Z(λ)∗ onto Ran Z(λ). Then we get S(λ) − I = U (S(λ) − I)U ∗ , (ii)

and (4.11) follows. This argument is borrowed from [28, Lemma 7.7.1]. Existence and completeness of wave operators is well known, see e.g. [28, Theorem 6.4.5]. As in the proof of part (i), we have the representation (4.12), (4.13) for a.e. λ ∈ σ ˆac (H0 ) (see e.g. [28, Section 5.5(3)]) and the required statement follows by the same argument as above.

Proof of Theorem 2.2. The existence of the derivatives F0 (λ) and F (λ) follows from Lemma 4.1. Thus, by Theorem 2.1, we obtain (2.11). By Lemma 4.4 and Lemma 4.5, we have α(λ) =

1 1 S(λ) − I = S(λ) − I 2 2

for a.e. λ ∈ σ ˆac (H0 ) ∩ Δ. Thus, we have (2.13) and therefore (1.1) for a.e. σac (H0 ), by Lemma 4.3, λ∈σ ˆac (H0 ) ∩ Δ. On the other hand, for a.e. λ ∈ Δ\ˆ we have α(λ) = 0. Thus, according to (2.15), the relations (2.13) and (1.1) hold true also for a.e. λ ∈ Δ\ˆ σac (H0 ). Proof of Theorems 2.3 and 2.4. By Lemma 4.2, the derivatives F0 (λ), F (λ) and the limits T0 (λ + i0), T (λ + i0) exist for a.e. λ ∈ R. Thus, the identity (2.11) follows from Theorem 2.1. The identities (2.13) and (1.1) follow for a.e. λ ∈ R as in the proof of Theorem 2.2.

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4.3. The Fredholm Property Proof of Theorem 2.6. 1. As in the proof of Lemma 4.4, we get that the operators I + T0 (λ + i0)J and I + JT0 (λ + i0) have bounded inverses and the identities (4.9), (4.10) hold true. 2. From (4.9), (4.10) we obtain I − A(λ)J = (I + T0 (λ + i0)J)−1 (I + A0 (λ)J)(I + T0 (λ + i0)∗ J)−1 .

3.

This proves that dim Ker(I − A(λ)J) = dim Ker(I + A0 (λ)J) and so (ii)⇔(iii). Let us prove that dim Ker(I + S(λ)) = dim Ker(I + A0 (λ)J).

(4.14)

Using the identity (4.9) and the fact that dim Ker(I + XY ) = dim Ker(I + Y X) for any bounded operators X and Y , we obtain: dim Ker(I + S(λ)) = dim Ker(I − iB0 (λ)1/2 J(I − T (λ + i0)J)B0 (λ)1/2 ) = dim Ker(I − iB0 (λ)1/2 J(I + T0 (λ + i0)J)−1 B0 (λ)1/2 ) = dim Ker(I − iB0 (λ)J(I + T0 (λ + i0)J)−1 ) = dim Ker(I + T0 (λ + i0)J − iB0 (λ)J) = dim Ker(I + A0 (λ)J), as required. Let us prove that (i)⇔(ii). By the definition (2.20) and by Theorem 2.1, it sufﬁces to prove that α(λ) < 1 if and only if Ker(I + A0 (λ)J) = {0}. Suppose that Ker(I + A0 (λ)J) = {0}. Then by (4.14), we have Ker(I + S(λ)) = {0}. Since S(λ) − I is compact, it follows that −1 ∈ / σ(S(λ)). Since S(λ) is unitary, we get S(λ) − I < 2. By (4.7), it follows that α(λ) < 1. Conversely, suppose that dim Ker(I + A0 (λ)J) > 0. Then dim Ker(I + S(λ)) > 0 and therefore S(λ)−I = 2. By (4.7), it follows that α(λ) = 1. 4.

5. Piecewise Continuous Functions ϕ We closely follow the proof used by S. Power in his description [16] of the essential spectrum of Hankel operators with piecewise continuous symbols. We use the shorthand notation δ(ϕ) = ϕ(H) − ϕ(H0 ). 5.1. Auxiliary Statements Lemma 5.1. Assume (2.1) and let ϕ1 , ϕ2 ∈ P C0 (R). Suppose that sing supp ϕ1 ∩ sing supp ϕ2 = ∅. Then the operator δ(ϕ1 )δ(ϕ2 ) is compact.

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Proof. 1. For j = 1, 2 one can represent ϕj as ϕj = ψj + ζj , where ζj ∈ C0 (R), ψj ∈ P C0 (R) and supp ψ1 ∩ supp ψ2 = ∅. By Lemma 3.4, the operators δ(ζ1 ) and δ(ζ2 ) are compact. We have δ(ϕ1 )δ(ϕ2 ) = (δ(ψ1 ) + δ(ζ1 ))(δ(ψ2 ) + δ(ζ2 )) 2.

and so it sufﬁces to prove that the operator δ(ψ1 )δ(ψ2 ) is compact. One can choose ω ∈ C0 (R) such that ψ1 ω = ψ1 and ωψ2 ≡ 0. Then ω(H0 )ψ2 (H0 ) = 0 and ψ1 (H)ψ2 (H0 ) = ψ1 (H)ω(H)ψ2 (H0 ) = ψ1 (H)(ω(H) − ω(H0 ))ψ2 (H0 ), and the operator in the r.h.s. is compact by Lemma 3.4. By the same argument, the operator ψ1 (H0 )ψ2 (H) is compact. It follows that the operator δ(ψ1 )δ(ψ2 ) = (ψ1 (H) − ψ1 (H0 ))(ψ2 (H) − ψ2 (H0 )) = −ψ1 (H0 )ψ2 (H) − ψ1 (H)ψ2 (H0 ) is compact, as required.

Lemma 5.2. Let An , n = 1, . . . , N , be bounded operators in a Hilbert space. Assume that An Am is compact for all n = m. Then (5.1) σess (A1 + · · · + AN ) ∪ {0} = ∪N j=1 σess (Aj ) . See e.g. [15, Section 10.1] for a proof via the Calkin algebra argument. We would like to emphasise that Lemma 5.2 holds true with the definition of the essential spectrum as stated in Sect. 2.5; it is in general false for some other definitions of the essential spectrum, see e.g. [24, Section XIII.4, Example 1]. 5.2. Proof of Theorem 2.5 We start by considering the case of ﬁnitely many discontinuities: Lemma 5.3. Assume the hypothesis of Theorem 2.5 and suppose in addition that the set sing supp ϕ is finite. Then the conclusion of Theorem 2.5 holds true. Proof. 1. First assume that ϕ has only one discontinuity, i.e. sing supp ϕ = {λ0 }. Denote ϕ(λ) = (ϕ(λ0 + 0) − ϕ(λ))/κλ0 (ϕ).

(5.2)

0 + 0) = 0. We can write ϕ as Then ϕ(λ 0 − 0) = 1, ϕ(λ ϕ(λ) = χ(−∞,λ0 ) (λ) + ζ(λ), where χ(−∞,λ0 ) (λ) is the characteristic function of (−∞, λ0 ) and ζ ∈ C(R) is such that the limits of ζ(λ) as λ → ±∞ exist. Then by Lemma 3.4, ζ(H) − ζ(H0 ) is compact, and so ϕ(H) − ϕ(H 0 ) = E(−∞, λ0 ) − E0 (−∞, λ0 ) + compact operator. By Theorem 2.2 and Weyl’s theorem on the invariance of the essential spectrum under the compact perturbations, we obtain − ϕ(H 0 )) = [−α(λ0 ), α(λ0 )]. σess (ϕ(H)

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Recalling the definition (5.2) of ϕ, ˜ we obtain σess (ϕ(H) − ϕ(H0 )) = [−α(λ0 )κλ0 (ϕ), α(λ0 )κλ0 (ϕ)]. 2.

(5.3)

Consider the general case; let sing supp ϕ = {λ1 , . . . , λN } ⊂ Δ. One can N represent ϕ = n=1 ϕn , where ϕn ∈ P C0 (R), sing supp ϕn = {λn } and κλn (ϕn ) = κλn (ϕ) for each n. Then δ(ϕ) =

N

δ(ϕn ),

n=1

and by Lemma 5.1, the operators δ(ϕn )δ(ϕm ) are compact for n = m. Applying Lemma 5.2 and the ﬁrst step of the proof, we get N σess (δ(ϕ))∪{0} = ∪N n=1 σess (δ(ϕn )) = ∪n=1 [−α(λn )κλn (ϕ), α(λn )κλn (ϕ)].

Since σess is a closed set, we get 0 ∈ σess (δ(ϕ)) and thus the required statement (2.19) follows. Proof of Theorem 2.5. 1.

Let

Λ0 = {λ ∈ Δ | |κλ (ϕ)| ≥ 1}, Λn = {λ ∈ Δ | 2−n ≤ |κλ (ϕ)| < 2−n+1 },

n = 1, 2, . . . .

The set Λn is ﬁnite for all n ≥ 0. It is easy to see that for each n ≥ 0 there exists a function ϕn ∈ P C0 (R) with sing supp ϕn = Λn , supp ϕn ⊂ Δ, and κλ (ϕn ) = κλ (ϕ) ∀λ ∈ Λn , (5.4) 1 ϕn ∞ = max |κλ (ϕ)| ≤ 2−n . 2 λ∈Λn With this choice, the series n≥0 ϕn converges absolutely and uniformly on R and deﬁnes a function f = n≥0 ϕn such that f ∈ P C0 (R) and def

2.

the function ζ = ϕ − f is in the class C0 (R). For a given N ∈ N, write ϕ = fN + gN + ζ,

fN =

N

ϕn ,

n=0

∞

gN =

ϕn .

n=N +1

By Lemma 5.1, the operator δ(ϕm )δ(ϕn ) is compact for n = m. By the estimate (5.4), the series in the r.h.s. of δ(ϕm )δ(gN ) =

∞

δ(ϕm )δ(ϕn )

n=N +1

converges in the operator norm, and so for any m ≤ N the operator δ(ϕm )δ(gN ) is also compact. Applying Lemma 5.2 to the decomposition

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δ(ϕ) = δ(fN ) + δ(gN ) + δ(ζ) and subsequently using Lemma 5.3, we get σess (δ(ϕ)) ∪ {0} = σess (δ(fN )) ∪ σess (δ(gN )) ∪ {0}

N σess (δ(ϕn )) ∪ σess (δ(gN )) ∪ {0} = n=0

⎛ =⎝

⎞ [−α(λ)κλ (ϕ), α(λ)κλ (ϕ)]⎠ ∪ σess (δ(gN )).

|κλ (ϕ)|≤21−N

Finally, by the estimate (5.4) we have δ(gN ) ≤ 2gN ∞ ≤ 21−N and therefore σess (δ(gN )) ⊂ {z ∈ C | |z| ≤ 21−N }. Since N can be taken arbitrary large, we obtain σess (δ(ϕ)) ∪ {0} = ∪λ∈Δ [−α(λ)κλ (ϕ), α(λ)κλ (ϕ)]. Since σess is a closed set, we get 0 ∈ σess (δ(ϕ)) and thus the required statement (2.19) follows. Acknowledgements The author is grateful to Nikolai Filonov, Serge Richard and Dmitri Yafaev for the critical reading of the manuscript and making a number of useful suggestions.

References [1] Aleksandrov, A., Peller V.: Operator H¨ older–Zygmund functions. Adv. Math. (to appear) [2] Aleksandrov, A., Peller, V.: Functions of operators under perturbations of class Sp . J. Funct. Anal. (to appear) [3] Avron, J., Seiler, R., Simon, B.: The index of a pair of projections. J. Funct. Anal. 120(1), 220–237 (1994) [4] Birman, M.Sh., Solomyak, M.Z.: Double Stieltjes operator integrals, III (in Russian) . Probl. Math. Phys., Leningrad. Univ. 6, 27–53 (1973) [5] Birman, M.Sh., Solomyak, M.Z.: Double operator integrals in a Hilbert space. Integr. Equ. Oper. Theory 47, 131–168 (2003) [6] Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Clarendon Press, Oxford (1987) [7] Halmos, P.R.: Two subspaces. Trans. Am. Math. Soc. 144, 381–389 (1969) [8] Howland, J.S.: Spectral theory of operators of Hankel type, II. Indiana Univ. Math. J. 41(2), 427–434 (1992) [9] Kato, T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1966) [10] Kostrykin, V., Makarov, K.: On Krein’s example. Proc. Am. Math. Soc. 136(6), 2067–2071 (2008) [11] Krein, M.G.: On the trace formula in perturbation theory (in Russian) Mat. Sb. 33(75) , no. 3, 597–626 (1953)

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[12] Nazarov, F., Peller, V.: Lipschitz functions of perturbed operators. C. R. Acad. Sci. Paris, Ser. I 347, 857–862 (2009) [13] Peller, V.V.: Hankel operators in perturbation theory of unitary and selfadjoint operators. Funct. Anal. Appl. 19, 111–123 (1985) [14] Peller, V.V.: Hankel operators in perturbation theory of unbounded self-adjoint operators. Analysis and partial differential equations, pp. 529–544. Lecture Notes in Pure and Appl. Math., vol. 122. Dekker, New York (1990) [15] Peller, V.V.: Hankel operators and their applications. Springer, Berlin (2003) [16] Power, S.: Hankel Operators on Hilbert Space. Pitman, London (1982) [17] Pushnitski, A.: The spectral shift function and the invariance principle. J. Funct. Anal. 183(2), 269–320 (2001) [18] Pushnitski, A.: The scattering matrix and the differences of spectral projections. Bull. Lond. Math. Soc. 40, 227–238 (2008) [19] Pushnitski, A.: The Birman-Schwinger principle on the continuous spectrum. Preprint, arXiv:0911.2134 [20] Pushnitski, A., Yafaev, D.: Spectral theory of discontinuous functions of selfadjoint operators and scattering theory. Preprint, arXiv:0907.1518 [21] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis. Academic Press, New York (1972) [22] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-adjointness. Academic Press, New York (1975) [23] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering Theory. Academic Press, New York (1979) [24] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978) [25] Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1970) [26] Simon, B.: Schr¨ odinger semigroups. Bull. AMS 7(3), 447–526 (1982) [27] Sobolev, A.V., Yafaev, D.R.: Spectral properties of an abstract scattering matrix. Proc. Steklov Inst. Math. 3, 159–189 (1991) [28] Yafaev D.R.: Mathematical Scattering Theory. General theory. American Mathematical Society, Providence (1992) Alexander Pushnitski Department of Mathematics King’s College London Strand, London WC2R 2LS, UK e-mail: [email protected] Received: October 19, 2009. Revised: October 31, 2009.

Integr. Equ. Oper. Theory 68 (2010), 101–113 DOI 10.1007/s00020-010-1792-9 Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Norms and Essential Norms of the Singular Integral Operator with Cauchy Kernel on Weighted Lebesgue Spaces Takahiko Nakazi and Takanori Yamamoto Abstract. Let α and β be bounded measurable functions on the unit circle T, and let L2 (W ) be a weighted L2 space on T. The singular integral operator Sα,β is defined by Sα,β f = αP f +βQf (f ∈ L2 (W )) where P is an analytic projection and Q = I − P is a co-analytic projection. In the previous paper, the essential norm of Sα,β are calculated in the case when W is a constant function. In this paper, the essential norm of Sα,β are estimated in the case when W is an A2 -weight. Mathematics Subject Classification (2000). Primary 45E10; Secondary 47B35. Keywords. Norm, essential norm, analytic projection, A2 -weight, Helson–Szeg˝ o weight, singular integral operator.

1. Introduction Let m denote the normalized Lebesgue measure dθ/2π on the unit circle T. For 1 ≤ p ≤ ∞, Lp = Lp (T, m) denotes the usual Lebesgue space on T and H p denotes the usual Hardy space on T. Let S be the singular integral operator defined by f (η) 1 dη (a.e. ζ ∈ T) (Sf )(ζ) = πi η−ζ T

where the integral is understood in the sense of Cauchy’s principal value (cf. [2, vol. I, p. 12]). If f is in L1 then (Sf )(ζ) exists for almost all ζ on T. Let P = (I + S)/2

and Q = (I − S)/2,

The first author was supported by Grant-in-Aid Scientific Research No. 20540148. The second author was supported by Research Grant in Hokkai-Gakuen University.

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where I denotes the identity operator. Then P z n = 0 if n < 0, and P z n = z n if n ≥ 0. P is called as an analytic projection or the Riesz projection, and Q is called as a co-analytic projection. We refer to any nonzero and nonnegative W ∈ L1 as a weight. We denote by L2 (W ) the weighted L2 -space with the norm ⎛ ⎞1/2 f W = ⎝ |f |2 W dm⎠ . T 2

If W = 1, then L (W ) becomes a usual L2 space. Then we write f W as f . We recall that W is an A2 -weight if P is a bounded projection of L2 (W ) onto H 2 (W ). In this case we always have that W > 0, W −1 is an A2 -weight, L2 (W ) ⊂ L1 and H 2 (W ) = H 1 ∩ L2 (W ). If α, β ∈ L∞ , then the singular integral operator Sα,β on L2 (W ) is defined by (f ∈ L2 (W )).

Sα,β f = αP f + βQf

Hence P = S1,0 and S = S1,−1 . Let us denote by Sα,β W the norm of Sα,β on L2 (W ). If W = 1, then we write Sα,β W as Sα,β . The essential norm Sα,β W,e is the distance to K(L2 (W )), the set of all compact operators on L2 (W ), that is, Sα,β W,e = inf{Sα,β + KW : K ∈ K(L2 (W ))}. If W = 1, then we write Sα,β W,e as Sα,β e . Problem 1. Establish the norm formula of the operator Sα,β on L2 (W ), where α, β ∈ L∞ , and W is an A2 -weight. Problem 2. Establish the essential norm formula of the operator Sα,β on L2 (W ), where α, β ∈ L∞ , and W is an A2 -weight. It is well known that SW = P W +

P 2W − 1.

If α, β are complex constants and W is an A2 -weight, then Problem 1 is solved by Feldman–Krupnik–Markus (cf. [2, vol. II, p. 213, Theorem 5.1, and p. 215, Lemma 5.3]: Theorem A (Norm Theorem). Let α, β be complex constants, and let W be an A2 -weight. Then

2

2 |α| + |β| |α| − |β| Sα,β W = γ + + γ+ , 2 2 where γ = |(α − β)/2|2 (P 2W − 1). Suppose W is an A2 -weight. Then the norms P W and SW are known as the following. Let h be an outer function in H 2 satisfying W = |h|2 , and let ¯ φ = h/h. Let d = dist(φ, H ∞ ) = inf∞ φ + g∞ , g∈H

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and let θ be the angle between the ranges of P and Q. Then it is known that θ > 0, d < 1, d = cos θ and 1 1+d 1 θ =√ . , and SW = cot = P W = 2 sin θ 2 1−d 1−d Hence in Theorem A, 2 α − β 2 d2 (P 2W − 1) = α − β γ = . 2 2 1 − d2 If α, β are complex constants and W is an A2 -weight, then Problem 2 is solved by Gohberg–Krupnik (cf. [2, vol. II, p. 213, Theorem 5.1]: Theorem B (Essential Norm Theorem). Let α, β be complex constants, and let W be an A2 -weight. Then Sα,β W,e = =

Sα,β + KW

2

2 |α| + |β| |α| − |β| γ+ + γ+ , 2 2 inf

K∈K(L2 (W ))

where γ = |(α − β)/2|2 (P 2W,e − 1). In [2, p. 216], by the Gelfand–Naimark theorem, Theorem B follows immediately from Theorem A. In [2], the operator aI + bS is considered. Since aI + bS = (a + b)P + (a − b)Q, the formula of Sα,β follows from the formula of aI + bS, where α = a + b and β = a − b. By the above theorems, if α, β are complex constants and W = 1, then Sα,β = Sα,β e = max{|α|, |β|}, because P = P e = 1. We recall that |α| + |β| |α| − |β| + max{|α|, |β|} = . 2 2 If α, β ∈ L∞ and W = 1, then Problem 1 is solved by Nakazi–Yamamoto [5]: Theorem C (Norm Theorem). Let α, β ∈ L∞ . Then

2 2

2 2 2

|α| + |β| |α| − |β|

Sα,β 2 = inf∞ + |αβ¯ − k|2 +

k∈H 2 2

.

∞

∞

If α, β ∈ L

and W = 1, then Problem 2 is solved by Nakazi [4]:

Theorem D (Essential Norm Theorem). Let α, β ∈ L∞ . Then Sα,β 2e =

Sα,β + K2

2 2

2 2

|α| + |β|2 |α| − |β|

+ |αβ¯ − k|2 + = inf

k∈H ∞ +C 2 2

inf

K∈K(L2 )

.

∞

2

Notice that Sα,β W,e = 0 if and only if Sα,β is compact on L (W ). By Theorem D and the main Theorem 3.1 in Sect. 3, Sα,β is compact on L2 (W ) if and only if Sα,β = 0.

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It is difficult to solve Problems 1 and 2 in the case when α, β are complex functions and W is also a function. We are interested in the relation between Theorems A and C, and the relation between Theorems B and D. The purpose of this paper is to estimate the norm Sα,β W using the norm Sφα,β , and to estimate the essential norm Sα,β W,e using the essential norm Sφα,β e . In Sect. 2, we estimate the norm Sα,β W using the norm Sφα,β . In Theorem 3, we estimate the essential norm Sα,β W,e using the essential norm Sφα,β e . The main theorems in this paper are Theorems 2.2 in Sect. 2, and Theorem 3.1 in Sect. 3.

2. Norms of Sα,β on L2 (W ) In [6], we proved the following Theorem 2.1 which gives the norm formula of Sα,β on L2 (W ) in the special case when αβ¯ ∈ H ∞ . We give the short proof of Theorem 2.1. If αβ¯ ∈ H ∞ and W is not a constant, then it is difficult to give the norm formula of Sα,β W . In the following Theorem 2.2, we estimate the norm Sα,β W of an operator Sα,β on L2 (W ) using the norm Sφα,β of an operator Sφα,β on L2 . By Theorem C,

2 2

2 2 2

|α| |α| + |β| − |β|

. + |φαβ¯ − k|2 + Sφα,β 2 = inf∞

k∈H 2 2

∞

We use Theorems 2.1 and 2.2 to prove the main Theorem 3.1 in Sect. 3. Theorem 2.1. Let α, β ∈ L∞ . Let φ and W be functions such that there exists ¯ and W = |h|2 . If αβ¯ belongs to an outer function h ∈ H 2 satisfying φ = h/h H ∞ and |α − β| > 0, then

2

2

|α| + |β| |α| − |β|

γk +

, inf + γk + Sα,β W =

∞ 2 2 k∈H , |φ−k|<1

∞

where

α − β 2 |φ − k|2 γk = . 2 1 − |φ − k|2

Proof. If α = β, then Sα,β W = Sα,α W = αIW = α∞ , and hence the formula holds. Suppose α = β. Let dm = dθ/2π. Suppose c is a constant satisfying c ≥ αP + βQW . Then |αf + β¯ g |2 W dm ≤ c2 |f + g¯|2 W dm, (f ∈ P, g ∈ zP), where P is the set of all analytic polynomials, and zP = {zf (z) : f ∈ P}. Let A = α/c and B = β/c. Then A = B, because α = β. Since αβ¯ ∈ H ∞ , it ¯ ∈ H ∞ . Then follows that AB |Af + B¯ g |2 W dm ≤ |f + g¯|2 W dm, (f ∈ P, g ∈ zP).

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Since W = |h|2 , it follows that W > 0. By the Cotlar–Sadosky lifting theorem [1], this implies that max{|A|, |B|} ≤ 1, and there exists a k1 ∈ H 1 such that ¯ |(1 − AB)W − k1 |2 ≤ (1 − |A|2 )(1 − |B|2 )W 2 . Suppose k1 = 0. Then |A−B|W = 0, and hence A = B, because W > 0. This is a contradiction, because A = B. Therefore k1 = 0. Since (1 − |A|2 )(1 − ¯ 2 , it follows that |(1 − AB)W ¯ ¯ , and hence − k1 | ≤ |1 − AB|W |B|2 ) ≤ |1 − AB| ¯ is an ¯ 2|1 − AB|W ≥ |k1 | > 0. Since max{|A|, |B|} ≤ 1, it follows that 1 − AB ∞ ∞ ¯ outer function because AB ∈ H . Hence there exists a k ∈ H such that A − B 2 (1 − |A|2 )(1 − |B|2 ) 2 |φ − k| ≤ =1− ¯2 ¯ . |1 − AB| 1 − AB Hence max{|α|, |β|} ≤ c and ¯ 2 c2 − αβ¯ 2 1 − AB 1 . ≤ = 1 − |φ − k|2 A−B c(α − β) Therefore max{|α|2 , |β|2 } ≤ c2 and

1 2 2 − 1 + |α| + |β| c2 + |αβ|2 ≥ 0 c4 − |α − β|2 1 − |φ − k|2 Hence c2 ≥

Therefore

|φ − k|2 2 2 + |α| + |β| |α − β|2 1 − |φ − k|2

2 1 |φ − k|2 2 + |β|2 + + |α| − 4|αβ|2 . |α − β|2 2 1 − |φ − k|2 1 2

2 α − β 2 |φ − k|2 |α| + |β| c≥ + 2 1 − |φ − k|2 2

2 α − β 2 |φ − k|2 |α| − |β| + + . 2 1 − |φ − k|2 2

The converse is also true. This completes the proof. By Theorem 2.1, Sα,β W

2

α − β 2

(P 2 − 1) + |α| + |β| ≤ W

2

2 ∞ ∞

2

α − β

|α| − |β| 2 2

,

+ (P W − 1) +

2 2 ∞

∞

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α − β 2

1

γk ≤ inf − 1

2 ∞ 2 |φ−k|<1 ∞ k∈H , |φ−k|<1 1 − |φ − k| ∞

2

2

α − β

α − β 1 2

= −1 =

2

2 (P W − 1). 1 − d2 ∞ ∞

inf ∞ ,

By Theorem A, if α, β are constants, then the equality holds. Let α be a complex constant. Then it is known that Sα,0 W = αP W = √

|α| , 1 − d2

and Sφα,0 = φαP = αP = |α|,

because P = 1. Hence 1 1 √ Sφα,0 ≤ Sα,0 W ≤ √ Sφα,0 . 1+d 1−d More generally, we have the following theorem, which is one of the main theorems. By Theorem C, if α, β ∈ L∞ , then

2 2

2 2 2

|α| + |β| |α| − |β|

. + |φαβ¯ − k|2 + Sφα,β 2 = inf∞

k∈H 2 2

∞

∞

Theorem 2.2. Let α, β ∈ L , and let W be an A2 -weight. Let h be an outer ¯ Let d = dist(φ, H ∞ ). function in H 2 such that W = |h|2 , and let φ = h/h. Then d < 1, and 1 1 √ Sφα,β ≤ Sα,β W ≤ √ Sφα,β . 1+d 1−d Proof. Let P be the set of all analytic polynomials, and zP = {zf (z) : f ∈ P}. Since h is an outer function in H 2 , it follows that hP = {hf : f ∈ P} is norm dense in H 2 , and zhP = {zhf : f ∈ P} is norm dense in zH 2 = H02 . Let F = hf and G = hg. Then |αf + β¯ g |2 W dm |αhf + βh¯ g |2 dm 2 = sup Sα,β W = sup |f + g¯|2 W dm |hf + h¯ g |2 dm f ∈P,g∈zP f,g |αhf + βφhg|2 dm |αF + βφG|2 dm = sup = sup 2 |hf + φhg| dm |F + φG|2 dm f,g F ∈hP, G∈zhP |φαF + βG|2 dm = sup |φF + G|2 dm F ∈H 2 , G∈zH 2

−1 |φαF + βG|2 dm |φF + G|2 dm = sup . |F + G|2 dm |F + G|2 dm F,G Hence 2

Sφα,β

−1 |φF + G|2 dm ≤ Sα,β 2W sup |F + G|2 dm F,G 2

≤ Sφα,β

−1 |φF + G|2 dm . inf F,G |F + G|2 dm

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By Theorem C,

2 2

2

|φ| + 1

|φF +G|2 dm |φ| − 1

sup + |φ − k|2 + = Sφ,1 = inf∞

2 k∈H 2 2 |F +G| dm F,G

∞

= 1 + inf∞ φ − k∞ = 1 + d. k∈H

By [5, Theorem 3.4], ⎛ ⎛ ⎞⎞ 2

2 2 |φF +G|2 dm |φ| + 1 −1 |φ| ⎠⎠ inf − |φ−k|2 + = sup ⎝ess inf ⎝ F,G T 2 2 |F +G|2 dm k∈H ∞ = sup (ess inf (1 − |φ − k|)) = sup (1 − φ − k∞ ) k∈H ∞

T

k∈H ∞

= 1 − inf∞ φ − k∞ = 1 − d. k∈H

Hence Sφα,β 2 Sφα,β 2 ≤ Sα,β 2W ≤ . 1+d 1−d This completes the proof.

3. Essential Norms of Sα,β on L2 (W ) In Theorem D, the essential norm Sα,β e of an operator Sα,β on L2 was established for α, β ∈ L∞ . If W is not a constant, then it is difficult to give the essential norm formula of Sα,β W,e for α, β ∈ L∞ . In the following main Theorem 3.1, we estimate the essential norm Sα,β W,e of an operator Sα,β on L2 (W ) using the essential norm Sφα,β e of an operator Sφα,β on L2 . By Theorem D,

2 2

2 2 2

|α| + |β| + |φαβ¯ − k|2 + |α| − |β|

. Sφα,β 2e = inf

k∈H ∞ +C 2 2

∞

Theorem 3.1. Let α, β ∈ L∞ , and let W be an A2 -weight. Let h be an outer ¯ Let d = dist(φ, H ∞ ). function in H 2 such that W = |h|2 , and let φ = h/h. Then d < 1, and √ 1−d γ Sφα,β e ≤ Sα,β W,e ≤ √ Sφα,β e , 1+d 1−d where 2 γ = lim sup ¯ z n P + z n QW,e ≤ √ . n→∞ 1 − d2 We use the following Lemma 3.2 to prove Theorem 3.1. Lemma 3.2. Let W be an A2 -weight. Let K be a compact operator on L2 (W ). Then lim K(z n P + z¯n Q)W = 0.

n→∞

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Proof. Let f and g be trigonometric polynomials. Since W is an A2 -weight, z n W f )P g ∈ Lp . Since it follows that W ∈ Lp for some p > 1, and hence (¯ n p n p z W f ) ∈ L . Let dm = dθ/2π. Since W −1 is z¯ W f ∈ L , it follows that P (¯ an A2 -weight, it follows that 2 1 1 n W dm = |P (¯ P (¯ z W f ) z n W f )|2 dm W W 1 ≤ P 21/W |¯ z n W f |2 dm W 2 2 = P 1/W |f | W dm < ∞. Since

Q(¯ z n W f )P gdm =

P (¯ z n W f )Qgdm = 0,

it follows that

(z n P )∗ f, g W = f, z n P g W = f (z n P g)W dm = (¯ z n W f )P gdm = P (¯ z n W f )P gdm 1 P (¯ z n W f )gW dm = P (¯ z n W f )gdm = W 1 n = P (¯ z W f ), g . W W

Hence (z n P )∗ f =

1 P (¯ z n W f ). W

Similarly n

∗

n

(¯ z Q) f, g W = f, z¯ Qg W = f (¯ z n Qg)W dm = (z n W f )Qgdm = Q(z n W f )Qgdm 1 n Q(z n W f )gW dm = Q(z W f )gdm = W 1 = Q(z n W f ), g . W W Hence (¯ z n Q)∗ f = If (W f )(z) =

∞

j=−∞ cj z

P (¯ znW f ) =

∞ j=n

j

1 Q(z n W f ). W

(|z| = 1), then

cj z j−n

and

Q(z n W f ) =

∞ j=n+1

c−j z n−j .

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Hence z n P (¯ znW f ) =

∞ j=n

cj z j

and z¯n Q(z n W f ) =

∞

c−j z −j .

j=n+1

By the Hunt–Muckenhoupt–Wheeden theorem [3], 2 2 ∞ ∞ j 1 −j 1 dm = lim c z c z lim j −j W dm = 0, n→∞ n→∞ j=n j=n+1 W because W −1 is an A2 -weight. Since |z| = 1, it follows that 2 1 n W dm P (¯ z W f ) |(z n P )∗ f |2 W dm = lim lim n→∞ n→∞ W 2 1 = lim dm z n W f )| |z n P (¯ n→∞ W 2 ∞ j 1 = lim dm = 0, cj z n→∞ j=n W and

lim

n→∞

2 1 n |(¯ z Q) f | W dm = lim W Q(z W f ) W dm n→∞ 2 1 = lim dm |¯ z n Q(z n W f )| n→∞ W 2 ∞ −j 1 = lim c z −j W dm = 0. n→∞ j=n+1 n

∗

2

z n Q)∗ → 0 as n → ∞ in the strong operaTherefore (z n P )∗ → 0 and (¯ tor topology. Since K is a compact operator on L2 (W ), K can be approximated by finite rank operators on L2 (W ). Hence we can assume that K is a rank one operator on L2 (W ). Then there exist h0 , h1 in L2 (W ) such that Kf = f, h0 W h1 , (f ∈ L2 (W )). Since K(z n P + z¯n Q)f W = |f, (z n P + z¯n Q)∗ h0 W | · h1 W ≤ f W (z n P + z¯n Q)∗ h0 W h1 W , it follows that K(z n P + z¯n Q)W ≤ (z n P + z¯n Q)∗ h0 W h1 W . This implies that K(z n P + z¯n Q)W → 0 as n → ∞. This completes the proof. Proof of Theorem 3.1 Proof. At first, we prove the left inequality. By Theorem 2.2, Szn φα,¯zn β Szn φα,¯zn β √ ≤ Szn α,¯zn β W ≤ √ . 1+d 1−d

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By Theorem C,

2 2

2 2 2

|α| |α| + |β| − |β|

+ |φαβ¯ − k|2 + Szn φα,¯zn β 2 = inf

2 2 k∈¯ z 2n H ∞

.

∞

∞

Let C = C(T) be a set of all continuous functions on T, and let H + C = {f + g : f ∈ H ∞ , g ∈ C}. Since H ∞ + C is closed, it follows from Theorem D that

2 2

2 2 2

|α| |α| +|β| − |β|

+ |φαβ¯ −k|2 + lim Szn φα,¯zn β 2 = inf

n→∞ k∈H ∞+C 2 2

∞

= Sφα,β 2e . Hence Sφα,β e Sφα,β e √ ≤ lim inf Szn α,¯zn β W ≤ lim sup Szn α,¯zn β W ≤ √ . n→∞ 1+d 1−d n→∞

Put U = Sz,¯z = zP + z¯Q, then U n = Szn ,¯zn = z n P + z¯n Q for any positive integer n. By Theorem 2.1,

n n

n| 2 n| 2

|z |z |+|¯ z | − |¯ z

γk +

U n W = inf + γk +

2 2 k∈H ∞ ,|φ−k|<1

∞

√

= inf

γk + 1 + γk k∈H ∞ ,|φ−k|<1 ∞

1 |φ − k|

≤ inf +

2 2 k∈H ∞ ,|φ−k|<1 1 − |φ − k| 1 − |φ − k| ∞

1 + |φ − k|

= inf

1 − |φ − k| k∈H ∞ ,|φ−k|<1 ∞ 1 + φ − k∞ 1+d , ≤ inf = 1 − φ − k∞ 1−d k∈H ∞ ,|φ−k|<1 where

n z − z¯n 2 |φ − k|2 |φ − k|2 γk = ≤ . 1 − |φ − k|2 2 1 − |φ − k|2

Let K be an arbitrary compact operator on L2 (W ). By Lemma 3.2, lim KU n W = lim KSzn ,¯zn W = lim K(z n P + z¯n Q)W = 0.

n→∞

Since

n→∞

n→∞

1+d Sα,β + KW ≥ Sα,β + KW U n W 1−d ≥ (Sα,β + K)U n W ≥ Szn α,¯zn β W − KU n W ,

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Norms of the Singular Integral Operator

it follows that

lim sup Szn α,¯zn β W ≤ n→∞

111

1+d Sα,β + KW . 1−d

Hence Sφα,β e √ ≤ lim sup Szn α,¯zn β W ≤ 1+d n→∞

1+d Sα,β W,e . 1−d

Therefore

√ 1−d Sφα,β e ≤ Sα,β W,e . 1+d This is the left inequality. Next, we prove the right inequality. If f (z) = ∞ j j=−∞ aj z (|z| = 1), then ⎞ ⎛ ∞ n P z n Qf = P ⎝ a−j z n−j ⎠ = a−j z n−j , j=1

and

⎛ Q¯ znP f = Q ⎝

j=1

∞

⎞ aj z j−n ⎠ =

j=0

n−1

aj z j−n .

j=0

Since W is an A2 -weight, it follows that P z n Q and Q¯ z n P are finite rank operators on L2 (W ). Hence z n P + z n Q)W,e = αz n P z¯n P + β z¯n Qz n QW,e . (αz n P + β z¯n Q)(¯ Since

⎛ z n P z¯n P f = z n P ⎝

∞

⎞ aj z j−n ⎠ = z n

j=0

and

⎛

z¯n Qz n Qf = z¯n Q ⎝

∞

∞

aj z j−n =

j=n

⎞ a−j z n−j ⎠ = z¯n

j=1

∞

∞

aj z j ,

j=n

a−j z n−j =

j=n+1

∞

a−j z −j ,

j=n+1

it follows that (P − z n P z¯n P )f =

n−1

aj z j

and

(Q − z¯n Qz n Q)f =

j=0

n

a−j z −j .

j=1 n

n

Since W is an A2 -weight, it follows that P − z P z¯ P and Q − z¯n Qz n Q are finite rank operators on L2 (W ). Hence Sα,β W,e = αP + βQW,e = αz n P z¯n P + β z¯n Qz n QW,e = (αz n P + β z¯n Q)(¯ z n P + z n Q)W,e . Hence z n P + z n QW,e Sα,β W,e ≤ αz n P + β z¯n QW ¯ ≤ Szn α,¯zn β W ¯ z n P + z n QW,e .

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By the proof of the left inequality, z n P + z n QW,e Sα,β W,e ≤ lim sup Szn α,¯zn β W · lim sup ¯ n→∞

n→∞

Sφα,β e ≤ √ z n P + z n QW,e . · lim sup ¯ 1−d n→∞ Let γ = lim sup ¯ z n P + z n QW,e . n→∞

Since 1 P W = QW = √ , 1 − d2 it follows that γ ≤ lim sup ¯ z n P + z n QW ≤ √ n→∞

2 . 1 − d2

This completes the proof. In Theorem 3.1, if W = 1, then φ = 1, d = 0, and z n P + z n Qe ≤ 1, γ = lim sup ¯ n→∞

because for n ≥ 0 ¯ z n P + z n Qe ≤ P z¯n P + Qz n Q = max{P z¯n P , Qz n Q} = 1. In [6], we proved that α∞ ≤ αP W

α

= inf

2 k∈H ∞ ,|φ−k|<1 1 − |φ − k|

≤ α∞ P W ,

∞

and that the second inequality cannot replace with equality when W (ζ) = |ζ + 1|1/2 , h(ζ) = (ζ + 1)1/4 , and α = χE , E = (−ε, ε) ⊂ T for sufficiently small ε > 0. If W = 1, then for α ∈ L∞ , the equality: αP = αP e = α∞ holds. Theorems C and D with β = 0 implies this equality. The next Corollary 3.3 follows immediately from Theorem 3.1 and the equality αP e = z n P + z n Qe = 1. Hence α∞ . If W = 1, then d = 0 and γ = lim supn→∞ ¯ Corollary 3.3 implies the equality αP e = α∞ . Corollary 3.3. Let α ∈ L∞ . Let φ and W an outer function h ∈ H 2 satisfying φ = an A2 -weight. Then √ 1−d α∞ ≤ αP W,e 1+d

be functions such that there exists ¯ h/h and W = |h|2 . Suppose W is γ ≤√ α∞ , 1−d

where 2 γ = lim sup ¯ z n P + z n QW,e ≤ √ . n→∞ 1 − d2

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113

Acknowledgment Part of this paper was written during visits of the second author to Kyungpook National University. He wishes to thank the faculty and administration of that institution for their warm hospitality. The authors would like to thank the referee for careful reading of the paper and finding some misprints.

References [1] Cotlar, M., Sadosky, C.: On the Helson–Szeg˝ o theorem and a related class of Toeplitz kernels. In: Harmonic Analysis in Euclidean Spaces, Proc. Sympos. Pure Math, vol. 35, Amer. Math. Soc., Providence, pp. 383–407 (1979) [2] Gohberg, I., Krupnik, N.: One-Dimensional Linear Singular Integral Equations, vols. I, II. Birkh¨ auser, Basel (1992) [3] Hunt, R., Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973) [4] Nakazi, T.: Essential norms of some singular integral operators. Arch. Math. 73, 439–441 (1999) [5] Nakazi, T., Yamamoto, T.: Norms of some singular integral operators and their inverse operators. J. Oper. Theory 40, 185–207 (1998) [6] Nakazi, T., Yamamoto, T.: Norms of some singular integral operators on weighted L2 spaces. J. Oper. Theory 50, 311–330 (2003) Takahiko Nakazi Department of Mathematics Hokusei Gakuen University Sapporo 004-8631, Japan e-mail: [email protected] Takanori Yamamoto Department of Mathematics Hokkai-Gakuen University Sapporo 062-8605, Japan e-mail: [email protected] Received: November 29, 2009. Revised: January 15, 2010.

Integr. Equ. Oper. Theory 68 (2010), 115–150 DOI 10.1007/s00020-010-1803-x Published online June 19, 2010 c The Author(s) This article is published with open access at Springerlink.com 2010

Integral Equations and Operator Theory

Krein Systems and Canonical Systems on a Finite Interval: Accelerants with a Jump Discontinuity at the Origin and Continuous Potentials D. Alpay, I. Gohberg (Z L), M. A. Kaashoek, L. Lerer and A. L. Sakhnovich Abstract. This paper is devoted to connections between accelerants and potentials of Krein systems and of canonical systems of Dirac type, both on a finite interval. It is shown that a continuous potential is always generated by an accelerant, provided the latter is continuous with a possible jump discontinuity at the origin. Moreover, the generating accelerant is uniquely determined by the potential. The results are illustrated on pseudo-exponential potentials. The paper is a continuation of the earlier paper of the authors (Alpay et al. in Modern Analysis and Applications. The Mark Krein Centenary Conference, vol. 2, pp. 19–36, OT 191. Birkh¨ auser, Basel, 2009) dealing with the direct problem for Krein systems. Mathematics Subject Classification (2010). Primary 34A55, 45D05; Secondary 47A45, 47B35, 93B15. Keywords. Krein systems, canonical systems, accelerant, continuous potential, pseudo-exponential potential, fundamental solution, inverse problems, semi-separable integral operators, triangular operators.

When Israel Gohberg, the second author of this paper, passed away on October 12, 2009, the work on this paper was finished, except for the last section of which only the first draft existed. The expression Z L after his name is used in Hebrew and means “of blessed memory”.

D. Alpay wishes to thank the Earl Katz family for endowing the chair which supported his research. The work of Leonid Lerer was supported by ISF—Israel Science Foundation, Grant No 121/09, and that of Alexander Sakhnovich by the Austrian Science Fund (FWF) under Grant No. Y330.

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1. Introduction Let T > 0, and let k be a scalar continuous function on the interval [−T, T] which is hermitian, that is, k(−t) = k(t) on −T ≤ t ≤ T. Assume that for each 0 < τ ≤ T the corresponding convolution integral operator Tτ on L2 (0, τ ), τ (Tτ f )(t) = f (t) −

k(t − s)f (s) ds,

0 ≤ t ≤ τ,

(1.1)

0

is invertible, and let γτ (t, s) be the corresponding resolvent kernel, i.e., τ γτ (t, s) −

k(t − ξ)γτ (ξ, s) dξ = k(t − s),

0 ≤ t, s ≤ τ.

(1.2)

0

Consider the entire functions

⎛

P(τ, λ) = eiλτ ⎝1 + τ P∗ (τ, λ) = 1 +

τ

⎞ e−iλx γτ (x, 0)dx⎠ ,

(1.3)

0

eiλx γτ (τ − x, τ )dx,

(1.4)

0

and put Y (τ, λ) = P(τ, λ) P∗ (τ, λ) . Then, as was proved by M.G. Krein in [8], the function Y (τ, λ) satisﬁes the differential system

∂ 0 a(τ ) 1 0 Y (τ, λ) = Y (τ, λ) iλ + , (1.5) 0 0 a(τ ) 0 ∂τ with a(τ ) = γτ (τ, 0) for τ ∈ (0, T]. The functions P(τ, λ) and P∗ (τ, λ) are usually referred to as Krein orthogonal functions. We call (1.5) a Krein system when, as in the previous paragraph, the function a is given by a(τ ) = γτ (τ, 0), where γτ (t, s) is the resolvent kernel corresponding to some k on [−T, T] with the properties described in the previous paragraph. In that case, following Krein, the function k is called an accelerant for (1.5), and we shall refer to a as the potential associated with the accelerant k. The result referred to above holds in greater generality, namely for systems with matrix-valued accelerants that are allowed to have a jump discontinuity at the origin. In fact, in [1] the following result is proved. Theorem 1.1. Let k be an r × r-matrix function, which is hermitian, i.e., k(−t) = k(t)∗ , and continuous on −T ≤ t ≤ T with possibly a jump discontinuity at the origin. Assume that for each 0 < τ ≤ T the corresponding integral operator Tτ on L2r (0, τ ) given by (1.1) is invertible, and let γτ (t, s) be the corresponding resolvent kernel as in (1.2). Put

Vol. 68 (2010)

Krein and Canonical Systems ⎛ P(τ, λ) = eiλτ ⎝Ir +

⎞ e−iλx γτ (x, 0)dx⎠

(1.6)

0

τ P∗ (τ, λ) = Ir +

τ

117

eiλx γτ (τ − x, τ )dx.

(1.7)

0

Then a(τ ) = γτ (0, τ ), with 0 < τ ≤ T, extends to a continuous function on [0, T] and Y (τ, λ) = P(τ, λ) P∗ (τ, λ) satisfies

Ir 0 0 a(τ ) ∂ Y (τ, λ) = Y (τ, λ) iλ . (1.8) + ∂τ 0 0 0 a(τ )∗ The phrase k is continuous on −T ≤ t ≤ T with possibly a jump discontinuity at the origin, which appears in the above theorem, means that the function k is continuous on the intervals −T ≤ t < 0 and 0 < t ≤ T and that the two limits limt↓0 k(t) and limt↑0 k(t) exist. The actual value of k at the origin does not play a role. From [6] we know that under the conditions in Theorem 1.1 the resolvent kernel function γτ (t, s) is continuous on the triangles on 0 ≤ s < t ≤ τ and 0 ≤ t < s ≤ τ , and that γτ (t, s) has a continuous extension on the closures of each of these triangles. Jumps may appear on the diagonal 0 ≤ s = x ≤ τ . In particular, the evaluation of γτ at the point (τ, 0), appearing in Theorem 1.1, is well-deﬁned. As for the scalar case we call (1.8) a Krein system when the potential a is obtained in the way described in Theorem 1.1, and in that case we say that k is an accelerant for (1.8). In this paper we deal, among other things, with the following inverse problem. Consider the system (1.8) and assume that the potential a is an r × r-matrix valued function continuous on [0, T]. Does it follow that (1.8) is a Krein system? In other words, does there exist an r × r-matrix valued accelerant k on [−T, T], with possibly a jump discontinuity at the origin, such that the potential corresponding to k is the given potential a? If we restrict to continuous accelerants, the answer is negative. For instance (see [1]), the potential 2i , τ ∈ [0, 1], a(τ ) = 1 + e−2iτ does not have a continuous accelerant. However, we shall prove that for the larger class of accelerants introduced here, the answer is aﬃrmative. Krein systems are closely related to canonical differential systems of Dirac type. In fact, if Y is a C2r×2r -valued solution of the system (1.8) with potential a, then the function ¯ ∗ U (τ, λ) = e−iτ λ Y (τ, −2λ) (1.9) is a solution of the canonical system

0 d − ij U (τ, λ) = λU (τ, λ) + dτ v(τ )∗

v(τ ) 0

U (τ, λ),

(1.10)

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where

j=

Ir 0

0 −Ir

IEOT

and v(τ ) = −ia(τ )

(0 ≤ τ ≤ T).

(1.11)

It will be convenient to state our main results in terms of a canonical system rather than a Krein system. In this paper we show that a continuous matrix-valued potential v is always generated by an accelerant (provided a jump discontinuity at the origin is allowed) and that an accelerant is uniquely determined by the potential, that is, if two accelerants generate the same potential, then they are equal. In fact, we shall prove the following theorem. Theorem 1.2. Consider the canonical system (1.10), and assume that its potential v is continuous on the interval [0, T]. Then, there is a unique r × r matrix function k, which is hermitian, i.e., k(−t) = k(t)∗ , and continuous on −T ≤ t ≤ T with possibly a jump discontinuity at the origin, such that the following holds: for each 0 < τ ≤ T the convolution operator τ (1.12) (Tτ f )(t) = f (t) − k(t − s)f (s) ds, 0 ≤ t ≤ τ, 0

is invertible on

L2r (0,

τ ), and the potential v of (1.10) is given by v(τ ) = −iγτ (τ, 0),

0 < τ ≤ T.

(1.13)

Here γτ (t, s) is the resolvent kernel corresponding to Tτ as in (1.2). In analogy with the theory of Krein systems, an r × r matrix function k with the properties described in the above theorem will be called an accelerant for the canonical system (1.10). In this case we also say that the potential v is generated by the accelerant k. Using this terminology, Theorem 1.2 just tells us that a canonical system with a continuous potential has a unique accelerant. Given a continuous matrix-valued potential v we shall also present a formula for the fundamental solution of the canonical system (1.10) in terms of the accelerant generating the potential v. The result (Theorem 2.1 in Sect. 2) can be viewed as an addition to Theorem 1.1 The statement in Theorem 1.2 about the uniqueness of the accelerant is known and has been proved in [1] using recent results about the continuous analogue of the resultant for certain entire matrix functions (see Theorem 1.3 in [1] for further details). In this paper we give a new proof of the uniqueness using a formula for the fundamental solution of the canonical system (1.10) in terms of a given accelerant, which is presented in Theorem 2.1. For the case when the potential v is bounded, bounded accelerants k have been constructed in [10] following the scheme outlined in Section 8.2 of [15] (see also [9]). In the present paper, to prove Theorem 1.2, the approach of [10] is speciﬁed and developed further for the case of continuous potentials. Also the material related to Theorem 2.1 below is inspired by and builds on results from Sections 3 and 4 in [10].

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The paper consists of six sections including this introduction. In Sect. 2 we derive a formula for the fundamental solution of the canonical system (1.10) in terms of its accelerant. The result is used in in Sect. 3 to give a new proof of the uniqueness of the accelerant given the potential as stated in Theorem 1.2. The next two sections complete the proof of Theorem 1.2. Section 4 has an auxiliary character and is interesting in its own right. We show that a lower triangular semi-separable integral operator from a certain class is similar to the operator of integration and that the corresponding similarity operator can be chosen in such a way that both this similarity operator and its inverse map functions with a continuous derivative into functions with a continuous derivative. This result is then used in Sect. 5 to prove Theorem 1.2. In the ﬁnal section the main result of Sect. 2 is speciﬁed further for pseudo-exponential potentials.

2. The Fundamental Solution Throughout this section k is a r × r matrix function on [−T, T], which is hermitian, i.e. k(−t) = k(t)∗ , and k is continuous on −T ≤ t ≤ T with possibly a jump discontinuity at the origin. We assume that k is an accelerant for the canonical system (1.10). The latter means that for each 0 < τ ≤ T the convolution operator (1.1) is invertible on L2r (0, τ ), and the potential v is the r × r continuous matrix function on [0, T] determined by k via the formula v(τ ) = −iγτ (τ, 0),

0 < τ ≤ T,

(2.1)

where γτ (t, s)is the corresponding resolvent kernel as in (1.2). We shall derive (explicitly in terms of the accelerant k) the fundamental solution u(x, λ) of (1.10) satisfying the initial condition

1 Ir −Ir ∗ . (2.2) u(0, λ) = Q , where Q = √ Ir 2 Ir For this purpose we need the following r × r matrix functions: ⎛ ⎞ x 1 (x, λ) = e2iλx ⎝Ir − 2 e−2iλt k(t) dt⎠ , 2 (x, λ) = e2iλx Ir .

(2.3)

0

Both 1 (·, λ) and 2 (·, λ) are deﬁned on [0, T]. Note that for 0 ≤ x ≤ T we have d 1 (x, λ) = 2iλ1 (x, λ) − 2k(x), dx

d 2 (x, λ) = 2iλ2 (x, λ). dx

(2.4)

The next theorem is the main result of this section. Theorem 2.1. Assume that the r × r matrix function k is an accelerant for the canonical system (1.10), and let γτ (t, s) be the corresponding resolvent kernel as in (1.2). For 0 ≤ τ ≤ T, λ ∈ C, and j = 1, 2 put

120

D. Alpay et al. ⎫ ⎧ τ ⎬ ⎨ 1 θj (τ, λ) = √ e−iτ λ j (τ, λ) + γτ (τ, s)j (s, λ) ds ⎭ ⎩ 2 0 ⎫ ⎧ τ ⎬ ⎨ 1 ωj (τ, λ) = √ e−iτ λ (−1)j Ir + γτ (0, s)j (s, λ) ds , ⎭ ⎩ 2

IEOT

(2.5)

(2.6)

0

where 1 (·, λ) and 2 (·, λ) are given by (2.3). Then the 2r ×2r matrix function u(τ, λ) defined by θ1 (τ, λ) θ2 (τ, λ) u(τ, λ) = , 0 ≤ τ ≤ T, (2.7) ω1 (τ, λ) ω2 (τ, λ) is the fundamental solution of (1.10) with initial condition (2.2). Using the definition of 2 (·, λ) in the second part of (2.3) we see that ⎛ ⎞ τ 1 iτ λ ⎝ θ2 (τ, λ) = √ e Ir + e−2iλs γτ (τ, τ − s) ds⎠, (2.8) 2 0 ⎛ ⎞ τ 1 −iτ λ ⎝ Ir + e2iλs γτ (0, s) ds⎠. (2.9) ω2 (τ, λ) = √ e 2 0

The formula for ω2 (τ, λ) is immediate from the definition and the one for θ2 (τ, λ) follows using the following calculation: ⎛ ⎞ τ 1 θ2 (τ, λ) = √ e−iτ λ ⎝e2iλτ Ir + e2iλs γτ (τ, s) ds⎠ 2 0 ⎛ ⎞ τ 1 = √ e−iτ λ ⎝e2iλτ Ir + e2iλ(τ −s) γτ (τ, τ − s) ds⎠ 2 0 ⎛ ⎞ τ 1 = √ eiτ λ ⎝Ir + e−2iλs γτ (τ, τ − s) ds⎠. 2 0

The expressions (2.8) and (2.9) show that the functions θ2 (τ, λ) and ω2 (τ, λ) are closely related to the Krein orthogonal entire matrix functions P(τ, λ) and P∗ (τ, λ) appearing in Theorem 1.1. In fact we have 1 ¯ ∗, θ2 (τ, λ) = √ e−iτ λ P∗ (τ, 2λ) 2

1 ¯ ∗. ω2 (τ, λ) = √ eiτ λ P(τ, 2λ) 2

Thus Theorem 2.1 can be seen as an addition to Theorem 1.1. To prove Theorem 2.1, we need some preliminaries. In the sequel we write T in place of TT . The fact that Tτ is invertible for each 0 < τ ≤ T is equivalent to T being strictly positive. The latter property implies that T factorizes as T = ΓΓ∗ , where Γ is an invertible lower triangular integral operator,

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121

x (Γf )(x) = f (x) +

γ− (x, t)f (s) ds,

0 ≤ x ≤ T,

(2.10)

× γ− (x, s)f (s) ds,

0 ≤ x ≤ T,

(2.11)

0

(Γ−1 f )(x) = f (x) +

x 0

× γ− (x, s)

being continuous on 0 ≤ s ≤ x ≤ T. We with both γ− (x, s) and shall refer to T = ΓΓ∗ as the LU -factorization of T . From [6] we also know that × (τ, s) = γτ (τ, s), γ−

0 ≤ s ≤ τ ≤ T.

(2.12)

We shall need the following three lemmas. Lemma 2.2. We have

(Γ−1 k)(τ ) = Tτ−1 (k|[ 0, τ ] ) (τ ) = γτ (τ, 0),

0 < τ ≤ T.

(2.13)

Proof. We ﬁrst prove the second equality in (2.13). Since γτ (t, s) is the resolvent kernel corresponding to Tτ , we know (cf., (1.2)) that τ γτ (t, s) −

k(t − ξ)γτ (ξ, s) dξ = k(t − s),

0 ≤ s ≤ t ≤ τ.

0

This equality holds a.e on 0 ≤ s ≤ t ≤ τ . But then, since both γτ (t, s) and k(t − s) are continuous on 0 ≤ s ≤ t ≤ τ , the above equality holds at each point of 0 ≤ s ≤ t ≤ τ . In particular, at the point (t, 0). Thus τ γτ (t, 0) −

k(t − α)γτ (α, 0) dα = k(t),

0 ≤ t ≤ τ.

0

This shows that Tτ γτ (·, 0) = k|[ 0, τ ] , and hence −1 Tτ (k|[ 0, τ ] ) (x) = γτ (x, 0), 0 ≤ x ≤ τ.

(2.14)

For x = τ this yields the second identity in (2.13). Next we prove the ﬁrst identity in (2.13). Fix 0 < τ ≤ T. Since k is continuous on [0, T], the function Γ−1 k is continuous on [0, T]. From the previous part of the proof we know that Tτ−1 (k|[ 0, τ ] ) is continuous on [0, τ ]. Hence for both functions the evaluation at τ is well-deﬁned. Moreover, −1

(Γ

τ k)(τ ) = k(τ ) +

× γ− (τ, s)k(s) ds,

0

(Tτ−1 (k|[ 0, τ ] )(τ ) = k(τ ) +

τ γτ (τ, s)k(s) ds. 0

According to (2.12) we have the ﬁrst equality in (2.13).

× (τ, s) γ−

= γτ (τ, s) for 0 ≤ s ≤ τ , which yields

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Lemma 2.3. Let f be a continuously differentiable Cr×m -valued function on the interval [0, T]. Then Γ−1 f is also continuously differentiable on [0, T] and d −1 −1 d Γ f (τ ) − Γ f (τ ) dτ dτ ⎛ ⎞ τ = γτ (τ, 0) ⎝f (0) + γτ (0, s)f (s) ds⎠ , 0 < τ ≤ T. (2.15) 0

Proof. Recall that

Γ

−1

τ

× γ− (τ, s)f (s) ds

f (τ ) = f (τ ) +

τ = f (τ ) +

0

γτ (τ, s)f (s) ds. 0

Next, using the generalized Krein–Sobolev identities in formulas (2.10) and (2.11) of [1], we have τ τ d d γτ (τ, s)f (s) ds = γτ (τ, τ − s)f (τ − s) ds dτ dτ 0

0

= γτ (τ, 0)f (0) + (α), where τ (α) = 0

d (γτ (τ, τ − s)f (τ − s)) ds dτ

τ γτ (τ, 0)γτ (0, τ − s)f (τ − s) ds

= 0

τ γτ (τ, τ − s)

+ 0

d f dτ

(τ − s) ds.

We conclude that τ d −1 d d Γ f (τ ) = f (τ ) + γτ (τ, s) f (s) ds dτ dτ dτ 0 ⎛ ⎞ τ + γτ (τ, 0) ⎝f (0) + γτ (0, s)f (s) ds⎠ =

d Γ−1 f dτ

(τ ) ⎛

+ γτ (τ, 0) ⎝f (0) +

0

τ

⎞ γτ (0, s)f (s) ds⎠ .

0 −1

Thus Γ

f is continuously differentiable and (2.15) holds.

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Lemma 2.4. For each g ∈ L2r (0, T) we have τ

−1

(Γ

∗

−1

k)(t) (Γ

τ g)(t) dt =

0

γτ (0, t)g(t) dt,

0 ≤ τ ≤ T.

(2.16)

0

Proof. The case when τ = 0 is trivial. Fix 0 < τ ≤ T. Let Πτ be the projection of L2r (0, T) onto L2r (0, τ ) deﬁned by Πτ f = f |[ 0, τ ] . Note that Π∗τ is the canonical embedding of L2r (0, τ ) into L2r (0, T). Put Pτ = Π∗τ Πτ . Then Pτ is the orthogonal projection of L2r (0, T) onto the subspace consisting of all functions in L2r (0, T) with support in [0, τ ]. Since Γ and Γ−1 are lower triangular, we have Πτ Γ = Πτ ΓPτ ,

Πτ Γ−1 = Πτ Γ−1 Pτ .

(2.17)

From the second identity in (2.17), the definition of Tτ , and the factorization TT = ΓΓ∗ , we see that Tτ = Πτ TT Π∗τ = Πτ ΓΓ∗ Π∗τ = Πτ ΓPτ ΓΠ∗τ = Πτ ΓΠ∗τ Πτ Γ∗ Π∗τ . Since, by lower triangularity, Πτ ΓΠ∗τ = (Πτ Γ−1 Π∗τ )−1 , we obtain Tτ−1 = Πτ Γ−∗ Pτ Γ−1 Π∗τ .

(2.18)

Tτ−1 (k|[ 0, τ ] ) = Πτ Γ−∗ Pτ Γ−1 Π∗τ (Πτ k) = Πτ Γ−∗ Pτ Γ−1 Pτ k.

(2.19)

If follows that

Now using (2.14), the lower triangularity of Γ−1 , and the above identities one computes that τ

−1

(Γ

∗

−1

k)(t) (Γ

T f )(t) dt = (Pτ Γ−1 k)(t)∗ (Pτ Γ−1 f )(t) dt

0

0

T = (Pτ Γ−1 Pτ k)(t)∗ (Pτ Γ−1 Pτ f )(t) dt 0

T = (Pτ Γ−∗ Pτ Pτ Γ−1 Pτ k)(t)∗ f (t) dt 0

T = (Π∗τ Tτ−1 Πτ k)(t)∗ f (t) dt 0

τ = 0

This proves (2.16).

Tτ−1 (k|[ 0, τ ] (t)∗ f (t) dt =

τ γτ (0, t)f (t) dt. 0

Using the identities (2.12) and (2.16) we see that the expressions for θj (τ, λ) and ωj (τ, λ) in (2.5) and (2.6), respectively, can be rewritten as

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follows: 1 θj (τ, λ) = √ e−iτ λ Γ−1 j (· , λ) (τ ), 2 1 −iτ λ ωj (τ, λ) = √ e 2 ⎧ ⎫ τ ⎨ ⎬ × (−1)j Ir + (Γ−1 k)(t)∗ Γ−1 j (· , λ) (t) dt . ⎩ ⎭

(2.20)

(2.21)

0

We are now ready to prove Theorem 2.1. Proof of Theorem 2.1. We split the proof into three parts. The ﬁrst part deals with the initial value condition; the two other parts concern the proof that u(τ, λ) satisﬁes the differential equation (1.10). Part 1. Using the expressions for θj and ωj in (2.20) and (2.21), respectively, and the fact that Γ−1 is a lower triangular integral operator (see (2.11)), we obtain the following identities 1 1 θj (0, λ) = √ j (0 , λ) = √ Ir , 2 2 Thus

⎡

√1 2

Ir

u(0, λ) = ⎣ − √12 Ir

1 ωj (0, λ) = √ (−1)j Ir 2 ⎤

Ir 1 ⎦= √ 1 √ Ir 2 −Ir 2 √1 2

Ir

Ir Ir

(j = 1, 2).

= Q∗ .

Hence u(τ, λ) has the desired value at τ = 0. To complete the proof it sufﬁces to prove the following differential expressions: d ωj (τ , λ) = −iλωj (τ , λ) − iv(τ )∗ θj (τ, λ), j = 1, 2, dτ d θj (τ , λ) = iλθj (τ , λ) + iv(τ )ωj (τ, λ), j = 1, 2. dτ

(2.22) (2.23)

The ﬁrst two identities will be proved in the next part and the other two in the ﬁnal part. Part 2. In this part we prove (2.22). Since k is continuous on [0, T], the fact that the kernel function of the lower triangular integral operator G−1 is continuous on 0 ≤ s ≤ x ≤ T implies that Γ−1 k is also continuous on [0, T]. Similarly, using the continuity of 1 (· , λ) and 2 (· , λ) on [0, T], we see that Γ−1 1 (· , λ) and Γ−1 2 (· , λ) are continuous on [0, T]. Thus the functions under the integrals in the definitions of ω1 (· , λ) and ω2 (· , λ) are continuous. This implies that ω1 (· , λ) and ω2 (· , λ) are continuously differentiable and d ωj (τ , λ) = −iλωj (τ , λ) dτ 1 + √ e−iτ λ (Γ−1 k)(τ )∗ Γ−1 j (· , λ) (τ ), 2

j = 1, 2.

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Using (2.13) and (2.1) we see that (Γ−1 k)(τ ) = iv(τ ). This together with the expression of θj in (2.20) shows that (2.22) holds. Part 3. In this part we prove (2.23). First note that d 1 θj (τ, λ) = −iλθj (τ, λ) + √ e−iτ λ dτ 2

d −1 Γ j (· , λ) (τ ), dτ

j = 1, 2. (2.24)

Applying Lemma 2.3 with f = j (· , λ), j = 1, 2, yields

d −1 d Γ j (· , λ) (τ ) = Γ−1 j (· , λ) (τ ) dτ dτ ⎛ + γτ (τ, 0) ⎝j (0 , λ) + 0 ≤ τ ≤ T.

τ

⎞ γτ (0, s)j (s, λ) ds⎠ ,

0

Using the formula for the potential v in (2.1) and the identity in (2.16) we obtain d −1 d Γ j (· , λ) (τ ) = Γ−1 j (· , λ) (τ ) + iv(τ ) dτ dτ τ + iv(τ ) (Γ−1 k)(t)∗ Γ−1 j (· , λ) (t) dt,

0 ≤ τ ≤ T.

0

Next we use the identities in (2.4) and (Γ−1 k)(τ ) = iv(τ ) to show that 1 d √ e−iτ λ Γ−1 1 (· , λ) (τ ) dτ 2 1 1 −iτ λ −1 Γ 1 (· , λ) (τ ) − 2 √ e−iτ λ Γ−1 k (τ ), = 2iλ √ e 2 2 √ = 2iλθ1 (τ, λ) − 2 e−iτ λ iv(τ ), and 1 d 1 √ e−iτ λ Γ−1 2 (· , λ) (τ ) = 2iλ √ e−iτ λ Γ−1 2 (· , λ) (τ ) dτ 2 2 = 2iλθ2 (τ, λ).

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Returning to (2.24), ﬁrst for j = 1 and next for j = 2, we obtain √ d θ1 (τ, λ) = −iλθ1 (τ, λ) + 2iλθ1 (τ, λ) − 2 e−iτ λ iv(τ ) dτ ⎛ ⎞ τ 1 + iv(τ ) √ e−iτ λ ⎝Ir + (Γ−1 k)(t)∗ Γ−1 1 (· , λ) (t) dt⎠ 2 0

1 = iλθ1 (τ, λ) + iv(τ ) √ e−iτ λ 2 ⎛ ⎞ τ × ⎝−2Ir + Ir + (Γ−1 k)(t)∗ Γ−1 j (· , λ) (t) dt⎠ 0

= iλθ1 (τ, λ) + iv(τ )ω1 (τ, λ), and d θ2 (τ, λ) = −iλθ2 (τ, λ) + 2iλθ2 (τ, λ) dτ ⎛ ⎞ τ 1 −iτ λ ⎝ + iv(τ ) √ e Ir + (Γ−1 k)(t)∗ Γ−1 2 (· , λ) (t) dt⎠ 2 0

= iλθ2 (τ, λ) + iv(τ )ω2 (τ, λ),

Thus (2.23) is proved.

3. Uniqueness of the Accelerant Let u(x, λ) be the fundamental solution of the canonical system (1.10) satisfying the initial condition (2.2), and put (3.1) θ(x) = Ir 0 u(x, 0), 0 ≤ x ≤ T. Since the potential v of (1.10) is assumed to be continuous, the function θ is continuously differentiable on [0, T]. With θ we associate a lower triangular semi-separable integral operator L acting on L2r (0, T), namely x (Lf )(x) = θ(x)J J=

0

0

Ir

Ir

0

θ(t)∗ f (t) dt,

0 ≤ x ≤ T,

(3.2)

.

(3.3)

Note that L depends only on (1.10); accelerants do not play a role yet. The operator L will play an important role in the proof of the uniqueness of the accelerant (in the present section), and also later on in the construction of the accelerant given a continuous potential (in Sect. 5). In this section k is an accelerant for the canonical system (1.10), and we will show that k is uniquely determined by the potential v. First we recall that the statement “T is a convolution operator with kernel function k” can

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be expressed in terms of an intertwining relation involving the operator of integration which is the operator A on the space L2r (0, T) deﬁned by x (Af )(x) = f (t) dt (0 ≤ x ≤ T). (3.4) 0

In fact, using Theorem 1.2 in Chapter 1 of [14], we know that AT + T A∗ = BJB ∗ , where J is deﬁned by (3.3), 1 B : C2r → L2r (0, T), By = √ (·)y (y ∈ C2r ). 2 Here is the r × 2r matrix function given by x (x) = h(x) Ir with h(x) = Ir − 2 k(t)dt, 0 ≤ x ≤ T.

(3.5) (3.6)

(3.7)

0

We shall need the following proposition and an additional lemma. Proposition 3.1. Let k be an accelerant for the canonical system (1.10), and let T = ΓΓ∗ be the LU -factrization of the corresponding convolution integral operator T . Then the operator L defined by (3.2) is similar to the operator of integration A; in fact, L = Γ−1 AΓ. Proof. We ﬁrst show that L + L∗ = Γ−1 BJB ∗ Γ−∗ . The fact that k is an accelerant for the canonical system (1.10) allows us to use the results of the previous section. Let 1 (x, λ) and 2 (x, λ) be the r×r matrix functions deﬁned by (2.3). Note that 1 (x, 0) is equal to h(x), where h is the function appearing in (3.7), and 2 (x, 0) = Ir . It follows that (x) = 1 (x, 0) 2 (x, 0) , 0 ≤ x ≤ T. But then we see from (3.1), (2.7), and (2.20) that 1 θ(x) = √ (Γ−1 )(x), 0 ≤ x ≤ T. 2

(3.8)

Using the definition of B in (3.6), the preceding identity yields Γ−1 By = θ( ·)y for each y ∈ C2r . As L is deﬁned by (3.2), we obtain L + L∗ = Γ−1 BJB ∗ Γ−∗ . Next, since T = ΓΓ∗ , the identity in (3.5) can be rewritten as AΓΓ∗ + ΓΓ∗ A∗ = BJB ∗ . Multiplying the latter identity from the left by Γ−1 and from the right by the operator Γ−∗ yields Γ−1 AΓ + (Γ−1 AΓ)∗ = Γ−1 BJB ∗ Γ−∗ . By the result of the ﬁrst paragraph, L + L∗ = Γ−1 BJB ∗ Γ−∗ . It follows that L − Γ−1 AΓ = (Γ−1 AΓ)∗ − L∗ .

(3.9)

Note that the operator in the left hand side of (3.9) is a lower triangular integral operator of the ﬁrst kind, while the operator in the right hand side of (3.9) is an upper triangular operator of the ﬁrst kind. Hence both sides are equal to the zero operator. Thus L = Γ−1 AΓ.

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Lemma 3.2. Let R be an operator on L2r (0, T) commuting with the operator of integration A given by (3.4). If, in addition, (Ru)(x) = u for each u ∈ Cr and each 0 ≤ x ≤ T, then R is the identity operator on L2r (0, T). Proof. Let β be the canonical embedding operator from Cr into L2r (0, T), that is, β is given by (βu)(x) = u for each u ∈ Cr and 0 ≤ x ≤ T. Then Rβ = β. Since R commutes with operator of integration A, we have RAn β = An Rβ = An β. Thus R acts as the identity operator on the closed linear hull ∞ n n n=0 Im A β. By induction one shows that Im β + Im Aβ + · · · + Im A β r consists of all C -valued polynomials of degree at most n. Sincethe set of all ∞ Cr -valued polynomials is dense in L2r (0, T), we conclude that n=0 Im An β coincides with L2r (0, T), and hence R is identity operator on L2r (0, T).

Theorem 3.3. The accelerant is uniquely determined by the potential. Proof. By specifying (2.8) for λ = 0 we see (using (2.12)) that

1√ 0 −1 2 (Γ Ir )(x) = θ2 (x, 0) = Ir 0 u(x, 0) , 0 ≤ x ≤ T. I 2 r

(3.10)

Thus Γ−1 Ir depends on the potential v only and not on the particular choice of the accelerant. Now ﬁx the potential v, and let k˜ be another accelerant determining v. Thus k˜ is a hermitian r × r matrix function on the interval −T ≤ t ≤ T, which is is continuous on the interval −T ≤ t ≤ T with possibly a jump discontinuity at the origin. Furthermore, the convolution integral operator T˜ deﬁned by T ˜ − s)f (s) ds, k(x

(T˜f )(x) = f (x) −

0 ≤ x ≤ T,

0

˜Γ ˜ ∗ be the LU -factorization is a strictly positive operator on L2r (0, T). Let Γ ˜ of T . Then Proposition 3.1, together with the fact that L depends on (1.10) ˜ −1 AΓ. ˜ In other words, the operator ΓΓ ˜ −1 comonly, shows that Γ−1 AΓ = Γ mutes with the operator A. The result of the ﬁrst paragraph of the proof yields ˜ −1 Ir . Thus the operator ΓΓ ˜ −1 commutes with A and (ΓΓ ˜ −1 u)(x) = Γ−1 Ir = Γ r u for each u ∈ C and 0 ≤ x ≤ T. According to Lemma 3.2 this implies that ˜ −1 is the identity operator on L2 (0, T). Hence Γ ˜ = Γ. But then T˜ = T , ΓΓ r and thus k˜ = k. This proves the uniqueness of the accelerant.

4. Semi-Separable Triangular Operators Similar to the Operator of Integration Throughout this section K is a semi-separable lower triangular integral operator on L2r (0, T), that is, the action of K is given by x (Kf )(x) = F (x) G(t)f (t) dt, f ∈ L2r (0, T). (4.1) 0

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Here F (·) and G(·) are matrix functions of sizes r × p and p × r, respectively, and their entries are square summable on the interval [0, T]. In fact, we shall assume that F (·) and G(·) are continuously differentiable on [0, T] and such that F (x)G(x) = Ir ,

0 ≤ x ≤ T.

(4.2)

A simple example of such an operator is the operator of integration A on L2r (0, T) deﬁned by (3.4). We shall see that any semi-separable lower triangular operator K satisfying the conditions referred to above is similar to the operator of integration A and with a similarity operator of a special kind. The precise result is presented in the next proposition. Proposition 4.1. Let F and G be continuously differentiable, and assume (4.2) holds. Then the operator K defined by (4.1) is similar to the operator of integration A. More precisely, K = EAE −1 where E is a lower triangular operator of the form x (4.3) (Ef )(x) = ρ(x)f (x) + e(x, t)f (t)dt, f ∈ L2r (0, T). 0

Here, e(x, t) is a continuous r × r matrix function on 0 ≤ t ≤ x ≤ T, which is zero at t = 0, and the r × r matrix function ρ is given by d ρ(x) = F (x)G(x)ρ(x), ρ(0) = Ir . (4.4) dx Moreover, the operators E ±1 map functions with a continuous derivative into functions with a continuous derivative. When F and G are boundedly differentiable and continuous derivatives are replaced by bounded derivatives, the above proposition is a particular case of Theorem 1 in [11]. The restriction to continuously differentiable F and G is the new element here. Since the above proposition plays an essential role in the proof of our main theorem, we will present a full proof. In order to prove Proposition 4.1 we ﬁrst make some heuristic remarks explaining the line of reasoning that we will follow. Assume we have an operator E on L2r (0, T) with all the properties described in Proposition 4.1. In particular, KE = EA. By rewriting this identity in terms of the kernel functions of the integral operators A, K, E, we get x x F (x)G(t)ρ(t) + F (x) G(s)e(s, t) ds = ρ(x) + e(x, s) ds. t

t

Taking t = 0 and using e(x, 0) = 0 for each 0 ≤ x ≤ T, we conclude that x (EIr )(x) = ρ(x) + e(x, s) ds = F (x)G(0), 0 ≤ x ≤ T. (4.5) 0

Here we view Ir as the r × r matrix function on [0, T] which is identically equal to the r × r identity matrix, and E is applied to Ir column wise.

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On the other hand, since K and A are Volterra operators (cf., Section 12.9 in [2]) the identity KE = EA implies that (I − λK)−1 E = E(I − λA)−1 for each λ ∈ C. Here, and in the sequel, I denotes the identity operator on L2r (0, T). As is well known, for each f in L2r (0, T) we have

(I − λA)−1 f (x) = f (x) + λ

x 0 ≤ x ≤ T.

eλ(x−t) f (t) dt,

(4.6)

0

With f (·) = Ir this yields (I − λA)−1 Ir (x) = eλx Ir ,

0 ≤ x ≤ T.

(4.7)

It follows (using the identity (4.5)) that x λx

e(x, t)eλt Ir dt

ρ(x)e Ir + 0

= E(e Ir ) (x) = E(I − λA)−1 Ir (x) = (I − λK)−1 EIr (x) = (I − λK)−1 F (·)G(0) (x),

λ·

0 ≤ x ≤ T.

Hence in order to ﬁnd the kernel function e(x, t) it is natural to solve the equation (I − λK)g(·, λ) = F (·)G(0) and to analyze its solution. This will be done in Lemmas 4.2 and 4.3 below. We begin with some preparations According to the general theory of semiseparable integral operators (see Chapter IX in [3]), the inverse of operator I − λK is given by

−1

(I − λK)

x

f (x) = f (x) +

η(x, t, λ)f (t)dt,

(4.8)

0

where and η(x, t, λ) = λF (x)u1 (x, λ)u1 (t, λ)−1 G(t), d u1 (x, λ) = λG(x)F (x)u1 (x, λ), dx u1 (0, λ) = Ir .

0 ≤ t ≤ x ≤ T,

0 ≤ x ≤ T,

(4.9) (4.10) (4.11)

We also need the r × r matrix function u 1 (x) deﬁned by d u 1 (x) = −G(x)F (x) u1 (x), dx

0 ≤ x ≤ T,

u 1 (0) = Ir .

(4.12)

We are now ready to prove the ﬁrst lemma. Lemma 4.2. Let F and G be continuously differentiable, and assume (4.2) holds. Let h be the r × r matrix function defined by h(x) = F (x)G(0) on 0 ≤ x ≤ T, and let ρ be the r × r matrix function given by (4.4). Put g(x, λ) = ρ(x)−1 (I − λK)−1 h (x), 0 ≤ x ≤ T, (4.13)

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where (I − λK)−1 is applied to h columnwise. Then g satisfies the following integro-differential equation d g(x, λ) − α(x) dx

x β(t)g(t, λ)dt − λg(x, λ) = 0,

g(0, λ) = Ir , (4.14)

0

where α and β are the continuous functions on [0, T] given by α(x) = ρ(x)−1 F (x) u1 (x), −1

β(t) = − u1 (t)

0 ≤ x ≤ T,

(G(t)F (t)G(t) + G (t)) ρ(t),

(4.15) 0 ≤ t ≤ T.

(4.16)

Proof. Put g(x, λ) = ρ(x)g(x, λ). Using (4.8)–(4.11), (4.13), and the definition of the matrix function h, we present g in the form g(x, λ)

x

= F (x)G(0) + λF (x)u1 (x, λ)

u1 (t, λ)−1 G(t)F (t)G(0)dt

0

x

d u1 (t, λ)−1 G(0) dt dt 0 = F (x)G(0) − F (x)u1 (x, λ) u1 (x, λ)−1 − Ir G(0) = F (x)u1 (x, λ)G(0). = F (x)G(0) − F (x)u1 (x, λ)

(4.17)

It follows that g(x, λ) = ρ(x)−1 F (x)u1 (x, λ)G(0).

(4.18)

Clearly g is differentiable and d g(x, λ) = ρ(x)−1 gx (x, λ) − ρ(x)−1 ρ (x)ρ(x)−1 g(x, λ) dx = ρ(x)−1 {λF (x)G(x)F (x)+F (x)−F (x)G(x)F (x)} u1 (x, λ)G(0) = λg(x, λ) + ρ(x)−1 F (x) (Ip − G(x)F (x)) u1 (x, λ)G(0). Here we took into account the identity (4.2). From (4.12) we see that d d u 1 (t)−1 = − u 1 (t) u u1 (t)−1 1 (t)−1 = u 1 (t)−1 G(t)F (t). dt dt Hence d u 1 (t)−1 (Ip − G(t)F (t)) u1 (t, λ) dt =u 1 (t)−1 G(t)F (t) (Ip − G(t)F (t)) u1 (t, λ) +u 1 (t)−1 (−G (t)F (t) − G(t)F (t)) u1 (t, λ) + λ u1 (t)−1 (Ip − G(t)F (t)) G(t)F (t)u1 (t, λ).

(4.19)

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Since (Ip − G(t)F (t))G(t) = 0 because of condition (4.2), we see that d u 1 (t)−1 (Ip − G(t)F (t)) u1 (t, λ) dt =u 1 (t)−1 (G(t)F (t) − G(t)F (t)G(t)F (t) − G (t)F (t) − G(t)F (t)) u1 (t, λ) = − u1 (t)−1 (G(t)F (t)G(t) + G (t)) F (t)u1 (t, λ). Using the definition of β in (4.16) and the identity (4.18), we obtain d u 1 (t)−1 (Ip − G(t)F (t)) u1 (t, λ) G(0) = β(t)g(t, λ). (4.20) dt Recall that (Ip − G(t)F (t))G(t) = 0 and (Ip − G(0)F (0))G(0) = 0, in particular. From integration by parts it follows that x β(t)g(t, λ) dt = u 1 (x)−1 (Ip − G(x)F (x)) u1 (x, λ)G(0) 0

x − (Ip − G(0)F (0)) G(0)−λ

u 1 (t)−1 (Ip −G(t)F (t)) G(t)

0

× F (t)u1 (t, λ)G(0)dt =u 1 (x)−1 (Ip − G(x)F (x)) u1 (x, λ)G(0). But then, using (4.19) and the definition of α in (4.15), we arrive at the identity (4.14). The following lemma provides an integral representation of g. Lemma 4.3. Let γ(x, t) be an r×r matrix function, continuous on the interval 0 ≤ t ≤ x ≤ T. Then the integro-differential equation x d g(x, λ) − γ(x, t)g(t, λ)dt − λg(x, λ) = 0, g(0, λ) = Ir , (4.21) dx 0

has a unique continuously differentiable solution g. Moreover, g is of the form x λx g(x, λ) = e Ir + eλt N (x, t) dt, 0 ≤ x ≤ T, (4.22) 0

where N (x, t) is continuous on 0 ≤ t ≤ x ≤ T and N (x, 0) = 0 for each 0 ≤ x ≤ T. Proof. Throughout the proof we ﬁx λ ∈ C. By definition a solution g(x, λ) of (4.21) is absolutely continuous on 0 ≤ x ≤ T. In that case, since γ(x, t) is continuous on 0 ≤ t ≤ x ≤ T, we see that x λg(x, λ) + γ(x, t)g(t, λ) ds is continuous on 0 ≤ x ≤ T. 0

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d g(x, λ) is also continuous on 0 ≤ x ≤ T. Thus any solution of But then dx (4.21) is automatically continuously differentiable. By integrating the equation in (4.21) over 0 ≤ x ≤ τ , where 0 ≤ τ ≤ T, we obtain the equation

g(·, λ) − λAg(·, λ) − ARg(·, λ) = Ir .

(4.23)

Here A is the operator of integration on L2r (0, T) deﬁned by (3.4) and R is the operator on L2r (0, T) given by x (Rf )(x) =

γ(x, t)f (t) dt,

f ∈ L2r (0, T).

0

Note that for each f in L2r (0, T) the function Af is absolutely continuous on [0, T]. It follows that any solution of (4.23) is absolutely continuous, and thus the problems (4.21) and (4.23) are equivalent. Using (4.6) we get

−1

(I − λA)

x

0 ≤ x ≤ T.

eλ(x−t) f (t) dt,

Af (x) =

(4.24)

0

From (4.24) and the definition of R we see that for each f in L2r (0, T)

−1

(I − λA)

x

γ (x, t; λ)f (t) dt,

ARf (x) =

0 ≤ x ≤ T,

0

where x γ (x, t; λ) =

eλ(x−s) γ(s, t) ds,

0 ≤ t ≤ x ≤ T.

t

It follows that I − (I − λA)−1 AR is an invertible operator on L2r (0, T). Hence the problem (4.23) has a unique solution in L2r (0, T), namely −1 (I − λA)−1 Ir . g(·, λ) = I − (I − λA)−1 AR

(4.25)

We conclude that Eq. (4.21) has a unique continuously differentiable solution. It remains to show that the solution g(·, λ) is of the form (4.22). To do this, we use (4.7) and rewrite (4.25) as g(·, λ) = eλ· Ir +

∞

(I − λA)−1 AR

k=1

k

eλ· Ir .

(4.26)

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Let us compute the ﬁrst term with k = 1. Using (4.24) and the definition of R, we get ⎛ r ⎞ x (I − λA)−1 AR eλ· Ir (x) = eλ(x−r) ⎝ γ(r, t)eλt Ir dt⎠ dr 0

x = 0

x =

⎛ ⎝ ⎛ ⎝

0

r

0

⎞

eλ(x+t−r) γ(r, t) dt⎠ dr

0

⎞

x

eλt γ(r, r + t − x)dt⎠ dr

x−r

x =

⎛

eλt ⎝

0

x

⎞ γ(r, r + t − x)dr⎠ dt

x−t

x

eλt γ1 (x, t) dt,

= 0

where

x γ(r, r + t − x)dr,

γ1 (x, t) =

0 ≤ t ≤ x ≤ T.

(4.27)

x−t

Next deﬁne matrix γk (t, s), k = 2, 3, . . ., recursively by x y γk+1 (x, t) = γ(y, s)γk (s, t + y − x) ds dy.

(4.28)

x−t y+t−x

Then, using similar calculations as for k = 1 above, one proves by induction that for each k ≥ 1 we have x ! k (I − λA)−1 AR eλ· Ir (x) = eλt γk (x, t) dt, 0 ≤ x ≤ T. (4.29) 0

Observe that for each k the function γk (x, t) is continuous on 0 ≤ t ≤ x ≤ T. Furthermore, as we see from (4.27) and (4.28), we have γk (x, t) ≤ ck

x2k−1 , (2k − 1)!

0 ≤ t ≤ x ≤ T,

k ≥ 1.

(4.30)

Here c is a constant independent of k. Finally, using (4.26), we conclude that (4.22) holds with N (x, t) =

∞

γk (x, t),

0 ≤ t ≤ x ≤ T.

k=1

By (4.30) the convergence in the preceding formula is uniform on the triangle 0 ≤ t ≤ x ≤ T. Since each of the terms γk (x, t) is continuous on this triangle, it follows that N (x, t) is continuous on 0 ≤ t ≤ x ≤ T as desired. Finally,

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from (4.27) and (4.28) it is clear that γk (x, 0) = 0 for each 0 ≤ x ≤ T and each positive integer k. But then N (·, 0) is identically equal to zero too. Proof of Proposition 4.1. We split the proof into three parts. In the ﬁrst part we deﬁne the operator E and establish the similarity KE = EA. In the two other parts we prove that E ±1 map functions with a continuous derivative into functions with a continuous derivative. Part 1. Let g(x, λ) be the matrix function deﬁned by (4.13). From Lemma 4.2 we know that g(x, λ) satisﬁes the integro-differential equation (4.14). But then we can apply Lemma 4.3 with γ(x, t) = α(x)β(t), where α(·) and β(·) are deﬁned by (4.15) and (4.16). It follows that g admits the representation x λx (4.31) g(x, λ) = e Ir + N (x, t)eλt Ir dt, 0 ≤ x ≤ T, 0

with N (x, t) being continuous on 0 ≤ t ≤ x ≤ T and with N (·, 0) identically equal to zero. Now let E be the operator on L2r (0, T) deﬁned by x (Ef )(x) = ρ(x)f (x) + ρ(x)N (x, t)f (t) dt, 0 ≤ x ≤ T. (4.32) 0

Here the r×r matrix function ρ is deﬁned by (4.4). Thus E has the form (4.3) with e(x, t) = ρ(x)N (x, t). Obviously e(x, t) is continuous on 0 ≤ t ≤ x ≤ T and e(·, 0) = 0 on [0, T]. We claim that this operator E has all the properties described in Proposition 4.1. From (4.7), (4.13), formula (4.32) applied to f = eλ· Ir , and the identity in (4.31) we see that E(I − λA)−1 Ir = E(eλ· Ir ) = ρ(·)g(·, λ) = (I − λK)−1 h, where h = F (·)G(0). Taking λ = 0 in the above identity, we obtain h = EIr . Therefore, (I − λK)−1 EIr = E(I − λA)−1 Ir .

(4.33)

From the series expansion in (4.33) it follows that K j EIr = EAj Ir ,

j = 0, 1, 2, . . . .

(4.34)

Therefore, for each j = 0, 1, 2, . . ., we have (KE)Aj Ir = K(EAj Ir ) = K j+1 EIr = EAj+1 Ir = (EA)Aj Ir . (4.35) As the closed linear span of the columns of the matrices {Aj Ir }∞ j=0 coincides 2 with Lr (0, T), the equalities in (4.35) yield KE = EA. Since E is invertible, we obtain K = EAE −1 , and hence K and A are similar. It remains to prove that E ±1 map functions with a continuous derivative into functions with a continuous derivative. Part 2. To show that E has this property, let f be any Cn -valued function on [0, T] with a continuous derivative. Then f (·) = (Ag)(·) + u, where g is the derivative of f and u is a constant r × r matrix. As we have seen in the previous paragraph, EIr = h = F (·)G(0). Thus Eu = F (·)G(0)u. According

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to our hypotheses, F (·) is continuously differentiable. Hence the same holds true for Eu. Next note that x (EAg)(x) = (KEg)(x) = F (x) G(t)(Eg)(t) dt. 0

Since ρ is continuous on [0, T] and e(x, t) is continuous on 0 ≤ t ≤ x ≤ T, we know that E maps continuous functions into continuous functions. In particular, Eg is continuous, and hence the above formula shows that EAg has a continuous derivative. Therefore, Ef = EAg + Eu is continuously differentiable. Part 3. Next, we prove that E −1 maps functions with a continuous derivative into functions with a continuous derivative. First notice that E −1 admits the representation (E

−1

−1

f )(x) = ρ(x)

x f (x) +

e× (x, t)f (t) dt,

f ∈ L2r (0, T).

(4.36)

0

As e(x, t) is continuous on 0 ≤ t ≤ x ≤ T, the same holds true for e× (x, t), and thus E −1 maps continuous function into continuous functions. In terms of the kernel functions the identity E −1 E = I means ×

−1

e (x, t)ρ(t) + ρ(x)

x e(x, t) +

e× (x, s)e(s, t) ds = 0,

0 ≤ t ≤ x ≤ T.

t

Recall that e(·, 0) ≡ 0. Thus by taking t = 0 in the preceding identity we obtain e× (x, 0) = 0 for 0 ≤ x ≤ T. We shall need the operator K1 on L2r (0, T) deﬁned by

x

(K1 f )(x) = F (x)

G(t) dt,

f ∈ L2r (0, T).

0

Here F is the derivative of F , which is a continuous function on [0, T]. Notice that K = A(I + K1 ). Since KE = EA, we have E −1 K = AE −1 which yields E −1 A = E −1 A(I + K1 )(I + K1 )−1 = E −1 K(I + K1 )−1 = AE −1 (I + K1 )−1 .

(4.37)

Since the kernel function F (x)G(t) of K1 is continuous, (I + K1 )−1 maps continuous functions into continuous functions. Now let f be a Cr -valued function on [0, T] with a continuous derivative. As in the previous part, we can represent f as f (·) = (Ag)(·) + u, where g is the derivative of f and u is a constant r × r matrix. According to (4.37) we have E −1 Ag = AE −1 (I + K1 )−1 g. Since both E −1 and (I + K1 )−1 map continuous functions into continuous functions, the function E −1 (I + K1 )−1 g is continuous. Thus E −1 Ag has a continuous derivative. Hence in order to prove that E −1 f has a continuous derivative, it sufﬁces to show that E −1 Ir has a continuous derivative. By rewriting the identity

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E −1 K = AE −1 in terms of the kernel functions of A, K, and E −1 we get x x −1 × −1 ρ(x) F (x)G(t) + e (x, s)F (s)dsG(t) = ρ(t) + e× (s, t)ds. t

t ×

By taking t = 0 and using e (x, 0) = 0 for 0 ≤ x ≤ T we obtain x −1 Ir − ρ(x) F (x)G(0) − e× (x, s)F (s)dsG(0) = 0.

(4.38)

0

Since F (0)G(0) = Ir , we can use (4.36) and (4.38) to show that (E −1 Ir )(x) = Ir − ρ(x)−1 (F (x) − F (0)) G(0) x − e× (x, s) (F (s) − F (0)) ds G(0).

(4.39)

0

Using (AF (·)G(0))(x) = ((F (x) − F (0))G(0), it follows from (4.39) and (4.36) that (E −1 Ir )(x) = Ir − (E −1 AF (·))(x)G(0). As the right-hand side of the latter identity has a continuous derivative, we obtain that E −1 Ir is continuously differentiable.

5. Construction of an Accelerant In this section we establish the main part of Theorem 1.2. Throughout the 2r × 2r matrix function u(x, λ) is the fundamental solution of the canonical system (1.10) normalized by 1 Ir −Ir ∗ . (5.1) u(0, λ) = Q , where Q = √ 2 Ir Ir The potential v of (1.10) is assumed to be continuous on [0, T]. Finally, j and J are signature matrices, j is deﬁned by (1.11) and J by (3.3). Our aim is to show that v is generated by an accelerant. In what follows θ and ω are the r ×2r matrix functions on [0, T] deﬁned by (5.2) θ(x) = Ir 0 u(x, 0), 0 ≤ x ≤ T, (5.3) ω(x) = 0 Ir u(x, 0), 0 ≤ x ≤ T. We begin with two lemmas. The ﬁrst will enable us to use Proposition 4.1. Lemma 5.1. Let θ be the r × 2r matrix function on [0, T] defined by (5.2). Then θ is continuously differentiable on [0, T], θ(x)Jθ(x)∗ = Ir

and

θ (x)Jθ(x)∗ = 0

(0 ≤ x ≤ T).

(5.4)

Proof. It is straightforward to check that Q deﬁned in (5.1) satisﬁes the identities Q∗ = Q−1 ,

QjQ∗ = J,

Q∗ JQ = j.

(5.5)

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Since u(x, λ) satisﬁes (1.10) and the potential v is continuous, the function u(x, λ) is continuously differentiable in x. Furthermore, again using that d u(x, λ)∗ ju(x, λ) = 0. Hence, taking into u(x, λ) satisﬁes (1.10), we have dx account (5.1) and (5.5), we derive u(x, λ)∗ ju(x, λ) = J,

u(x, λ)Ju(x, λ)∗ = j.

(5.6)

The fact that u(x, λ) is continuously differentiable in x, implies that θ is continuously differentiable on [0, T]. Furthermore, from (5.6) and (1.10) we see that for each 0 ≤ x ≤ T we have ∗ ∗ Ir = Ir , (5.7) θ(x)Jθ(x) = Ir 0 u(x, 0)Ju(x, 0) 0 d ∗ ∗ Ir = 0. (5.8) u(x, 0) Ju(x, 0) θ (x)Jθ(x) = Ir 0 dx 0

Thus the identities in (5.4) hold.

Lemma 5.2. Let ω be the r × 2r matrix function defined by (5.3), and let θ be as in (5.2). Then ω is continuously differentiable on [0, T], and for each 0 ≤ x ≤ T we have 1 θ(x)Jω(x)∗ = 0, ω (x)Jω(x)∗ = 0, ω(0) = √ −Ir Ir . (5.9) 2 Moreover, the three identities in (5.9) determine ω uniquely. Finally, ω (x)Jθ(x)∗ = −iv(x)∗ ,

0 ≤ x ≤ T.

(5.10)

Proof. Since u(x, λ) is continuously differentiable in x on [0, T], the same holds true for ω(x). From the second identity in (5.6) and the definitions of θ and ω in (5.2) and in (5.3), respectively, we get 0 0 θ(x)Jω(x)∗ = Ir 0 u(x, 0)Ju(x, 0)∗ = Ir 0 j = 0. Ir Ir Analogously, using (1.10), ω (x)Jω(x) = 0

∗

= 0

d ∗ 0 Ir u(x, 0) Ju(x, 0) dx Ir 0 0 iv(x) Ir j = 0. ∗ Ir −iv(x) 0

Thus the ﬁrst two identities in (5.9) are proved. The third follows directly the normalizing condition (5.1). To prove that the three identities in (5.9) determine ω uniquely, note that the second identity in (5.9) implies that d ∗ (ω(x)Jω(x)∗ ) = ω (x)Jω(x)∗ + ω(x)J (ω (x)) = 0. dx

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Thus, using the third identity in (5.9), we obtain ω(x)Jω(x)∗ = −Ir . The latter identity, together with the ﬁrst identity in (5.9), yields

0 u(x, 0)Jω(x)∗ = . (5.11) −Ir Let ω ˜ be another continuously differentiable function on [0, T] such that the three identities in (5.9) hold with ω ˜ in place of ω. Repeating the above reasoning for ω ˜ in place of ω we see that (5.11) holds for ω ˜ in place of ω. ˜ (x)∗ ) = 0. But the matrices u(x, 0) and J are nonThus u(x, 0)J(ω(x)∗ − ω singular. Thus ω ˜ (x) = ω(x) for each x ∈ [0, T]. It follows that ω is uniquely determined by the identities in (5.9). To prove the ﬁnal identity (5.10) we use (1.10) and the definitions of θ and ω in (5.2) and (5.3). This yields

d I u(x, 0) Ju(x, 0)∗ r ω (x)Jθ(x)∗ = 0 Ir 0 dx

0 iv(x) Ir = 0 Ir = −iv(x)∗ . 0 0 −iv(x)∗

Hence (5.10) is proved.

In what follows it will be convenient to use the following notation:

Ir 0 θ0,1 (x) = θ(x) (0 ≤ x ≤ T); (5.12) , θ0,2 (x) = θ(x) Ir 0

Ir 0 ω0,1 (x) = ω(x) (0 ≤ x ≤ T). (5.13) , ω0,2 (x) = ω(x) Ir 0 Thus

θ(x) θ0,1 (x) u(x, 0) = = ω(x) ω0,1 (x)

θ0,2 (x) , ω0,2 (x)

0 ≤ x ≤ T.

We now return to the operator L deﬁned by (3.2). Thus L is the lower triangular semi-separable integral operator on L2r (0, T) deﬁned by x (Lf )(x) = θ(x)J

θ(t)∗ f (t) dt,

0 ≤ x ≤ T.

(5.14)

0

Here θ is as in (5.2) (cf., (3.1)) and J as in (3.3). Recall (see the ﬁrst paragraph of Sect. 3) that the definition of L does not involve accelerants and depends on (1.10) only. However, if the potential v of (1.10) is given by an accelerant, then Proposition 3.1 tells us that L is similar to the operator of integration with a similarity operator of a special kind. The next proposition goes in the reverse direction. Proposition 5.3. Assume that the operator L defined by (5.14) is similar to the operator of integration, L = Λ−1 AΛ, where Λ and Λ−1 have the following

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properties. Both Λ and Λ−1 are lower triangular operators, x ρ(x, t)f (s) ds,

(Λf )(x) = f (x) +

0 ≤ x ≤ T,

(5.15)

0

(Λ−1 f )(x) = f (x) +

x

ρ× (x, s)f (s) ds,

0 ≤ x ≤ T,

(5.16)

0 ×

with ρ(x, s) and ρ (x, s) being continuous r × r matrix functions on the triangles 0 ≤ s ≤ x ≤ T. Furthermore, we assume that Λ and Λ−1 map continuously differentiable functions into continuously differentiable functions, and 1 1 (Λθ0,1 )(0) = √ Ir, √ (Λ−1 Ir )(x) = θ0,2 (x) (0 ≤ x ≤ T). (5.17) 2 2 Then the r × r matrix function k given by ⎧ d ⎨ − √1 dx (Λθ0,1 )(x), for 0 < x ≤ T, 2 k(x) = ⎩ k(−x)∗ , for −T ≤ x < 0.

(5.18)

is an accelerant and k generates the potential v. The above result will allow us to complete the proof of Theorem 1.2. In fact, using Proposition 4.1, we shall show that given L as above a similarity operator Λ with the properties described in Proposition 5.3 always exists. Proof. Let k be deﬁned by (5.18). Clearly, k is hermitian on [−T, T]. Since θ0,1 is continuously differentiable, the fact that Λ maps continuously differentiable functions into continuously differentiable functions implies that k is continuous on [−T, T] with a possible jump discontinuity at the origin. The proof that k is an accelerant and generates the potential v will be split into two parts. Part 1. In this part we show that k is an accelerant. Let Tτ be the operator on L2r (0, τ ) given by τ (Tτ f )(t) = f (t) −

k(t − s)f (s) ds,

0 ≤ t ≤ τ.

(5.19)

0

To prove that k is an accelerant, we have to show that the operator TT is strictly positive on L2r (0, τ ). To establish the latter fact we prove the following identity: T = TT = ΛΛ∗ .

(5.20)

Note that the right hand side of (5.20) is an LU -factorization. In order to establish (5.20), recall that L = Λ−1 AΛ, where A is the operator of integration. Take f ∈ L2r (0, τ ). Using the similarity relation

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L = Λ−1 AΛ it follows that Λ

−1

AΛf + Λ

−1

AΛ

∗

T

∗

f = Lf + L f = θ(·)J

θ(t)∗ f (t) dt.

(5.21)

0

By multiplying (5.21) from the left by Λ and replacing f by Λ∗ f we obtain T AΛΛ f + ΛΛ A f = (Λθ)(·)J (Λθ)(t)∗ f (t) dt. ∗

∗

∗

(5.22)

0

Thus the selfadjoint operator S = ΛΛ∗ , which acts on L2r (0, τ ), satisﬁes the identity ASf + SA∗ f = (Λθ)(·)J

T (Λθ)(t)∗ f (t) dt,

f ∈ L2r (0, τ ).

(5.23)

0

Now, with k given by (5.18), let s be the r × r matrix function deﬁned by s(x) =

1 Ir − 2

x k(t)dt

0 < x ≤ T.

(5.24)

0

From the√ﬁrst identity in (5.17) and the definition of k in (5.18) we see that Λθ0,1 = 2 s. By√applying Λ to both sides of the second identity in (5.17) we obtain Λθ0,2 = ( 2)−1 Ir . Summarizing we have 1 Λθ = √ 2s(·) Ir . 2

(5.25)

Using the later identity in the right hand side of (5.23) we obtain T (Λθ)(·)J (Λθ)(t)∗ f (t) dt 0

1 2s(x) Ir J = 2

=

1 2s(x) Ir 2

T

0

T

Ir f (t) dt 2s(t)∗

0

T

T f (t) dt +

= s(x) 0

2s(t)∗ f (t) dt Ir

0

s(t)∗ f (t) dt,

0 ≤ x ≤ T.

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But then (5.23) can be rewritten as T

∗

(ASf + SA f )(x) =

(s(x) + s(t)∗ ) f (t) dt,

0 ≤ x ≤ T.

(5.26)

0

According to Theorem 2.2 in Chapter 1 of [14] (see also [7] and [12]), the equation (5.26) has a unique solution which is given by d (Sf )(x) = dx

T s(x − t)f (t) dt,

s(−x) = −s(x)∗

(0 < x ≤ T).

0

(5.27) (Note Theorem 2.2 in Chapter 1 of [14] is stated for scalar kernel functions, but the result also holds for matrix-valued kernel functions [13]. In fact, to get the result for matrix-valued kernel functions one just writes S as a r × r matrix with operator entries and applies the scalar-valued result to each of these entries.) From (5.27) and (5.24) we see that S = TT , and thus (5.20) is proved. In particular, k is an accelerant. Part 2. Let v˜ be the potential generated by the accelerant k, where k is as in the previous part. In this part we show that v = v˜. Consider the canonical system (1.10) with the potential v being replaced by v˜. Let u ˜(x, λ) be the corresponding fundamental solution normalized at x = 0 by u ˜(0, λ) = Q∗ , where Q is as in (2.2). Put ˜ ˜(x, 0), ω ˜ (x) = 0 Ir u ˜(x, 0 (0 ≤ x ≤ T). θ(x) = Ir 0 u From (5.10) we know that ω ˜ J θ˜∗ = −i˜ v∗ ,

0 ≤ x ≤ T. (5.28) Thus to prove v = v˜ it sufﬁces to show that θ = θ˜ and ω = ω ˜. ˜ Since k is an accelerant generating the potenWe ﬁrst show that θ = θ. tial v˜, we can apply the results of Sects. 2 and 3 to the canonical system (1.10) with v˜ in place of v. In particular, using (3.8) in the present setting, we see that ⎤ ⎡ x 1 ˜ where (x) ˜ = ⎣Ir − 2 k(t) dt Ir ⎦ . θ˜ = √ Λ−1 , 2 0

Here Λ is the lower triangular integral operator which appears in the LU factorization (5.20) of the convolution operator T = TT deﬁned by k via ˜ (5.19). By (5.24) we have (x) = 2s(x) Ir , and hence, using (5.25), we obtain θ˜ = θ. Next we prove that ω ˜ = ω. By applying Lemma 5.2 to the canonical system (1.10) with v˜ in place of v, we have 1√ ˜ ˜ (x)J ω ˜ (x)∗ = 0, ω ˜ (0) = 2 −Ir Ir . θ(x)J ω ˜ (x)∗ = 0, ω 2 ˜ However, θ = θ. Thus (5.9) holds with ω ˜ in place of ω. But then we can use the uniqueness statement in Lemma 5.2 to show that ω ˜ = ω.

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We have now proved that v = v˜, and hence k is an accelerant generating the potential v. Completing the proof of Theorem 1.2. Let L be the lower triangular semiseparable integral operator deﬁned by (3.2); see also (5.14). In order to complete the proof of Theorem 1.2 it sufﬁces to show that L is similar to the operator A of integration, L = Λ−1 AΛ, where Λ has all the properties stated in Proposition 5.3. For this purpose we use Proposition 4.1 with K = L,

F (x) = θ(x),

G(x) = Jθ(x)∗

(0 ≤ x ≤ T).

(5.29)

By Lemma 5.1, the functions F and G in (5.29) are continuously differentiable on [0, T] and condition (4.2) is satisﬁed. Furthermore, the second identity in (5.4) implies that for F and G in (5.29) the solution ρ of the differential equation (4.4) is identically equal to Ir . Thus, by Proposition 4.1, L = EAE −1 ,

(5.30)

where A is the operator of integration deﬁned by (3.4) and E on L2r (0, T) is a lower triangular integral operator of the form x (Ef )(x) = f (x) +

e(x, t)f (t)dt,

f ∈ L2r (0, T).

(5.31)

0

Moreover, we know that e(x, t) is a continuous r × r matrix function on 0 ≤ t ≤ x ≤ T, which is zero at t = 0, and the operators E ±1 map functions with a continuous derivative into functions with a continuous derivative. To construct the lower triangular integral operator Λ we need (apart from the operator E) an additional normalizing lower triangular operator. This operator is the lower triangular convolution operator E0 deﬁned by x e0 (x − t)f (t)dt,

(E0 f )(x) = θ0,2 (0)f (x) +

where

(5.32)

0

d (E −1 θ0,2 )(x). e0 (x) := dx

(5.33)

Recall that θ0,2 is deﬁned by the second identity in (5.12). Since θ is continuously differentiable (see Lemma 5.1), the same holds true for θ0,2 . Using the fact that E −1 maps functions with a continuous derivative into functions with a continuous derivative, we conclude that e0 is continuous (in fact, continuously differentiable). Lemma 5.4. Let E0 be the operator on L2r (0, T) defined by (5.32), and let A be the operator of integration defined by (3.4). Then E0 A = AE0

and

(E0 Ir )(x) = (E −1 θ0,2 )(x)

(0 ≤ x ≤ T).

(5.34)

Furthermore, E0 is invertible and E0±1 map functions with a continuous derivative into functions with a continuous derivative.

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Proof. Since E0 is a lower triangular convolution integral operator, E0 commutes with the operator of integration. Thus the ﬁrst identity in (5.34) holds. From (5.31) with f = θ0,2 we see that (E −1 θ0,2 )(0) = θ0,2 (0). By using the latter identity, (5.32), and (5.33) we obtain x x (E0 Ir )(x) = θ0,2 (0) + e0 (x − t) dt = θ0,2 (0) + e0 (t) dt 0

x = θ0,2 (0) + 0

0

d −1 (E θ0,2 )(t) dt dt

= θ0,2 (0) + (E −1 θ0,2 )(x) − θ0,2 (0) = (E −1 θ0,2 )(x), which yields the second identity in (5.34). According to (5.2), (5.12), and the initial condition in (5.1), we have 1 θ0,1 (0) θ0,2 (0) = Ir 0 u(0, 0) = √ Ir Ir . (5.35) 2 √ In particular, θ0,2 (0) = Ir / 2, and so E0 is invertible. Furthermore, E0−1 is of the form x −1 −1 (E0 f )(x) = θ2 (0) f (x) + e× 0 ≤ x ≤ T, (5.36) 0 (x − t)f (t) dt, 0

e× 0 (x)

being continuous on 0 ≤ x ≤ T. Next, let f be any Cr -valued function on [0, T] with a continuous derivative. Write f as f (·) = (Ag)(·) + u, where g is the derivative of f and u is a constant r × r matrix. Then E0 f = E0 Ag + E0 u = AE0 g + E0 u. Since e0 and g are continuous functions, E0 g is continuous, and thus E0 Ag is continuously differentiable. Hence in order to prove that E0 f is continuously differentiable, it sufﬁces to show that E0 u has this property. The latter can be derived from the second identity in (5.34) and the properties of E. A more direct argument is as follows. From (5.32) we see that x x (E0 u)(x) = θ2 (0)u + e0 (x − t)u dt = θ2 (0)u + e0 (t)u dt. with

0

0

Since e0 is continuous, this implies that E0 u is continuously differentiable as desired. In a similar way, using that E0−1 commutes with A and that E0−1 is −1 given by (5.36) with e× 0 being continuous, one shows that E0 maps functions with a continuous derivative into functions with a continuous derivative. For latter purposes we note that (E0−1 E −1 θ0,1 )(0) = Ir .

(5.37)

To see this, observe that by (5.36) for any continuous Cr -valued function f we have (E0−1 f )(0) = θ0,2 (0)−1 f (0). We apply this identity to f = E −1 θ0,1 . We know that θ0,1 is continuously differentiable, and hence E −1 θ0,1 has the same property. In particular, E −1 θ0,1 is continuous. Using (4.36) and the

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fact that in this case ρ deﬁned by (4.4) is identically equal to Ir , we see that (E −1 θ0,1 )(0) = θ0,1 (0). Thus (E0−1 E −1 θ0,1 )(0) = θ0,2 (0)−1 (E −1 θ0,1 )(0) = θ0,2 (0)−1 θ0,1 (0). But then (5.35) yields (5.37). Now deﬁne 1 Λ = √ E0−1 E −1 . 2

(5.38)

We claim that Λ given by (5.38) satisﬁes all the conditions on Λ stated in Proposition 5.3. Indeed, from (5.30) and the ﬁrst identity in (5.34) we see that L = ΛAΛ−1 . Furthermore, Λ and Λ−1 are lower triangular integral operators of the form (5.15) and (5.16), respectively, and their respective kernel functions are continuous on the triangles 0 ≤ s ≤ t ≤ T, because the kernel functions of E ±1 and E0±1 have these properties. Since E ±1 and E0±1 map functions with a continuous derivative into functions with a continuous derivative, the same holds true for Λ and Λ−1 . It remains to check the identities in (5.17). The ﬁrst identity in (5.17) follows from the definition of Λ in (5.38) and the equality in (5.37). Finally, we use the second equality in (5.34). The latter can be rewritten as E0−1 E −1 θ0,2 = Ir . Using definition of Λ in (5.38), this yields the second identity in (5.17). Thus Λ given by (5.38) satisﬁes all the conditions on Λ appearing in Proposition 5.3. Hence the potential v is generated by an accelerant, as desired.

6. Pseudo-Exponential Potentials In this section we consider the class of so-called pseudo-exponential potentials, which has been introduced in [4]; see also [5]. The aim is to show how Theorem 2.1 can be used to present an alternative proof of the basic formula for the fundamental solution given in Theorem 4.2 of [4]; see also Section 2 in [5]. We begin with some notation. Fix an integer n > 0 and a triple of parameter matrices: an n × n matrix B and n × r matrices Φ1 and Φ2 . Recall that the triple B, Φ1 , Φ2 is called admissible whenever B ∗ − B = iΦ2 Φ∗2 .

(6.1)

Throughout Φ is the n × r matrix given by Φ = Φ1 + iΦ2 . Now let B, Φ1 and Φ2 be an admissible triple, and put ∗

k(t) = −2Φ∗1 e2itB Φ,

k(−t) = k(t)∗ ,

t > 0.

(6.2)

By taking adjoints, a minor modiﬁcation of the proof of Proposition 5.2 in [1] shows that the function k is an accelerant on each interval [−T, T], and the corresponding potential is given by ∗

v(τ ) = 2iΦ∗1 eiτ A Σ(τ )−1 eiτ A Φ,

A = B − Φ1 Φ∗2 ,

(6.3)

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where t Σ(t) = In +

with Π(t) = e−itA Φ1

Π(s)Π(s)∗ ds,

−eitA Φ . (6.4)

0

Note that with Φ1 = γ1 and Φ2 = γ2 , we have Φ = γ1 + iγ2 , and in this case v in (6.3) is just equal to v given by (4.6) in [4]. The following result is a variant of Theorem 4.2 in [4]. Proposition 6.1. Let B, Φ1 and Φ2 be an admissible triple, and let v be the potential defined by (6.3). Then the fundamental solution u(x, λ) of the canonical system (1.10) satisfying the initial condition (2.2) is given by u(τ, λ) = wA,Π (τ, λ)eiτ λj wA,Π (0, λ)−1 Q∗ ,

(6.5)

where j is the 2r × 2r matrix in the left hand side of (1.11) and wA,Π (τ, λ) = I2r + ijΠ(τ )∗ Σ(τ )−1 (λIn − A)−1 Π(τ ).

(6.6)

The proof of Proposition 6.1 given below is very different from the proof of Theorem 4.2 in [1]. Here we shall use that the potential v in (6.3) is generated by the accelerant k in (6.2). This fact will allow us to employ the formula for the fundamental solution given in Theorem 2.1. We shall only prove equality (6.5) for the block ω2 (τ, λ) of u(τ, λ) (see (2.7)); the representation of the other blocks can be proved in a similar way. Proof. We shall show that ω2 = ω "2 , where ω "2 denotes the right lower block on the right-hand side of (6.5). In Theorem 2.1 the block ω2 (τ, λ is given (cf., (2.9)) by ⎫ ⎧ τ ⎬ ⎨ 1 (6.7) ω2 (τ, λ) = √ e−iτ λ Ir + e2isλ γτ (0, s)ds . ⎭ ⎩ 2 0

Here γτ (t, s) is the resolvent kernel corresponding to the accelerant k. Using adjoints, the same line of reasoning as in the proof of Proposition 5.2 in [1], shows that

× ∗ Φ1 , (6.8) γτ (0, s) = −2Φ∗ e−iτ A Σ(τ )−1 e−iτ A In 0 ei(s−τ )AM Φ2 where

A× M

= −2

A

−Φ1 Φ∗1

0

A∗

.

In what follows we shall use the identity

× ∗ I In 0 e−iτ AM n = eiτ A Σ(τ )eiτ A . iIn

(6.9)

(6.10)

Here A× M and Σ(τ ) are as in (6.9) and (6.4), respectively. Note that (6.10) is the analogue of formula (4.7) in [4].

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By substituting (6.8) in (6.7) we get # ∗ 1 ω2 (τ, λ) = √ e−iτ λ Ir + 2iΦ∗ e−iτ A Σ(τ )−1 e−iτ A In 0 2 ! Φ $ −1 2iτ λ 1 −iτ A× M e × 2λI2n + A× I − e . 2n M Φ2

(6.11)

Since A× M is given by (6.9), we can rewrite (6.11) in the form % ∗ 1 ω2 (τ, λ) = √ e−iτ λ Ir − iΦ∗ e−iτ A Σ(τ )−1 e−iτ A (A − λIn )−1 2 ! $ 2iτ λ Φ1 −iτ A× ∗ ∗ −1 M e × In Φ1 Φ1 (A −λIn ) I2n −e . (6.12) Φ2 ! × × . From (6.9) and (6.10) it Partition e−iτ AM into n × n blocks e−iτ AM kj

follows that ×

e−iτ AM

!

!11 ×

e−iτ AM

21

= e2iτ A , = 0,

×

e−iτ AM ! ×

e−iτ AM

12

!

∗

22

= e2iτ A ,

! ∗

= i e2iτ A − eiτ A Σ(τ )eiτ A

(6.13) . (6.14)

Taking into account (6.12)–(6.14) we arrive at % ∗ 1 ω2 (τ, λ) = √ eiτ λ −iΦ∗ e−iτ A Σ(τ )−1 e−iτ A (A − λIn )−1 Φ1 2 & 1 × Ir + Φ∗1 (A∗ − λIn )−1 Φ2 + √ e−iτ λ 2 % ∗ −iτ A∗ −1 −iτ A × Ir + iΦ e Σ(τ ) e (A − λIn )−1 e2iτ A Φ1 ∗

∗

+ Φ1 Φ∗1 (A∗ −λIn )−1 e2iτ A Φ2 +i e2iτ A −eiτ A Σ(τ )eiτ A

!

!' Φ2

.

(6.15) Now, consider the right lower block ω "2 of the right-hand side of (6.5). The transfer matrix function wA,Π (τ, λ) has the property (see, e.g., [15]): wA,Π (τ, λ)∗ jwA,Π (τ, λ) = j. In particular, we have wA,Π (0, λ)−1 = jwA,Π (0, λ)∗ j. Hence, using (2.2), (6.4), and (6.6), we can write ω "2 (τ, λ) ∗ 1 0 Ir − iΦ∗ e−iτ A Σ(τ )−1 (A − λIn )−1 e−iτ A Φ1 =√ 2

∗ Ir Φ1 ∗ −1 −Φ Φ ×eiτ λj I2r + ij − λI ) (A . 1 n −Φ∗ Ir

−eiτ A Φ

!

(6.16)

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Formula (6.15) has the form 1 1 ω2 (τ, λ) = √ eiτ λ c+ (τ, λ) + √ e−iτ λ c− (τ, λ), (6.17) 2 2 where c± are the expressions between curly braces contained in (6.15). Formula (6.16) can be rewritten in a similar form 1 1 ω "2 (τ, λ) = √ eiτ λ " c+ (τ, λ) + √ e−iτ λ " c− (τ, λ), (6.18) 2 2 where ∗

" c+ (τ, λ) = −iΦ∗ e−iτ A Σ(τ )−1 (A − λIn )−1 e−iτ A Φ1 × Ir + Φ∗1 (A∗ − λIn )−1 Φ2 ,

(6.19)

! " c− (τ, λ) = Ir + iΦ∗ e−iτ A Σ(τ )−1 (A − λIn )−1 eiτ A Φ × Ir + Φ∗ (A∗ − λIn )−1 Φ2 . ∗

(6.20)

In (6.19) and (6.20) we used the equality Φ1 − Φ = −iΦ2 ; see the second parc+ . To prove agraph of this section. Comparing (6.15) and (6.19) yields c+ = " c− we shall need the equality AΣ(τ )−Σ(τ )A∗ = iΠ(τ )jΠ(τ )∗ , that that c− = " is, equality (1.22) from [4] rewritten in our present notations. Equivalently, we have ∗

∗

Σ(τ )(A∗ − λIn ) + ie−iτ A Φ1 Φ∗1 eiτ A − ieiτ A ΦΦ∗ e−iτ A = (A − λIn )Σ(τ ). 2iτ A

Now, use (6.15), (6.20), and e

∗

2iτ A

(Φ1 + iΦ2 ) = e

∗

−1

" c− (τ, λ) − c− (τ, λ) = Φ (A − λIn ) ∗ −iτ A∗

−iΦ e

−1

Σ(τ )

(6.21)

Φ to get

Φ2

−1

(A − λIn )

! ∗ ∗ × −eiτ A ΦΦ∗ e−iτ A + e−iτ A Φ1 Φ∗1 eiτ A − iΣ(τ )(A∗ − λIn ) ∗

×eiτ A (A∗ − λIn )−1 Φ2 .

(6.22)

Finally, we substitute (6.21) into (6.22). This yields " c− (τ, λ) = c− (τ, λ). "2 . Hence we have " c± = c± , and formulas (6.17) and (6.18) imply ω2 = ω 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.

References [1] Alpay, D., Gohberg, I., Kaashoek, M.A., Lerer, L., Sakhnovich, A.L.: Krein systems. In: Modern Analysis and Applications. The Mark Krein Centenary Conference, vol. 2, OT 191, pp. 19–36. Birkh¨ auser, Basel (2009) [2] Gohberg, I., Goldberg, S., Kaashoek, M.A.: Basic Classes of Linear Operators. Birkh¨ auser, Basel (2003)

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[3] Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of Linear Operators, vol. I. Birkh¨ auser, Basel (1990) [4] Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.: Canonical sytems with rational spectral densities: explicit formulas and applications. Math. Nach. 194, 93–125 (1998) [5] Gohberg, I., Kaashoek, M.A., Sakhnovich, A.L.: Scattering problems for a canonical system with a pseudo-exponential potential. Asymptotic Anal. 29, 1–38 (2002) [6] Gohberg, I., Koltracht, I.: Numerical solution of integral equations, fast algorithms and Krein–Sobolev equations. Numer. Math. 47, 237–288 (1985) [7] Koltracht, I., Kon, B., Lerer, L.: Inversion of structured operators. Integral Equ. Oper. Theory 20, 410–480 (1994) [8] Krein, M.G.: On the theory of accelerants and S-matrices of canonical differential systems. Dokl. Akad. Nauk SSSR (N.S.) 111, 1167–1170 (1956) [9] Sakhnovich, A.L.: Asymptotic behavior of spectral functions of an S-node. Soviet Math. (Iz. VUZ) 32, 92–105 (1988) [10] Sakhnovich, A.L.: Dirac type and canonical systems: spectral and Weyl-Titchmarsh functions, direct and inverse problems. Inverse Probl. 18, 331–348 (2002) [11] Sakhnovich, L.A.: Spectral analysis of Volterra’s operators defined in the space of vector-functions L2m (0, l). Ukr. Mat. J. 16, 259–268 (1964) [12] Sakhnovich, L.A.: Equations with a difference kernel on a finite interval. Russian Math. Surv. 35, 81–152 (1980) [13] Sakhnovich, L.A.: Systems of equations with difference kernels. Ukr. Math. J. 32, 44–50 (1980) [14] Sakhnovich, L.A.: Integral equations with difference kernels on finite intervals. OT 84. Birkh¨ auser, Basel (1996) [15] Sakhnovich, L.A.: Spectral Theory of Canonical Differential Systems. Method of Operator Identities, OT 107. Birkh¨ auser, Basel (1999)

D. Alpay Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva 84105 Israel e-mail: [email protected] I. Gohberg (Z L), M. A. Kaashoek (B) Afdeling Wiskunde Faculteit der Exacte Wetenschappen VU University Amsterdam De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands e-mail: [email protected] L. Lerer Department of Mathematics Technion, Israel Institute of Technology Haifa 32000, Israel e-mail: [email protected]

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A. L. Sakhnovich Fakult¨ at f¨ ur Mathematik Universit¨ at Wien Nordbergstrasse 15, 1090 Vienna, Austria e-mail: al [email protected] Received: December 23, 2009. Revised: May 1, 2010.

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Integr. Equ. Oper. Theory 68 (2010), 151–161 DOI 10.1007/s00020-010-1823-6 Published online July 20, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

On Singular Integral Operators with Rough Kernel Along Surface Yong Ding, Qingying Xue and Kˆozˆo Yabuta Abstract. In this note we give a simple method to transfer the eﬀect of the surface to the radial function in the kernel of singular integral along surface. Using this idea, we give some continuity of the singular integrals along surface with Hardy space function kernels on some function spaces, such as Lp (Rn ), Lp (Rn , ω), Triebel–Lizorkin spaces F˙ps,q (Rn ), Besov spaces B˙ ps,q (Rn ), generalized Morrey spaces Lp,φ (Rn ) and Herz spaces K˙ pα,q (Rn ). Our results improve and extend substantially some known results on the singular integral operators along surface. Mathematics Subject Classification (2010). Primary 42B20; Secondary 42B25. Keywords. Singular integral, rough kernel, along surface.

1. Introduction The purpose of this note is to report that one can substantially improve some known results concerning the Lp and weighted Lp boundedness of singular integral operators with rough kernel along surface, and extend some results about commutators between BMO functions and singular integral operators. Let Rn , n ≥ 2, be the n-dimensional Euclidean space and S n−1 be the unit sphere in Rn with area element dσ(x ) on S n−1 . Let Ω(x)|x|−n be a homogeneous function of degree −n on Rn , with Ω ∈ L1 (S n−1 ) and Ω(x )dσ(x ) = 0, (1.1) S n−1

The first and second named authors were supported partly by NSF of China (Grant: 10931001 and 10701010), SRFDP of China (Grant: 20090003110018), CPDRFSFP (Grant: 200902070), Beijing Natural Science Foundation (Grant: 1102023), Zhejiang Natural Science Foundation (Grant: Y7080325) and SRF for ROCS, SEM. The third-named author was partly supported by Grant-in-Aid for Scientific Research (C) Nr. 20540195, Japan Society for the Promotion of Science.

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where x = x/|x| for any x = 0. Suppose Φ is a nonnegative (or nonpositive) Φ(t) and monotonic C 1 function on (0, ∞) such that ϕ(t) := tΦ (t) is bounded. Suppose also b is a function on (0, ∞) satisfying 1 R

R |b(t)|γ dt ≤ Cγ

(1.2)

0

for some γ > 1 and Cγ > 0, where Cγ is independent of R > 0. We say that b ∈ Δγ if b satisﬁes (1.2) (Δ∞ = L∞ (R+ )). Obviously, for 1 < r < q < ∞, Δ∞ ⊂ Δq ⊂ Δr ⊂ Δ1 . For these Φ, b and Ω, we deﬁne the singular integral operator TΩ,Φ,b by TΩ,Φ,b (f )(x) = p. v. b(|y|)Ω(y )|y|−n f (x − Φ(|y|)y )dy Rn

= lim

ε→0, A→∞ε<|y|
b(|y|)Ω(y )|y|−n f (x − Φ(|y|)y )dy,

(1.3)

where y = y/|y| ∈ S n−1 and f ∈ S (Rn ). For the sake of simplicity, we denote TΩ,Φ,b = TΩ if Φ(t) = t and b ≡ 1, and TΩ,Φ,b = TΩ,b if Φ(t) = t. In 1956, using the method of rotation, Calder´ on and Zygmund [2] proved that TΩ is bounded on Lp for 1 < p < ∞ when Ω is odd in L1 (S n−1 ), or even in L log+ L(S n−1 ) satisfying (1.1). In 1996, Seeger [13] showed that TΩ is of weak type (1, 1) if Ω ∈ L log+ L(S n−1 ) with the mean zero condition (1.1) for dimension n ≥ 2. In 1979, Ricci and Weiss [12] and Connett [4] proved independently that TΩ is still bounded on Lp for 1 < p < ∞ if Ω ∈ H 1 (S n−1 ), where and in the sequel, H 1 (S n−1 ) denotes the Hardy space on the unit sphere (see [12] or [8] for its definition). The following fact is well known: there are the containing relationships on the unit sphere S n−1 : C ∞ Lq (q > 1) L log+ L H 1 L1 . In [11], Grafakos and Stefanov gave a nice survey, which contains a thorough discussion of the history of the operator TΩ . On the other hand, in 1979, Feﬀerman [10] proved that TΩ,b is a bounded operator on Lp for 1 < p < ∞ if b ∈ Δ∞ and Ω satisﬁes the Lipschitz condition on S n−1 . In 1986, Duoandikoetxea and Rubio de Francia [7] improved the ∗ (see Sect. 3.1 results in [10] to prove that TΩ,b and the maximal operator TΩ,b p for the definition) are both bounded on L for 1 1. In 1997, Fan and Pan [8] gave the Lp boundedness of the operator TΩ,Φ,b (see Theorem D and Remark in [8]): Theorem A. Suppose Φ satisfies the following conditions: |Φ(t)| ≤ C1 |t|d ,

|Φ (t)| ≤ C2 |t|d−2

and

C3 |t|d−1 ≥ |Φ (t)| ≥ C4 |t|d−1 (1.4)

for some d = 0 and t > 0. If b ∈ Δγ for some γ > 1 and Ω ∈ H 1 (S n−1 ) satisfies (1.1), then the singular integral operator TΩ,Φ,b defined by (1.3) is a

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bounded operator on Lp (Rn ) for |1/p − 1/2| < min(1/2, 1/γ ), where and in the sequel, γ satisfies 1/γ + 1/γ = 1. We ﬁnd that one can transfer the eﬀect of Φ to the b part. Using this idea, we obtain the same Lp boundedness under more weak assumption on Φ. Theorem 1.1. Suppose Φ is a nonnegative (or nonpositive) and monotonic Φ(t) C 1 function on (0, ∞) such that ϕ(t) := tΦ (t) is bounded. Let TΩ,Φ,b be the singular integral operator defined by (1.3) with b ∈ Δγ for some γ > 1. If Ω ∈ H 1 (S n−1 ) satisfies (1.1), then TΩ,Φ,b is a bounded linear operator on Lp (Rn ) for p: |1/p − 1/2| < min(1/2, 1/γ ). Remark 1.2. Note that the assumptions on Φ in Theorem A imply our assumptions on Φ, and we impose no assumption on Φ . Moreover, under our assumptions on Φ, the following facts are obvious (see [14]): (i) limt→0 Φ(t) = 0 and limt→∞ |Φ(t)| = ∞ if Φ is nonnegative and increasing, or nonpositive and decreasing; (ii) limt→0 |Φ(t)| = ∞ and limt→∞ Φ(t) = 0 if Φ is nonnegative and decreasing, or nonpositive and increasing. Similarly, we can weaken the assumptions in the theorems in the paper by Fan et al. [9], where weighted norm inequalities for TΩ,Φ,b are treated. In order to state our results, we ﬁrst recall the definitions of certain weights. Definition 1.3. Suppose that ω(t) ≥ 0 and ω ∈ L1loc (R+ ). For 1 0 such that for any interval I ⊂ R+ , ⎛ ⎞⎛ ⎞p−1 1 1 ⎝ ω(r)dr⎠ ⎝ ω(r)−1/(p−1) dr⎠ ≤ C < ∞. (1.5) |I| |I| I

I

If there is a constant C > 0 such that ω ∗ (r) ≤ Cω(r)

for a.e.

r ∈ R+ ,

(1.6)

∗

where ω denotes the standard Hardy–Littlewood maximal function of ω on R+ , then we say ω ∈ A1 (R+ ). Definition 1.4. If ω(x) = ν1 (|x|)ν2 (|x|)1−p (x ∈ Rn ), where either νi ∈ A1 (R+ ) is decreasing or νi2 ∈ A1 (R+ ), i = 1, 2, then we say ω ∈ A˜p (R+ ). Definition 1.5. For 1 0, ω ∈ L1loc (R+ ) and ω 2 ∈ Ap (R+ )}. Let AIp (Rn ) be the weight class deﬁned by using all n-dimensional intervals with sides parallel to coordinate axes, and Ap (Rn ) be the Muckenhoupt Ap weight class. In what follows, for 1 ≤ p < ∞, any measurable function f and any weight ω, we deﬁne ⎛ ⎞1/p f Lp (ω) := ⎝ |f (x)|p ω(x)dx⎠ , Rn

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and the weighted Lp spaces associated to the weight ω by Lp (Rn , ω(x)dx) = Lp (ω) := {f ; f Lp (ω) < ∞}. We know by [5] that A¯p (R+ ) ⊆ A˜p (R+ ), and by [6] that the Hardy–Littlewood maximal operator M is bounded on Lp (ω) for ω ∈ A˜p (R+ ), and thus A˜p (R+ ) ⊂ Ap (Rn ). We let further A˜Ip = A˜p ∩ AIp and A¯Ip = A¯p ∩ AIp . Theorem 1.6. Suppose Φ is a nonnegative (or nonpositive) and monotonic C 1 Φ(t) function on (0, ∞) such that ϕ(t) := tΦ (t) is bounded. Let b ∈ Δγ for some γ ≥ 2. If Ω ∈ H 1 (S n−1 ) satisfies (1.1), then TΩ,Φ,b is a bounded operator on Lp (ω), provided ω ∈ A˜Ip/γ (R+ ) with γ ≤ p < ∞ and p > 1. Remark 1.7. In [9], the authors set the following assumptions to get the conclusion in the above theorem. In the case Φ(t) ≥ 0 and strictly increasing, (a) Φ(2t) ≥ λΦ(t) for all t > 0 and some λ > 1, (b) Φ(2t) ≤ cΦ(t) for all t > 0 and some c > 0, (c) Φ (t) ≥ C1 Φ(t)/t for all t > 0 and some C1 > 0. Our result says that only the assumption (c) is needed. The proofs of the theorems are given in the second section. In Sect. 3, we will give several further results, which are the extensions of the known conclusions, including the truncated maximal operator, weighted inequalities and commutators. Moreover, in Sect. 3, we give also the boundedness of TΩ,Φ,b on Triebel–Lizorkin spaces, Besov spaces, Morrey spaces and Herz spaces, respectively. Throughout this paper, the letter C will denote a positive constant that may vary at each occurence but is independent of the essential variables.

2. Proofs of Theorems 1.1 and 1.6 To prove the theorems, we prepare the following two lemmas. Lemma 2.1. Let Φ and ϕ be the same as in Theorem 1.1. If b ∈ Δγ for some γ ≥ 1, then 1 R

R

γ |b(|Φ|−1 (t))ϕ(|Φ|−1 (t))|γ dt ≤ Cγ ( ϕ γ−1 ∞ + ϕ ∞ ),

R > 0, (2.1)

0

that is, b(|Φ|−1 )ϕ(|Φ|−1 ) ∈ Δγ . Proof. We prove the lemma only in the case where Φ is positive and increasing, since in the other cases one can prove in a quite similar way. For the function ϕ(Φ−1 (t))b(Φ−1 (t)), by the change of variable t = Φ(r), integration by parts and again by the change of variable r = Φ−1 (t), we get

Vol. 68 (2010) 1 R

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155

|ϕ(Φ−1 (t))b(Φ−1 (t))|γ dt

0

1 = R ≤

=

Φ−1 (R)

|ϕ(r)b(r)|γ Φ (r)dr

0

ϕ γ−1 ∞ R ϕ γ−1 ∞ R

ϕ γ−1 = −1∞ Φ (R)

Φ−1 (R)

0

Φ(r) |b(r)|γ Φ (r)dr Φ (r)r

Φ−1 (R)

|b(r)|γ 0

Φ(r) dr r

Φ−1 (R)

0

≤ Cγ ϕ γ−1 ∞ +

= Cγ ϕ γ−1 ∞ +

ϕ γ−1 ∞ |b(s)|γ ds− R

Cγ ϕ γ−1 ∞ R Cγ ϕ γ−1 ∞ R

Φ−1 (R)

0

R 0

⎛ r ⎞ ⎝ |b(s)|γ ds⎠ Φ (r)r−Φ(r) dr r2

Φ−1 (R)

0

0

Φ(r) dr r

t

dt

Φ−1 (t)

Φ (Φ−1 (t))

≤ Cγ ϕ γ−1 ∞ (1 + ϕ ∞ ).

This completes the proof of Lemma 2.1.

Remark 2.2. We note that in our case, |Φ|−1 (t) = Φ−1 (t) if Φ is nonnegative, and |Φ|−1 (t) = Φ−1 (−t) if Φ is nonpositive. Lemma 2.3. Let Φ and ϕ be the same as in Theorem 1.1. Then TΩ,Φ,b f (x) ⎧ TΩ,ϕ(Φ−1 )b(Φ−1 ) f (x), ⎪ ⎪ ⎪ ⎨−T Ω,ϕ(Φ−1 )b(Φ−1 ) f (x), = ⎪ TΩ,ϕ(Φ −1 (−·))b(Φ−1 (−·)) f (x), ˜ ⎪ ⎪ ⎩ −TΩ,ϕ(Φ −1 (−·))b(Φ−1 (−·)) f (x), ˜

if if if if

Φ Φ Φ Φ

is is is is

nonnegative and increasing, nonnegative and decreasing, nonpositive and decreasing, nonpositive and increasing, (2.2)

˜ where Ω(y) = Ω(−y). Proof. We discuss only the case where Φ is nonnegative and increasing, since in the other cases one can prove in a quite similar way. Applying the change

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of variable t = Φ(r), we have Ω(y ) b(|y|)f (x − Φ(|y|)y )dy |y|n ε<|y|

A

= S n−1 ε

= S n−1

= S n−1

⎛ ⎜ ⎝

Ω(y ) b(r)f (x − Φ(r)y )drdσ(y ) r Φ(A)

⎞ −1

b(Φ

Φ(ε)

⎛ ⎜ ⎝

Φ(A)

t (t))Ω(y )f (x − ty ) ⎟ dt⎠ dσ(y ) −1 t Φ (t)Φ (Φ−1 (t)) ⎞

−1

b(Φ

Φ(ε)

= Φ(ε)<|y|<Φ(A)

(t))Ω(y )f (x − ty ) ⎟ ϕ(Φ−1 (t))dt⎠ dσ(y ) t

Ω(y ) ϕ(Φ−1 (|y|))b(Φ−1 (|y|))f (x − y)dy, |y|n

and Lemma 2.3 follows.

(2.3)

Proofs of Theorems 1.1 and 1.6. Using Lemma 2.1, Lemma 2.3 and Theorem A for the special case Φ(t) = t, we get Theorem 1.1. Also, using Lemma 2.1 Lemma 2.3 and Theorems 1 and 2 in [9] for the special case Φ(t) = t, we may get the desired assertion in our Theorem 1.6. By Theorems 1.1 and 1.6, we obtain the following two corollaries. Corollary 2.4. Let 1 < p ≤ γ, γ ≥ 2, p = ∞, and w1/(1−p) ∈ A˜p /γ (R+ ). If Φ, b, Ω are the same as in Theorem 1.6, then TΩ,Φ,b is bounded on Lp (w). This follows by duality and Theorem 1.6. Corollary 2.5. Let Φ, b, Ω be the same as in Theorem 1.6. For p0 ≥ γ , p1 ∈ (1, γ ) and t ∈ [0, 1], let r(t) = tp0 /(p1 (1 − t) + tp0 ) and pt = p0 p1 / (p1 (1−t)+tp0 ). Suppose w ∈ A˜p/γ (R+ ), then TΩ,Φ,b is bounded on Lpt (wr(t) ). Proof. We can get this corollary by interpolating with change of measures between Theorems 1.1 and 1.6.

3. Several Further Applications In this section, we will give some extensions of known results by applying Lemmas 2.1 and 2.3. 3.1. The Maximal Singular Integrals ∗ Firstly, we will study the truncated maximal operator TΩ,Φ,b of TΩ,Φ,b . For any 0 < ε < A, we deﬁne ε,A TΩ,Φ,b f (x) = b(|y|)|y|−n Ω(y )f (x − Φ(|y|)y )dy, ε<|y|
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and ε,A ∗ TΩ,Φ,b f (x) = sup |TΩ,Φ,b f (x)|. 0<ε
∗ ∗ In the sequel, we denote TΩ,Φ,b = TΩ,b when Φ(t) = t for simplicity. As is well∗ known, the boundedness of TΩ,Φ,b on Lp (ω) implies the almost everywhere ε,A existence of the limit limε→0,A→∞ TΩ,Φ,b f (x), the principal value deﬁning p TΩ,Φ,b for f ∈ L (ω).

Theorem 3.1. Suppose Ω ∈ H 1 (S n−1 ) satisfies (1.1), b ∈ Δγ for some γ ≥ 2 ∗ is bounded on Lp (ω), and Φ be the same as in Theorem 1.1. Then TΩ,Φ,b I ˜ provided ω ∈ Ap/γ with max{1, γ } < p < ∞. Proof. By Lemmas 2.1 and 2.3, we have b(|Φ|−1 )ϕ(|Φ|−1 ) ∈ Δγ with γ ≥ 2 ∗ ∗ = TΩ,ϕ(|Φ| and TΩ,Φ,b −1 )b(|Φ|−1 ) . Hence, using Theorem 3 in [9] for the special case Φ(t) = t we get the desired conclusion. 3.2. Weighted Inequalities for the Weight |x|α Secondly, applying Lemma 2.1, Lemma 2.3 and Theorem 4 in [9] for Φ(t) = t, we may get the following weighted boundedness of TΩ,Φ,b with power weights |x|α and α ∈ R: Theorem 3.2. Suppose Ω ∈ H 1 (S n−1 ) satisfies (1.1), b ∈ Δγ for some γ ≥ 2 and Φ(t) be the same as in Theorem 1.1. Then TΩ,Φ,b is bounded on Lp (|x|α ), provided α ∈ (−1, p/γ − 1) with max{1, γ } < p < ∞. Interpolating with change of measures between Theorems 1.1 and 3.2 we can obtain the following Corollary 3.3. Let Φ, b, Ω be the same as in Theorem 1.6. For p0 ≥ γ , p1 ∈ (1, γ ) and t ∈ [0, 1], let pt = p0 p1 /(p1 (1 − t) + tp0 ). Then TΩ,Φ,b is bounded on Lpt (|x|α dx), provided that αpt /p1 ∈ (−1, p0 /γ − 1). 3.3. Boundedness of Commutators Thirdly, we discuss the boundedness of commutator. Deﬁne the k-th commutator of TΩ,Φ,b and a function h by Ω(y ) k [h, TΩ,Φ,b ] (f )(x) = p. v. b(|y|)(h(x) |y|n Rn

− h(x − Φ(|y|)y ))k f (x − Φ(|y|)y )dy. For the kth commutator [h, TΩ,Φ,b ]k we obtain the following theorem, Theorem 3.4. Suppose Ω ∈ Lq (S n−1 )(q > 1) satisfies (1.1), q ≤ p < ∞ and b ∈ L∞ (R+ ) and h ∈ BMO(Rn ). Let Φ be the same function as in Theorem 1. Then [h, TΩ,Φ,b ]k is bounded on Lp (w) for w ∈ Ap/q . Proof. We discuss only the case where Φ is nonnegative and increasing, since in the other cases one can prove in a quite similar way. In a way similar to

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the proof of Lemma 2.3, we easily get k [h, TΩ,Φ,b ] (f )(x) = lim

ε→0, A→∞Φ(ε)<|y|<Φ(A)

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Ω(y ) |y|n

×ϕ(Φ−1 (|y|))b(Φ−1 (|y|))(h(x) − h(y))k f (x − y)dy. From the assumptions on Φ and b, it follows that the function ϕ(Φ−1 (|y|)) × b(Φ−1 (|y|)) is a radial and bounded function. Hence, by the remark to Theorem 5 in [6, p. 878] we obtain that TΩ,Φ,b is bounded from Lp (w) to Lp (w) for w ∈ Ap/q . Thus, by using Theorem 2.13 in [1] we can deduce that [h, TΩ,Φ,b ]k is Lp (w) bounded for w ∈ Ap/q . 3.4. Boundedness on Triebel–Lizorkin and Besov Spaces Recently, Chen and Ding [3] showed that the rough singular integral TΩ,b with Ω ∈ H 1 (S n−1 ) and b ∈ Δ∞ is bounded on the Triebel–Lizorkin spaces and Besov spaces. For 0 < p, q ≤ ∞(p = ∞) and s ∈ R, the homogeneous Triebel–Lizorkin space F˙ ps,q (Rn ) is deﬁned by ⎧ ⎫ ⎛ ⎞1/q

⎪

⎪ ⎨ ⎬

−jsq q⎠

⎝ F˙ ps,q (Rn ) = f ∈ S (Rn ) : f F˙ps,q = 2 |Ψ ∗ f | < ∞ j

p

⎪ ⎪ ⎩ ⎭ L j∈Z

(3.1) and the homogeneous Besov space B˙ ps,q (Rn ) is deﬁned by ⎧ ⎫ ⎞1/q ⎛ ⎪ ⎪ ⎨ ⎬ B˙ ps,q (Rn ) = f ∈ S (Rn ) : f B˙ ps,q = ⎝ 2−jsq Ψj ∗ f qLp ⎠ <∞ , ⎪ ⎪ ⎩ ⎭ j∈Z (3.2) j (ξ) = φ(2j ξ) where S (Rn ) denotes the tempered distribution class on Rn , Ψ ∞ n for j ∈ Z and φ ∈ Cc (R ) satisﬁes the following conditions: (i) 0 ≤ φ(x) ≤ 1; (ii) supp(φ) ⊂ {x : 1/2 ≤ |x| ≤ 2}; (iii) φ(x) > c > 0 if 3/5 ≤ |x| ≤ 5/3. The inhomogeneous versions of Triebel–Lizorkin space and Besov spaces, which are denoted by Fps,q (Rn ) and Bps,q (Rn ), respectively, are obtained by adding the term Θ∗f p to the right side of (3.1) or (3.2) with j∈Z replaced ∞ ⊂ {ξ : |ξ| ≤ 2}, Θ(x) by , where Θ ∈ S (Rn ), supp Θ > c > 0 if j≥1

|x| ≤ 5/3. Using Lemmas 2.1 and 2.3, and by the conclusions in Theorems 1 and 2 in [3] with the remark (see [3, p. 500]), we get immediately the boundedness of TΩ,Φ,b on the Triebel–Lizorkin space and Besov spaces:

Theorem 3.5. Suppose Ω ∈ H 1 (S n−1 ) satisfies (1.1), b ∈ Δ∞ and Φ is the same as in Theorem 1.1. Then

Vol. 68 (2010) (a) (b) (c) (d)

TΩ,Φ,b f F˙ps,q TΩ,Φ,b f Fps,q TΩ,Φ,b f B˙ ps,q TΩ,Φ,b f Bps,q

On Singular Integral Along Surface ≤ C f F˙ps,q ≤ C f Fps,q ≤ C f B˙ ps,q ≤ C f Bps,q

for for for for

159

1 < p, q < ∞; 1 < p, q < ∞ and s > 0; 1 < p, q < ∞; 1 < p, q < ∞ and s > 0.

3.5. Boundedness on Morrey Spaces In this section, we study the boundedness of TΩ,Φ,b in the following generalized Morrey spaces. Definition 3.6. Let φ(r) be a positive increasing function on (0, ∞) and satisfy φ(2r) ≤ Dφ(r)(r > 0) for some 1 ≤ D < 2n . Let 1 ≤ p < ∞. We denote by Lp,φ = Lp,φ (Rn ) the space of locally integrable functions f for which |f (x)|p dx ≤ C p φ(r) Br (x0 )

for all x0 ∈ R and r > 0, where Br (x0 ) is the ball with center x0 and radius r > 0. n

Combining Lemma 2.1, Lemma 2.3 with Theorem 5 in [9] for Φ(t) = t, we obtain the boundedness of TΩ,Φ,b on the generalized Morrey spaces Lp,φ : Theorem 3.7. Let φ be a positive increasing function on (0, ∞) and satisfy φ(2r) ≤ Dφ(r)(r > 0) for some 1 ≤ D < 2n . Suppose Ω ∈ H 1 (S n−1 ) satisfies (1.1), b ∈ Δγ for some γ ≥ 2 and Φ be the same as in Theorem 1.1. Then TΩ,Φ,b is a bounded operator on Lp,φ for max{1, γ } < p < ∞. 3.6. Boundedness on Herz Spaces Finally, we consider the case of Herz spaces. For simplicity, we only discuss the boundedness of TΩ,Φ,b in homogeneous Herz spaces here. Let Bk = {x ∈ Rn ; |x| ≤ 2k } and Ck = Bk \Bk−1 for k ∈ Z. Let χk denote the characteristic function of the set Ck . Definition 3.8. Let α ∈ R and 0 < p, q < ∞. The homogeneous Herz space K˙ pα,q (Rn ) is deﬁned by K˙ pα,q (Rn ) = {f ∈ Lploc (Rn \{0}); f K˙ pα,q (Rn ) < ∞}, ∞ where f K˙ pα,q (Rn ) = { k=−∞ 2kαq f χk qLp (Rn ) }1/q . Theorem 3.9. Suppose Ω ∈ H 1 (S n−1 ) satisfies (1.1), 1 γ . Let Φ(t) be the same as in Theorem 1.1. Then TΩ,Φ,b is a bounded linear operator in K˙ pα,q (Rn ), provided 0 < q < ∞ and α ∈ (−1/p, 1/γ − 1/p). Proof. Since b(|Φ|−1 )ϕ(|Φ|−1 ) ∈ Δγ with γ ≥ 2 by Lemma 2.1 and Lemma 2.3, we get the desired conclusion by Theorem 6 in [9] for the case Φ(t) = t. Acknowledgements The authors would like to express their gratitude to the referees for their valuable comments.

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References [1] Alvarez, J., Bagby, R., Kurtz, D., P´erez, C.: Weighted estimates for commutators of linear operators. Stud. Math. 104, 195–209 (1993) [2] Calder´ on, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956) [3] Chen, Y., Ding, Y.: Rough singular integral operators on Triebel–Lizorkin spaces and Besov spaces. J. Math. Anal. Appl. 347, 493–501 (2008) [4] Connett, W.: Singular integrals near L1 . Proc. Symp. Pure Math. Am. Math. Soc. 35(Part I), 163–165 (1979) [5] Ding, Y., Lu, S.: Weighted Lp -boundedness for higher order commutators of oscillatory singular integrals. Tohoku Math. J. 48, 437–449 (1996) [6] Duoandikoetxea, J.: Weighted norm inequalities for homogeneous singular integrals. Trans. Am. Math. Soc. 336, 869–880 (1993) [7] Duoandikoetxea, J., Rubiode Francia, J.L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541–561 (1986) [8] Fan, D., Pan, Y.: A singular integral operators with rough kernel. Proc. Am. Math. Soc. 125, 3695–3703 (1997) [9] Fan, D., Pan, Y., Yang, D.: A weighted norm inequality for rough singular integrals. Tohoku Math. J. 51, 141–161 (1999) [10] Feﬀerman, R.: A note on singular integrals. Proc. Am. Math. Soc. 74, 266–270 (1979) [11] Grafakos, L., Stefanov, A.: Convolution Calder´ on-Zygmund singular integral operators with rough kernels, Analysis of Divergence (Orono, ME, 1997), pp. 119–143, Appl. Numer. Harmon. Anal., Birkh¨ auser Boston, Boston, MA, USA (1999) [12] Ricci, F., Weiss, G.: A characterization of H 1 (Σn−1 ). Proc. Symp. Pure Math. 35(part I), 289–294 (1979) [13] Seeger, A.: Singular integral operators with rough convolution kernels. J. Am. Math. Soc. 9, 95–105 (1996) [14] Xue, Q., Yabuta, K.: Correction and Addition to: “L2 -boundedness of Marcinkiewicz integrals along surfaces with variable kernels”. Sci. Math. Jpn. 65, 291–298 (2007)

Yong Ding (B) and Qingying Xue School of Mathematical Sciences Laboratory of Mathematics and Complex Systems Ministry of Education Beijing Normal University Beijing 100875, China e-mail: [email protected] Qingying Xue Institute of Applied Physics and Computational Mathematics PO Box 8009, Beijing 100088, China e-mail: [email protected]

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On Singular Integral Along Surface

Kˆ ozˆ o Yabuta School of Science and Technology Kwansei Gakuin University Gakuen 2-1 Sanda 669-1337 Japan e-mail: [email protected] Received: June 1, 2009. Revised: June 25, 2010.

161

Integr. Equ. Oper. Theory 68 (2010), 163–191 DOI 10.1007/s00020-010-1785-8 Published online March 23, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Discrete-Time Multi-Scale Systems Daniel Alpay and Mamadou Mboup Abstract. We introduce multi-scale ﬁltering by the way of certain double convolution systems. We prove stability theorems for these systems and make connections with function theory in the polydisc. Finally, we compare the framework developed here with the white noise space framework, within which a similar class of double convolution systems has been deﬁned earlier. Mathematics Subject Classification (2000). Primary 94A12, 47N70, 46E22; Secondary 93D25, 42A70. Keywords. Discrete-scale transformation, scale invariance, linear systems, self-similarity, reproducing kernels.

Contents 1. Introduction 164 2. Scaling Operator for Discrete-Time Signals 167 3. Discrete-Scale Invariant Systems and Signals 171 4. The Trigonometric Moment Problem 174 5. The Case of One Generator 176 6. The Trigonometric Moment Problem in Compact Semialgebraic Sets179 7. The Case of a Finite Number of Generators 180 8. BIBO Stability 182 9. Dissipative Systems 184 185 10. 1 -2 Bounded Systems 11. The White Noise Space Setting and a Table 187 References 189 D. Alpay thanks the Earl Katz family for endowing the chair which supported his research. This research is part of the European Science Foundation Networking Program HCAA, and was supported in part by the Israel Science Foundation grant 1023/07.

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1. Introduction In the present work we study a new type of double convolution system, which arises in the theory of multi-scale systems. We use the approach of the second named author presented at the Mathematical Theory of Networks and Systems conference in 2006 in Kyoto (see [28]). Let az + b a b γ= ∈ SU (1, 1) and let γ(z) = c d cz + d be the corresponding automorphism of the open unit disc D. Following [28], consider the map 1 f (γ(z)) Tγ (f )(z) = cz + d where f is analytic in D. Then Tγ f is also analytic in D. Let f (z) =

∞

fn z n

and

(Tγ f )(z) =

n=0

∞

fn,γ z n

n=0

be the Taylor expansions of f and Tγ f respectively. Let N = {1, 2, . . .}

and

N0 = {0, 1, 2, . . .} .

To simplify the notation, we henceforth write fn (γ) in place of fn,γ . For a hyperbolic transformation γ, the map which associates to the sequence {fn }n∈N0 the sequence {fn (γ)}n∈N0 will be called the scaling operation (see the precise definition in Sect. 2). Thus, the signal {fn (γ)}n∈N0 deﬁnes the scale shift of {fn }n∈N0 at scale αγ , the associated multiplier of γ as given in equation (2.3). This definition was motivated in [28] by the study of the selfsimilarity property [27]. This property, which appears in many engineering applications, in particular on high quality LAN ethernet network trafﬁc (see [25]), may be seen as a weighted form of stationarity in scale. Now the scale shift does not admit a clear cut definition for discrete-time signals. Consider now a discrete subgroup Γ of SU (1, 1), which represents the scales we will use to study the signals and systems. One associates to the sequence {fn }n∈N0 its scale transform {fn (γ)}n∈N0 ,γ∈Γ , which is a function of n ∈ N0 and γ ∈ Γ. We introduce linear systems as expressions of the form yn =

n

hn−m um ,

(1.1)

m=0

where denotes the convolution in Γ. Thus, ⎛ ⎞ n ⎝ yn (γ) = hn−m (γ ◦ ϕ−1 )um (ϕ)⎠ , m=0

γ ∈ Γ.

(1.2)

ϕ∈Γ

be its dual group (the group of unimodular Take Γ Abelian and let Γ characters). Then the Fourier transform (with respect to Γ) of both sides of (1.2) and the Z transform yield Y (z, σ) = H(z, σ)U (z, σ),

z ∈ D, σ ∈ Γ,

(1.3)

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where Y (z, σ) =

∞

z n y n (σ) and U (z, σ) =

n=0

∞

znu n (σ),

(1.4)

n=0

and where, for every z ∈ D the equality in (1.3) is μ -a.e. ( μ is the Haar measure on Γ). The function H(z, σ) can be seen as the transfer function of the discrete-time scale-invariant system. Formula (3.8) suggests to deﬁne and study

n depend on σ in hierarchies of transfer functions, for which the functions h some pre-assigned way (for instance, when they are polynomials in σ), or when the function H(z, σ) is a rational function of z or of σ. When Γ is trivial we recover causal discrete time-invariant linear systems of the form yn =

n

hn−m um ,

n = 0, 1, . . .

(1.5)

m=0

where (hn ) is the impulse response and where the input sequence (um ) and output sequence (ym ) belong to some sequences spaces. The function H(z, σ) reduces then to the classical transfer function. We study for these systems in particular the notions of dissipativity and of bounded input bounded output (BIBO) stability. See for instance [1] for a survey. We now brieﬂy discuss the parallels and differences between the present work and other studies. In [4] (see also [3,5]), the ﬁrst named author together with David Levanony considered another example of double convolution system, when both hn and un are random variables, which belong to the white noise space, or more generally to the Kondratiev space. The product hn−m um in (1.5) is then replaced by the Wick product. The Wick product takes the form of a convolution with respect to an appropriate basis, and we get a double convolution system. Using the Hermite transform, one can deﬁne a generalized transfer function, which is a function analytic in z and in a countable number of other variables (these variables take into account the randomness). The white noise space setting is reviewed in the last section of this paper, with purpose the comparison between the present paper and [4]. There are parallel and analogies between the theory of linear stochastic systems presented in [4] and the theory developed here. These parallels are pointed out in the sequel, and serve as guide and motivation for some of the proofs in the present paper. Still, there are some differences between the statements and the proofs of the stability theorems in [4] and the proofs given here. These differences are pointed out in the text. Our approach is different from that of wavelets. Indeed, the scale transform is the starting point of our approach (initiated in [28]) to multi-scale analysis in discrete time. In opposition to wavelets, we propose a transform which has on the same level both the time and the scale aspect. Let us now

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elaborate on the differences of our approach to wavelets. Recall that in continuous time the wavelet transform of a signal f is deﬁned by ∗ t−u 1 dt, W f (u, s) = f (t) √ ψ s s R

where s is the scale parameter and ψ is the mother wavelet; see [26, pp. 78–79]. The discrete time version of the transform is obtained, for discrete scales s = aj , by discretizing the above integral as in ∗ fk ψk−n,j , W f [n, aj ] = n

k

where ψn,j = √1aj ψ aj and fk = f (k), assuming a sampling period normalized by 1 (see [26, pp. 88–89]). The decomposition provides an appropriate mathematical tool for signal analysis. In particular, it makes it possible to extract the components of a given discrete-time signal at a given scale on a discrete grid. However, the question of deﬁning the scale shift operator (dilation) for purely discrete-time signals is dodged somehow. Another point of departure from our approach is that the wavelet transform (either continuous- or discrete-time) has only one convolution as compared to the double time and scale convolution considered in the present work (see equation (1.2) above). Finally we mention that in the case which we will consider, the group Γ will be indexed by Zp , but the resulting setting is completely different from the one of classical N D theory. It is also different from the settings described in for instance the works [11–13]. We now turn to the outline of this paper. It consists of 10 sections besides the introduction. In Sect. 2 we review the approach to discrete multiscale systems presented in [28]. In Sect. 3 we deﬁne the systems which we will study in the paper, and the related notion of transfer function. Sect. 4 is of a review nature. We discuss the trigonometric moment problem and related reproducing kernel Hilbert spaces of the type introduced by de Branges and Rovnyak. In Sect. 5 we consider the case where the sub-group has one generator and is inﬁnite. We use the classical one dimensional moment problem to associate to the Haar measure of the dual group a uniquely deﬁned measure on the unit circle. This allows us to use function theory on the disc and on the bidisc. In Sect. 6 we review some deep results of Mihai Putinar, see [29], on positive polynomials on compact semi-algebraic sets and their use to solve the trigonometric moment problem on the polydisc. In Sect. 7 we consider the case where Γ is no more cyclic but has a ﬁnite number of generators. Although the results in Sect. 5 are particular cases of the ones in Sect. 7 we have chosen to present both for ease of exposition. The next three sections consider stability results: bounded input bounded output (BIBO) stability is considered in Sect. 8, dissipative systems are studied in Sect. 9 and Sect. 10 is devoted to the counterpart of 1 − 2 stability. In the last section we review the white noise space setting, and present a table and some

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remarks, which point out the analogies between the setting in [4] and the present work. To conclude this introduction, we note that a general theory of double convolution systems is in preparation, [6]. We also note that some of the results presented here have been announced in [8].

2. Scaling Operator for Discrete-Time Signals We brieﬂy summarize the approach to multi-scale systems presented in [28]. We ﬁrst note the following: If F (s), (s) 0, denotes the Laplace transform √ of a continuous-time signal f (t), t 0, then, for any α = 1/β > 0, αF (αs) is the Laplace transform of f (βt). Therefore, time scaling has a similar form in the frequency domain. As opposed to the continuous-time case, time scaling is not clearly deﬁned in the discrete-time setting. Nevertheless, the preceding remark is the key step to deﬁne a scaling operator for discrete-time signal. Consider the M¨ obius transformation π eiθ − s , |θ| < Gθ (s) = −iθ e +s 2 which maps conformally the open right half-plane C+ onto the open unit disc. To recall our definition of the scaling operator (see [28] and also [9]), we note that the scale shift in C+ , Sα : s → Sα (s) = αs,

α>0

translates in the unit disc, via Gθ (s), into the hyperbolic transformation γ{α} (z) = (Gθ ◦ Sα ◦ G−1 θ )(z) =

(eiθ + αe−iθ ) z + (1 − α) . (1 − α)z + (e−iθ + αeiθ )

(2.1)

Any such transformation maps the open unit disc (resp. the unit circle) into itself. Now, the most general fractional linear transformation which maps the open unit disc (resp. the unit circle) into itself has the form γ1 z + γ2 , |γ1 |2 − |γ2 |2 = 1. (2.2) γ(z) = ∗ γ2 z + γ1∗ If |(γ1 )| > 1, then the transformation is hyperbolic [16] and it takes the form z − ξ1 γ(z) − ξ1 = αγ , αγ > 0, (2.3) γ(z) − ξ2 z − ξ2 √ λ∗ [(γ1 )]2 −1+i(γ1 ) λγ where ξ1 = = γ ∗ and ξ2 = − γγ∗ are the two ﬁxed points. γ2∗ 2 2 The constant αγ is called the multiplier of the transformation, see [16, p. 15], and is given by (γ1 ) − [(γ1 )]2 − 1 . (2.4) αγ = (γ1 ) + [(γ1 )]2 − 1 Noting that |ξ1 | = |ξ2 | = 1, one may rearrange (2.3) to obtain λγ − λ∗γ eiξγ z λγ − λ∗γ eiξγ γ(z) = α , γ 1 + eiξγ γ(z) 1 + eiξγ z

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where eiξγ = iθγ

e

=

λγ |λγ | ,

λγ γ2 .

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Dividing both sides of this equality by |λγ | and setting

we recover (2.1) up to a rotation: iξγ z). eiξγ γ(z) = (Gθγ ◦ Sαγ ◦ G−1 θγ )(e

(2.5)

Any hyperbolic transformation γ of the form (2.2) is thus (conformally) equivalent to a scale shift Sαγ in C+ . In the sequel, we will be interested in Abelian subgroups of hyperbolic transformations, but the remainder of this section deals with general fractional linear transformations az + b ϕ(z) = , with ad − bc = 1, cz + d from the open unit disc onto itself. To each such transformation, we associate (in bold letters) a b ϕ= ∈ SU (1, 1), c d and if m is a nonnegative integer, we deﬁne (Tϕ f )(z) =

1 f (ϕ(z)). (cz + d)m

Such transformations appear in a number of places. For instance, a function f such that (Tϕ f ) = σ(ϕ)f for all ϕ in a Fuchsian group1 is called character-automorphic of weight m, with character σ (see e.g. [31]). We mention in particular [18, §5], where they also appear (with the transpose ϕt instead of ϕ) in the setting of integral geometry and harmonic analysis of the complex projective line. We may also mention the theory of modular forms (then, the entries are in Z); See for instance [17, Chapter 6]. However, in the present study, we consider only transformations with weight 1, that is m is ﬁxed in all the sequel to m = 1. The reason is that we require Tϕ to be unitary in the Hardy class H2 (D) (see Theorem 2.3). The following result is straightforward, and its proof will be omitted. It is mentioned in [18, p. 434] (with the opposite order of the transformation because of the transposition used there). Lemma 2.1. Let ϕ1 and ϕ2 belong to SU (1, 1). Then Tϕ2 ◦ Tϕ1 = Tϕ1 ϕ2 ,

(2.6)

and in particular for every ϕ ∈ SU (1, 1) it holds that Tϕ−1 = (Tϕ )−1 .

(2.7)

Let f be analytic in the open unit disc. Then Tϕ f is also analytic in the open unit disc. The mapping Tϕ induces a mapping from the space of 1

A Fuchsian group is a properly discontinuous group (the identity transformation is isolated) each of whose transformation maps D, T and C\D onto themselves [23].

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sequences of coefﬁcients of power seriesof functions analytic in a neighbor∞ hood of the origin into itself: if f (z) = n=0 z n xn is the power series expansion at the origin of the function f analytic in D, then (Tϕ f )(z) =

∞

xn,ϕ z n .

n=0

In view of (2.7) we have: Proposition 2.2. Let ϕ ∈ SU (1, 1), and let {an }n∈N0 be a sequence of complex numbers such that lim supn−→∞ |an+1 |1/(n+1) ≤ 1. Then there exists a sequence of complex numbers {bn }n∈N0 such that lim sup |bn+1 |1/(n+1) ≤ 1, n−→∞

and

Proof. Let a(z) = formula

an = bn,ϕ ,

∞

n=0

n = 0, 1, . . .

an z n . It sufﬁces to deﬁne a series bn,ϕ via the

∞

bn,ϕ z n = (Tϕ−1 a)(z),

n=0

and use (2.7).

Theorem 2.3. The operator Tϕ is unitary from H2 (D) onto itself with norm equal to 1. It is also continuous from H∞ (D) into itself with norm equal to 1/(|d| − |c|). Proof. First recall the formula 1 − ϕ(z)ϕ(w)∗ 1 = 1 − zw∗ (cz + d)(cw + d)∗

(2.8)

where z, w are in the domain of definition of ϕ. Furthermore, recall that a function f deﬁned in D is analytic there and belongs to H2 (D), with f H2 (D) ≤ 1, if and only if the kernel 1 − f (z)f (w)∗ 1 − zw∗ is positive in D; see for instance [1, Theorem 2.6.6]. We now compute Δ(z, w) =

1 − (Tϕ f )(z)(Tϕ f (w))∗ 1 − zw∗

for z, w ∈ D. Using (2.8) we can write: Δ(z, w) =

1 (1 − −

ϕ(z)ϕ(w)∗ )(cz

+ d)(cw + d)∗

1 f (ϕ(z))f (ϕ(w))∗ (cz + d)(cw + d)∗

(2.9)

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1 (cz + d)(cw + d)∗ 1 ∗ × − f (ϕ(z))f (ϕ(w)) . 1 − ϕ(z)ϕ(w)∗

The kernel 1 − f (ϕ(z)f (ϕ(w))∗ (1 − ϕ(z)ϕ(w)∗ ) is positive in D since the kernel (2.9) is positive there. It follows that Δ(z, w) is positive in the open unit disc, and thus, by [1, Theorem 2.6.6], the function Tϕ (f ) belongs to H2 (D) and has norm less or equal to 1. To prove that the norm is indeed equal to 1 we use (2.6) and (2.7), which imply that 1 ≤ Tϕ · Tϕ−1 , which, together with the fact that both Tϕ and Tϕ−1 have norm less that 1 implies that Tϕ = 1. We now show that Tϕ is unitary. Let f ∈ H2 (D). We have: f H2 (D) = Tϕ−1 Tϕ (f ) H2 (D) ≤ Tϕ (f ) H2 (D) ≤ f H2 (D) , since both Tϕ and Tϕ−1 are contractive. It follows that Tϕ is unitary. The second claim is easily veriﬁed.

We therefore associate to the signal x = {xn }n∈N0 ∈ 2 the signal indexed by N0 × SU (1, 1). In the sequel, we will simplify {xn,ϕ }n∈N0 ϕ∈SU (1,1)

the notation by writing xn (ϕ) in place of xn,ϕ . For ﬁxed n, {xn (ϕ)}, with ϕ ∈ SU (1, 1), represents a scale signal, that is, the observation of the signal {xn } at time n through the scales ϕ ∈ SU (1, 1). In the sequel, we will consider (by convention) the zooming as corresponding to the “positive” scales. Definition 2.4. The scale-causal projection of {xn (ϕ)}, ϕ ∈ SU (1, 1) is given by the restriction of {xn (ϕ)} to the scales ϕ for which the multiplier is strictly less than one: αϕ < 1, where αϕ is the multiplier of the transformation, and is given by (2.4) (see also (2.3)). Now on, we consider only discrete Abelian subgroups of hyperbolic transformations. Definition 2.5. Given a discrete Abelian subgroup Γ of SU (1, 1) we denote by Γ+ the set of transformations consisting of the identity and of the scales ϕ for which the multiplier is strictly less than one: αϕ < 1. The system (1.1) will be scale-causal if the elements hn ∈ 2 (Γ+ ). Given γ and ϕ two elements of Γ, we will say that γ succeeds ϕ and will note ϕ γ, if γ ◦ ϕ−1 ∈ Γ+ that is: ϕγ

⇐⇒ αγ◦ϕ−1 1.

Proposition 2.6. The relation deﬁnes a total order in Γ.

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Proof. Since we assume that Γ is Abelian, all the transformations must have the same ﬁxed points. The parameters ξγ and θγ in (2.5) are therefore constant. The proof then follows upon noting that the multiplier αγ◦ϕ is given by: αγ◦ϕ = αγ αϕ . With this order we obtain a bijection γ → (γ)

(2.10)

between Γ and Z, and one can identify 2 (Γ) and 2 (Z) and 2 (Γ+ ) and 2 (N0 ). Remark 2.7. Using the isomorphism we introduce the following definition: Definition 2.8. The function u : Γ+ → C has ﬁnite support if

N (u) = max {(γ) such that u(γ) = 0} < ∞, where is the bijection deﬁned by (2.10). The support of the function u is the interval [0, N (u)] ⊂ N0 . The results of this section remain valid if we replace the Hardy space H2 (D) by its vector-valued version H2 (D) ⊗ H, where H is some Hilbert space. One can deﬁne in particular the scaling of random sequences when H is a probability space. Remark 2.9. In [28, equation (17)] another kind of systems is considered, with only one convolution. The main issue there is the notion of scaleinvariance in a stronger form, which will not be considered here; see also [7] for related work. Finally, we note that one could consider systems non-causal with respect to n, that is of the form yn = hn−m um . Z

Thus, there are really four possibilities for the various stability theorems we present, depending on whether we have time causality or not, and scalecausality or not. In this paper we only give part of all possible results.

3. Discrete-Scale Invariant Systems and Signals The scaling operators Tϕ form a group of operators from the Hardy space H2 (D) onto itself. Recall that we have discretized the scale axis and we have restricted ϕ to a discrete subgroup Γ of SU (1, 1). Also we take Γ Abelian (cyclic) and consisting of hyperbolic transformations. Recall again that Γ stands for the dual of Γ: it is formed by the set of functions σ : Γ → T such that σ(ι) = 1 and ∀ γ, ϕ, σ(γ ◦ ϕ) = σ(γ)σ(ϕ),

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are called where ι stands for the identity transformation. The elements of Γ characters of the group Γ (see [19]). We denote by μ the Haar measure of Γ, which is compact by the Pontryagin duality [19]. We recall the definition of the Fourier transform on Γ and of its inverse: x(γ)σ(γ)∗ , x (σ) = γ∈Γ

x (σ)σ(γ)d μ(σ).

x(γ) = Γ

The Haar measure d μ is normalized so that Plancherel’s theorem holds: 2 2 f 2 (Γ) = |f (γ)| = |f (σ)|2 d μ(σ) = f 2L2 (dμ) . γ∈Γ

Γ

See [15, Theorem 8.4.2 p. 123]. Definition 3.1. A signal will be a sequence {un (·)}n∈N0 of elements of 2 (Γ), and such that the condition sup un (·) 2 (Γ) < ∞

(3.1)

n=0,1,...

holds. A scale-causal signal will be a sequence {un (·)}n∈N0 of elements of 2 (Γ+ ), and such that the condition sup un (·) 2 (Γ+ ) < ∞

(3.2)

n=0,1,...

holds. In the sequel, we will impose the following stronger norm constrains on a signal, besides (3.1) or (3.2), namely: ∞

un (·) 22 (Γ) < ∞,

(3.3)

un (·) 2 (Γ) < ∞,

(3.4)

n=0

or ∞ n=0

and similarly for scale-causal signals. We note the following: a dissipative ﬁlter cannot be eﬀective at all scales. At some stage, details cannot be seen. These intuitive facts are made more precise in the following proposition. Proposition 3.2. (1)

Assume that the supports of the un are uniformly bounded. Then, (3.3) is in force. (2) Assume that the support of un is inﬁnite for all n. Then, the sum on the left side of (3.3) diverges.

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Proof. Let be the bijection (2.10), and let N be such that the support of all the functions γ → un (γ) is inside [0, N ]. Then, ∞

un (·) 22 (Γ) =

n=0

N ∞

|un (γ)|2

n=0 (γ)=0 N ∞

=

|un (γ)|2

(γ)=0 n=0

≤ N u 2 (N0 ) , since (see Theorem 2.3) the maps Tγ are unitary from H2 (D) onto itself. The second claim is proved similarly. An example of (un ) satisfying Condition (1) of Proposition (3.2) has been presented in the paper [28], where the corresponding group Γ is Fuchsian. This was used therein, to deﬁne the scale unit-pulse signal. A similar condition was also considered by Yuditskii [31] in the description of the direct integral of spaces of character-automorphic functions. Definition 3.3. An impulse response (resp. a scale-causal impulse response) will be a sequence {hn (·)}n∈N0 of elements of 2 (Γ) (resp. of 2 (Γ+ )) such that for every n ∈ N0 , the multiplication operator Mhn :

u → hn u,

n = 0, 1, . . .

(3.5)

is bounded from 2 (Γ) into itself (resp. from 2 (Γ+ ) into itself) and such that sup hn 2 (Γ) < ∞

(resp.

n=0,1,...

sup hn 2 (Γ+ ) < ∞).

(3.6)

n=0,1,...

We note that we do not require the operator norms of the operators Mhn to be uniformly bounded in n. There is no direct connections between the norm hn 2 (Γ) and the operator norm of Mhn . Condition (3.6) is needed for expression (3.8) below to make sense. The systems that we consider here are deﬁned by (1.1), that is, by the double convolution (1.2), that we recall below: −1 hn−m (γ ◦ δ )um (δ) . (3.7) yn (γ) = m∈Z

δ∈Γ

In view of (3.6) the series H(z, σ) =

∞

z n hn (σ)

(3.8)

n=0

converges in the L2 (d μ) norm for every z ∈ D. Taking the Fourier transform (with respect to Γ) of both sides of (1.2) we obtain y n (σ) =

n m=0

xm (σ), hn−m (σ)

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μ) sense. Taking now the Z transform we where the equality is in the L2 (d get (1.3): Y (z, σ) = H(z, σ)U (z, σ), where Y (z, σ) is deﬁned by (1.4): ∞ ∞ z n y n (σ) and U (z, σ) = znu n (σ), Y (z, σ) = n=0

n=0

and where, for every z ∈ D the equality in (1.3) is μ -a.e. The function H(z, σ) can be seen as the transfer function of the discrete-time scale-invariant system. Formula (3.8) suggests to deﬁne and study

n depend on σ in hierarchies of transfer functions, for which the functions h some pre-assigned way (for instance, when they are polynomials in σ), or when the function H(z, σ) is a rational function of z or of σ. In the next two sections, under the hypothesis that the sub-group Γ has a ﬁnite number, say p, of generators, we will associate to the system (1.1) an analytic function of p + 1 variables, which we will call the generalized transfer function of the system.

4. The Trigonometric Moment Problem We ﬁrst gather some well known facts on the trigonometric moment problem in form of a theorem. Theorem 4.1. Given an inﬁnite sequence . . . , t−1 , t0 , t1 , . . . of complex numbers such that t−n = t∗n ,

n = 0, 1, . . . ,

there exists a positive measure dν on [0, 2π) such that 2π tn =

e−inθ dν(θ),

n ∈ N0 ,

0

if and only if all the Toeplitz matrices TN = (tn−m )n,m=0,...,N are non-negative. See for instance [24, Theorem 2.7 p. 66]. The measure is then unique (when normalized). We also recall that the sequence (tn ) and the measure dν are related by 2π iθ ∞ e +z n dν(θ), tn z = t0 + 2 eiθ − z n=1 0

and thus the function Φ(z) = t0 + 2

∞ n=1

2π n

tn z = 0

eiθ + z dν(θ) eiθ − z

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is analytic and has a positive real part in the open unit disc. Using Stieltjes inversion formula, one can recover ν from Φ via the formula b Re{Φ(reiθ )}dθ = ν(b) − ν(a− ),

lim

r→1 a

where we assume that ν is right continuous. We also recall that the function Φ(z) + Φ(w)∗ = 2(1 − zw∗ )

2π (eiθ 0

dν(θ) − z)(eiθ − w)∗

is positive for z, w ∈ C\T, where T denotes the unit circle. We denote by L+ (Φ) the associated reproducing kernel Hilbert space when z and w are restricted to the open unit disc. The following result has ﬁrst been proved by de Branges and Shulman [14]. In the statement, H2 (dν) denotes the closed linear span in L2 (dν) of the functions z m for m ≥ 0. Theorem 4.2. The space L+ (Φ) consists of the functions of the form h(z) =

2π 0

h(eit )eit dν(t), eit − z

h ∈ H2 (dν),

with norm h L+ (Φ) = h H2 (dν) . We now recall some results on the structure of the space H2 (dν). Theorem 4.3. Assume that H2 (dν) = L2 (dν). Then H2 (dν) is a reproducing kernel Hilbert space, and its reproducing kernel is of the form A(z)A(w)∗ − B(z)B(w)∗ , 1 − zw∗ where A(z) and B(z) are functions analytic oﬀ the unit circle. Proof. We assume that H2 (dν) = L2 (dν), and let h0 ∈ L2 (dν) H2 (dν). Let α ∈ C\T be such that 2π 0

h0 (eiθ )∗ dν(θ) = 0. eiθ − α

Let p be a polynomial; then Rα p ∈ H2 (dν), where (Rα p)(z) =

p(z) − p(α) . z−α

Then Rα p , h0 H2 (dν) = 0,

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and therefore we obtain

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2π 0

p(α) =

p(eiθ )h0 (eiθ )∗ dν(θ) eiθ −α 2π h0 (eiθ )∗ dν(θ) 0 eiθ −α

.

Therefore the map p → p(α) is continuous on the polynomials, and extends to a continuous map to H2 (dν). Therefore H2 (dν) is a reproducing kernel Hilbert subspace of L2 (dν). The proof is then ﬁnished by using [2, Theorem 3.1 p. 600].

5. The Case of One Generator In this section, we consider the case of a cyclic group Γ, generated by a hyperbolic transformation γ 0 ∈ SU (1, 1). Any transformation in Γ is thus of the

form γ0m = γ0 ◦ . . . ◦ γ0 , m ∈ Z. m times

Theorem 5.1. There exists a positive measure dν(θ) on [0, 2π) such that

2π σ(γ0m )d μ(σ)

Γ

eimθ dν(θ),

=

m ∈ Z.

(5.1)

0

Proof. We use Theorem 4.1. Let tm = σ(γ0m )∗ d μ(σ),

m ∈ Z.

Γ

we have Since |σ(γ0 )| = 1 for all σ ∈ Γ t−m = σ(γ0 )∗ σ(γ0m )d μ(σ) = σ(γ0m ), σ(γ0 )L2 (dμ) , Γ

and therefore all the Toeplitz matrices TN = (t−m ),m=0,...N are non-negative. It follows from Theorem 4.1 that there exists a uniquely deﬁned measure dν such that 2π tm =

e−imθ dν(θ),

m = 0, 1, 2, . . . ,

0

and hence we obtain (5.1).

Remark 5.2. The proof of the previous theorem formalizes the intuitive idea that one can make the “change of variable” σ(γ0 ) = eiθ(σ) .

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Theorem 5.3. The linear map I which to σ(γ0m ) associates the function z m : I(σ(γ0m )) = z m ,

m ∈ Z,

(5.2)

μ) into L2 (dν). is an isomorphism from L2 (d Proof. For a function f of the form f (σ) =

M

N, M ∈ N0

cn σ(γ0n ) where

and

cn ∈ C,

(5.3)

−N

we have

f 2L2 (dμ) =

cn c∗m tm−n

n,m=−N,...,M

=

cn c∗m

n,m=−N,...,M

2π |

= 0

M

2π

e−i(m−n)θ dν(θ)

0

cn einθ |2 dν(θ)

n=−N

= I(f ) 2L2 (dν) . The result follows by continuity since such f are dense in L2 (d μ). To verify this last claim we note the following: By Plancherel’s theorem, the map from μ) which to the sequence which consists only of zeros, except 2 (Γ) onto L2 (d the n-th element which is equal to 1, associates the function σ(γ0 )n , extends to a unitary map. We will be interested in particular in the positive powers of γ 0 , which correspond to zooming (we consider that the multiplier of γ 0 , i.e. the associated scale αγ0 , is less than 1). Definition 5.4. We denote by H2 (d μ) the closure in L2 (d μ) of the functions σ(γ0 )n , n = 0, 1, 2, . . .. Similarly, we denote by H2 (dν) the closure in L2 (dν) of the functions z n , n = 0, 1, 2, . . .. Note that it may happen that L2 (d μ) = H2 (d μ). Following [4] we introduce the next definition. Definition 5.5. The map I will be called the Hermite transform. is compact and therefore Recall that Γ L2 (d μ) ⊂ L1 (d μ).

(5.4)

μ) does not belong to In general the product of two elements f and g in L2 (d μ), and one cannot deﬁne I(f g), let alone compare it with the product L2 (d I(f )I(g). On the other hand, we will need in the sequel only the case where at least one of the elements in the product f g deﬁnes a bounded multipliμ) into itself; see Definition 3.3 and the proof of cation operator from L2 (d Theorem 9.3 for instance. This is exploited in the next theorem.

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μ) such that the operator of multiplication by Theorem 5.6. Let f ∈ L2 (d μ) into itself. Then for every g in f deﬁnes a bounded operator from L2 (d μ) it holds that: L2 (d I(f g) = I(f )I(g).

(5.5)

Proof. We note that the multiplicative property (5.5) holds for f and g of the form (5.3). To prove the theorem we ﬁrst assume that g is of the form (5.3), and consider a sequence (pn ) of elements of the form (5.3), converging μ) norm. The function g is in particular bounded, and so to f in the L2 (d μ), and we can write: f g ∈ L2 (d f g − pn g L2 (dμ) ≤ K f − pn L2 (dμ) , μ)- norm to f g. where K > 0 is such that |g| ≤ K. Thus, f pn tends in L2 (d The function I(g) is bounded, and I(f ) ∈ L2 (dν). Therefore: I(f g) − I(f )I(g) L2 (dν) ≤ I(f g) − I(pn g) L2 (dν) + I(pn g) − I(f )I(g) L2 (dν) = f g − pn g L2 (dμ) + I(pn )I(g) − I(f )I(g) L2 (dν) ≤ f g − pn g L2 (dμ) + K1 pn − f L2 (dμ) , where K1 > 0 is such that |I(g)| ≤ K1 , and where we have used that I(pn g) = I(pn )I(g) since both pn and g are of the form (5.3). Hence, we obtain (5.5) for f and g as asserted. μ) be such that f g ∈ L2 (d μ). Then, I(f g) is well Let now f, g ∈ L2 (d deﬁned. Let (qn ) be a sequence of elements of the form (5.3), converging to g in the L2 (d μ) norm. Then, by the preceding argument, I(f qn ) = I(f )I(qn ),

∀n ∈ N.

In view of this equation and of the inclusion (5.4) we can write: I(f g) − I(f )I(g) L1 (dν) ≤ I(f g) − I(f )I(qn ) L1 (dν) + I(f )I(qn ) − I(f )I(g) L1 (dν) = I(f (g − qn )) L1 (dν) + I(f )I(qn − g) L1 (dν) . ν ), By Cauchy–Schwarz inequality and by the isometry property of I on L2 (d we have that I(f (g − qn )) L1 (dν) ≤ ν((0, 2π]) · I(f (g − qn )) L2 (dν) = ν((0, 2π]) · f (g − qn ) L2 (dμ) . Since multiplication by f is assumed to deﬁne a bounded operator from μ) into itself, there exists a constant C > 0 such that L2 (d f (g − qn ) L2 (dμ) ≤ C g − qn L2 (dμ) → 0 as n → ∞.

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Using once more the Cauchy–Schwarz inequality we have: I(f )I(qn − g) L1 (dν) ≤ I(f ) L2 (dν) · I(qn ) − I(g) L2 (ν) = f L2 (dμ) · qn − g L2 (dμ) →0

as n → ∞.

The claim follows.

At this stage we need a change of notation; since two (and, in the next section, p + 1) complex variables appear, we denote by z1 (and by z1 , . . . , zp in the following section) the variables related to the Hermite transform, and keep z for the Z-transform variable (this notation differs from the one in [4], where the Z-transform variable is denoted by ζ). Definition 5.7. The function H (z, z1 ) =

∞

n )(z1 ) z n I(h

(5.6)

n=0

is called the generalized transfer function of the system. Taking the Hermite transform on both sides of (1.3), or, equivalently, taking the Z transform and the Hermite transform on both sides of (3.7), we obtain Y (z, z1 ) = H (z, z1 )U (z, z1 ), ∞ n un )(z1 ), and similarly for Y (z, z1 ). The funcwhere U (z, z1 ) = n=0 z I(

tion H is analytic in a neighborhood of (0, 0) ∈ C2 . It is of interest to relate the properties of H and of the system. This is done in Sections 9 and 10 of the paper. We ﬁrst study, in the next section, the case where Γ has a ﬁnite number of generators.

6. The Trigonometric Moment Problem in Compact Semialgebraic Sets In [29] a solution is given to the K-moment problem when K is a compact semi-algebraic set. The material is quite deep, and cannot be easily summarized in a short overview here. The purpose of this section is to serve as a guide to the reader to the topic. The starting point is a semi-algebraic subset of Rn , deﬁned by the positivity of κ polynomials K = {x ∈ Rn ; pj (x) ≥ 0 , j = 1, . . . κ} . Because of the application we have in mind in the next section, we will assume: Hypothesis 6.1. a) n is even and we set n = 2p b) The polynomials pj are of even degree and their highest degree homogeneous parts have only the origin as common zero. One denotes by C+ (K) the cone of polynomials positive on K. In [29, Theorem 1.4, p. 972] it is proved that, under Hypothesis 6.1, positive polynomials on K belong to the additive cone

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C = Σ2 + p1 Σ2 + · · · + pκ Σ2 , where Σ2 denotes the convex cone generated by all squares of polynomials in C[x]. The key result of [29] is: Theorem 6.2 ([29, Lemma 3.2 p. 978]). A functional L on R[x] which is positive on C is of the form L(P ) = P (x)dν(x), P ∈ R[x], K

where dν is a positive measure on K. This result gives the solution to the moment problem on K. Let α,m with , m ∈ Np0 be complex numbers. One deﬁnes a map L from R[x] into R ﬁrst on monomials by L((z z ∗m )) = (α,m ) and

L((z z ∗m )) = (α,m ),

and then extends the map L by linearity to R[x]. Then there exists a positive measure on K such that z z ∗m dν(x) = α,m K

if and only if L satisﬁes the following conditions: L(|P (z, z ∗ )|2 ) ≥ 0,

∀P ∈ C[x],

L(pj (x)|s(z)| ) ≥ 0,

∀s ∈ C[z],

2

∗

L(pj (x)|P (z, z )| ) ≥ 0, 2

j = 1, . . . p,

(6.1)

j = p + 1, . . . κ.

7. The Case of a Finite Number of Generators We now assume that the Abelian group Γ has a ﬁnite number, say p, of generators, which we will denote by γ1 , . . . , γp . We assume that they are independent in the sense that if γ1n1 ◦ · · · ◦ γpnp = ι for some integers n1 , . . . np ∈ Z, then n1 = · · · = np = 0. In particular, each generator is of the form (2.1), γi (z) = γ{αi } (z) = (Gθ ◦ Sαi ◦ G−1 θ )(z) with θ ﬁxed, and where the set {αi }pi=1 generates a free discrete subgroup of the multiplicative group of positive real numbers. We use in a free way the multi-index notation, and recall that T denotes the unit circle. Theorem 7.1. There is a positive measure dν on the distinguished boundary of the polydisc such that n1 np σ(γ1 ) · · · σ(γp )d μ(σ) = ein1 θ1 · · · einp θp dν(θ1 , . . . , θp ). Γ

Tp

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To prove this theorem we specialize the results of the preceding section to the case of Tp . It is a compact algebraic set, with n = 2p, κ = 2p and polynomials P1 (x) = 1 − |z1 |2 , . . . , Pp (x) = 1 − |zp |2 , and Pp+1 (x) = |z1 |2 − 1, . . . , P2p (x) = |zp |2 − 1. Proof of Theorem 7.1. We deﬁne a linear form on polynomials in the variables z1 , . . . , zp , z1∗ , . . . , zp∗ by α ∗β μ(σ). L(z z ) = (σ(γ1 ))α1 −β1 · · · (σ(γp ))αp −βp d Γ

Let p be a polynomial in the variables z1 , . . . , zp , z1∗ , . . . , zp∗ . We write for short p(z, z ∗ ) = p(z1 , . . . , zp , z1∗ , . . . , zp∗ ). Let p(z, z ∗ ) = cα,β z α z ∗β . Then L(p(z, z ∗ )) = cα,β (σ(γ1 ))α1 −β1 · · · (σ(γp ))αp −βp d μ(σ), Γ

and therefore we have L(p(z, z ∗ )) = p(σ(γ1 ), σ(γ2 ), . . . , σ(γ1 )∗ , σ(γ2 )∗ , . . .)d μ(σ).

(7.1)

Γ ∗

Since |p(z, z )| is still a polynomial in z and z ∗ ,

L (1 − zj zj∗ )p(z, z ∗ ) = 0, j = 1, 2, . . . p,

L |p(z, z ∗ )|2 ≥ 0, 2

and hence the conditions (6.1) hold, and one can apply Theorem 6.2.

(7.2)

Remark 7.2. The fact that the characters are of modulus 1 allows to prove (7.1) and (7.2). It does not seem possible to relate our problems with another moment problem when p > 1 (for instance on the ball of Cp ). Definition 7.3. The Hermite transform of the element hα σ(γ α ) f (σ) = α

is I(f )(z) =

hα z α .

α

Theorem 7.4. Let f ∈ L2 (d μ) be such that the operator of multiplication by f deﬁnes a bounded operator from L2 (d μ) into itself. Then, for every μ): g ∈ L2 (d I(f g) = I(f )I(g).

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The proof is the same as for p = 1 (see the proof of Theorem 5.6). As in Definition 5.7, the function of p + 1 variables H (z, z1 , . . . , zp ) =

∞

n )(z1 , . . . , zp ) z n I(h

n=0

is called the generalized transfer function of the system.

8. BIBO Stability The system (1.1) will be called bounded input bounded output (BIBO) if there is an M > 0 such that for every {un (γ)} such that sup un (·) 2 (Γ) < ∞

(8.1)

n∈N0

the output is such that {yn (γ)}γ∈Γ ∈ 2 (Γ), n = 0, 1, . . ., and it holds that sup yn (·) 2 (Γ) ≤ M sup un (·) 2 (Γ) .

n∈N0

(8.2)

n∈N0

The following theorem gives a characterization of BIBO systems. The proof follows the proof of [4, Theorem 3.2]. We note the following difference between the two theorems: in [4] the multiplication operators, that is the counterparts of the operators Mhn deﬁned here using the Wick product, are automatically bounded. As explained there, this is due to V˚ age’s inequality (see [22, Proposition 3.3.2 p. 118] and (3.1) in [4], and Section 11 below). Here we do not have an analogue of this inequality. Theorem 8.1. The system (1.1) is BIBO if and only if the following two conditions hold: (a)

The multiplication operators (3.5) u → hn u,

Mhn :

n = 0, 1, . . .

are bounded from 2 (Γ) into itself. (b) For all v(·) ∈ 2 (Γ+ ) with v(·) 2 (Γ) = 1 it holds that ∞

M∗hn (v) 2 (Γ) ≤ M

(8.3)

n=0

for some strictly positive constant M . Proof. That the condition (8.3) is sufﬁcient is readily seen. Indeed, take v ∈ 2 (Γ) with v(·) 2 (Γ) = 1. From (1.1) we have: yn , v2 (Γ) =

n m=0

um , M∗hn−m v2 (Γ) ,

n = 0, 1, . . . ,

(8.4)

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and hence |yn , v2 (Γ) | ≤

n

um (·) 2 (Γ) M∗hn−m v 2 (Γ)

m=0

≤

sup um (·) 2 (Γ)

n

m=0,...n

M∗hn−m v 2 (Γ)

m=0

≤ M sup um (·) 2 (Γ) . m∈N0

We obtain (8.3) by taking v = yn / yn 2 (Γ) when yn = 0. We now show that (8.3) is necessary. We assume that the system is bounded input and bounded output. We ﬁrst note that the multiplication operators Mhn are necessarily bounded. Indeed, assume that (8.2) is in force and take u0 = u ∈ 2 (Γ) and un = 0 for n > 0. Then, yn = hn u = Mhn (u),

n = 0, 1, . . . ,

and it follows from (8.2) that Mhn ≤ M for n = 0, 1, . . . Let us now consider an input sequence (un ) which satisﬁes (3.1). For a given n and v choose if M∗hn−m v = 0,

um = 0 and um =

M∗hn−m v

M∗hn−m v 2 (Γ)

otherwise.

We obtain from (8.4) and (8.2) that n

M∗hn−m v 2 (Γ) ≤ M,

m=0

from which we get (8.3).

We now make a number of remarks: ﬁrst, condition (8.2) is implied by the stronger, but easier to deal with, condition ∞

Mhn ≤ M.

(8.5)

n=0

When Γ is the trivial subgroup of SU (1, 1), conditions (8.3) or (8.5) reduce to the classical condition ∞ |hn | < ∞. n=0

Finally, other versions of this theorem could be given, with non causal systems with respect to the variable n (as in [4]), or with scale-causal signals. We state the last one. The proof is the same as the proof of Theorem 8.1. Theorem 8.2. The system (1.1) is scale-causal and BIBO if and only if the following two conditions hold:

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The multiplication operators (3.5) u → hn u,

Mhn :

n = 0, 1, . . .

are bounded from 2 (Γ+ ) into itself. (b) For all v(·) ∈ 2 (Γ) with v(·) 2 (Γ+ ) = 1 it holds that ∞

M∗hn (v) 2 (Γ+ ) ≤ M.

n=0

9. Dissipative Systems We will call the system (1.1) dissipative if for every input sequence (un ) such that ∞ un (·) 22 (Γ) < ∞ n=0

it holds that ∞

yn (·) 22 (Γ) ≤

n=0

∞

un (·) 22 (Γ) .

(9.1)

n=0

Theorem 9.1. The system is dissipative if and only if the L(2 (Γ))-valued function S(z) =

∞

z n Mhn

n=0

is analytic and contractive in the open unit disc. Proof. Equations (9.1) expresses that the block Toeplitz operator ⎛ ⎞ Mh0 0 0 ··· ⎜Mh1 Mh0 0 · · ·⎟ ⎜ ⎟ ⎜ .. ⎟ .. ⎝ . ⎠ . is a contraction from 2 (2 (Γ)) into itself, and this is equivalent to the asserted condition on S. We consider the case of scale-causal signals (see Definition 2.4). Definition 9.2. The system (1.1) will be called scale-causal dissipative if the following conditions hold: (1) (2)

The operators Mhn are bounded from 2 (Γ+ ) into itself. Condition (9.1) holds, with 2 (Γ) replaced by 2 (Γ+ ).

Recall that we have denoted by H2 (dν) the closure in L2 (dν) of the powers z α , where all the components of α are greater or equal to 0. Taking the Fourier and Hermite transforms we have:

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Theorem 9.3. The system is scale-causal dissipative if and only the function ∞

n )(z1 , . . . , zp ) z n I(h (9.2) H (z, z1 , . . . , zp ) = n=0

is contractive from H2 (D) ⊗ H2 (dν) into itself. Furthermore, if the space H2 (dν) is a reproducing kernel Hilbert space, say with reproducing kernel K(z1 , . . . , w1 , . . .), condition (9.2) is equivalent to the positivity of the kernel 1 − H (z, z1 , . . .)H (w, w1 , . . .)∗ K(z1 , . . . , w1 , . . .) 1 − zw∗

(9.3)

in Dp+1 . Proof. Since the operators Mhn are assumed bounded, we have hn−m um ∈ 2 (Γ+ ),

m = 0, . . . , n,

for all entries um ∈ 2 (Γ+ ). Thus

μ), h n−m u m ∈ H2 (d and we may apply Theorem 5.6. We can write: m m hn−m um = I(hn−m )I(u

m ) n=0

2 (Γ+ )

n=0

. H2 (dν)

Thus the dissipativity is translated into the contractivity of the block Toeplitz operator ⎞ ⎛ 0 0 ··· Mh 0 ⎜M

Mh 0 0 · · ·⎟ ⎟ ⎜ h1 ⎟ ⎜ . .. ⎠ ⎝ .. . from 2 (H2 (dν)) into itself, and hence the claim on H . To prove the second claim, we remark that H2 (D) ⊗ H2 (dν) is the reproducing kernel Hilbert space with reproducing kernel 1 K(z1 , · · · , w1 , · · · ). 1 − zw∗ This comes from the fact that the reproducing kernel of a tensor product of reproducing kernel Hilbert spaces is the product of the reproducing kernels; see [10,30]. Condition (9.3) follows then from the well-known characterization of bounded multipliers in reproducing kernel Hilbert spaces; see for instance [1], and the references therein.

10. 1 -2 Bounded Systems The system (1.1) will be called 1 -2 bounded if there is a M > 0 such that for all inputs (un ) satisfying ∞ un (·) 2 (Γ) < ∞, n=0

186 we have

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∞

1/2 yn (·) 22 (Γ)

n=0

≤M

∞

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un (·) 2 (Γ) .

n=0

Taking the Fourier transform, this condition can be rewritten as: 1/2 ∞ ∞ 2

yn L2 (dμ) ≤ M

un L2 (dμ) , n=0

(10.1)

n=0

The system (1.1) will be called scale-causal 1 -2 bounded if it is moreover scale-causal, that is, if the operators Mhn are bounded from 2 (Γ+ ) into itself. Condition (10.1) then becomes: 1/2 ∞ ∞ 2

yn H2 (dμ) ≤ M

un H2 (dμ) (10.2) , n=0

n=0

from which we obtain, much in the same way as in [4], the following result. For completeness we present a proof. Theorem 10.1. A necessary and sufﬁcient condition for the system (1.1) to be scalar-causal and 1 -2 bounded is that the function ∞

n (σ) ∈ H2 (D) ⊗ H2 (d H(z, σ) = znh μ), (10.3) n=0

or, equivalently, that the transfer function ∞

n )(z1 ) ∈ H2 (D) ⊗ H2 (dν). H (z, z1 ) = z n I(h

(10.4)

n=0

Proof. To see that condition (10.3) is necessary, it sufﬁces to take the sequence 1 if n = 0 u

n = 0 if n = 0. Then

y

n = hn

n = 0, 1, . . . ,

μ). Conversely, and condition (10.2) implies that H(z, σ) ∈ H2 (D) ⊗ H2 (d μ). From the expression assume that the function H(z, σ) ∈ H2 (D) ⊗ H2 (d (1.4), and using the Cauchy–Schwarz inequality on ∞ Y (z, σ) = H(z, σ)U (z, σ) = H(z, σ) (z n u

n ) , n=0

we have Y (z, σ) H2 (D)⊗H2 (dμ) ≤ ≤

∞ n=0 ∞ n=0

H(z, σ)z n u

n H2 (D)⊗H2 (dμ) z n u

n H2 (D)⊗H2 (dμ) H(z, σ)| H2 (D)⊗H2 (dμ) .

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But we have that z n u

n H2 (D)⊗H2 (dμ) un H2 (dμ) =

, and so we obtain (10.2) with M = H(z, σ)| H2 (D)⊗H2 (dμ) . The equivalence with condition (10.4) follows by taking the Hermite transform. When H2 (dν) = L2 (dν) (recall that Γ is ﬁnitely generated), (10.4) can be translated into reproducing kernel conditions. In particular, in the cyclic case, we have: Theorem 10.2. Assume that H2 (dν) = L2 (dν), and let A(z1 )A(w1 )∗ − B(z1 )B(w1 )∗ 1 − z1 w1∗ be the reproducing kernel of H2 (dν). The system (1.1) is scale-causal and 1 − 2 bounded if and only if there is a M > 0 such that the kernel A(z1 )A(w1 )∗ − B(z1 )B(w1 )∗ − M H (z, z1 )H (w, w1 )∗ (1 − zw∗ )(1 − z1 w1∗ ) is positive in the bidisc. As in the case of equation (9.3), this comes from the characterization of the reproducing kernel of a tensor product of reproducing kernel Hilbert spaces.

11. The White Noise Space Setting and a Table Another kind of double convolution system, with a setting quite similar to the setting presented here, has been developed in [4], and relies on Hida’s theory of the white noise space (see [20–22] for the latter). We now review the main features of Hida’s theory and of [4]. The starting point in Hida’s theory is the function K(s1 − s2 ) = e−

s1 −s2 2 L2 (R) 2

,

which is positive in the sense of reproducing kernels for s1 , s2 in the Schwartz space S of real valued rapidly vanishing functions. By the Bochner-Minlos theorem there exists a probability measure P on the dual space S of real valued tempered distributions such that s2 L2 (R) e− 2 = ei w,s dP (w), s ∈ S, S

where we have denoted by w, s the duality between S and S . The white noise space is deﬁned to be the real Hilbert space L2 (S , F, P ), where F denotes the underlying Borelian sigma-algebra. Among all orthonormal basis of L2 (S , F, P ), there is one which plays a special role; it is constructed from the Hermite functions, and is indexed by the set of inﬁnite sequences (α1 , α2 , . . .) indexed by N, and with values in N0 , and for which αj = 0 for all j at the exception of at most a ﬁnite number

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of j. See [22, Deﬁnition 2.2.1 p. 19]. We will denote by Hα (with α ∈ ) the elements of this basis. An element fα Hα , fα ∈ R, (11.1) F = α∈

belongs to the white noise space if α!fα2 < ∞. α∈

The Wick product is deﬁned by Hα ♦Hβ = Hα+β ,

α, β ∈ .

The white noise space is not stable under the Wick product, and there is the need to introduce a nuclear space, called the Kondratiev space, within which the Wick product is stable. The Kondratiev space of stochastic distributions is the inductive limit of the real Hilbert spaces Hk of formal sums of the form (11.1) for which fα2 (2N)−qα < ∞ α

for some q ∈ N, where we use the notation def.

(2N)α = 2α1 × 4α2 × 6α3 · · · One can also consider the complexiﬁed versions of these spaces. We also recall V˚ age’s inequality (see [22, Proposition 3.3.2 p. 118]): Fix some integer l > 0, and let k > l + 1. Consider h ∈ Hl and u ∈ Hk . Then, h♦u ∈ Hk and h♦u k ≤ A(k − l) h l u k , where A(k − l) =

(2N)(l−k)α

α∈

is a ﬁnite number. We can now introduce the systems considered in [4]. A system will be characterized by a sequence {hn }n∈Z of elements in Hl for some l ∈ N, and a signal will be a sequence of elements in one of the spaces Hk , with k > l + 1. Input-output relations are expressions of the form hn−m ♦xm , n ∈ Z. yn = m∈Z

Note that in view of V˚ age’s inequality the output sequence consists also of elements of Hk . Furthermore, decomposing this equation along the basis Hα we obtain the double convolution system hα−β (n − m)uβ (m), n ∈ Z. yα (n) = m∈Z β≤α

The map I which to Hα associates the polynomial z α is called the Hermite transform. It is such that I(f ♦g) = I(f )I(g),

∀f, g ∈ S−1 .

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Note that under the Hermite transform the white noise space is mapped onto the reproducing kernel Hilbert space with reproducing kernel e z,w2 , that is, onto the Fock space. We now give the table presenting the parallels between the white noise space case (as applied in the paper [4]), and the present multi-scale case. The reader might want to look at a similar table in [3], where the analogies between the white noise space case and the hyper-holomorphic case are presented. The setting Underlying space Hermite transform

Stochastic case The white noise space I(Hα ) = z α

Underlying reproducing kernel Hilbert space Key tool used

The Fock space

The product Double convolution

Minlos theorem (to build the white noise space) Wick product yα (n) = m∈Z β≤α hα−β (n − m)uβ (m)

Multi-scale case 2 (Γ) I(σ(γ)α ) = z α (Γ: ﬁnitely generated) The space H2 (dν)

Moment problem on the polydisc (to build the Hermite transform) Convolution with respect to Γ. yn (γ) = n ϕ∈Γ m=0 hn−m (γ ◦ ϕ−1 )um (ϕ)

is a convolution in 2 (Γ). Remark 11.1. The pointwise product in L2 (dμ) Strictly speaking, it would be better to deﬁne the Hermite transform as the composition of the Fourier transform and of the map Hα → z α . Acknowledgements It is a pleasure to thank Professor Mihai Putinar for explaining to us the solution of the moment problem in the case of the polydisc.

References [1] Alpay, D.: The Schur algorithm, reproducing kernel spaces and system theory. In: SMF/AMS Texts and Monographs, vol. 5, American Mathematical Society, Providence, RI (2001) Translated from the 1998 French original by Stephen S. Wilson [2] Alpay, D., Dym, H.: Hilbert spaces of analytic functions, inverse scattering and operator models, I. Integr. Equ. Oper. Theory 7, 589–641 (1984) [3] Alpay, D., Levanony, D.: Rational functions associated to the white noise space and related topics. Potential Anal. 29, 195–220 (2008) [4] Alpay, D., Levanony, D.: Linear stochastic systems: a white noise approach. Acta Appl. Math. (to appear) [5] Alpay, D., Levanony, D., Pinhas, A.: Linear State space theory in the white noise space setting (Preprint)

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[6] Alpay, D., Levanony, D., Mboup, M.: Double convolution systems (in preparation) [7] Alpay, D., Mboup, M.: A characterization of Schur multipliers between character-automorphic Hardy spaces. Integ. Equ. Oper. Theory 62, 455–463 (2008) [8] Alpay, D., Mboup, M.: Transform´ee en ´echelle de signaux stationnaires. Comptes-Rendus math´ematiques (Paris), vol. 347, Issues 11–12, June 2009, pp. 603– 608 [9] Alpay, D., Mboup, M.: A natural transfer function space for linear discrete time-invariant and scale-invariant systems. In: Proceedings of NDS09, Thessaloniki, Greece, June 29–July 1, 2009 [10] Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 227– 404 (1950) [11] Ball, J., Bolotnikov, V.: Boundary interpolation for contractive-valued functions on circular domains in Cn . In: Current Trends in Operator Theory and its Applications, Oper. Theory Adv. Appl., vol. 149, pp. 107–132. Birkh¨ auser, Basel (2004) [12] Ball, J., Trent, T., Vinnikov, V.: Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces. In: Proceedings of Conference in Honor of the 60–th Birthday of M.A. Kaashoek, Operator Theory: Advances and Applications, vol. 122, pp. 89–138. Birkhauser (2001) [13] Ball, J., Vinnikov, V.: Functional models for representations of the Cuntz algebra. In: Operator Theory, Systems Theory and Scattering Theory: Multidimensional Generalizations, Oper. Theory Adv. Appl., vol. 157, pp. 1–60. Birkh¨ auser, Basel (2005) [14] de Branges, L., Shulman, L.A.: Perturbation theory of unitary operators. J. Math. Anal. Appl. 23, 294–326 (1968) [15] Deitmar, A.: A ﬁrst course in harmonic analysis. Universitext, 2nd edn. Springer (2005) [16] Ford, L.R.: Automorphic functions, 2nd edn. Chelsea, New-York (1915) [17] Freitag, E., Busam, R.: Complex Analysis. Springer (2005) [18] Guelfand, I.M., Graev, M.I., Vilenkin, N.Ja.: Les distributions. Tome 5. G´eom´etrie int´egrale et th´eorie des repr´esentations. Dunod, Paris (1970) [19] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. I/II, Springer, Berlin, G¨ ottingen Heidelberg (1963/1970) [20] Hida, T., Kuo, H., Potthoﬀ, J., Streit, L.: White noise. An inﬁnite-dimensional calculus. Mathematics and its Applications, vol. 253, Kluwer, Dordrecht (1993) [21] Hida, T.: White noise analysis: part I. Theory in progress. Taiwan. J. Math. 7, 541–556 (2003) [22] Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic partial differential equations. In: Probability and its Applications. Birkh¨ auser Boston Inc., Boston, MA (1996) [23] Katok, S.: Fuchsian groups. Chicago Lecture Notes in Mathematics, University of Chicago Press (1992) [24] Kre˘ın, M.G., Nudelman, A.A.: The Markov moment problem and extremal problems. In: Translations of Mathematical Monographs, vol. 50, American Mathematical Society, Providence, RI (1977) [25] Leland, W., Taqqu, M., Willinger, W., Wilson, D.: On the self-similar nature of Ethernet trafﬁc (extended version). IEEE/ACM Trans. Netw. 2(1), 1–15 (1994)

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´ [26] Mallat, S.: Une exploration des signaux en ondelettes. Les ´editions de l’Ecole Polytechnique (2000) [27] Mandelbrot, B.B., Van Ness, W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10(4), 422–437 (1968) [28] Mboup, M.: A character-automorphic Hardy spaces approach to discrete-time scale-invariant systems. In: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, July 24–28, 2006, pp. 183–188 (2006) [29] Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993) [30] Saitoh, S.: Theory of reproducing kernels and its applications. In: Longman scientiﬁc and technical, vol. 189 (1988) [31] Yuditskii, P.: Two remarks on Fuchsian groups of Widom type. In: Operator Theory: Advances and Applications, vol. 123, pp. 527–537. Birkhauser (2001) Daniel Alpay Department of mathematics Ben-Gurion University of the Negev POB 653, 84105 Beersheba Israel e-mail: [email protected] Mamadou Mboup CReSTIC-UFR des Sciences Exactes et Naturelles, Universit´e de Reims Champagne-Ardenne BP 1039, Moulin de la Housse 51687 Reims Cedex 2 France e-mail: [email protected] Received: September 24, 2009. Revised: January 24, 2010.

Integr. Equ. Oper. Theory 68 (2010), 193–205 DOI 10.1007/s00020-010-1814-7 Published online August 3, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Multipliers for p-Bessel Sequences in Banach Spaces Asghar Rahimi and Peter Balazs Abstract. Multipliers have been recently introduced as operators for Bessel sequences and frames in Hilbert spaces. These operators are defined by a fixed multiplication pattern (the symbol) which is inserted between the analysis and synthesis operators. In this paper, we will generalize the concept of Bessel multipliers for p-Bessel and p-Riesz sequences in Banach spaces. It will be shown that bounded symbols lead to bounded operators. Symbols converging to zero induce compact operators. Furthermore, we will give sufficient conditions for multipliers to be nuclear operators. Finally, we will show the continuous dependency of the multipliers on their parameters. Mathematics Subject Classification (2010). Primary 42C15; Secondary 41A58, 47A58. Keywords. p-Bessel sequence, p-Riesz sequence, (p, q)-Bessel multiplier, (r, p, q)-nuclear operators.

1. Introduction In [28], Schatten provided a detailed study of ideals of compact operators using their singular decomposition. He investigated the operators of the form λ ϕk ⊗ψk where (φk ) and (ψk ) are orthonormal families. In [4] the orthok k normal families were replaced with Bessel and frame sequences to define Bessel and frame multipliers. Definition 1.1. Let H1 and H2 be Hilbert spaces, let (ψk ) ⊆ H1 and (φk ) ⊆ H2 be Bessel sequences. Fix m = (mk ) ∈ l∞ (K). The operator Mm,(φk ),(ψk ) : H1 → H2 defined by Mm,(φk ),(ψk ) (f ) = mk f, ψk φk k

is called the Bessel multiplier for the Bessel sequences (ψk ) and (φk ). The sequence m is called the symbol of M.

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Several basic properties of these operators were investigated in [4]. For a theoretical approach it is very natural to extend this notion and consider such operators in more general settings. For p-Bessel sequences in Banach spaces is leading to interesting results in functional analysis and operator theory. We are going to show only theoretical properties. Nevertheless it should be mentioned, that multipliers are not only interesting from a theoretical point of view, see e.g. [5,6,13], but they are also used in applications, in particular in the fields of audio and acoustics. The first frame multipliers investigated were Gabor (frame) multipliers [14]. In signal processing they are used under the name ‘Gabor filters’ [20] as a particular choice to implement a time-variant filter. In computational auditory scene analysis they are known by the name ‘time-frequency masks’ [31] and are used to extract single sound source out of a mixture of sounds in a way linked to human auditory perception. In real-time implementations of filtering systems, they approximate time-invariant filters [8] as they are easily implementable. On the other hand, as a particular way to implement time-variant filters, they are used for example for sound morphing [12] or psychoacoustical modeling [9]. In general the idea for a Gabor (or wavelet) multiplier is to amplify or attenuate parts of audio signal, which can be separated in the time-frequency plane. Clearly Banach space theory is right at the foundation of functional analysis and operator theory, and as such is relevant for theory. But it recently also has become more and more important for time-frequency analysis, see e.g. [16]. It is used in engineering applications in compressed sensing, refer e.g. to [25], as well as in audio or image sampling [2]. Applications in wireless communication can be envisioned [17,21]. Therefore we hope that the results in this paper are not only interesting from a theoretical point of view but can be applied in the not-too-far future, for example by combining multipliers with the concept of sparsity and persistence [19]. In this paper, we define and investigate multipliers in Banach spaces. In Sect. 2, we will give the basic definitions and known results needed. In Sect. 3 we will give basic results for multipliers for p-Bessel sequences. In particular we will show that multipliers with bounded symbols are well-defined bounded operators with unconditional convergence and that symbols converging to zero correspond to compact operators. Section 4 will look at sufficient conditions for multipliers to be (r, p, q)-nuclear. Finally, in Sect. 5, we will look at how the multipliers depend on the given parameters, i.e. the analysis and synthesis sequences as well as the symbol. We will show that this dependence is continuous, using a similarity of sequences in an lp sense.

2. Preliminaries We will only consider reflexive Banach spaces. We will assume that p, q > 0 are real numbers such that p1 + 1q = 1. For any separable Banach space, we can define p-frame and p-Bessel sequences [3,11] as follows:

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Definition 2.1. A countable family (gi )i∈I ⊆ X ∗ is a p-frame for the Banach space X (1 0 exist such that p1 p |gi (f )| ≤ B f X for all f ∈ X. A f X ≤ i∈I

It is called a p-Bessel sequence with bound B if the second inequality holds. For p-Bessel sequences we can define the analysis operator U : X → lp with U (f ) = (gi (f )). Following the definition we see that U ≤ B. Further more, let T : lq → X ∗ be the synthesis operator defined by T ((di )) = i di gi . Proposition 2.2. [11] (gi ) ⊆ X ∗ is a p-Bessel sequence with bound B if and q ∗ only if T is a well-defined (hence bounded) operator from l into X and T ≤ B. In this case T ((di )) = i di gi converges unconditionally. Furthermore gi ≤ B. Definition 2.3. Let Y be a Banach space. A family (gi )i∈I ⊂ Y is a q-Riesz sequence (1 < q < ∞) for Y if constants A, B > 0 exist such that for all finite scalar sequence (di ), q1 q1 q q |di | ≤ di gi Y ≤ B |di | . (2.1) A i∈I

i∈I

i∈I

The family is called a q-Riesz basis (1 < q < ∞) for Y if it fulfills (2.1) and span{gi }i∈I = Y . It immediately follows from the definition that, if (gi )i∈I is a q-Riesz basis then A ≤ gi Y ≤ B for all i ∈ I. Any q-Riesz basis for X ∗ is a p-frame for X [11]. The following proposition shows a connection between these two notions similar to the case for Hilbert spaces: Proposition 2.4 [11]. Let (gi )i∈I ⊂ X ∗ be a p-frame for X. Then the following are equivalent: 1. 2. 3.

basis for X ∗ . (gi )i∈I is a q-Riesz q di gi = 0, then di = 0 for all i ∈ I. If (di )i∈I ∈ l and (gi )i∈I has a biorthogonal sequence (fi )i∈I ⊂ X, i.e., a family for which gi (fj ) = δi,j (Kronecker delta), for all i, j ∈ I.

Theorem 2.5 [11]. Let (gi )i∈I ⊂ X ∗ be a q-Riesz basis for X ∗ with bounds A, B. Then there exists a unique p-Riesz basis (fi )i∈I ⊂ X for which gi (f )fi and g = fi (g)gi f= i

i

∗

for all f ∈ X and g ∈ X . The bounds of (fi )i∈I are 1/B and 1/A, and it is biorthogonal to (gi ). Definition 2.6. We will call the unique sequence of Theorem 2.5 the dual of gi ). (gi ) and denote it by (˜

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2.1. Perturbation of p-Bessel Sequences Similar to the case for Hilbert spaces perturbation results for Banach spaces are possible. Theorem 2.7 [10]. Let U : X → Y be a bounded operator, X0 a dense subspace of X and V : X → Y a linear mapping. If for λ1 , μ > 0 and 0 ≤ λ2 < 1 U x − V x ≤ λ1 U x + λ2 V x + μ x , for all x ∈ X0 , then V is a bounded linear operator. Corollary 2.8. Let (ψk ) ⊆ X ∗ be a p-Bessel sequence. p 1. If (φk ) ⊆ X ∗ is a sequence with ( k ψk − φk X ∗ )1/p < μ < ∞, then (φk ) is a p-Bessel sequences with bound B + μ. (l) 2. Let (φ k ) be a sequence such that for all ε there exists an Nε with p (l) (l) ( k ψk − φk )1/p < ε for all l ≥ N . Then the sequence (φk ) X∗ is a Bessel sequence and for all l ≥ N Uφ(l) − U(ψk ) < ε and Tφ(l) − T(ψk ) < ε. k

Op

Op

k

Proof. For any c ∈ p with finite support, we have TΨ c − TΦ c = ci (ψi − φi ) i ≤ |ci | ψi − φi i

≤

1/q q

|ci |

·

i

1/p ψi − φi

p

i

≤ c q μ. Furthermore UΨ f − UΦ f = (ψi (f ) − φi (f )) p 1/p p = |ψi (f ) − φi (f )| i

≤

1/p ψi −

i

p φi X ∗

f X

≤ μ f X . We can apply Theorem 2.7 for λ1 = λ2 = 0 and use Proposition 2.2. Part (2) can be proved in an analogue way.

For a full treatment of perturbation for frames and Bessel sequences in Banach spaces refer to [29].

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(l)

Definition 2.9. Let (ψk )k∈K ⊆ X ∗ and (ψk )k∈K ⊆ X ∗ be a sequence of (l) elements for all l ∈ N. The sequences (ψk ) are said to converge to (ψk ) in (l)

lp

an lp -sense, denoted by (ψk ) −→ (ψk ), if for any ε > 0 there exists Nε > 0 1 (l) such that ( k ψk − ψk pX ∗ ) p < ε, for all l ≥ Nε . This is related to the notions of ‘quadratic closeness’ [32], and ‘Bessel norm’ [4].

3. Multipliers for p-Bessel Sequences Lemma 3.1. Let (ψk ) ⊆ X1∗ be a p-Bessel sequence for X1 with bound B1 , let (φk ) ⊆ X2 be a q-Bessel sequence for X2∗ with bound B2 , let m ∈ l∞ . The operator Mm,(φk ),(ψk ) : X1 → X2 defined by Mm,(φk ),(ψk ) (f ) = mk ψk (f )φk . k

is well defined. This sum converges unconditionally and M Op ≤ B2 B1 · m ∞ . Proof. As ∀f ∈ X1 (mk · ψk (f )) ∈ lp , M converges unconditionally and is well defined by Proposition 2.2. For n > 0 we have n n m ψ (f )φ ≤ mk ψk (f )φk X2 k k k k=1

X2

k=1

≤ m ∞

n

p1 p

|ψk (f )|

k=1

sup h≤1

n

q1 q

|φk (h)|

k=1

≤ m ∞ · B1 f X1 . sup (B2 h X2∗ ) h=1

= m ∞ · B1 · B2 f X1 . So the multiplier is bounded with bound m ∞ · B1 · B2 .

Using the representation Mm,(φk ),(ψk ) = Tφk Dm Uψk gives a more direct way to prove the above bound. Here Dm is the diagonal operator on ∞ defined by Dm (ξi ) = (mi ξi ). Using the above Lemma, we can define: Definition 3.2. Let (ψk ) ⊆ X1∗ be a p-Bessel sequence for X1 and let (φk ) ⊆ X2 be a q-Bessel sequence for X2∗ . Let m ∈ l∞ . The operator Mm,(φk ),(ψk ) : X1 → X2 , defined by Mm,(φk ),(ψk ) (f ) = mk ψk (f )φk k

is called (p,q)-Bessel multiplier. The sequence m is called the symbol of M.

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Proposition 3.3. Let (ψk ) ⊆ X1∗ be a p-Bessel sequence for X1 with no zero elements, let (φk ) ⊆ X2 be a p-Riesz sequence for X2 and let m ∈ l∞ . Then the mapping m → Mm,(φk ),(ψk ) is injective from l∞ into B(X1 , X2 ). Proof. Suppose Mm = Mm , then k mk ψk (f )φk = k mk ψk (f )φk for all f . As (φk ) is a p-Riesz basis for its span, mk ψk (f ) = mk ψk (f ) for all f, k. For every k there exists f such that ψk (f ) = 0, which implies that mk = mk . Proposition 3.4. Let (ψk ) ⊆ X1∗ be a q-Riesz basis for X1∗ with bounds A1 and B1 , let (φk ) ⊆ X2 be a q-frame for X2∗ with bounds A2 and B2 and let m ∈ l∞ . Then A1 A2 m ∞ ≤ Mm,(φk ),(ψk ) Op ≤ B1 B2 m ∞ . Particularly M is bounded if and only if m is bounded. Proof. Lemma 3.1 gives the upper bound. Proposition 2.4 states that (ψk ) has a biorthogonal sequence (fi ) ⊆ X1 , i.e. ψk (fi ) = δk,i . (fi ) is also a Riesz basis with bounds B11 , A11 , and so 1 1 B1 ≤ fi ≤ A1 for all i ∈ I. For arbitrary i ∈ I, we have Mf Mfi mk ψk (fi )φk ≥ sup = sup M op = sup fi i∈I fi i∈I f ∈X1 f mi φi fi i∈I φi = sup |mi | fi i∈I ≥ A1 A2 m ∞ . = sup

So A1 A2 m ∞ ≤ M op .

The following proposition shows that under certain condition on m the multiplier can be invertible,1 the inverse being the multiplier with the inverted symbol, similar to a result in [7] for Hilbert spaces. Proposition 3.5. Let (ψk ) ⊆ X1∗ be a q-Riesz basis for X1∗ , (φk ) ⊆ X2 be a p-Riesz basis for X2∗ . Let m be semi-normalized (i.e. 0 < inf | mk |≤ sup | mk | < +∞). Then Mm,(φk ),(ψk ) is invertible and (Mm,(φk ),(ψk ) )−1 = M( 1

1 mk

˜k ),(φ ˜k ) . ),(ψ

For a detailed study of invertible multiplier (on Hilbert spaces) see [30].

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Multipliers for p-Bessel Sequences

Proof. It is clear that ( m1k ) ∈ ∞ and thus M( For f ∈ X1 M(

1 mk

˜k ),(φ ˜k ) ),(ψ

◦ Mm,(φk ),(ψk ) f = M(

1 mk

199

˜k ),(φ ˜k ) ),(ψ

1 mk

˜k ),(φ ˜k ) ),(ψ

is well-defined.

mk ψk (f )φk

k

1 = mk ψk (f )φk ψ˜i φ˜i m i i k 1 = mk ψk (f )φ˜i (φk )ψ˜i mi i k = ψi (f )ψ˜i i

= f. That Mm,(φk ),(ψk ) ◦ M(

1 mk

˜k ),(φ ˜k ) f ),(ψ

= f for all f ∈ X2 can be shown in an

analogous way. Hence, −1 Mm,(φk ),(ψk ) = M(

1 mk

˜k ),(φ ˜k ) . ),(ψ

For Banach spaces it is well known that the limit of finite rank operators (in the operator norm) is a compact operator (although this is not an equivalent conditions as is the case for Hilbert spaces). We are using this property in: Lemma 3.6. Let (ψk ) ⊆ X1∗ be a p-Bessel sequence for X1 with bound B1 , let (φk ) ⊆ X2 be a q-Bessel sequence for X2∗ with bound B2 . If m ∈ c0 then Mm,(φk ),(ψk ) is compact. Proof. For a given m ∈ c0 , let m(N ) = (m0 , m1 , . . . , mN −1 , 0, 0, . . . ). The symbol m is converging to zero, so for all > 0 there is a N such that m − m(N ) ≤ for all N ≥ N . As m ∈ l∞ by Lemma 3.1 we have for all ∞ N ≥ N Mm,(φ ),(ψ ) − Mm(N ) ,(φ ),(ψ ) = M (N ) k k k k (m−m ),(φk ),(ψk ) Op Op ≤ m − m(N ) · B2 · B1 ≤ · B2 · B1 .

∞

Therefore Mm(N ) ,(φk ),(ψk ) is converging to Mm,(φk ),(ψk ) in the operator norm. As Mm(N ) ,(φk ),(ψk ) is clearly a finite rank operator, we have shown the result. For two normed spaces X and Y , B(X, Y ) denotes the set of all linear bounded operators from X to Y . Let W , X, Y and Z be normed spaces. For elements y ∈ Y and ω ∈ X ∗ define an operator y ⊗ ω ∈ B(X, Y ) by (y ⊗ ω)(z) = ω(z)y

for all z ∈ X.

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For arbitrary S ∈ B(W, X), T ∈ B(Y, Z), y ∈ Y , ω ∈ X ∗ , z ∈ Z and τ ∈ Y ∗ these operators satisfy (z ⊗ τ )(y ⊗ ω) = τ (y) · z ⊗ ω T (y ⊗ ω) = T (y) ⊗ ω (y ⊗ ω)S = y ⊗ S ∗ (ω) (y ⊗ ω)∗ = ω ⊗ κ(y) y ⊗ ω Op = y Y ω X ∗ where κ : Y → Y ∗∗ is the canonical injection defined by κ(y)(η) = η(y) for all y ∈ Y, η ∈ Y ∗ . The above notations are borrowed from [22]. By using the above notations, we can write the (p, q)-Bessel multiplier in the form Mm,(φk ),(ψk ) = mk φ k ⊗ ψ k . k

It is easy to see that M∗m,(φk ),(ψk ) =

mk ψk ⊗ κ(φk ) = Mm,(ψk ),(κ(φk )) .

k

Putting the above results together, we obtain the following theorem which is a generalization of one of the results in [4] for Banach spaces. Theorem 3.7. Let M = Mm,(φk ),(ψk ) be a (p, q)-Bessel multiplier for the p-Bessel sequence (ψk ) ⊆ X1∗ , the q-Bessel sequence (φk ) ⊆ X2 with bounds B1 and B2 . Then, the following hold. 1.

If m ∈ l∞ , M is a well defined bounded operator with M Op ≤ B2 B1 · m ∞ . Furthermore, the sum mk ψk (f )φk converges unconditionally for all k

2. 3.

f ∈ X1 . M∗m,(φk ),(ψk ) = k mk ψk ⊗ κ(φk ) = Mm,(ψk ),(κ(φk )) . If m ∈ c0 , M is a compact operator.

4. Nuclear Operators in Banach Spaces The theory of trace-class operators in Hilbert spaces was created in 1936 by Murray and Von Neumann. In the earlier Fifties, Alexander Grothendieck [18] and Ruston [26,27] independently extended this concept to operators acting in Banach spaces. Trace-class operators on Banach spaces are called nuclear operators. This idea is generalized in [23]: Let 0 < p ≤ ∞. A family x = (xi )i∈I ⊆ X, where xi ∈ X for i ∈ I, is called weakly p-summable if (x∗ (xi )) ∈ p (I) whenever x∗ ∈ X ∗ . We put wp (xi ) := sup{ x∗ (xi ) p : x∗ ≤ 1}.

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The class of all weakly p-summable sequences on X is denoted by Wp (X). Clearly wp (xi ) < ∞ ( by Banach–Steinhaus Theorem ). From the above notations, it is clear that if (gi )i∈I ⊂ X ∗ is a p-Bessel sequence for X then (gi ) ∈ Wp (X ∗ ).2 ≥

1 p1

+

with (σi ) ∈ , ∈ Wq (X ), and (yi ) ∈ Wp (Y ) where p11 + p1 = In the case r = ∞ let us suppose that (σi ) ∈ c0 . We put

1 q1

+ q1 = 1.

Definition 4.1 [23]. Let 0 < r ≤ ∞, 1 ≤ p1 , q1 ≤ ∞, and 1 + operator S ∈ B(X, Y ) is called (r, p1 , q1 )-nuclear if ∞ S= σi x∗i ⊗ yi

1 r

1 q1 .

An

i=1 r

(x∗i )

∗

N(r,p1 ,q1 ) (S) := inf { (σi ) r · wq (x∗i ) · wp (yi )} , where the infimum is taken over all so-called (r, p1 , q1 )-nuclear representations described above. Theorem 4.2 [23]. An operator S ∈ B(X, Y ) is (r, p1 , q1 )-nuclear if and only if there exist operators F , D and E with S = F DE, such that Dσ ∈ B(q , p1 ) is a diagonal operator of the form Dσ (ξi ) = (σi ξi ) with (σi ) ∈ r if 0 < r < ∞ and (σi ) ∈ c0 if r = ∞. Furthermore, E ∈ B(X, q ) and F ∈ B(p1 , Y ). In this case, N(r,p1 ,q1 ) (S) := inf E (σi ) r F , where the infimum is taken over all possible factorizations. From Theorem 4.2 and the above notations, we can easily conclude the next result for multipliers using Mm,(φk ),(ψk ) = Tφk Dm Uψk as a decomposition of M. Corollary 4.3. Let (ψk ) ⊆ X1∗ be a p-Bessel sequence for X1 with bound B1 , let (φk ) ⊆ X2 be a q-Bessel sequence for X2∗ with bound B2 . Let r > 0 and m ∈ r . Then Mm,(φk ),(ψk ) is a (r, p, q)-nuclear operator with N(r,p,q) (M) ≤ B1 B2 m r .

5. Changing the Ingredients Results from [4] can be generalized to p-Bessel sequences: Theorem 5.1. Let M = Mm,(φk ),(ψk ) be a (p, q)-Bessel multiplier for the p-Bessel sequences (ψk ) ⊆ X1∗ , the q-Bessel sequence (φk ) ⊆ X2 with bounds B1 and B2 . Let p1 , q1 ≥ 1 be such that p11 + q11 = 1 allowing p1 , q1 = ∞. Then the operator M depends continuously on m, (ψi ) and (φi ), in the following (l) (l) sense: Let (ψi ) ⊆ X1∗ and (φi ) ⊆ X2 be Bessel sequences3 indexed by l ∈ I. 2

As mentioned in the introduction we only consider reflexive Banach spaces. Please note that for a convergence of p-Bessel sequences in an lp -sense we would get the Bessel property by Corollary 2.8 for big enough l.

3

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1.

Let m(l) → m in lp1 . Then Mm(l) ,(ψi ),(φi ) − Mm,(ψi ),(φi ) Op → 0.

2.

(l)

Let m ∈ lp1 and let the sequences (ψi ) converge to (ψi ) in an lq1 -sense. Then for l → ∞ Mm,(ψ(l) ),(φi ) − Mm,(ψi ),(φi ) → 0. Op

i

3.

(l) (φi )

Let m ∈ lp1 and let the sequences converge to (φi ) in an lq1 -sense. Then for l → ∞ Mm,(ψi ),(φi ) − Mm,(ψi ),(φ(l) ) → 0. Op

i

4.

IEOT

(l)

(l)

Let m(l) → m in lp1 and let the sequences (ψi ) respectively (φi ) converge to (ψi ) respectively (φi ) in an lq1 -sense. Then for l → ∞ Mm(l) ,(ψ(l) ),(φ(l) ) − Mm,ψi ,φi → 0. i

i

Op

Proof. 1. By Theorem 3.7 Mm(l) ,(ψ ),(φ ) − Mm,(ψ ),(φ ) = M (l) k k k k (m −m),(ψk ),(φk ) Op Op ≤ m(l) − m B1 B2 ∞ ≤ m(l) − m p1 B1 B2 ≤ B1 B2 . 2.

for l > N . For l > N

(l) mk ψk ⊗i φk mk ψk ⊗i φk −

Op

(l) = mk ψk − ψk ⊗i φk Op √ (l) ≤ |mk | ψk − ψk ∗ B2 k

X1

q1 1/q1 √ (l) ≤ B2 mp1 ψk −ψk ∗ X1 √ ≤ B2 mp1 ε.

3. 4.

Use corresponding arguments as in (2). Mm(l) ,(ψ(l) ),(φ(l) ) −Mm,(ψk ),(φk ) ≤ Mm(l) ,(ψ(l) ),(φ(l) ) − Mm,(ψ(l) ),(φ(l) ) k k k k k k + Mm,(ψ(l) ),(φ(l) ) − Mm,(ψ ),(φ(l) ) k k k k + Mm,(ψ ),(φ(l) ) − Mm,(ψk ),(φk ) k k ≤ ε B1 B2 + m p1 ε B2 + m B1 ε

=ε · B1 B2 + m p1 B2 + B1 for l bigger than the maximum N needed for the convergence conditions.

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6. Outlook and Perspectives We have shown that the concept of multipliers can be extended to p-frames in Banach spaces. It can also be done for other settings. For g-frames a paper was already accepted [24]. In the future we will consider to extend this notion to other setting, for example for Frechet frames, matrix valued frames, pgframes [1] or continuous frames. In particular the last notion can be interesting also for application as in this setting the question, how continuous and discrete frame multipliers can be related, is of relevance. This would be an interesting result for the link of STFT and Gabor multipliers. Such a connection is particular interesting in relating a physical model, which normally is continuous, using multipliers to the implemented algorithm, which is discrete and finite-dimensional. For the future work the relation of (p, q, r)-nuclear operators with Gelfand triple may be investigated. For applications wavelet, Gabor and frames of translates are very important classes of frames. Most of these systems can be described as localized frames [15]. For multipliers of localized frames, which are currently investigated, the results of this paper are directly applicable and so can become more relevant for signal processing algorithms. We further hope that the results in this paper can be directly useful for applications in signal processing, as both Banach space methods and multipliers become more and more important for applications, as mentioned in the introduction. Acknowledgements Some of the results in this paper were obtained during the first author’s visit at the Acoustics Research Institute, Austrian Academy of Sciences, Austria. He thanks this institute for their hospitality. This work was partly supported by the WWTF project MULAC (‘Frame Multipliers: Theory and Application in Acoustics; MA07-025). The authors would like to thank Diana Stoeva for her discussions and comments. Also, the authors would like to thank referees for their suggestions.

References [1] Abodllahpour, M.R., Faroughi, M.H., Rahimi, A.: pg-Frames in Banach spaces. Methods Funct. Anal. Toplol. 13(3), 201–210 (2007) [2] Aldroubi, A., Gr¨ ochenig, K.: Non-uniform sampling and reconstruction in shiftinvariant spaces. SIAM Rev. 43, 585–620 (2001) [3] Aldroubi, A., Sun, Q., Tang, W.-S.: p-Frames and shift invariant subspaces of Lp . J. Fourier Anal. Appl. 7(1), 1–21 (2001) [4] Balazs, P.: Basic definition and properties of Bessel multipliers. J. Math. Anal. Appl. 325(1), 571–585 (2007) [5] Balazs, P.: Hilbert–Schmidt operators and frames—classification, best approximation by multipliers and algorithms. Int. J. Wavelets Multiresolut. Inf. Processing 6(2), 315–330 (2008)

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[6] Balazs, P.: Matrix-representation of operators using frames. Sampling Theory Signal Image Processing (STSIP) 7(1), 39–54 (2008) [7] Balazs, P., Antoine, J.-P., Grybos, A.: Weighted and controlled frames: mutual relationship and first numerical properties. Int. J. Wavelets Multiresolut. Inf. Processing 8(1), 109–132 (2010) [8] Balazs, P., Deutsch, W.A., Noll, A., Rennison, J., White, J.: STx Programmer Guide, Version: 3.6.2. Acoustics Research Institute, Austrian Academy of Sciences (2005) [9] Balazs, P., Laback, B., Eckel, G., Deutsch, W.A.: Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking. IEEE Trans. Audio Speech Lang. Processing 18(1), 34–49 (2010) [10] Casazza, P.G., Christensen, O.: Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl. 3(5), 543–557 (1997) [11] Christensen, O., Stoeva, D.: p-Frames in separable Banach spaces. Adv. Comput. Math. 18(2-4), 117–126 (2003) [12] Depalle, Ph., Kronland-Martinet, R., Torr´esani, B.: Time-frequency multipliers for sound synthesis. In: Proceedings of the Wavelet XII conference, SPIE annual Symposium, San Diego, August 2007 [13] D¨ orfler, M., Torr´esani, B.: Representation of operators in the time-frequency domain and generalized gabor multipliers. J. Fourier Anal. Appl. 16(2), 261– 293 (2010) [14] Feichtinger, H.G., Nowak, K.: A First Survey of Gabor Multipliers, chap. 5, pp. 99–128. Birkh¨ auser Boston (2003) [15] Fornasier, M., Gr¨ ochenig, K.: Intrinsic localization of frames. Constr. Approx. 22, 395–415 (2005) [16] Gr¨ ochenig, K., Heil, C.: Modulation spaces and pseudodifferential operators. Integral Equ. Oper. Theory 34(4), 439–457 (1999) [17] Gr¨ ochenig, K., Rzeszotnik, Z.: Banach algebras of pseudodifferential operators and their almost diagonalization. Ann. Inst. Fourier (Grenoble), pp. 2279–2314 (2008) [18] Grothendieck, A.: Produits tensoriels topologiques et espace nucl´eaires. Mem. Am. Math. Soc. 16 (1955) [19] Kowalski, M., Torr´esani, B.: Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients. Signal Image Video Processing 3(3):251–264 (2009). doi:10.1007/s11760-008-0076-1 [20] Matz, G., Hlawatsch, F.: Linear time-frequency filters: On-line Algorithms and applications. In: Papandreou-Suppappola, A. (eds.) Application in Time-Frequency Signal Processing, chap. 6, pp. 205–271. CRC Press, Boca Raton (2002) [21] Matz, G., Schafhuber, D., Gr¨ ochenig, K., Hartmann, M., Hlawatsch, F.: Analysis, optimization, and implementation of low-interference wireless multicarrier systems. IEEE Trans. Wireless Commun. 6(4), 1–11 (2007) [22] Palmer, T.: Banach Algebras and the General Theory of *-Algebras. Algebras and Banach Algebras (Encyclopedia of Mathematics and its Applications), vol. 1 (1995) [23] Pietsch, A.: Operator Ideals. North-Holland, Amsterdam (1980) [24] Rahimi, A.: Multipliers of generalized frames. Bull. Iranian Math. Soc. (2010, in press)

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[25] Rauhut, H., Schnass, K., Vandergheynst, P.: Compressed sensing and redundant dictionaries. IEEE Trans. Inform. Theory 54(5), 2210–2219 (2008) [26] Ruston, A.F.: Direct products of banach spaces and linear functional equations. Proc. Lond. Math. Soc. (3) 1, 327–348 (1951) [27] Ruston, A.F.: On the fredholm theory of integral equations for operators belonging to the trace class of general banach spaces. Proc. Lond. Math. Soc, (2), pp. 109–124 (1951) [28] Schatten, R.: Norm Ideals of Completely Continuous Operators. Springer, Berlin (1960) [29] Stoeva, D.: Perturbation of Frames in Banach spaces (submitted) [30] Stoeva, D., Balazs, P.: Unconditional Convergence and Invertibility of Multipliers (2010, submitted) [31] Wang, D., Brown, G.: Computational Auditory Scene Analysis: Principles, Algorithms, and Applications. Wiley-IEEE Press, New York (2006) [32] Young, R.M.: An Introduction to Nonharmonic Fourier Series. Acedmic Press, London (1980) Asghar Rahimi (B) Department of Mathematics University of Maragheh Maragheh, Iran e-mail: [email protected] Peter Balazs Acoustics Research Institute, Austrian Academy of Sciences Wohllebengasse 12-14, 1040 Vienna, Austria e-mail: [email protected] Received: October 11, 2009. Revised: May 12, 2010.

Integr. Equ. Oper. Theory 68 (2010), 207–227 DOI 10.1007/s00020-010-1799-2 Published online June 11, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Bounded Mild Solutions for Semilinear Integro Differential Equations in Banach Spaces Carlos Lizama and Gaston M. N’Gu´er´ekata Abstract. In this paper we study the structure of various classes of spaces of vector-valued functions M(R; X) ranging between periodic functions and bounded continuous functions. Some of these functions are introduced here for the first time. We propose a general operator theoretical approach to study a class of semilinear integro-differential equations. The results obtained are new and they recover, extend or improve variety of recent works. Mathematics Subject Classification (2010). Primary 45N05; Secondary 43A60. Keywords. Linear and semilinear integro-differential equations, regularized operator families, bounded mild solutions.

1. Introduction The rapid development of the theory of integro-differential equations in inﬁnite-dimensional spaces has been strongly promoted by the large number of applications in physics, engineering and biology. Abstract integro-differential equations are still in a state of ﬂux, with new basic results continuously emerging. Questions like existence of solutions, continuous dependence, perturbations, and general asymptotic behavior are at present an active area of research. In this paper, we consider the following integro-differential equation

t

u (t) = Au(t) +

a(t − s)Au(s)ds + f (t, u(t)),

−∞

C. Lizama was partially supported by Project FONDECYT 1100485.

(1.1)

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where A is a closed linear operator deﬁned in a Banach space X and a ∈ L1loc (R+ ) is an scalar-valued kernel. Equations of this kind arise, for example, in the study of heat ﬂow in materials of fading memory type as well as some equations of population dynamics. For more information on this subject see the papers [9,13,20,39] and the monograph [40] (particularly Chapter II, Section 9) and the references therein. Suppose that we know something about the asymptotic behavior of the forcing function f (t, x). For example, f could be bounded or asymptotically periodic. What conditions do we need on the operator A and the kernel a in order to conclude that the solution u of Eq. (1.1) exists and has the same asymptotic behavior as f ?. Among others, Da Prato and Lunardi [13] studied this problem for Eq. (1.1) in the linear case (see also [8,14]) under several conditions on A and a(·). The results of Da Prato et.al. were then used by Sforza [42] to derive global existence and uniqueness results for the associated semilinear problem. A key assumption in all the above mentioned works is that A generates an analytic semigroup (not necessarily strongly continuous). However, they also treat more general operator valued kernels. In this paper, we will answer the stated problem in various spaces of vector-valued functions M(R; X) ranging between periodic functions and bounded continuous functions. Our approach provide a uniﬁed treatment for most of the more important classes of vector-valued functions that have recently appeared in the literature like almost automorphic, pseudo-almost automorphic, asymptotically periodic and almost periodic, to mention a few examples of the spaces to be considered. We ﬁrst present and analyze the structure of a hierarchy of spaces of functions M(R; X) between periodic functions and bounded continuous functions. We note that some of these spaces, which appear naturally in our study, are introduced here for the ﬁrst time. Then we propose an operator theoretical approach to study the structure of bounded solutions for Eq. (1.1) in these function spaces. It reveals the way in that regularized families of bounded linear operators [31] can be used in order to produce ad hoc concepts of mild solutions for abstract integro-differential equations. Finally, we apply the proposed method to the semilinear history value problem (1.1) giving new results as well as recovering, improving and extending a vast class of recent studies. We hope that the methods outlined in this paper can serve as guidelines to obtain similar results for other abstract differential equations that are of recent interest, like, e.g., fractional differential equations, delay equations, etc.

2. Preliminaries: The Function Spaces Let X be a Banach space. We denote BC(X) := f : R → X : f is continuous, ||f ||∞ := sup ||f (t)|| < ∞ . t∈R

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Let Pω (X) := {f ∈ BC(X) : ∃ ω > 0, f (t + ω) = f (t) ∀t ∈ R} be the space of all vector-valued periodic functions. For the space of almost periodic functions (in the sense of Bohr), we set AP (X) which consists of all functions f ∈ BC(X) such that for each > 0 there exists a ω > 0 such that every subinterval of R of length ω contains at least one point τ such that ||f (t + τ ) − f (t)||∞ ≤ . This definition is equivalent to the so-called Bochner’s criterion (cf. [34, Theorem 3.1.8]), namely, f ∈ AP (X) if and only if for every sequence of reals (sn ) there exists a subsequence (sn ) such that (f (· + sn )) is uniformly convergent on R. Almost periodic functions are uniformly continuous √ on R. (cf. [34, Theorem 3.1.4]). A simple example is f (t) = 2 + sin(t) + sin( 2t). Observe that AP (X) is a Banach space with the norm || · ||∞ and Pω (X) ⊂ AP (X). Note that the function f (t) above is not periodic, thus the inclusion is strict. The space of compact almost automorphic functions will be denoted by AAc (X). Recall that a continuous bounded function f belongs to AAc (X) if and only if for every sequence (sn ) of real numbers there exists a subsequence (sn ) ⊂ (sn ) such that limt→∞ f (t + sn ) =: f (t) and limt→∞ f (t − sn ) = f (t) uniformly over compact subsets of R. Clearly the function f above is continuous on R. Therefore f is uniformly continuous [33]. In other words compact almost automorphic functions are uniformly continuous on R. We have that AAc (X) is a Banach space under the norm || · ||∞ and Pω (X) ⊂ AP (X) ⊂ AAc (X) ⊂ BC(X). The space of almost automorphic functions is deﬁned as follows AA(X) := {f ∈ BC(X) : for all (sn ), exists (sn ) ⊂ (sn ) s.t. limt→∞ f (t + sn ) =: f (t) and limt→∞ f (t − sn ) = f (t)∀t ∈ R}, provided with the norm || · ||∞ . 1 √ As a typical example, we can take f (t) = sin( 2+sin(t)+sin( ). We have 2t) that AA(X) is a Banach space with the norm || · ||∞ and the following inclusions hold: Pω (X) ⊂ AP (X) ⊂ AAc (X) ⊂ AA(X) ⊂ BC(X). 1 √ ) is not uniformly conNote that the function f (t) = sin( 2+sin(t)+sin( 2t) tinuous, so the inclusion AAc (X) ⊂ AA(X) is strict. Let F1 = {Pω (X), AP (X), AAc (X), AA(X)} and Ω ∈ F1 . Then we have

Theorem 2.1. Assume f, f1 , f2 ∈ Ω. Then we have • • •

f1 + f2 ∈ Ω, λf ∈ Ω, for any scalar λ fτ (t) := f (t + τ ) ∈ Ω for any τ ∈ R

Proof. See for instance [34].

Theorem 2.2. For any f ∈ Ω, the range Rf of f is relatively compact in X.

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Proof. Let Ω ∈ F1 , then since Ω ⊂ AA(X), we can conclude in view of [34, Theorem 2.1.3.]. Theorem 2.3. Let (fn ) ⊂ Ω, such that fn → f uniformly on R. Then f ∈ Ω. Proof. The case Ω = Pω is trivial. Indeed, let (fn ) ⊂ Pω , such that fn → f uniformly on R. Then for any > 0, there exists N such that fn (t)−f (t) < 2 for any n > N and t ∈ R. Thus f (t + ω) − f (t) ≤ f (t + ω) − fn (t + ω) + fn (t + ω) − fn (t) + fn (t) − f (t) < , for any t ∈ R. Which shows that f (t + ω = f (t), for any t ∈ R. For Ω = AP (X) (resp. Ω = AA(X)), see [34, Theorem 3.1.4] (resp. Theorem 2.1.10). The case Ω = AAc (X) is similar to the one of AA(X). So we omit it. t Theorem 2.4. Assume that f ∈ Ω and let F (t) := 0 f (s)ds. Then F ∈ Ω if and only if Rf is relatively compact in X. Proof. For Ω = AP (X) (resp. Ω = AA(X)), see [34, Theorem 3.2.6] (resp. Theorem 2.4.4). The case Ω = AAc (X) is similar to the one of AA(X). So we omit it. Remark 2.5. Note that if X does not contain a subspace isomorphic to c0 (for instance X is a uniformly convex Banach space), the above theorem is called Kadet’s theorem [28, Theorem 2, p. 86] and it reads: If f is ω-periodic (resp. almost periodic), then F is ω-periodic (resp. almost periodic) if and only if it is bounded. Kadet’s theorem is valid for all periodic, almost periodic and almost automorphic sequences [38]. Now we consider the set C0 (X) := {f ∈ BC(X) : lim ||f (t)|| = 0}, |t|→∞

and deﬁne the space of asymptotically periodic functions as APω (X) := Pω (X) ⊕ C0 (X). Analogously, we deﬁne the space of asymptotically almost periodic functions AAP (X) := AP (X) ⊕ C0 (X), the space of asymptotically compact almost automorphic functions, AAAc (X) := AAc (X) ⊕ C0 (X), and the space of asymptotically almost automorphic functions AAA(X) := AA(X) ⊕ C0 (X). We have the following natural proper inclusions APω (X) ⊂ AAP (X) ⊂ AAAc (X) ⊂ AAA(X) ⊂ BC(X).

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Remark 2.6. We observe that APω (X) = SAPω (X). where SAPω (X) := {f ∈ BC(X) : ∃ω > 0 ||f (t + ω) − f (t)|| → 0 as t → ∞}. This fact was only recently proved in [27], providing a counterexample to the assertion given in [23, Lemma 2.1]. This way, in general we only have APω (X) ⊂ SAPω (X). The class of functions in SAPω (X) is called S-asymptotically ω-periodic (see [27] for a discussion of qualitative properties of this class of functions). We next consider the following set ⎧ ⎫ T ⎨ ⎬ 1 ||f (s)||ds = 0 , P0 (X) := f ∈ BC(X) : lim T →∞ 2T ⎩ ⎭ −T

and deﬁne the following classes of spaces: The space of pseudo-periodic functions P Pω (X) := Pω (X) ⊕ P0 (X), the space of pseudo almost periodic functions P AP (X) := AP (X) ⊕ P0 (X), the space of pseudo compact almost automorphic functions P AAc (X) := AAc (X) ⊕ P0 (X), and the space of pseudo almost automorphic functions P AA(X) := AA(X) ⊕ P0 (X). As before, we also have the following relationship between them P Pω (X) ⊂ P AP (X) ⊂ P AAc (X) ⊂ P AA(X) ⊂ BC(X). Since C0 (X) ⊂ P0 (X), we have the following diagram that summarizes the different classes of subspaces deﬁned above AA(X) ⇒ AAA(X) ⇒ P AA(X) ⇑ ⇑ ⇑ AAc (X) ⇒ AAAc (X) ⇒ P AAc (X) ⇑ ⇑ ⇑ AP (X) ⇒ AAP (X) ⇒ P AP (X) ⇑ ⇑ ⇑ Pω (X) ⇒ APω (X) ⇒ P Pω (X) ⇓ SAPω (X)

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Remark 2.7. The definition of almost periodic functions was introduced by Bohr [6]. Compact almost automorphic functions was introduced by Fink [21] after previous work of S. Bochner, who introduced the concept of almost automorphic functions (see [5]). Asymptotically periodic functions appears by the ﬁrst time in works of de Bruijn [15] whereas asymptotically almost periodic functions was introduced by Fr´echet [22]. The concept of asymptotically almost automorphic functions was deﬁned by N’Gu´er´ekata [35]. Pseudo periodic functions are treated, apparently for the ﬁrst time, in the article by Yuan [44]. Pseudo almost periodic functions are introduced in the literature by Zhang [45]. Finally, the concept of pseudo almost automorphic functions was only recently introduced by Liang et al. [30]. The concepts of asymptotically compact almost automorphic functions as well as pseudo compact almost automorphic functions appears here by the ﬁrst time. The fact that the space P AA(X) is complete under the sup-norm was only recently proved, see [43].

3. The Linear Case In this section, we study bounded solutions for the linear integro-differential equation t a(t − s)Au(s)ds + f (t), t ∈ R. (3.1) u (t) = Au(t) + −∞

Recall that a function u ∈ C 1 (R; X) is called a strong solution of Eq. (3.1) on R if u ∈ C(R; D(A)) and Eq. (3.1) holds on R. If merely u(t) ∈ X instead of the domain of A, we say that u is a mild solution of the linear equation (3.1). Conditions under which a mild solution implies a strong one has been studied in Pr¨ uss [40]. We shall denote by M(R, X), or simply M(X), any of the spaces deﬁned in the previous section. We deﬁne the set M(R × X; X) which consists of all functions f : R × X → X such that f (·, x) ∈ M(R, X) uniformly for each x ∈ K, where K is any bounded subset of X. Given f ∈ M(R × X; X), we ask for conditions under which there exists a solution u ∈ M(R, X). We remark that, in general, the form in which integro-differential equations arise in applications is given by Eq. (3.1) on the line, and the problem

t a(t − s)Av(s)ds + g(t),

v (t) = Av(t) +

t ≥ 0,

(3.2)

0

arises from Eq. (3.1) as a history value problem. When considering problems with forces f ∈ M(R, X), the equation to consider is (3.1), since Eq. (3.2) is only time invariant but not translation invariant, only Eq. (3.1) enjoys the latter property. In this context the important question arises whether the solutions v(t) of Eq. (3.2) and u(t) of Eq. (3.1) are asymptotic to each other, i.e., whether u(t) − v(t) → 0 as t → ∞, whenever f (t) → g(t) → 0 as t → ∞.

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Under reasonable assumptions this turns to be the case (cf. [3]), and therefore the term limiting equation of (3.2) makes sense for Eq. (3.1). We recall that the Laplace transform of a function f ∈ L1loc (R+ , X) is given by ∞ L(f )(λ) := fˆ(λ) :=

e−λt f (t)dt,

Reλ > ω,

0

where the integral is absolutely convergent for Reλ > ω. Furthermore, we denote by B(X) the space of bounded linear operators from X into X endowed with the norm of operators, and the notation ρ(A) stands for the resolvent set of A. In order to give an operator theoretical approach to Eq. (3.1) we recall the following definition (cf. [40]) (see also Remark 3.5 below for the motivation). Definition 3.1. Let b ∈ L1loc (R) be given. Let A be a closed and linear operator with domain D(A) deﬁned on a Banach space X. We call A the generator of a solution operator (or resolvent family) if there exists μ ∈ R and a strongly 1 : Reλ > μ} ⊂ ρ(A) continuous function S : R+ → B(X) such that { ˆb(λ) and ∞ 1 1 −1 ( − A) x = e−λt S(t)xdt, Reλ > μ, x ∈ X. λˆb(λ) ˆb(λ) 0

In this case, S(t) is called the solution operator generated by A. In the scalar case, there is a large bibliography which studies the concept of resolvent, we refer to the monograph by Gripenberg et al. [24], and references therein. We emphasize the fact that because of the uniqueness of the Laplace transform, in the case b(t) ≡ 1 the family S(t) corresponds to a C0 semigroup. We note that solution operators, as well as resolvent families, are a particular case of (b, k)-regularized families introduced in [31]. According to [29] a solution operator S(t) corresponds to a (1, b)-regularized family. Definition 3.2. [40] A strongly measurable family of operators ∞{T (t)}t≥0 ⊂ B(X) is called uniformly integrable (or strongly integrable) if 0 ||T (t)||dt < ∞. ∞ In what follows, we will denote T := 0 ||T (t)||dt < ∞ for any uniformly integrable family of such operators {T (t)}t≥0 . Note that exponentially stable C0 -semigroups are examples of uniformly integrable families of operators. The following is our main result on maximal regularity under convolution of the above mentioned spaces. It corresponds to a summary, with new, and in some cases, a slight extension and improvement of recent results given by a number of authors (cf. [1,3,18,25,27,32–34]).

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Theorem 3.3. Let {S(t)}t≥0 ⊂ B(X) be a uniformly integrable and strongly continuous family. If f belongs to one of the spaces M(X), and w(t) is given by t S(t − s)f (s) ds

w(t) =

(3.3)

−∞

then w belongs to the same space as f . Proof. We ﬁrst consider periodic functions. Given f ∈ Pω (X), with a simple change of variables, we have t+ω

w(t + ω) − w(t) =

t

S(t + ω − s)f (s)ds − −∞

S(t − s)f (s)ds = 0.

−∞

We now consider the space of almost periodic functions AP (X). So if f ∈ AP (X), then by hypotheses, for each > 0 there exists a T > 0 such that every subinterval of R of length T contains at least one point h such that supt∈R ||f (t + h) − f (t)|| ≤ . We have t sup ||w(t + h) − w(t)|| = sup || t∈R

t∈R

S(t − s)[f (s + h) − f (s)]ds||

−∞

≤ ||S|| sup ||f (t + h) − f (t)|| ≤ ||S||, t∈R

and therefore, w has the same property as f , i.e., it is almost periodic. AAc (X) : Let (σn )n∈N be a sequence of real numbers. Since f ∈ AAc (X) there exist a subsequence (sn )n∈N , and a continuous function v ∈ BC(X) such that f (t + sn ) converges to v(t) and v(t − sn ) converges to f (t) uniformly on compact subsets of R. Since t+s n

t

S(t + sn − s)f (s)ds =

w(t + sn ) = −∞

S(t − s)f (s + sn )ds,

(3.4)

−∞

using the Lebesgue’s dominated convergence theorem, we obtain that t w(t + sn ) converges to z(t) = −∞ S(t − s)v(s)ds as n → ∞ for each t ∈ R. Furthermore, the preceding convergence is uniform on compact subsets of R. To show this assertion, we take a compact set K = [−a, a]. For ε > 0, we choose Lε > 0 and Nε ∈ N such that ∞ ||S(s)||ds ≤ ε, Lε

f (s + sn ) − v(s) ≤ ε,

n ≥ Nε ,

s ∈ [−L, L],

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where L = Lε + a. For t ∈ K, we now can estimate t

w(t + sn ) − z(t) ≤

||S(t − s)|| f (s + sn ) − v(s) ds −∞

−L ≤

||S(t − s)|| f (s + sn ) − v(s) ds −∞

t ||S(t − s)|| f (s + sn ) − v(s) ds

+ −L

∞

≤ 2 f ∞

∞ ||S(s)||ds + ε

||S(s)||ds 0

t+L

≤ ε (2 f ∞ + ||S||) , which proves that the convergence is independent of t ∈ K. Repeating this argument, one can show that z(t − sn ) converges to w(t) as n → ∞ uniformly for t in compact subsets of R. This completes the proof in case of the space AAc (X). AA(X) : Let (sn ) ⊂ R be an arbitrary sequence. Since f ∈ AA(X) there exists a subsequence (sn ) of (sn ) such that lim f (t + sn ) = v(t),

for all t ∈ R

lim v(t − sn ) = f (t),

for all t ∈ R.

n→∞

and n→∞

From Eq. (3.4), note that

w(t + sn ) ≤ ||S|| f ∞ and by continuity of S(·)x we have S(t − σ)f (σ + sn ) → S(t − σ)v(σ), as n → ∞ for each σ ∈ R ﬁxed and any t ≥ σ. Then by the Lebesgue’s dominated convergence theorem, we obtain that w(t + sn ) converges to z(t) = t S(t − s)v(s)ds as n → ∞ for each t ∈ R. Similarly we can show that −∞ z(t − sn ) → w(t)

as n → ∞,

for all t ∈ R,

and the proof is complete. SAPω (X) : We have t+ω

w(t + ω) − w(t) =

t

S(t + ω − s)f (s)ds − −∞

S(t − s)f (s)ds 0

t S(t − s)[f (s + ω) − f (s)]ds.

= −∞

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For each > 0, there is a positive constant L such that ||f (t+ω)−f (t)|| ≤ , for every t ≥ L . Under these conditions, for t ≥ L , we can estimate t ||w(t + ω) − w(t)|| ≤

||S(t − s)[f (s + ω) − f (s)]||ds −∞

L ||S(t − s)[f (s + ω) − f (s)]||ds

≤ −∞

t ||S(t − s)[f (s + ω) − f (s)]||ds

+ L

L ≤ 2||f ||∞

t ||S(t − s)||ds +

−∞ ∞

= 2||f ||∞

||S(t − s)||ds

L

∞ ||S(s)||ds +

||S(s)||ds 0

t−L

which permits to conclude that w(t + ω) − w(t) → 0 as t → ∞. Now we will study the asymptotic behavior of the solutions. Let h ∈ C0 (X) and deﬁne t S(t − s)h(s)ds.

H(t) =

(3.5)

−∞

Let > 0 be given. There exist T > 0 such that ||h(s)|| < for all s > T and hence we can write T

t S(t − s)h(s) ds +

H(t) = −∞

S(t − s)h(s) ds. T

Then T ||H(t)|| ≤

t ||S(t − s)||||h(s)|| ds +

−∞

∞

≤ ||h||∞

||S(t − s)|| ds T

||S(v)||dv + ||S||,

t−T

and we conclude that H(t) → 0 as t → ∞. It permits us to infer the conclusion of the theorem for the spaces AP (X), AAP (X), AAAc (X) and AAA(X).

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Vanishing mean value: Let h ∈ P0 (X) and deﬁne H(t) as in Eq. (3.5). For R > 0 we have ⎡ ⎤ R R t 1 1 ⎣ ||H(t)||dt ≤ ||S(t − s)||||h(s)||ds⎦ dt 2R 2R −R

−R

−∞

⎡ ⎤ R ∞ 1 ⎣ ||S(s)||||h(t − s)||ds⎦ dt ≤ 2R −R

∞ = 0

0

⎡

1 ||S(s)|| ⎣ 2R

R

⎤ ||h(t − s)||dt⎦ ds

−R

Note that the set P0 (X) is translation-invariant. Hence, using the Lebesgue’s dominated convergence theorem, we obtain from the above inequality that R 1 2R −R ||H(t)||dt → 0 as R → ∞. We conclude that the spaces P Pω (X), P AP (X), P AAc (X) and P AA(X) have the maximal regularity property under the convolution deﬁned by Eq. (3.3). The following consequence is the main result of this section. Theorem 3.4. Assume that A generates a uniformly integrable (1, 1 − (1 ∗ a))regularized family S(t) on the Banach space X. Then for each f ∈ M(X) there is a unique mild solution u ∈ M(X) of Eq. (3.1). Proof. Let b(t) := 1 −

t 0

a(s)ds and deﬁne t S(t − s)f (s)ds,

u(t) := −∞

where S satisﬁes the resolvent equation t b(t − s)AS(s)xds,

S(t)x = x +

x ∈ X,

t ≥ 0.

(3.6)

0

In particular, since b(t) is differentiable, the above equation shows that for each x ∈ X, S (t)x exists and

t

S (t)x = AS(t)x −

a(t − s)AS(s)xds,

t ≥ 0.

(3.7)

0

It remains to prove that u deﬁned as above is a mild solution for Eq. (3.1). In fact, since S (t)x exists and A is closed, using Fubini’s theorem we obtain

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u (t) = S(0)f (t) +

t

IEOT

S (t − s)f (s)ds

−∞

t = f (t) +

AS(t − s)f (s)ds

−∞

t t−s a(t − s − τ )AS(τ )f (s)dτ ds − −∞ 0

t t a(t − v)AS(v − s)f (s)dvds

= f (t) + Au(t) − −∞ s

t v a(t − v)AS(v − s)f (s)dsdv

= f (t) + Au(t) − −∞ −∞

t

v a(t − v)

= f (t) + Au(t) − −∞

AS(v − s)f (s)dsdv

−∞

t = f (t) + Au(t) −

a(t − v)Au(v)dv.

−∞

Remark 3.5. The idea behind of the above theorem and its proof is the following: Given an abstract linear equation [in this case Eq. (3.2) as limiting equation of (3.1)] we take formally the Laplace transform and obtain F (λ)ˆ u(λ) = fˆ(λ) + initial conditions. For example, F (λ) = (λ − A − a ˆ(λ)A) in this case. Then, we deﬁne an ad hoc strongly continuous family of bounded and linear operators S(t) for the given abstract linear equation as those that satisfy ˆ F (λ)S(λ) = I, and ˆ S(λ)F (λ) = I. Then, we directly prove that the (mild) solution of Eq. (3.1) have the convot lution structure u(t) = −∞ S(t − s)f (s)ds. For instance, in case of Eq. (3.1) we ﬁnd that S(t) should formally satisfy −1 1 1 −1 ˆ S(λ) = (λ − A − a ˆ(λ)A) = −A . λ(1 + a ˆ(λ)) 1 + a ˆ(λ) Comparing with Definition 3.1 (or more generally the definition of (b, k)-regularized families, cf. [31]) we ﬁnd b(t) = 1 + a(t) (and k(t) = 1, resp.), so that the right ad hoc family to consider in this case, corresponds to

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those of resolvent families [40]. After this procedure, we simply check directly that the proposed “mild” solution solves, in fact, the abstract linear evolution equation under study. The definition of “mild” solution for the nonlinear case is then straightforward (see e.g., Definition 4.2 below). In case a(t) ≡ 0 we obtain the following corollary. Corollary 3.6. Assume that A generates an uniformly integrable semigroup S(t) on the Banach space X. Then for each f ∈ M(X) there is a unique mild solution u ∈ M(X) of the equation u (t) = Au(t) + f (t),

t ∈ R.

(3.8)

Remark 3.7. The above corollary generalizes [37, Theorem 3.1] when Ω = AA(X) and recovers [19, Theorem 2.7] when B = g = 0; Ω = AAA(X) and the remainder cases are new results. Example. Let A = −ρI where ρ > 0 and a(t) ≡ 0. Then S(t) = e−ρt I and we conclude that for each f ∈ M(X) the equation u (t) = −ρu(t) + f (t), t ∈ R, (3.9) t has the unique strong solution u(t) = −∞ e−ρ(t−s) f (s)ds, which belongs to M(X). In practice it may not be easy to check in Theorem 3.4 the conditions of uniform integrability or even the hypothesis of A being the generator of an (1, 1 − (1 ∗ a))-regularized family, therefore we state below a more direct criterion. Recall that a C ∞ -function g : R+ → R is called completely monotonic if (−1)n g (n) (t) ≥ 0 for all t ≥ 0 and n ∈ Z+ . Theorem 3.8. Suppose A generates an analytic C0 -semigroup, bounded on 1 some sector Σ(0, θ), and ∞ is invertible, let a ∈ Lloc (R+ ) be completely monotonic and such that 0 a(s)ds < 1. Then for each f ∈ M(X) there is a unique mild solution u ∈ M(X) of Eq. (3.1). Proof. Direct consequence of Theorem 3.4 and [40, Corollary 10.1].

Remark 3.9. Kernels a(t) satisfying the condition of the above theorem can be easily found using Bernstein’s theorem, which characterizes completely monotonic functions as Laplace transforms of positive measures supported 1 on R+ . A simple example following this method is a(t) = (t+α) 2 for α > 1. Example. Suppose f (·, x) ∈ M(L2 (Ω)) for each ﬁxed x ∈ Ω. Let α > 1. The problem t ut (t, x) = Δu(t, x) + −∞

1 Δu(s, x)ds + f (t, x) (t − s + α)2

admits a mild solution u which belongs to M(L2 (Ω)).

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4. The Semilinear Problem It is well known that the study of composition of two functions with special properties plays a key role in discussing the existence of solutions to semilinear equations. Our ﬁrst main result in this section, review and give some new composition theorems on the spaces deﬁned in the second section. Deﬁne the Nemytskii’s superposition operator N (ϕ)(·) := f (·, ϕ(·)) for ϕ ∈ M(X). From here on, M(X) will denote one the following spaces Pω (X), APω (X), P Pω (X), SAPω (X), AP (X), AAP (X), P AP (X), AA(X), AAA(X), P AA(X). We also denote C0 (R × X, X) = {h ∈ BC(R × X, X) : lim ||f (t, x)|| = 0 uniformly on t→∞

any subset of X} and ⎧ ⎨ P0 (R × X, X) = h ∈ BC(R × X, X) : h(·, x) ∈ BC(X) for all x ∈ X, and ⎩

lim

R→∞

1 2R

R ||h(t, x)||dt = 0 uniformly in x ∈ X −R

⎫ ⎬ ⎭

.

Theorem 4.1. Let f ∈ M(R × X, X) be given and assume that there exists a constant Lf > 0 such that ||f (t, x) − f (t, y)|| ≤ Lf ||x − y||,

(4.1)

for all t ∈ R, x, y ∈ X. Let ϕ ∈ M(X). Then N (ϕ) belongs to the same space as ϕ. Proof. For almost periodic functions, AP (X), the proof was ﬁrst provided in [1, Proposition 1]. See also [4, Lemma 3.4] and references therein. For the space AAP (X) our result is a consequence of [4, Lemma 8.3]. The case of P AP (X) is consequence of [29, Theorem 2.1]. See also [16, Theorem 3.4]. For almost automorphic functions, AA(X), a proof is given in [34, Theorem 2.2.4]. In case of AAA(X) the proof is contained in [30, Theorem 2.3]. In case of P AA(X) the result is consequence of [30, Theorem 2.4]. For the space SAPω (X) the result appear in [27, Lemma 4.1] and implicitly before in [26]. The remaining cases Pω (X); APω (X); P Pω (X); are proved as follows: Let Ω = Pω (X); assume that ϕ ∈ Ω and f (·, x) ∈ Ω uniformly for each x ∈ K, where K is any bounded subset of X. Then for all t ∈ R we have

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f (t + ω, ϕ(t + ω)) − f (t, ϕ(t)) = f (t + ω, ϕ(t + ω)) − f (t + ω, ϕ(t)) +f (t + ω, ϕ(t) − f (t, ϕ(t))

≤ f (t + ω, ϕ(t + ω)) − f (t + ω, ϕ(t))

+ f (t + ω, ϕ(t) − f (t, ϕ(t))

≤ L ϕ(t + ω) − φ(t)

= 0. Thus N (ϕ) ∈ Ω. Let Ω = APω (X) and assume that ϕ ∈ Ω and f (·, x) ∈ Ω uniformly for each x ∈ K, where K is any bounded subset of X. We can write f = g + h where g ∈ Pω (R×X, X), h ∈ C0 (R+ ×X, X) and ϕ = α+β where ϕ ∈ Pω (X) and β ∈ C0 (R+ , X). Now we write f (t, ϕ(t)) = f (t, ϕ(t)) − f (t, α(t)) + g(t, α(t)) + h(t, α(t)). Observe that by the above g(·, α(·)) ∈ Pω (X). Now I(t) := f (t, ϕ(t)) − f (t, α(t)) ≤ Lf ϕ(t) − α(t) = Lf β(t)

which shows that lim I(t) = 0.

t→∞

Also if we let K = {α(t) : t ∈ R} which is compact and bounded, then we obtain that lim h(t, α(t)) = 0.

t→∞

Thus N (ϕ) ∈ Ω. Let Ω = P Pω (X) and write f (t, ϕ(t)) = g(t, α(t)) + f (t, ϕ(t)) − g(t, α(t)) = g(t, α(t)) + f (t, ϕ(t)) − f (t, α(t)) + h(t, α(t)). As above g(·, α(·)) ∈ Pω (X). Now as in the Proof of Theorem 2.1 f (t, ϕ(t)) − f (t, α(t)), and h(t, α(t)) are in P0 (X). In what follows we study existence and uniqueness of solutions in M(X) for the semilinear integro-differential equation, t u (t) = Au(t) + a(t − s)Au(s)ds + f (t, u(t)). (4.2) −∞

Definition 4.2. A function u : R → X is said to be a mild solution to Eq. (4.2) if there exists a strongly continuous family of bounded and linear operators on X such that the function s → S(t − s)f (s, u(s)) is integrable on (−∞, t) for each t ∈ R and t S(t − s)f (s, u(s))ds, (4.3) u(t) = −∞

for each t ∈ R.

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We next give several theorems on existence of mild solutions for the semilinear problem. We begin with the following simple result. Theorem 4.3. Assume that A generates an uniformly integrable (1, 1−(1∗a))regularized family S(t) on the Banach space X. Let f ∈ M(R×X, X) be given and assume that f satisfy ||f (t, x) − f (t, y)|| ≤ Lf ||x − y||,

(4.4)

for all t ∈ R. Then Eq. (4.2) has a unique mild solution u ∈ M(X) whenever Lf < ||S||−1 . Proof. We deﬁne the operator F : M(X) → M(X) by t S(t − s)f (s, ϕ(s)) ds,

(F ϕ)(t) :=

t ∈ R.

(4.5)

−∞

In view of Theorems 3.3 and 4.1, F is well deﬁned. Then for ϕ1 , ϕ2 ∈ M(X) we have: t S(t − s)[f (s, ϕ1 (s)) − f (s, ϕ2 (s))]ds

F ϕ1 − F ϕ2 ∞ = sup t∈R −∞ ∞

≤ L sup t∈R

S(τ )

ϕ1 (t − τ ) − ϕ2 (t − τ ) dτ 0

≤ L ϕ1 − ϕ2 ∞ S . This proves that F is a contraction, so by the Banach ﬁxed point theorem there exists a unique u ∈ M(X), such that F u = u, that is u(t) = t S(t − s)f (s, u(s))ds. Since clearly u is a mild solution of Eq. (4.2) (cf. −∞ also the proof of Theorem 3.4), the proof is complete. The following consequence is immediate. Corollary 4.4. Suppose A generates an analytic C0 -semigroup, bounded on 1 some sector Σ(0, θ), and ∞ is invertible, let a ∈ Lloc (R+ ) be completely monotonic and such that 0 a(s)ds < 1. Let f ∈ M(R × X, X) be given and assume that f satisfy ||f (t, x) − f (t, y)|| ≤ Lf ||x − y||,

(4.6)

for all t ∈ R. Then there exists η > 0 such that Eq. (4.2) has a unique mild solution u ∈ M(X) whenever Lf < η. A different Lipschitz condition is considered in the following result. Recall that a strongly continuous family {S(t)}t≥0 ⊂ B(X) is said to be uniformly bounded if there exists a constant M > 0 such that S(t) ≤ M for all t ≥ 0.

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Theorem 4.5. Assume that A generates a bounded and uniformly integrable (1, 1 − (1 ∗ a))-regularized family S(t) on the Banach space X. Let f ∈ M(R × X, X) be given and assume that f satisfy ||f (t, x) − f (t, y)|| ≤ Lf (t)||x − y||,

t ∈ R,

(4.7)

1

where Lf ∈ L (R) ∩ BC(R). Then Eq. (4.2) has a unique mild solution u ∈ M(X). Proof. We deﬁne the operator F as in Eq. (4.5). Clearly under conditions on Lf , F ϕ ∈ M(X) if ϕ ∈ M(X). Now let ϕ1 , ϕ2 be in M(X). We can estimate t S(t − s)[f (s, ϕ1 (s)) − f (s, ϕ2 (s))]ds ||(F ϕ1 )(t) − (F ϕ2 )(t)|| = −∞

t ≤M

Lf (s) ϕ1 (s) − ϕ2 (s) ds −∞

Repeating the argument, we get ||(F n ϕ1 )(t) − (F n ϕ2 )(t)|| t s ≤M

sn−2

···

n −∞ −∞

Lf (s)Lf (s1 ) · · · Lf (sn−1 ) × −∞

×||ϕ1 (sn−1 ) − ϕ2 (sn−1 )||dsn−1 · · · ds1 ds ⎛ t ⎞n n M ⎝ ≤ Lf (τ )dτ ⎠ ||ϕ1 − ϕ2 ||∞ n! −∞

≤

(M ||Lf ||1 )n ||ϕ1 − ϕ2 ||∞ . n!

(M ||L || )n

f 1 Since < 1 for n sufﬁciently large, applying the contraction princin! ple we conclude that F has a unique ﬁxed point u ∈ M(X) which completes the proof.

Of course, an immediate consequence under the condition that A generates a bounded analytic semigroup, like Corollary 4.4, also holds. The particular case a(t) ≡ 0 reads as follows. Corollary 4.6. Suppose A generates an analytic C0 -semigroup, bounded on some sector Σ(0, θ), and invertible. Let f ∈ M(R×X, X) be given and assume that f satisfies ||f (t, x) − f (t, y)|| ≤ Lf (t)||x − y||,

t ∈ R,

(4.8)

1

where Lf ∈ L (R) ∩ BC(R). Then equation u (t) = Au(t) + f (t, u(t)), has a unique mild solution u ∈ M(X).

(4.9)

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We note that conditions of type (4.8) has been previously considered in the literature (see [10] and references therein). Now we consider a more general case of equations introducing a new class of functions L which do not necessarily belong to L1 (R). We have the following result. Theorem 4.7. Assume that A generates a bounded and uniformly integrable (1, 1 − (1 ∗ a))-regularized family S(t) on the Banach space X. Let f ∈ M(R×X, X) be given and assume that f satisfy the Lipschitz condition (4.8) t where Lf ∈ BC(R) and the integral −∞ Lf (s)ds exists for all t ∈ R. Then Eq. (4.2) has a unique mild solution u ∈ M(X). Proof. Deﬁne a new norm |||ϕ|| := supt∈R {v(t)||ϕ(t)||}, where v(t) := e−k

t

−∞

Lf (s)ds

and k is a ﬁxed positive constant greater than M := supt∈R ||S(t)||. Let ϕ1 , ϕ2 be in M(X), then we have v(t)||(F ϕ1 )(t) − (F ϕ2 )(t)|| t = v(t) S(t − s)[f (s, ϕ1 (s)) − f (s, ϕ2 (s))]ds −∞

t ≤M

v(t)Lf (s) ϕ1 (s) − ϕ2 (s) ds −∞

t ≤M

v(t)v(s)−1 Lf (s)v(s) ϕ1 (s) − ϕ2 (s) ds

−∞

t ≤ M |||ϕ1 (s) − ϕ2 (s)|||

v(t)v(s)−1 Lf (s)ds

−∞

M |||ϕ1 (s) − ϕ2 (s)||| = k

t kek

s t

Lf (τ )dτ

Lf (s)ds

−∞

M |||ϕ1 (s) − ϕ2 (s)||| = k

t

−∞

d k s Lf (τ )dτ ds e t ds

t

M [1 − e−k −∞ Lf (τ )dτ ]|||ϕ1 (s) − ϕ2 (s)||| k M |||ϕ1 (s) − ϕ2 (s)|||. ≤ k

=

Hence, since M/k < 1, F has a unique ﬁxed point u ∈ M(X).

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References [1] Amir, B., Maniar, L.: Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain. Ann. Math. Blaise Pascal 6(1), 1–11 (1999) [2] Andres, J., Bersani, A.M., Grande, R.F.: Hierarchy of almost periodic function spaces. Rend. Mat. Ser. VII Roma 26, 121–188 (2006) [3] Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69(11), 3692–3705 (2008) [4] Blot, J., Cieutat, P., N’Gu´er´ekata, G.M., Pennequin, D.: Superposition operators between various almost periodic function spaces and applications. Commun. Math. Anal. 6(1), 42–70 (2009) [5] Bochner, S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. USA 48, 2039–2043 (1962) [6] Bohr, H.: Zur theorie der fast periodischen funktionen. (German) I. Eine verallgemeinerung der theorie der fourierreihen. Acta Math. 45(1), 29–127 (1925) [7] Bugajewski, D., N’Gu´er´ekata, G.M.: On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces. Nonlinear Anal. 59, 1333–1345 (2004) [8] Cl´ement, Ph., Da Prato, G.: Existence and regularity results for an integral equation with infinite delay in a Banach space. Integr. Equ. Oper. Theory 11, 480–500 (1988) [9] Coleman, B.D., Gurtin, M.E.: Equipresence and constitutive equation for rigid heat conductors. Z. Angew. Math. Phys. 18, 199–208 (1967) [10] Cuevas, C., Lizama, C.: Almost automorphic solutions to a class of semilinear fractional differential equations. Appl. Math. Lett. 21, 1315–1319 (2008) [11] Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal. 37, 297–308 (1970) [12] Dafermos, C.M.: An abstract Volterra equation with applications to linear viscoelasticity. J. Differ. Equ. 7, 554–569 (1970) [13] Da Prato, G., Lunardi, A.: Periodic solutions for linear integrodifferential equations with infinite delay in Banach spaces. Differential Equations in Banach spaces. Lect. Notes Math. 1223, 49–60 (1985) [14] Da Prato, G., Lunardi, A.: Solvability on the real line of a class of linear Volterra integrodifferential equations of parabolic type. Ann. Math. Pura Appl. 4, 67–117 (1988) [15] De Bruijn, N.G.: The asymptotically periodic behavior of the solutions of some linear functional equations. Am. J. Math. 71, 313–330 (1949) [16] Diagana, T.: weighted pseudo almost automorphic functions and applications. C. R. Acad. Sci. Paris Ser. I 343, 643–646 (2006) [17] Diagana, T., Hern´ andez, E., Dos Santos, J.P.C.: Existence of asymptotically almost automorphic solutions to some abstract partial neutral integrodifferential equations. Nonlinear Anal. 71, 248–257 (2009) [18] Diagana, T., Henr´ıquez, H., Hern´ andez, E.: Almost automorphic mild solutions to some partial neutral functional–differential equations and applications. Nonlinear Anal. 69(5-6), 1485–1493 (2008)

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[19] Ding, H.-S., Liang, J., Xiao, T.-J.: Asymptotically almost automorphic solutions for some integro-differential equations with nonlocal conditions. J. Math. Anal. Appl. 338(1), 141–151 (2008) [20] Faˇsangov´ a, E., Pr¨ uss, J.: Asymptotic behaviour of a semilinear viscoelastic beam model. Arch. Math. (Basel) 77, 488–497 (2001) [21] Fink, A.M.: Almost automorphic and almost periodic solutions which minimize functionals. Tˆ ohoku Math. J. 20(2), 323–332 (1968) [22] Fr´echet, M.: Les fonctions asymptotiquement presque-periodiques continues (French). C. R. Acad. Sci. Paris 213, 520–522 (1941) [23] Gao, H., Wang, K., Wei, F., Ding, X.: Massera-type theorem and asymptotically periodic logistic equations. Nonlinear Anal. Real World Appl. 7, 1268– 1283 (2006) [24] Gripenberg, G., Londen, S.-O., Staffans, O.: Volterra Integral And Functional Equations. In: Encyclopedia of Mathematics and its Applications, vol. 34. Cambridge University Press, Cambridge (1990) [25] Henr´ıquez, H.R., Lizama, C.: Compact almost periodic solutions to integral equations with infinite delay. Nonlinear Anal. 71, 6029–6037 (2009) [26] Henr´ıquez, H.R., Pierri, M., T´ aboas, P.: Existence of S-asymptotically ω-periodic solutions for abstract neutral equations. Bull. Aust. Math. Soc. 78, 365–382 (2008) [27] Henr´ıquez, H.R., Pierri, M., T´ aboas, P.: On S-asymptotically ω-periodic functions on Banach spaces and applications. J. Math. Anal. Appl. 343(2), 1119– 1130 (2008) [28] Levitan, B.M., Zhikov, V.V.: Almost Periodic Functions and Differential Equations. Moscow University House, Moscow (1978) (English Translation by Cambridge University Press, 1982) [29] Li, H., Huang, F., Li, J.: Composition of pseudo almost periodic functions and semilinear differential equations. J. Math. Anal. Appl. 255, 436–446 (2001) [30] Liang, J., Zhang, J., Xiao, T.J.: Composition of pseudo almost automorphic and asymptotically almost automorphic functions. J. Math. Anal. Appl. 340(2), 1493–1499 (2008) [31] Lizama, C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000) [32] Mophou, G.M., N’Gu´er´ekata, G.M.: On some classes of almost automorphic functions and applications to fractional differential equations (2010, to appear) [33] N’Gu´er´ekata, G.M.: Topics in Almost Automorphy. Springer, New York (2005) [34] N’Gu´er´ekata, G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer, New York (2001) [35] N’Gu´er´ekata, G.M.: Quelques remarques sur les fonctions asymptotiquement presque automorphes (French). Ann. Sci. Math. Quebec 7(2), 185–191 (1983) [36] N’Gu´er´ekata, G.M.: Comments on almost automorphic and almost periodic functions in Banach spaces. Far East J. Math. Sci. (FJMS) 17(3), 337–344 (2005) [37] N’Gu´er´ekata, G.M.: Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations. Semigroup Forum 69, 80–86 (2004)

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[38] Nguyen, V.M., Naito, T., N’Gu´er´ekata, G.M.: A spectral countability condition for almost automorphy of solutions of differential equations. Proc. Am. Math. Soc. 134, 3257–3266 (2006) [39] Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–304 (1971) [40] Pr¨ uss, J.: Evolutionary integral equations and applications. In: Monographs in Mathematics, vol. 87. Birkh¨ auser, Boston (1993) [41] Renardy, M., Hrusa, W.J., Nohel, J.A.: Mathematical problems in viscoelasticity. In: Pitman Monographs Pure Applied Mathematics, vol. 35. Longman Sci. Tech., Harlow, Essex (1988) [42] Sforza, D.: Existence in the large for a semilinear integrodifferential equation with infinite delay. J. Differ. Equ. 120, 289–303 (1995) [43] Xiao, T.J., Liang, J., Zhang, J.: Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces. Semigroup Forum 76(3), 518– 524 (2008) [44] Yuan, R.: Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument. Nonlinear Anal. 41(7–8), 871–890 (2000) [45] Zhang, C.Y.: Integration of vector-valued pseudo-almost periodic functions. Proc. Am. Math. Soc. 121(1), 167–174 (1994) Carlos Lizama Departamento de Matem´ atica, Facultad de Ciencias Universidad de Santiago de Chile Casilla 307-Correo 2, Santiago, Chile e-mail: [email protected] Gaston M. N’Gu´er´ekata (B) Department of Mathematics Morgan State University Baltimore, MD 21251, USA e-mail: Gaston.N’[email protected] Received: October 30, 2009. Revised: March 4, 2010.

Integr. Equ. Oper. Theory 68 (2010), 229–241 DOI 10.1007/s00020-010-1790-y Published online March 23, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Operators Commuting with the Volterra Operator are not Weakly Supercyclic Stanislav Shkarin Abstract. We prove that any bounded linear operator on Lp [0, 1] for 1 ≤ p < ∞, commuting with the Volterra operator V , is not weakly supercyclic, which answers aﬃrmatively a question raised by L´eon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic type condition on an operator which prevents it from being weakly supercyclic and is satisﬁed for any operator commuting with V . Mathematics Subject Classification (2000). Primary 47A16; Secondary 37A25. Keywords. Supercyclic operators, weakly supercyclic operators, Volterra operator.

1. Introduction All vector spaces are assumed to be over K being either the ﬁeld C of complex numbers or the ﬁeld R of real numbers. As usual, Z+ is the set of non-negative integers and N is the set of positive integers. For a Banach space X, symbol L(X) stands for the space of bounded linear operators on X and X ∗ is the space of continuous linear functionals on X. For T, S ∈ L(X), we write [T, S] = T S − ST and by C(T ) we denote the centralizer of T : C(T ) = {S ∈ L(X) : [T, S] = 0}. We say that Y is a Banach space embedded into a Banach space X if Y is a linear subspace of X endowed with its own norm, which deﬁnes a topology on Y (maybe non-strictly) stronger than the one inherited from X and turns Y into a Banach space. For instance, C[0, 1] with the sup-norm is a Banach space embedded into L1 [0, 1]. We say that T ∈ L(X) is supercyclic (respectively, weakly supercyclic) if there exists x ∈ X such that the projective orbit Opr (x, T ) = {zT n x : z ∈ K, n ∈ Z+ } is dense in X (respectively, dense in X with the weak topology). In this case x is said to be a supercyclic vector (respectively, a weakly supercyclic vector) for T . Supercyclicity was introduced by Hilden The author would like to thank A. Montes and F. Leon for their interest and comments.

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and Wallen [13] and studied intensely since then. We refer to the survey [21] for the details. Weak supercyclicity was introduced by Sanders [26] and studied in, for instance [16–19,22,24,27,29]. Gallardo and Montes [10], answering a question raised by Salas, demonstrated that the Volterra operator x (1.1) V f (x) = f (t) dt, 0

acting on Lp [0, 1] for 1 ≤ p < ∞, is non-supercyclic. In [22,16] it is shown that V is not weakly supercyclic. In [28] it is proved that for 1 < p < ∞ and any non-zero f ∈ Lp [0, 1], the sequence V n f /V n f is weakly convergent to 0 in Lp [0, 1]. In [4,16] it is demonstrated that certain operators on Lp [0, 1], commuting with V , are not weakly supercyclic. Recently, Eveson [9] proved that every operator of the shape V α T for α > 0 is non-supercyclic, where T is an invertible operator on L2 [0, 1] commuting with V and V α is the Riemann–Liouville operator. L´eon-Saavedra and Piqueras-Lerena [16] raised a question whether any bounded operator on Lp [0, 1] for 1 ≤ p < ∞, commuting with V , is not weakly supercyclic. In the present article we close the issue by answering this question aﬃrmatively. Theorem 1.1. Let 1 ≤ p < ∞ and T ∈ L(Lp [0, 1]) be such that T V = V T . Then T is not weakly supercyclic. Our approach has nothing in common with the ones from [4,9,16]. In [16] it is shown that positive operators commuting with V are not weakly supercyclic and, naturally, the proof employs positivity argument. In [4] it is shown that convolution operators on Lp [0, 1] are not weakly supercyclic provided the convolution kernel has a nice asymptotic behavior at zero and the proof is based upon upper and lower estimates of the orbits. The same idea is used in [9]. We, on the other hand, ﬁnd an algebraic type condition on an operator preventing it from being weakly supercyclic and demonstrate that operators, commuting with V , satisfy this condition. In order to formulate it in full generality we need to introduce the following class of Banach spaces. Let Y be the class of Banach spaces X such that for any sequence {xn }n∈Z+ in X satisfying n = O(xn ) as n → ∞, the set {xn : n ∈ Z+ } is closed in the weak topology (=weakly closed). Remark 1.2. It immediately follows that if X ∈ Y and A ⊆ Z+ is inﬁnite, then any subset of X of the shape {xn : n ∈ A} is weakly closed provided n = O(xn ) as n → ∞, n ∈ A. It is also clear that a (closed linear) subspace of a Banach space from Y also belongs to Y. In order to prove Theorem 1.1, we need to know that Hilbert spaces belong to Y. The latter follows from the next lemma, which appears as Proposition 5.2 in [29]. Lemma 1.3. Let {xn }n∈Z+ be a sequence in a Hilbert space H such that xn −2 < ∞. Then {xn : n ∈ Z+ } is weakly closed in H. In particular, any Hilbert space belongs to Y.

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Remark 1.4. It is worth noting that the above lemma is proved in [29] by applying Ball’s solution [1] of the complex plank problem. Lemma 1.3 admits a natural stronger form. Bayart and Matheron [3, Theorem 10.5] showed that if X is a Banach space of non-trivial q is the type of X ∗ and type [20], −a xn < ∞ for some a ∈ (1, q), {xn }n∈Z+ is a sequence in X satisfying then the set {xn : n ∈ Z+ } is weakly closed. This result generalizes the case X = p with p1 + 1q = 1 proved in [29] (the proofs are based upon similar arguments). It follows that any Banach space of non-trivial type [20] belongs to Y. In particular, since spaces Lp [0, 1] have type min{p, 2} for 1 ≤ p < ∞, Lp [0, 1] ∈ Y for 1 < p < ∞. According to Kadets [14], see also [3, Chapter 10], any X ∈ Y has ﬁnite cotype [20]. It still remains unclear whether 1 or L1 [0, 1] belong to Y. Theorem 1.5. Let X be a Banach space and T ∈ L(X) be such that (1.5a )

there exists M ∈ L(X) satisfying [T, [T, M ]] = 0 and such that for any cyclic vector u for T , there are B, C ∈ C(T ) for which C[T, M ] = 0 and CM u = Bu.

Then T is not supercyclic. If additionally (1.5b )

there is R ∈ C(T ) such that R(X) is dense in X and R(X) ⊆ Y , where Y ∈ Y and Y is a Banach space, embedded into X,

then T is not weakly supercyclic. In the case X ∈ Y, condition (1.5b ) is automatically satisﬁed with R = I. Thus, we have the following corollary. Corollary 1.6. Let X ∈ Y and T ∈ L(X). If (1.5a ) is satisfied, then T is not weakly supercyclic. We would like to formulate a useful application of Theorem 1.5, dealing with operators on commutative Banach algebras. Note that we do not require a Banach algebra to be unital. For an element a of a commutative Banach algebra X, symbol Ma stands for the multiplication operator Ma ∈ L(X), Ma b = ab. We say that a commutative Banach algebra X is non-degenerate if Ma = 0 implies a = 0. Equivalently, X is non-degenerate if the intersection of kernels of all multiplication operators is {0}. Theorem 1.7. Let X be a non-degenerate commutative Banach algebra, Λ be a commutative subalgebra of L(X) containing {Ma : a ∈ X} and M ∈ L(X) be such that [M, S] ∈ Λ for each S ∈ Λ. Then any T ∈ Λ satisfying [T, M ] = 0 is not supercyclic. If additionally, there is R ∈ Λ with dense range, taking values in a Banach space Y ∈ Y, embedded into X, then any T ∈ Λ satisfying [T, M ] = 0 is not weakly supercyclic.

2. Proof of Main Results The following lemma is a generalization of Lemma 5.5 from [29].

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Lemma 2.1. Let {xn }n∈Z+ be a sequence in a topological vector space X, y ∈ X and f be a continuous linear functional on X such that f (y) = 1. Assume also that y belongs to the closureof the set Ω = {zxn : z ∈ K, n ∈ Z+ }. Then xn : n ∈ Z+ , f (xn ) = 0 . y belongs to the closure of N = f (x n) Proof. Since y ∈ / ker f and ker f is closed in X, y belongs to the closure u is continuous of Ω\ker f . Since the map F : X\ker f → X, F (u) = f (u) and y is in the closure of Ω\ker f , we see that F (y) = y is in the closure of F (Ω\ker f ) = N , as required. The next lemma is an immediate corollary of the Angle Criterion of supercyclicity [11]. For the sake of completeness we show that it also follows from Lemma 2.1. Lemma 2.2. Let X be a Banach space, dim X > 1, x ∈ X and T ∈ L(X). Assume also that there is f ∈ X ∗ \{0} such that f (T n x) = o(T n x) as n → ∞. Then x is not a supercyclic vector for T . Proof. Since dim X > 1, we can pick y ∈ X such that y ∈ / Opr (x, T ) and f (y) = 1. Assume that x is a supercyclic vector for T . Then y is in the cloT nx sure of Opr (x, T ). By Lemma 2.1, y is in the closure of N = un = f (T n x) : n n n n ∈ Z+ , f (T x) = 0 . Since f (T x) = o(T x), un → ∞ as n → ∞. Hence N is closed and therefore y ∈ N ⊂ Opr (x, T ). We have arrived to a contradiction. Lemma 2.3. Let {xn }n∈Z+ be a sequence of elements of a Banach space X ∈ Y of dimension > 1 such that there is a non-zero u ∈ X ∗ satisfying |u(xn )| = O(n−1 xn ) as n → ∞.

(2.1)

of the set Ω = {zxn : z ∈ K, n ∈ Z+ } is norm Then the weak closure Ω nowhere dense in X. In particular, Ω is not weakly dense in X. contains a non-empty norm open set Proof. Assume the contrary. Then Ω W . Since u = 0, dim X ≥ 2 and Ω is a countable union of one-dimensional subspaces of X, we can pick y ∈ W \Ω such that b = u(y) = 0. Let f = b−1 u. Then f ∈ X ∗ and f (y) = 1. By Lemma 2.1, y belongs to the weak closure of xn . According to (2.1), the set N = {yn : n ∈ Z+ , f (xn ) = 0}, where yn = f (x n) n = O(yn ) as n → ∞. Since X ∈ Y, Remark 1.2 implies that N is weakly closed in H. Hence y ∈ N ⊆ Ω. We have arrived to a contradiction. The following lemma is the key tool in the proof of Theorem 1.5. Lemma 2.4. Let X be a Banach space, x ∈ X and T ∈ L(X). Assume also that M ∈ L(X) and C, B, R ∈ C(T ) are such that S = [T, M ] ∈ C(T ) and CM Rx = BRx. Then |S ∗ C ∗ h(RT n x)| ≤ T (B − CM )∗ h(n + 1)−1 RT n x for any n ∈ Z+ and h ∈ X ∗ .

(2.2)

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Proof. Since T M − M T = S commutes with T , an elementary inductive argument shows that T n M − M T n = nST n−1 ∗

∗

for each n ∈ N.

(2.3)

∗

Let h ∈ X , g = S C h and y = Rx. Then for each n ∈ Z+ , we have g(T n y) = S ∗ C ∗ h(T n y) = h(CST n y). Using (2.3) and the equality CT = T C, we obtain (n+1)g(T n y) = (n+1)h(CST n y) = h(CT n+1 M y − CM T n+1 y) = h(T n+1 CM y − CM T n+1 y). Since CM y = By and BT = T B, we get (n+1)g(T n y) = h(T n+1 By − CM T n+1 y) = h(BT n+1 y − CM T n+1 y) = (B − CM )∗ h(T n+1 y). The above display immediately implies that (n + 1)|S ∗ C ∗ h(T n y)| = (n + 1)|g(T n y)| ≤ (B − CM )∗ hT n+1 y ≤ T (B − CM )∗ hT n y for any n ∈ Z+ . Since y = Rx and T R = RT , we see that T n y = T n Rx = RT n x and (2.2) follows. 2.1. Proof of Theorem 1.5 First, assume that only condition (1.5a ) is satisﬁed. Let M ∈ L(X) be the operator provided by (1.5a ) and S = [T, M ]. Then S ∈ C(T ). Assume that T has a supercyclic vector u ∈ X. By (1.5a ), we can pick B, C ∈ C(T ) such that CS = 0 and CM u = Bu. All conditions of Lemma 2.4 with R = I and x = u are satisﬁed. Since CS = 0, we have (CS)∗ = S ∗ C ∗ = 0 and we can pick h ∈ X ∗ such that g = S ∗ C ∗ h = 0. According to Lemma 2.4, g(T n u) = O(n−1 T n u) = o(T n u)

as n → ∞.

By Lemma 2.2, u cannot be a supercyclic vector for T . We have arrived to a contradiction. Assume now that (1.5a ) and (1.5b ) are satisﬁed. Let M and R be operators provided by (1.5a ) and (1.5b ). Then S = [T, M ] ∈ C(T ). Since a subspace of an element of Y belongs to Y, we, replacing Y by the closure of R(X) in Y , we can assume that R(X) is dense in Y . Assume that T has a weakly supercyclic vector x ∈ X. Then x is cyclic for T . Since R(X) is dense in X and R commutes with T , u = Rx is also a cyclic vector for T . By (1.5a ), we can pick B, C ∈ C(T ) such that CS = 0 and CM u = Bu. That is, CM Rx = BRx. Thus all conditions of Lemma 2.4 are satisﬁed. Since CS = 0, we have (CS)∗ = S ∗ C ∗ = 0 and we can pick h ∈ X ∗ such that g = S ∗ C ∗ h = 0. By the closed graph theorem, R : X → Y is continuous. Since x is a weakly supercyclic vector for T , Opr (x, T ) is weakly dense in X. Since R : X → Y is continuous and has dense range, Ω = R(Opr (x, T )) is weakly dense in Y . According to Lemma 2.4, g(RT n x) = O(n−1 RT n x) = O(n−1 RT n xY ) as n → ∞. By Lemma 2.3, {zRT n x : z ∈ K, n ∈ Z+ } = Ω is not weakly dense in Y . This contradiction completes the proof.

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2.2. Proof of Theorem 1.7 Let T ∈ Λ and S = [T, M ] = 0. Since Λ is closed under the operator A → [A, M ], we have S ∈ Λ. Since Λ is commutative, [T, S] = [T, [T, M ]] = 0. Let now u ∈ X be a cyclic vector for T , v = M u and C = Mu , B = Mv . Since Ma ∈ Λ for any a ∈ X and Λ is commutative, we have B, C ∈ C(T ). Moreover, since X is commutative, we have Cv = uv = vu = Bu. Next, we shall show that C is injective. Let y ∈ X be such that Cy = 0. That is, uy = yu = 0. Since T commutes with My , we get yT n u = T n (yu) = 0 for each n ∈ Z+ . Since u is cyclic for T , we have yx = 0 for each x ∈ X. Since X is non-degenerate, y = 0. Hence C is injective. Since S = 0, CS = C[T, M ] = 0. Thus (1.5a ) is satisﬁed. By Theorem 1.5, T is non-supercyclic. Assume additionally that there is R ∈ Λ with dense range, taking values in a Banach space Y ∈ Y, embedded into X. Since Λ is commutative, R ∈ C(T ) and (1.5b ) is satisﬁed. By Theorem 1.5, T is not weakly supercyclic. 2.3. Proof of Theorem 1.1 For 1 ≤ p ≤ ∞, consider the multiplication operator M on Lp [0, 1]: M f (x) = xf (x).

(2.4)

Lemma 2.5. Let 1 ≤ p < ∞ and V, M be the operators on Lp [0, 1] defined in (1.1) and (2.4). Then C(V ) ∩ C(M ) = {cI : c ∈ K}. Proof. Clearly {cI : c ∈ K} ⊆ C(V ) ∩ C(M ). Assume that T ∈ C(V ) ∩ C(M ). Then T M n = M n T for each n ∈ Z+ . Applying this operator equalities to the function 1, being identically 1, we see that T un = un f , where f = T 1 and un (x) = xn . Hence T p = pf for any polynomial p. Since the space of polynomials is dense in Lp [0, 1], we see that f ∈ L∞ [0, 1] and T is the operator of multiplication by f . Since T commutes with V , we have M f = T u1 = T V 1 = V T 1 = V f . Since only constant functions g satisfy the equality M g = V g, f is constant. Hence T = cI for some c ∈ K. This proves the required equality. Lemma 2.6. Let 1 ≤ p < ∞, T ∈ L(Lp [0, 1]), T V = V T and S = T M − M T . Then SV = V S. Proof. Clearly SV − V S = T M V − M T V − V T M + V M T = T (M V − V M ) − (M V − V M )T.

Since M V − V M = V 2 , we have SV − V S = T V 2 − V 2 T = 0.

The Volterra algebra is the Banach space L1 [0, 1] with the convolution multiplication x f g(x) = f (t)g(x − t) dt. 0

According to well-known properties of convolution, the above integral converges for almost all x ∈ [0, 1], f g ∈ L1 [0, 1] and f g1 ≤ f 1 g1 . It also follows that is associative and commutative: f g = g f and (f g) h = f (g h) for any f, g, h ∈ L1 [0, 1]. Moreover, for each p ∈ [1, ∞],

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f gp ≤ f 1 gp for any f ∈ L1 [0, 1] and g ∈ Lp [0, 1]. In particular turns each Lp [0, 1] into a commutative Banach algebra. It is also easy to see that the Volterra operator acts according to the formula V f = f 1. Thus, V is an injective multiplication operator and therefore each (Lp [0, 1], ) is non-degenerate. We will also need the following result of Erdos [7,8], see [23] for a different proof. Lemma 2.7. For 1 ≤ p < ∞, the subalgebra C(V ) of L(Lp [0, 1]) is commutative. We are ready to prove Theorem 1.1. Let 1 ≤ p < ∞. Then the Banach algebra Xp , being the Banach space Lp [0, 1] with the multiplication is commutative and non-degenerate. By Lemma 2.7, C(V ) is a commutative subalgebra of L(Xp ). Since Xp is commutative and V is a multiplication (by 1) operator on Xp , C(V ) contains all the multiplication operators. By Lemma 2.6, QM − M Q ∈ C(V ) for any Q ∈ C(V ), where M is deﬁned by (2.4). Clearly, R = V 2 ∈ C(V ) has dense range and takes values in the Hilbert space W01,2 [0, 1] being the Sobolev space of absolutely continuous functions on [0, 1] vanishing at 0 with square integrable ﬁrst derivative. Since W01,2 [0, 1] is embedded into Lp [0, 1] and any Hilbert space belongs to Y according to Lemma 1.3, Theorem 1.7 implies that any T ∈ C(V ) such that [T, M ] = 0 is not weakly supercyclic. Let now T ∈ C(V ) and [T, M ] = 0. By Lemma 2.5, T is a scalar multiple of the identity and therefore is non-cyclic. Thus any T ∈ C(V ) is not weakly supercyclic. The proof of Theorem 1.1 is complete.

3. Multiplication Operators on Banach Algebras By Theorem 1.1, multiplication operators on the Volterra algebra are not weakly supercyclic. Motivated by this observation, we would like to raise the following question. Problem 3.1. Can a multiplication operator on a commutative Banach algebra be supercyclic or at least weakly supercyclic? Does anything change if we add a scalar multiple of the identity to a multiplication operator? The rest of the section is devoted to the discussion of this problem. Recall that a continuous linear operator T on a topological vector space X is called hypercyclic if there is x ∈ X for which the orbit O(x, T ) = {T n x : n ∈ Z+ } is dense in X. If X is a Banach space, then T is called weakly hypercyclic if it is hypercyclic as an operator on X with the weak topology. The next example shows that in the non-commutative setting the answer to the above question is aﬃrmative. Example 3.2. Let H be the Hilbert space of Hilbert–Schmidt operators on 2 . With respect to the composition multiplication, H is a Banach algebra. Let also S ∈ H be deﬁned by its action on the basic vectors as follows: Se0 = 0, Sen = n−1 en−1 if n ≥ 1. Consider the left multiplication by S operator Φ ∈ L(H), Φ(T ) = ST . Then Φ is supercyclic and I + Φ is hypercyclic.

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Proof. Since ker Φn ∩ Φn (H) = {T ∈ H : T (2 ) ⊆ {span {e0 , . . . , en−1 }}, the union of ker Φn ∩ Φn (H) is dense in H. By [3, Theorem 2.2], the last property implies hypercyclicity of I + Φ. Note that hypercyclicity of I + Φ can also be easily derived from the main result of [5]. Supercyclicity of Φ follows also from the Supercyclicity Criterion [21]. The following two propositions highlight the difﬁculties of Problem 3.1. Proposition 3.3. Let X be a commutative complex Banach algebra. Then for any a ∈ X, the operator Ma is not weakly hypercyclic. Proof. Let a ∈ X. First, consider the case when there exists a non-zero character κ : X → C. It is easy to see that κ is an eigenvector of Ma∗ . Since the point spectrum of the dual of any weakly hypercyclic operator is empty [3], Ma is not weakly hypercyclic. It remains to consider the case when there are no non-zero characters on X. Since a commutative complex Banach algebra with no non-zero characters is radical [12], X is a radical algebra. Hence the spectrum of a is {0} and therefore Ma is quasinilpotent. It follows that each orbit of Ma is a sequence norm-convergent to 0. Thus, Ma cannot be weakly hypercyclic. Proposition 3.4. Let X be a commutative complex Banach algebra of dimension > 1. Assume also that there exists a ∈ X such that Ma is weakly supercyclic. Then X is radical. Proof. Assume that X is non-radical. Then there exists a non-zero character κ : X → C. Hence H = ker κ is an Ma -invariant closed hyperplane in X. As well-known, if a supercyclic operator on a topological vector space has an invariant closed hyperplane, then the restriction of some its scalar multiple to this hyperplane is hypercyclic, see, for instance [11,30]. Thus, replacing a by λa for some non-zero λ ∈ C, if necessary, we can assume that the restriction of Ma to the closed subalgebra H is weakly hypercyclic. We have arrived to a contradiction with Proposition 3.3. Finally, let X be a non-degenerate commutative Banach algebra and Λ = {cI + Ma : c ∈ K, a ∈ X}. Recall that a derivation on X is M ∈ L(X) satisfying M (ab) = (M a)b + a(M b). It is easy to see that for any derivation M on X the operator A → [M, A] preserves Λ. Moreover, M commutes with Ma if and only if M a = 0. Thus, by Theorem 1.7, cI + Ma is non-supercyclic provided there is a derivation on X, which does not annihilate a. Combining this with the second part of Theorem 1.5, we have the following proposition. Proposition 3.5. Let X be a non-degenerate commutative Banach algebra and a ∈ X. Assume also that there exists a derivation M on X for which M a = 0. Then cI + Ma is non-supercyclic for any c ∈ K. If additionally there exists R ∈ L(X) with dense range commuting with Ma and taking values in a Banach space Y ∈ Y embedded into X, then cI + Ma is not weakly supercyclic. From Propositions 3.4 and 3.5 it follows that the only place to look for an aﬃrmative answer to Problem 3.1 are radical commutative Banach algebras

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with few derivations. We conclude this section by the following observation dealing with the non-commutative case. Recall that a bounded linear operator on a Banach space X is called densely cyclic if the set of its cyclic vectors is dense in X. It is well-known and easy to see that each weakly supercyclic operator is densely cyclic. Proposition 3.6. Let X be a unital non-commutative Banach algebra. Then Ma is not densely cyclic for any a ∈ X. In particular, each Ma is not weakly supercyclic. Proof. Assume the contrary. Then there is a ∈ X for which Ma is densely cyclic. Since the set Inv(X) of invertible elements of X is non-empty and open, there is a cyclic vector b ∈ Inv(X) for Ma . Cyclicity of b for Ma is exactly the density of Y b in X, where Y is the unital subalgebra of X generated by a. Since R ∈ L(X), Ru = ub−1 is invertible, R(Y b) = Y is dense in X. Since Y is abelian, so is X. This contradiction completes the proof.

4. Concluding Remarks 1. The following example shows that the second part of condition (1.5a ) is essential. Example 4.1. Let {en }n∈Z+ be the canonical orthonormal basis in 2 and T, M ∈ L(2 ) be deﬁned by: T e0 = e0 , T en = en + n−1 en−1 for n ≥ 1, M e0 = 0 and M en = en−1 for n ≥ 1. Then T is hypercyclic, [T, M ] has dense range and [T, [T, M ]] = 0. Proof. Since T is the sum of the identity operator and a backward weighted shift, T is hypercyclic according to Salas [25]. Computing the values of operators on basic vectors, one can easily see that [T, M ] = (T − I)2 and therefore [T, [T, M ]] = 0. Finally, it is straightforward to see that the range of [T, M ] = (T −I)2 contains all basic vectors. Hence [T, M ] has dense range. 2. Since any operator from C(V ) fails to be weakly supercyclic, the following question becomes interesting. Assume that the Volterra operator V acts on L2 [0, 1]. Problem 4.2. Can we find operators A, B ∈ C(V ) and f ∈ L2 [0, 1] for which the set {zAn B m f : z ∈ K, m, n ∈ Z+ } is dense (or at least weakly dense) in L2 [0, 1]? In other words, can a 2-generated subsemigroup of C(V ) be supercyclic or at least weakly supercyclic? 3. The following notion was introduced by Enﬂo [6]. Let X be a Banach space and n ∈ N. We say that T ∈ L(X) is cyclic with support n if there is x ∈ X such that the set {a1 T k1 x + . . . + an T kn x : aj ∈ K, kj ∈ Z+ } is dense in X. Clearly cyclicity with support 1 is exactly supercyclicity. In [17] it is claimed that there are no known examples of an operator on an inﬁnite dimensional Banach space, cyclic with support n ≥ 2 and non-supercyclic. The next example shows that such operators do exist. It is a modiﬁcation of an example constructed in [2]. Consider the functions un : T → T, un (z) = z n

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for n ∈ Z, where T = {z ∈ C : |z| = 1}. Recall that a non-empty compact subset K of T is a Kronecker set if {un : n ∈ Z+ } is dense in C(K, T) with respect to the metric d(f, g) = max{|f (z) − g(z)| : z ∈ T}. It is well-known that there are inﬁnite Kronecker sets and even Kronecker sets with no isolated points. In the latter case the Banach space C(K) is inﬁnite dimensional. Example 4.3. Let K ⊂ T be a Kronecker set of cardinality > 1 and T ∈ L(C(K)), T f (z) = zf (z). Then T is not weakly supercyclic, while {a(T k 1 + T m 1) : a ≥ 0, k, m ∈ Z+ } is dense in C(K). Proof. Since K is a Kronecker set, K has empty interior in T and therefore the set W = {f ∈ C(K) : f (K) is ﬁnite} is dense in C(K). Let g ∈ W . Using the fact that any z ∈ C with |z| ≤ 2 is a sum of two numbers from T, we can ﬁnd g1 , g2 ∈ W such that |g1 | = |g2 | = 1 and g = r(g1 +g2 ), where r = g/2. Since gj ∈ C(K, T) and K is the Kronecker set, there are strictly increasing sequences {kn }n∈Z+ and {mn }n∈Z+ such that fkn = T kn 1 converges uniformly to g1 and fmn = T mn 1 converges uniformly to g2 as n → ∞. Hence r(T kn 1+T mn 1) converges to g in C(K). Since g is an arbitrary element of W and W is dense in C(K), we see that {aT k 1+ aT m 1 : a ∈ [0, ∞), k, m ∈ Z+ } is dense in C(K). Now we show that T is not weakly supercyclic. Indeed, let f ∈ C(K), f = 0. Pick s, t ∈ K such that s = t and f (s) = 0. Consider the set G = {g ∈ C(K) : |g(t)f (s)| > |g(s)f (t)|}. Clearly G is non-empty and weakly open in C(K). On the other hand, it is easy to see that for any element g of Opr (f, T ), |g(t)f (s)| = |g(s)f (t)|. It follows that Opr (f, T ) does not meet G and therefore cannot be weakly dense in C(K). Hence T is not weakly supercyclic. 4.1. Related Operators that are not Weakly Supercyclic In [15] it is observed that for any p ∈ (1, ∞) the Ces`aro operator 1 Cf (x) = x

x f (t) dt, 0

acting on Lp [0, 1], is hypercyclic. Clearly C is the composition of the Volterra operator and the (unbounded) operator of multiplication by the function α(x) = x−1 . The fact that α is non-integrable turns out to be the reason for this phenomenon. Proposition 4.4. Let 1 ≤ p < ∞, α ∈ L1 [0, 1], α ≥ 0 and assume that the formula x Tα f (x) = α(x) f (t) dt 0

defines a bounded linear operator on Lp [0, 1]. Then Tα is not weakly supercyclic. We prove the above proposition by applying the Comparison Principle for weak supercyclicity. Namely, assume that X and Y are Banach spaces,

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T ∈ L(X) and S ∈ L(Y ) are such that there exists a bounded linear operator J : X → Y with dense range satisfying JT = SJ. Then weak supercyclicity of T implies weak supercyclicity of S. It follows from the equality Opr (Jx, S) = J(Opr (x, T )), which implies that Jx is a weakly supercyclic vector for S provided x is a weakly supercyclic vector for T . Proof of Proposition 4.4. If α vanishes on a set of positive measure, then the range of Tα is non-dense and therefore Tα is not weakly supercyclic [26]. Thus, we can assume that α(x) > 0 almost everywhere. Hence the continuous function h = V α is strictly increasing. Multiplying α by a positive constant, if necessary, we can without loss of generality, assume that h(1) = 1. Since h(0) = 0, h provides an increasing autohomeomorphism of the interval [0, 1]. Let ϕ : [0, 1] → [0, 1] be the inverse of h: ϕ = h−1 . For any f ∈ Lp [0, 1], we, using the change of variables t = ϕ−1 (s), obtain 1 1 1 |f (ϕ(t))| |f (s)| −1 dt = (ϕ ) (s) ds = |f (s)| ds = f 1 ≤ f p . α(ϕ(t)) α(s) 0

0

0

f (ϕ(x)) α(ϕ(x))

Hence the formula Jf (x) = deﬁnes a bounded linear operator from Lp [0, 1] to L1 [0, 1]. Using the same change of variables, it is straightforward to see that JTα = V J, where V is the Volterra operator acting on L1 [0, 1]. Since J has dense range and V is not weakly supercyclic, we applying the Comparison Principle see that Tα is not weakly supercyclic. In particular, from Proposition 4.4 it follows that for any s > −1 the operator x s Rs f (x) = x f (t) dt, 0

acting on Lp [0, 1] for 1 ≤ p < ∞, is not weakly supercyclic. On the other hand, R−1 coincides with the hypercyclic Ces`aro operator C. In a similar way one can treat weighted Volterra operators with a positive weight. Let 1 ≤ p, q ≤ ∞ be such that p1 + 1q = 1 and α ∈ Lq [0, 1]. Then it is easy to see that formula x Sα f (x) = α(t)f (t) dt 0

deﬁnes a bounded linear operator on Lp [0, 1]. Proposition 4.5. Let 1 ≤ p < ∞, 1 < q ≤ ∞ be such that p1 + 1q = 1 and α ∈ Lq [0, 1] be almost everywhere positive. Then the operator Sα is not weakly supercyclic. Proof. Consider the bounded linear operator Mα : Lp [0, 1] → L1 [0, 1], Mα f (x) = α(x)f (x). Since α is positive almost everywhere, Mα has dense range. Since α ∈ L1 [0, 1], the operator Tα deﬁned in Proposition 4.4 acts

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boundedly on L1 [0, 1]. From the definitions of Tα , Mα and Sα it immediately follows that Mα Sα = Tα Mα . By Proposition 4.4, Tα acting on L1 [0, 1] is not weakly supercyclic. Applying the Comparison Principle, we see that Sα is not weakly supercyclic.

References [1] Ball, K.: The complex plank problem. Bull. Lond. Math. Soc 33, 433–442 (2001) [2] Bayart, F., Matheron, E.: Hyponormal operators, weighted shifts and weak forms of supercyclicity. Proc. Edinb. Math. Soc. 49, 1–15 (2006) [3] Bayart, F., Matheron, E.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009) [4] Bermudo, S., Montes-Rodr´ıguez, A., Shkarin, S.: Orbits of operators commuting with the Volterra operator. J. Math. Pures. Appl. 89, 145–173 (2008) [5] Bonet, J., Mart´ınez-Gim´enez, F., Peris, A.: Universal and chaotic multipliers on spaces of operators. J. Math. Anal. Appl. 297, 599–611 (2004) [6] Enﬂo, P.: On invariant subspace problem for Banach spaces. Acta Math. 158, 213–313 (1987) [7] Erdos, J.: Operators with abelian commutants. J. Lond. Math. Soc. 9, 637–640 (1975) [8] Erdos, J.: The commutant of the Volterra operator. Integr. Equ. Oper. Theory 5, 127–130 (1982) [9] Eveson, S.: Non-supercyclicity of Volterra convolution and related operators. Integr. Equ. Oper. Theory 62, 585–589 (2008) [10] Gallardo-Guti´errez, E., Montes-Rodr´ıguez, A.: The Volterra operator is not supercyclic. Integr. Equ. Oper. Theory 50, 211–216 (2004) [11] Gallardo-Guti´errez, E., Montes-Rodr´ıguez, A.: The role of the angle in supercyclic behavior. J. Funct. Anal. 203, 27–43 (2003) [12] Helemskii, A.: Banach and Locally Convex Algebras. Oxford University Press, New York (1993) [13] Hilden, H., Wallen, L.: Some cyclic and non-cyclic vectors of certain operators. Indiana Math. J. 23, 557–565 (1973) [14] Kadets, V.: Weak cluster points of a sequence and coverings by cylinders. Mat. Fiz. Anal. Geom. 11, 161–168 (2004) [15] L´eon-Saavedra, F., Piqueras-Lerena, A.: Cyclic properties of Volterra Operator. Pac. J. Math. 211, 157–162 (2003) [16] L´eon-Saavedra, F., Piqueras-Lerena, A.: Cyclic properties of Volterra operator II. Preprint [17] L´eon-Saavedra, F., Piqueras-Lerena, A.: Positivity in the theory of supercyclic operators. In: Perspectives in Operator Theory. Banach Center Publ., vol. 75, pp. 221–232. Polish Acad. Sci., Warsaw (2007) [18] L´eon-Saavedra, F., Piqueras-Lerena, A.: On weak positive supercyclicity. Isr. J. Math. 167, 303–313 (2008) [19] L´eon-Saavedra, F., Piqueras-Lerena, A.: Orbits of Cesaro type operators. Math Nachr. 282, 764–773 (2009)

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[20] Maurey, B.: Type, cotype and K-convexity. In: Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1299–1332. North-Holland, Amsterdam (2003) [21] Montes-Rodr´ıguez, A., Salas, H.: Supercyclic subspaces: spectral theory and weighted shifts. Adv. Math. 163, 74–134 (2001) [22] Montes-Rodr´ıguez, A., Shkarin, S.: Non-weakly supercyclic operators. J. Oper. Theory 58, 39–62 (2007) [23] Montes-Rodr´ıguez, A., Shakrin, S.: New results on a classical operator. Contemp. Math. 393, 139–158 (2006) [24] Prajitura, G.: Limits of weakly hypercyclic and supercyclic operators. Glasg. Math. J. 47, 225–260 (2005) [25] Salas, H.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347, 993–1004 (1995) [26] Sanders, R.: Weakly supercyclic operators. J. Math. Anal. Appl. 292, 148–159 (2004) [27] Sanders, R.: An isometric bilateral shift that is weakly supercyclic. Integr. Equ. Oper. Theory 53, 547–552 (2005) [28] Shkarin, S.: Antisupercyclic operators and orbits of the Volterra operator. J. Lond. Math. Soc. 73, 506–528 (2006) [29] Shkarin, S.: Non-sequential weak supercyclicity and hypercyclicity. J. Funct. Anal. 242, 37–77 (2007) [30] Shkarin, S.: Universal elements for non-linear operators and their applications. J. Math. Anal. Appl. 348, 193–210 (2008) Stanislav Shkarin Department of Pure Mathematics Queen’s University Belfast University Road Belfast BT7 1NN UK e-mail: [email protected] Received: November 2, 2009. Revised: January 15, 2010.

Integr. Equ. Oper. Theory 68 (2010), 243–254 DOI 10.1007/s00020-010-1800-0 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

On m-Accretivity of Perturbed Bochner Laplacian in Lp Spaces on Riemannian Manifolds Ognjen Milatovic Abstract. We consider a differential expression H = ∇∗ ∇ + V , where ∇ is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufﬁcient condition for H to have an m-accretive realization in the space Lp (E), where 1 < p < +∞. We study the same problem for the operator ΔM + V in Lp (M ), where 1 < p < ∞, ΔM is the scalar Laplacian on a complete Riemannian manifold M , and V is a locally integrable function on M . Mathematics Subject Classiﬁcation (2010). Primary 35P05, 58J50; Secondary 47B25, 81Q10. Keywords. Bochner Laplacian, bounded geometry, m-accretive, Riemannian manifold, Schr¨ odinger operator.

1. Setting Let (M, g) be a C ∞ Riemannian manifold without boundary, with metric g = (gjk ) and dim M = n. We will assume that M is connected and oriented. By dμ we will denote the Riemannian volume element of M . Let E be a Hermitian vector bundle over M , and let 1 ≤ p < +∞. By Lp (E) we denote the completion of Cc∞ (E) with respect to the norm ⎛ ⎞1/p up := ⎝ |u(x)|p dμ(x)⎠ , M

where | · | is the ﬁberwise norm in the ﬁber Ex . In what follows, by Tx∗ M and T ∗ M we will denote the cotangent space of M at x ∈ M and cotangent bundle of M respectively. Additionally, by C ∞ (E) we denote the space of smooth sections of E, by Cc∞ (E)—the space

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of smooth compactly supported sections of E, by D (E)—distributional sections of E, by C ∞ (M )—the space of complex-valued smooth functions on M , by Cc∞ (M )—the space of complex-valued smooth compactly supported functions on M , and by Ω1 (M )—the space of complex-valued smooth 1-forms on M . By ·, · we will denote the anti-duality of the pair (Lp (E), Lp (E)), where 1 ≤ p < +∞ and 1/p + 1/p = 1, and the anti-duality of the pair (D (E), Cc∞ (E)). Let ∇ : C ∞ (E) → C ∞ (T ∗ M ⊗ E) be a Hermitian connection on E. We consider a Schr¨ odinger-type differential expression H = ∇∗ ∇ + V, where V is a locally integrable section of the bundle End E of endomorphisms of E. Here, ∇∗ : C ∞ (T ∗ M ⊗ E) → C ∞ (E) is a differential operator which is formally adjoint to ∇ with respect to the usual scalar product in L2 (E). The operator ∇∗ ∇ : C ∞ (E) → C ∞ (E) is called Bochner Laplacian. A special case of this operator is the scalar Laplacian ΔM := d∗ d, where d : C ∞ (M ) → Ω1 (M ) is the standard differential. Another special case of operBochner Laplacian is the magnetic Laplacian ΔA := d∗A dA , where the √ ator dA : C ∞ (M ) → Ω1 (M ) is deﬁned by dA u := du + iuA. Here, i = −1, and A (called the magnetic potential) is a real-valued smooth 1-form on M . 1.1. Operators Associated to H Let 1 < p < +∞. We deﬁne the maximal operator Hp,max in Lp (E) associated to H by the formula Hp,max u = Hu with domain Dom(Hp,max ) = {u ∈ Lp (E) : V u ∈ L1loc (E), ∇∗ ∇u + V u ∈ Lp (E)}. ∗

(1.1)

∗

Here, the term ∇ ∇u in ∇ ∇u + V u is understood in distributional sense. In general, Dom(Hp,max ) does not contain Cc∞ (E), but it does if V ∈ p Lloc (End E). In this case, we can deﬁne Hp,min := Hp,max |Cc∞ (E) . Remark 1.1. Using the same definitions as in Sect. 1.1, we can also deﬁne Hp,max and Hp,min for p = 1 and p = ∞. However, we will not use those operators in this paper. In the sequel, we use the notations (Re V )(x) :=

V (x)+(V (x))∗ , 2

(Im V )(x) :=

V (x)−(V (x))∗ , 2i

√

x ∈ M, (1.2)

where i = −1 and (V (x))∗ denotes the adjoint of the linear operator V (x) : Ex → Ex (in the sense of linear algebra). Assumption (A1). Assume that (Re V )(x) ≥ 0 for all x ∈ M . In the sequel, by A we denote the closure of a closable operator A. Remark 1.2. In what follows, we will use the term “manifold of bounded geometry”. For the corresponding definition, see [9, Section A.1.1] or [3, Section 1.1]. In particular, a manifold of bounded geometry is complete.

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We now state the main results. Theorem 1.3. Assume that (M, g) is a manifold of bounded geometry. Assume that 1 < p < +∞ and V ∈ Lploc (End E). Additionally, assume that the assumption (A1) is satisfied. Then Hp,min generates a contraction semigroup on Lp (E). In particular, Hp,min is an m-accretive operator. Theorem 1.4. Assume that (M, g) is a manifold of bounded geometry. Assume that 1 < p < +∞. Additionally, assume that V = qI, where I is the identity endomorphism of E, and q ∈ Lploc (M ) is a complex-valued function such that Re q ≥ 0 and | Im q| ≤ L(Re q), where L ≥ 0 is a constant. Then Hp,min = Hp,max . The following corollary is an immediate consequence of Theorems 1.3 and 1.4. Corollary 1.5. Assume that (M, g) is a manifold of bounded geometry. Additionally, assume that 1 < p < +∞ and q : M → C is as in hypotheses of Theorem 1.4. Let ΔA be as in Sect. 1 and let Hp,min and Hp,max be the operators associated to H = ΔA + q as in Sect. 1.1. Then the following properties hold: 1. 2.

The operator Hp,min generates a contraction semigroup on Lp (M ). In particular, Hp,min is m-accretive. Hp,min = Hp,max .

In the following result, we do not assume bounded geometry on (M, g); however, we impose stronger requirements on the potential. Assumption (A2). Assume that (i) q ∈ Lsloc (M ), where s > max{p, n/2} and 1 < p < +∞. (ii) q(x) ≥ 0 for all x ∈ M . Theorem 1.6. Assume that (M, g) is a complete Riemannian manifold. Additionally, assume that the assumption (A2) is satisfied. Let ΔM be as in Sect. 1 and let Hp,min be the operator associated to H = ΔM +q as in Sect. 1.1. Then Hp,min generates a contraction semigroup on Lp (M ). In particular, Hp,min is m-accretive. Remark 1.7. Kato [6, Part A] studied the differential expression −Δ + V , where Δ is the standard Laplacian on Rn with standard metric and measure. Assuming 0 ≤ V ∈ Lploc (Rn ), Kato proved the results of Theorem 1.3 and Theorem 1.4 for all 1 ≤ p < +∞. Theorems 1.3 and 1.4 generalize the corresponding results of [7] which were proven for the differential expression ΔM + V in Lp (M ), where M is a manifold of bounded geometry, 1 < p < +∞ and 0 ≤ V ∈ Lploc (M ). Theorem 1.6 generalizes a result of Strichartz [10] about the m-accretivity of the closure of ΔM |Cc∞ (M ) in Lp (M ), where 1 < p < ∞ and M is a complete Riemannian manifold.

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2. Proof of Theorem 1.3 We begin with a few preliminary following lemmas. Lemma 2.1. Assume that (M, g) is a complete Riemannian manifold. Assume that 1 < p < ∞ and the Assumption (A1) is satisfied. Then, the operator Hp,min satisfies the following inequality: Re(Hp,min u, u|u|p−2 ) ≥ 0,

for all u ∈ Cc∞ (E),

(2.1)

2

where (·, ·) is the usual inner product in L (E). Proof. By [10] (or by [3, Chapter 2, Lemma 5.4]) it follows that Re(∇∗ ∇u, u|u|p−2 ) ≥ 0,

for all u ∈ Cc∞ (E).

(2.2)

By Assumption (A1) we get Re(V u, u|u|p−2 ) ≥ 0,

for all u ∈ Cc∞ (E).

The inequality (2.1) follows immediately from (2.2) and (2.3).

(2.3)

Remark 2.2. By definition of accretivity for an operator in Banach space (see, for instance [8, Section X.8]), the inequality (2.1) means that the operator Hp,min is accretive in Lp (E). By an abstract fact (see the remark preceding Theorem X.48 in [8]), the operator Hp,min is closable and Hp,min is accretive in Lp (E). Thus, the following inequality holds: ReHp,min u, u|u|p−2 ≥ 0,

for all u ∈ Dom(Hp,min ),

(2.4)

where ·, · is the anti-duality of the pair (Lp (E), Lp (E)) with 1/p+1/p = 1. 2.1. Distributional inequality Assume that 1 0, and consider the following distributional inequality: ( ΔM + λ ) u = ν ≥ 0,

u ∈ Lp (M ),

(2.5)

where the inequality ν ≥ 0 means that ν is a positive distribution, i.e. ν, φ ≥ 0 for any 0 ≤ φ ∈ Cc∞ (M ). Lemma 2.3. Assume that (M, g) is a manifold of bounded geometry. Assume that 1 < p < +∞. Assume that u ∈ Lp (M ) satisfies (2.5). Then u ≥ 0 (almost everywhere or, equivalently, as a distribution). Remark 2.4. For the proof of Lemma 2.3, see [7, Section 4]. In the case p = 2, Lemma 2.3 was proven in [2, Appendix B]. 2.2. Kato’s inequality We will use the following variant of Kato’s inequality for Bochner Laplacian (for the proof, see Theorem 5.7 in [2]). Lemma 2.5. Assume that (M, g) is a Riemannian manifold. Assume that E is a Hermitian vector bundle over M and ∇ is a Hermitian connection on E. Assume that w ∈ L1loc (E) and ∇∗ ∇w ∈ L1loc (E). Then the following distributional inequality holds: ΔM |w| ≤ Re∇∗ ∇w, sign wEx ,

(2.6)

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where

sign w(x) =

w(x) |w(x)|

if w(x) = 0 ,

0

otherwise.

247

and ·, ·Ex is the fiberwise inner product in Ex . Remark 2.6. The original version of Kato’s inequality was proven in Kato [4]. In Lemmas 2.7, 2.8, and 2.9 below we assume that 1 0. Then, Ran(Hp,min + λ) is dense in Lp (E).

Proof. Let v ∈ (Lp (E))∗ = Lp (E), where 1/p + 1/p = 1, be a continuous linear functional annihilating (λ + Hp,min )Cc∞ (E): (λ + Hp,min )u, v = 0,

for all u ∈ Cc∞ (E),

(2.7)

p

p

where ·, · denotes the anti-duality of the pair (L (E), L (E)). From (2.7) we get the following equality of distributional sections: ¯ + ∇∗ ∇ + V ∗ )v = 0, (λ where V ∗ is as in (1.2). Since by hypothesis V ∗ ∈ Lploc (End E) and since v ∈ Lp (E), by H¨ older’s ¯ we get ∇∗ ∇v ∈ inequality we have V ∗ v ∈ L1loc (E). Since ∇∗ ∇v = −V ∗ v − λv, L1loc (E). By Kato’s inequality and since (Re V ∗ )(x) = (Re V )(x) ≥ 0, we have ¯ − V ∗ v, sign vE ≤ −(Re λ)|v|, ¯ ΔM |v| ≤ Re∇∗ ∇v, sign vE = Re−λv x

x

and, hence, ¯ ≤ 0. (ΔM + Re λ)|v|

¯ = Re λ > 0, Since |v| ∈ Lp (M ) (with 1 0. Then, the following inequality holds: γup ≤ (λ + Hp,min )up ,

for all u ∈ Dom(Hp,min ).

(2.8)

Proof. By (2.4) it follows that Re(Hp,min + λ)u, u|u|p−2 ≥ γu, u|u|p−2 ,

(2.9)

for all u ∈ Dom(Hp,min ), where ·, · denotes the anti-duality of the pair (Lp (E), Lp (E)) with 1/p + 1/p = 1. By H¨older’s inequality, from (2.9) we get (Hp,min + λ)up u|u|p−2 p ≥ Re(Hp,min + λ)u, u|u|p−2 ≥ γu, u|u|p−2 , for all u ∈ Dom(Hp,min ). p/p and u, u|u|p−2 = upp and p − p/p = Since u|u|p−2 p = up p/p

1, dividing both sides of the last inequality by up obtain (2.8).

, we immediately

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Lemma 2.9. Let λ ∈ C and Re λ > 0. Then, Ran(Hp,min + λ) = Lp (E). Proof. Let f ∈ Lp (E). By Lemma 2.7 there exists a sequence uk ∈ Cc∞ (E) such that (Hp,min + λ)uk → f in Lp (E). By (2.8) it follows that uk is a Cauchy sequence in Lp (E), and, hence uk → u in Lp (E). By the definition of a closed operator (see, for example, [5, Section III.5.3]), it follows that u ∈ Dom(Hp,min + λ) and (Hp,min + λ)u = f . Proof of Theorem 1.3. By Remark 2.2 the operator Hp,min is accretive in Lp (E), and by Lemma 2.9 we have Ran(Hp,min + λ) = Lp (E) for all λ ∈ C such that Re λ > 0. Thus, by [8, Theorem X.48] it follows that Hp,min is the generator of a contraction semigroup on Lp (E). Now, by the remark preceding Theorem X.49 in [8], it follows that Hp,min is m-accretive.

3. Proof of Theorem 1.4 In the sequel, we will adopt certain arguments of Kato [6, Part A] to our setting. Lemma 3.1. Let (M, g) be a Riemannian manifold. Assume that 1 < p < +∞. Assume that V = qI, where I is the identity endomorphism of E, and q ∈ L1loc (M ) with Re q ≥ 0. Additionally, assume that u ∈ Dom(Hp,max ) and λ ∈ C. Let f := (Hp,max + λ)u. Then, the following distributional inequality holds: (Re λ + ΔM + Re q)|u| ≤ |f |.

(3.1)

Proof. Since u ∈ Dom(Hp,max ) it follows that V u ∈ L1loc (E) and Hp,max u ∈ Lp (E) ⊂ L1loc (E). Thus u ∈ L1loc (E) and ∇∗ ∇u ∈ L1loc (E). Since V u = qu, by Kato’s inequality (2.6) we have (Re λ + ΔM + Re q)|u| ≤ Re(λ + ∇∗ ∇ + V )u, sign uEx = Ref, sign uEx ≤ |f |,

which gives (3.1).

Lemma 3.2. Let (M, g) be a Riemannian manifold, and let 1 < p < +∞. Assume that V = qI, where I is the identity endomorphism of E, and q ∈ L1loc (M ) is a complex-valued function such that Re q ≥ 0 and | Im q| ≤ L(Re q), where L ≥ 0 is a constant. Then the operator Hp,max is closed. Proof. Let uk ∈ Dom(Hp,max ) be a sequence such that, as k → +∞, uk → u,

fk := Hp,max uk = ∇∗ ∇uk + V uk → f

in Lp (E).

(3.2)

We need to show that u ∈ Dom(Hp,max ) and Hp,max u = f . By passing to subsequences, we may assume that the convergence in (3.2) is also pointwise almost everywhere.

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The distributional inequality (3.1) holds if we replace u by uk − ul , f by fk − fl and λ by 0. With these replacements, we apply a test function 0 ≤ φ ∈ Cc∞ (M ) to (3.1) and get 0 ≤ (Re q)|uk − ul |, φ ≤ |fk − fl |, φ − ΔM |uk − ul |, φ,

(3.3)

where ·, · denotes the anti-duality of the pair (D (M ), Cc∞ (M )). Using integration by parts in the second term on the right hand side of the second inequality in (3.3), we get 0 ≤ (Re q)|uk − ul |, φ ≤ |fk − fl |, φ − |uk − ul |, ΔM φ.

(3.4)

Letting k, l → +∞, the right hand side of the second inequality in (3.4) tends to 0 by (3.2). Thus (Re q)uk φ is a Cauchy sequence in L1 (E). Since, by hypothesis, |(Im q)| ≤ L(Re q), it follows that (Im q)uk φ is a Cauchy sequence in L1 (E). Thus quk φ = V uk φ is a Cauchy sequence in L1 (E), and its limit must be equal to V uφ. Since φ ∈ Cc∞ (M ) may have an arbitrarily large support, it follows that V u ∈ L1loc (E). Thus V uk → V u in L1loc (E) and hence in D (E). Since uk → u in Lp (E) (and, hence in L1loc (E)), we get ∇∗ ∇uk → ∇∗ ∇u in D (E). Thus, fk = ∇∗ ∇uk + V uk → ∇∗ ∇u + V u in D (E). Since fk → f in Lp (E) ⊂ D (E), we obtain ∇∗ ∇u + V u = f ∈ Lp (E). This shows that u ∈ Dom(Hp,max ) and Hp,max u = f . Hence, Hp,max is closed. 3.1. Operators Associated to ΔM Let 1 < p < +∞. We deﬁne the maximal operator Ap,max in Lp (M ) associated to ΔM by the formula Ap,max u = ΔM u with the domain Dom(Ap,max ) = {u ∈ Lp (M ) : ΔM u ∈ Lp (M )}.

(3.5)

We deﬁne Ap,min := Ap,max |Cc∞ (M ) . Let (M, g) be a complete Riemannian manifold and 1 < p < +∞. By the proof of [10, Theorem 3.5] (or the proof of [3, Chapter 2, Theorem 5.8]), it follows that Ap,min generates a contraction semigroup on Lp (M ). By an abstract fact (see [8, Theorem X.47(a)]) this is equivalent to the following statement: 1 (−∞, 0) ⊂ ρ(Ap,min ) and (λ + Ap,min )−1 ≤ , for all λ > 0, λ (3.6) where · denotes the operator norm (for a bounded linear operator Lp (M ) → Lp (M )) and ρ(A) denotes the resolvent set of an operator A. Remark 3.3. Assume that (M, g) has bounded geometry, and 1 < p < +∞. Then by [9, Proposition 4.1] it follows that Ap,max = Ap,min . Thus, the statement (3.6) holds with Ap,min replaced by Ap,max . Lemma 3.4. Let (M, g) be a manifold of bounded geometry, and let 1 0. Then the following properties hold:

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for all u ∈ Dom(Hp,max ), we have γup ≤ (λ + Hp,max )up ;

2.

p

(3.7) p

the operator λ + Hp,max : Dom(Hp,max ) ⊂ L (E) → L (E) is injective.

Proof. We ﬁrst prove property 1. Let u ∈ Dom(Hp,max ) and f := (λ + Hp,max )u. By the definition of Dom(Hp,max ), we have f ∈ Lp (E), where 1 < p < +∞. Since Re q ≥ 0 and since V u = qu ∈ L1loc (E), from (3.1) we get the following distributional inequality: (γ + ΔM )|u| ≤ |f |.

(3.8)

By Remark 3.3 it follows that (γ + Ap,max )−1 : Lp (M ) → Lp (M ) is a bounded linear operator. Let us rewrite (3.8) as (γ + ΔM )[(γ + Ap,max )−1 |f | − |u|] ≥ 0. Note that (γ + Ap,max )−1 |f | ∈ Lp (M )

and |u| ∈ Lp (M ).

Thus, ((γ + Ap,max )−1 |f | − |u|) ∈ Lp (M ), and, hence, by Lemma 2.3 we have (γ + Ap,max )−1 |f | − |u| ≥ 0, i.e. |u| ≤ (γ + Ap,max )−1 |f |.

(3.9)

By (3.6) and by Remark 3.3 it follows that 1 (γ + Ap,max )−1 |f |p ≤ f p . γ

(3.10)

By (3.9) and (3.10) we have up ≤ (γ + Ap,max )−1 |f |p ≤

1 f p , γ

and (3.7) is proven. We now prove property 2. Assume that u ∈ Dom(Hp,max ) and (λ + Hp,max )u = 0. Using (3.7) with f = 0, we get up = 0, and hence u = 0. This shows that λ + Hp,max is injective. Proof of Theorem 1.4. Assume that λ ∈ C satisﬁes Re λ > 0. By Lemma 3.4 the operator λ + Hp,max : Dom(Hp,max ) ⊂ Lp (E) → Lp (E) is injective. Since Hp,min ⊂ Hp,max and since Hp,max is closed (see Lemma 3.2), it follows that Hp,min ⊂ Hp,max . By Lemma 2.9 we have Ran(Hp,min + λ) = Lp (E); hence, Ran(Hp,max + λ) = Lp (E). Therefore, the inverse (Hp,max + λ)−1 is deﬁned on the whole Lp (E). Now by (3.7) it follows that (−∞, 0) ⊂ ρ(Hp,max ) and (γ + Hp,max )−1 ≤

1 , γ

for all γ > 0.

Thus, by [8, Theorem X.47(a)] the operator Hp,max generates a contraction semigroup on Lp (E). By the remark preceding Theorem X.49 in [8], it follows that Hp,max is m-accretive. Since Hp,min is m-accretive (see Theorem 1.3)

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and since Hp,min ⊂ Hp,max , by definition of m-accretivity it follows that Hp,min = Hp,max .

4. Proof of Theorem 1.6 Unless speciﬁed otherwise, throughout this section we assume that the hypotheses of Theorem 1.6 are satisﬁed. 4.1. Cut-Oﬀ Functions Assume that (M, g) is a complete Riemannian manifold. Let x0 ∈ M and let Br (x0 ) denote the open ball (corresponding to metric g) of radius r centered at x0 . Then for all R > r > 0, there exists a Lipschitz (hence, differentiable almost everywhere) function φr,R on M such that 1. 2. 3.

0 ≤ φr,R (x) ≤ 1, supp φr,R ⊂ BR (x0 ), φr,R (x) = 1 on Br (x0 ); limr,R→∞ φr,R (x) = 1; |dφr,R (x)| ≤ c/(R − r) almost everywhere, where c is a constant independent of R and r, and | · | denotes the length of the cotangent vector dφr,R (x) ∈ Tx∗ M .

For a construction of the family φr,R see, for instance [3, Chapter 1, Lemma 1.28]. 4.2. Local Sobolev Spaces Let p ≥ 1, and let m ≥ 0 be an integer. In the sequel, we will use local Sobolev m,p m,p spaces Wloc (M ). The space Wloc (M ) consists of functions u ∈ Lploc (M ) such that their derivatives of order ≤ m in local coordinates also belong to Lploc in these coordinates. Lemma 4.1. Assume that q(x) satisfies (i) of Assumption (A2). Let p = p/(p − 1). Assume that u ∈ Lp (M ) and (ΔM + q)u = κu, where κ ∈ R. Then 2,p (M ). u ∈ Wloc 1 Proof. Since q ∈ Lsloc (M ), by H¨ older inequality it follows that qu ∈ Lbloc (M ), where 1 1 1 = + . b1 s p

Since s > p and 1/p + 1/p = 1, it follows that 1 < b1 < p . Since d∗ du = κu − qu,

(4.1)

1 (M ). we get d∗ du ∈ Lbloc Deﬁne t1 := p . Since 1 < b1 < p , using (4.1) and elliptic regularity (see, 2,b1 for instance [11, Section 6.5]), we get u ∈ Wloc (M ). By Sobolev imbedding theorem (see, for instance, the ﬁrst part of Theorem 2.10 in [1]), we have 2,b1 2 Wloc (M ) ⊂ Ltloc (M ), where

1 1 2 1 2 1 = − = + − . t2 b1 n t1 s n

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2 (M ), with t2 > t1 . Thus, by H¨ older inequalSince s > n/2, we obtain u ∈ Ltloc b2 ity, we get qu ∈ Lloc (M ), where

1 1 1 2 1 2 1 1 1 1 = + = + + − = + − . b2 s t2 s t1 s n b1 s n

(4.2)

2 From the ﬁrst equality in (4.2) we have b2 < t2 . Since u ∈ Ltloc (M ) and b2 b 2 (M ). Since s > n/2, from the third qu ∈ Lloc (M ), by (4.1) we get d∗ du ∈ Lloc equality in (4.2) we have b2 > b1 > 1. Using elliptic regularity in (4.1), we 2,b2 obtain u ∈ Wloc (M ). 2,bk Proceeding in this way, we obtain the relations u ∈ Wloc (M ), k = 1, 2, . . ., with 1 2 1 1 = + − . bk+1 bk s n

Using the above procedure, at some step we will get the inequality bk > p .

In the next lemma we use the technique of Strichartz [10] (see also Eichhorn [3, Chapter 2, Lemma 5.2]). Lemma 4.2. Assume that q(x) satisfies Assumption (A2). Let p = p/(p − 1). Assume that u ∈ Lp (M ) and ΔM u + qu = κu, where κ < 0. Then u = 0. Proof. Since q is real-valued and since u satisﬁes ΔM u + qu = κu, without loss of generality we may assume that u is real-valued. Let > 0 be a small parameter. Deﬁne f (t) := tp −2 for t ≥ 1 and f (t) := ( + t2 )(p −2)/2 for t ≤ 1 − . For 1 − ≤ t ≤ 1, we arrange f (t) so that 0 ≤ f ∈ C ∞ (R) and f (t) + tf (t) ≥ Cf (t),

(4.3)

where C is a positive constant independent of . (Note that the inequality (4.3) holds with C = p − 1 in the interval t ≥ 1, and with C = min{p − 1, 1} in the interval t ≤ 1 − .) 2,p By Lemma 4.1 it follows that u ∈ Wloc (M ). Using integration by parts (see, for instance [2, Lemma 8.8]) together with q ≥ 0 and f ≥ 0, we have 0 > κ(φ2r,R f (|u|)u, u) = (φ2r,R f (|u|)u, d∗ du) + (φ2r,R f (|u|)u, qu) ≥ (φ2r,R f (|u|)u, d∗ du) = 2(φr,R f (|u|)(dφr,R )u, du) + (φ2r,R f (|u|)(d|u|)u, du) + (φ2r,R f (|u|)du, du). Since u is real-valued, we have ud|u| = ud

(4.4) √ u2 = |u|du.

(4.5)

Using (4.4) and (4.5), we obtain 0 > 2(φr,R f (|u|)(dφr,R )u, du) + (φ2r,R f (|u|)|u|du, du) + (φ2r,R f (|u|)du, du).

(4.6)

From (4.6) we get (φ2r,R (f (|u|)|u| + f (|u|))du, du) ≤ 2|(φr,R f (|u|)(dφr,R )u, du)|.

(4.7)

Vol. 68 (2010)

On m-Accretivity of Perturbed Bochner Laplacian

Using (4.3) we have (φ2r,R (f (|u|)|u| + f (|u|))du, du) ≥ C

φ2r,R f (|u|)|du|2 dμ.

253

(4.8)

M

By Cauchy–Schwarz inequality we get 2|(φr,R f (|u|)(dφr,R )u, du)| ⎞1/2 ⎛ ⎛ ⎜ ≤ 2dφr,R L∞ ⎝ φ2r,R f (|u|)|du|2 dμ⎠ ·⎝ M

⎞1/2

⎟ f (|u|)|u|2 dμ⎠

.

supp φr,R

(4.9)

From (4.7), (4.8) and (4.9) we obtain C φ2r,R f (|u|)|du|2 dμ M

⎛

≤ 2dφr,R L∞ ⎝

⎞1/2 ⎛ ⎜ φ2r,R f (|u|)|du|2 dμ⎠ · ⎝

M

⎞1/2 ⎟ f (|u|)|u|2 dμ⎠

supp φr,R

which leads to φ2r,R f (|u|)|du|2 dμ ≤ 4C −2 dφr,R 2L∞ M

f (|u|)|u|2 dμ. (4.10)

supp φr,R

Letting → 0+ in (4.10) and using the definition of f (t), we get φ2r,R |u|p −2 |du|2 dμ ≤ 4C −2 dφr,R 2L∞ |u|p dμ. M

(4.11)

supp φr,R

Letting R → +∞ in (4.11), using the properties of φr,R from Sect. 4.1 above, and recalling the hypothesis |u|p ∈ L1 (M ), it follows that the right hand side of (4.11) goes to zero. Hence, we obtain |u|p −2 |du|2 dμ = 0. Br

Therefore, du = 0 on the set S on which u = 0; hence, d∗ du = 0 on S. By hypothesis we know that d∗ du + qu = κu; hence, qu = κu on S. Therefore, (q − κ)u = 0. Since q ≥ 0 and κ < 0, we get u = 0 on S. Proof of Theorem 1.6. The arguments from the proof of Theorem 1.3 above apply without changes, with one exception: the proof of Lemma 2.7 used the bounded geometry assumption. Thus, it remains to prove (without assuming bounded geometry) that for all λ > 0, the following is true: Ran(Hp,min + λ) is dense in Lp (M ). Let v ∈ (Lp (M ))∗ = Lp (M ), where 1/p + 1/p = 1, be a continuous linear functional annihilating (λ + Hp,min )Cc∞ (M ):

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for all u ∈ Cc∞ (M ), p

(4.12)

p

where ·, · denotes the anti-duality of the pair (L (M ), L (M )). From (4.12) we get the following distributional equality: (λ + d∗ d + q)v = 0. By Lemma 4.2 we get v = 0. This shows that Ran(Hp,min + λ) is dense in Lp (M ).

References [1] Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998) [2] Braverman, M., Milatovic, O., Shubin, M.: Essential self-adjointness of Schr¨ odinger type operators on manifolds. Russian Math. Surv. 57, 641–692 (2002) [3] Eichhorn, J.: Global Analysis on Open Manifolds. Nova Science Publishers, Hauppauge (2007) [4] Kato, T.: Schr¨ odinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972) [5] Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1980) odinger operators with a singular potential. In: [6] Kato, T. : Lp -theory of Schr¨ Nagel, R., Schlotterbeck, U., Wolﬀ, M.P.H. (eds.) Aspects of Positivity in Functional Analysis, pp. 63–78. North-Holland, Amsterdam (1986) [7] Milatovic, O.: On m-accretive Schr¨ odinger operators in Lp -spaces on manifolds of bounded geometry. J. Math. Anal. Appl. 324, 762–772 (2006) [8] Reed, M., Simon, B.: Methods of Modern Mathematical Physics I, II: Functional Analysis. Fourier Analysis, Self-adjointness. Academic Press, London (1975) [9] Shubin, M.A.: Spectral theory of elliptic operators on noncompact manifolds. Ast´erisque 207, 35–108 (1992) [10] Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52, 48–79 (1983) [11] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth, Leipzig (1995) Ognjen Milatovic (B) Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA e-mail: [email protected] Received: November 6, 2009. Revised: April 12, 2010.

Integr. Equ. Oper. Theory 68 (2010), 255–262 DOI 10.1007/s00020-010-1815-6 Published online July 15, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

On the Unital C*-Algebras Generated by Certain Subnormal Tuples Ameer Athavale Abstract. We consider an important class of subnormal operator m-tuples Mp (p = m, m + 1, . . .) that is associated with a class of reproducing kernel Hilbert spaces Hp (with Mm being the multiplication tuple on the Hardy space of the open unit ball B2m in Cm and Mm+1 being the multiplication tuple on the Bergman space of B2m ). Given any two ˜ p ) : p ≥ m}, C*-algebras A and B from the collection {C ∗ (Mp ), C ∗ (M ˜ p) where C ∗ (Mp ) is the unital C*-algebra generated by Mp and C ∗ (M ˜ p of Mp , we verify that A the unital C*-algebra generated by the dual M and B are either *-isomorphic or that there is no homotopy equivalence between A and B. For example, while C ∗ (Mm ) and C ∗ (Mm+1 ) are well˜ m ) and C ∗ (M ˜ m+1 ) are not known to be *-isomorphic, we find that C ∗ (M ˜ m) even homotopy equivalent; on the other hand, C ∗ (Mm ) and C ∗ (M are indeed *-isomorphic. Our arguments rely on the BDF-theory and K-theory. Mathematics Subject Classification (2010). Primary 47B20. Keywords. Subnormal, Ext, K0 , K1 .

1. Introduction All the Hilbert spaces to follow are complex, inﬁnite-dimensional, and separable. Given a Hilbert space H, we use B(H) to denote the algebra of bounded linear operators on H. An m-tuple S = (S1 , . . . , Sm ) of commuting operators Si in B(H) is said to be subnormal if there exist a Hilbert space J containing H and an m-tuple N = (N1 , . . . , Nm ) of commuting normal operators Ni in B(J ) such that Ni H ⊂ H and Ni /H = Si for 1 ≤ i ≤ m. Every subnormal operator tuple has a ‘minimal’ normal extension that is unique up to unitary equivalence (see [15]). If N = (N1 , . . . , Nm ) (with Ni in B(J )) is the minimal normal extension of a subnormal tuple S = (S1 , . . . , Sm ) (with Si in B(H)), and H⊥ ≡ J H is the orthocomplement of H in J , then one deﬁnes the dual S˜ of S to be the subnormal tuple S˜ = (S˜1 , . . . , S˜m )

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where S˜i = Ni∗ /H⊥ . The notion of the dual of a subnormal operator was ﬁrst explored systematically by Conway in [10]. If B2m is the open unit ball in Cm , consider the positive definite kernels 1 2m × B2m and the κp (λ, ω) = (1−λ¯ 1 ω1 −···− ¯ m ωm )p (p = m, m + 1, . . .) on B λ associated reproducing kernel Hilbert spaces Hp . Modulo constants, κm and κm+1 are respectively the well-known Szeg¨o and Bergman kernels associated respectively with the Hardy space Hm of B2m and the Bergman space Hm+1 of B2m . The operator tuples Mp = (Mp,z1 , . . . , Mp,zm ), where Mp,zi is the operator on Hp of multiplication by the coordinate function zi , are subnormal and play an important role in the model theory of subnormal operator m-tuples having their joint Taylor spectra contained in the closure B2m of B2m (see [1,3–6,16,19]). If S = (S1 , . . . , Sm ) is an m-tuple of commuting operators Si in B(H), then we use C ∗ (S) to denote the C*-subalgebra of B(H) generated by the set {S1 , . . . , Sm , IH }, where IH is the identity operator on H; C ∗ (S) will be referred to as the unital C*-algebra generated by S. It is a classical result of Coburn [9] that C ∗ (Mm ) and C ∗ (Mm+1 ) are *-isomorphic. Using elemen˜ m ) and C ∗ (M ˜ m+1 ) tary K-theory we verify in Section 3 below that C ∗ (M are not even homotopy equivalent, while using the Brown-Douglas-Fillmore ˜ m ) are theory (BDF-theory for short) we establish that C ∗ (Mm ) and C ∗ (M indeed *-isomorphic; in fact, we investigate such relationships for any two ˜ p ) : p ≥ m}. The Refs. of the C*-algebras in the collection {C ∗ (Mp ),C ∗ (M [12,18] adequately cover the facts related to the BDF-theory and K-theory of C*-algebras as used in the sequel.

2. Some Operator-Theoretic Preliminaries An m-tuple S = (S1 , . . . , Sm ) of operators Si in B(H) is said to be almost commuting if Si Sj − Sj Si are compact operators for 1 ≤ i, j ≤ m, and is said to be essentially normal if, in addition, Si∗ Sj −Sj Si∗ are compact operators for 1 ≤ i, j ≤ m. For the notion of the Taylor spectrum σ(S) of an m-tuple S of commuting operators in B(H) as well as for the notion of an almost commuting tuple S being Fredholm, the reader is referred to [11]. The Fredholm index ind(S) of an almost commuting Fredholm tuple S is deﬁned in [11] to be the negative of the Euler characteristic of a certain chain Koszul complex associated with S, and we follow that convention here. A point (λ1 , . . . , λm ) ∈ Cm is in the essential Taylor spectrum σe (S) of an almost commuting tuple S if and only if (S1 − λ1 , . . . , Sm − λm ) is not Fredholm. If S is an m-tuple of commuting operators in B(H), then σe (S) and σ(S) are compact subsets of Cm with σe (S) ⊂ σ(S). For a compact subset X of Cm , we use C(X) to denote the C*-algebra of complex continuous functions on X. Let a Hilbert space H be ﬁxed and let K(H) be the C*-algebra of compact operators on H. An extension (E, ψ) of C(X) by K(H) is a short exact sequence ι

ψ

0 −→ K(H) −→ E −→ C(X) −→ 0

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where E is a C*-subalgebra of B(H) containing K(H), ψ is a *-homomorphism, and ι is the inclusion map of K(H) into E. Given extensions (E1 , ψ1 ) and (E2 , ψ2 ) of C(X) by K(H), (E2 , ψ2 ) is said to be equivalent to (E1 , ψ1 ) if there exists a unitary map U from H onto H such that E2 = U E1 U ∗ and ψ1 (A) = ψ2 (U AU ∗ ) for all A in E1 ; this deﬁnes an equivalence relation on the set of all extensions of C(X) by K(H). The set of equivalence classes (E, ψ) can be endowed with the structure of an abelian group and is written as Ext(X). If S2m−1 is the unit sphere (which is the topological boundary of B2m in Cm ), then Ext(S2m−1 ) is known to be isomorphic to Z, the group of integers; also, Ext(X) = 0 if X is contractible. If S = (S1 , . . . , Sm ) is an m-tuple of operators in B(H) and T = (T1 , . . . , Tm ) an m-tuple of operators in B(J ), then we say that S and T are compalent (under U ) if there exist compact operators K1 , . . . , Km in B(J ) and a unitary map U from H onto J such that Ti = U Si U ∗ + Ki for 1 ≤ i ≤ m. Compalent almost commuting operator tuples have equal essential Taylor spectra [11]. If S = (S1 , . . . , Sm ) is an essentially normal tuple of operators in B(H), and if π is the canonical projection of B(H) onto the Calkin C*-algebra Q(H) = B(H)/K(H), then π(S) = (π(S1 ), . . . , π(Sm )) is an m-tuple of commuting normal elements in Q(H). If C ∗ (π(S)) is the C*-algebra generated by the set {π(S1 ), . . . , π(Sm ), π(IH )}, then it follows by the Gelfand–Naimark theory that there is a unique *-isomorphism φS of C ∗ (π(S)) onto C(σe (S)) mapping π(Si ) to zi and mapping π(IH ) to the constant function 1 on σe (S); clearly then ι

φS ◦π

0 −→ K(H) −→ C ∗ (S) + K(H) −→ C(σe (S)) −→ 0 is an extension of C(σe (S)) by K(H), which we denote by ES . If ES and ET are the extensions corresponding to essentially normal tuples S and T of operators in B(H) for which σe (S) = σe (T ), then it can be veriﬁed that ES and ET are equivalent if and only if S and T are compalent. If T happens to be an (essentially normal) m-tuple of operators in B(J ) for some other Hilbert space J with σe (S) = σe (T ), then in any discussion related to the compalence (or otherwise) of tuples S and T , a reference to ET should really be interpreted as a reference to ET , where T is the tuple (V ∗ T1 V, . . . , V ∗ Tm V ) for a ﬁxed unitary V from H onto J ; clearly, S and T are compalent if and only if ES and ET are equivalent. An m-tuple S = (S1 , . . . , Sm ) of operators in B(H) is said to be irreducible if the only closed subspaces of H that reduce all Si are {0} and H (which is the same as requiring that any of C ∗ (S) and W ∗ (S) be an irreducible C*-algebra, where W ∗ (S) is the von Neumann algebra generated by {S1 , . . . , Sm }). A subnormal m-tuple S = (S1 , . . . , Sm ) operators in B(H) is said to be pure if there is no non-trivial closed subspace L of H which is reducing for all Si and for which Si /L are normal. The commutant of a subset A of B(H) will be denoted by A . The proof of the next proposition employs the same ideas as in [10].

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Proposition 2.1. If S is a pure subnormal tuple of operators in B(H), then S is irreducible if and only if S˜ is irreducible. Proof. We ﬁrst observe that Theorem 7 of [7] (along with its corollary there) generalizes in a straightforward way to subnormal operator tuples (refer to Lemma 3 of [15]). That in turn leads to a straightforward generalization of Theorem 8 of [7] so that, if N (with Ni in B(J )) is the minimal normal extenJ sion of S and PH the orthogonal projection of J onto H, one has W ∗ (S) and J ∗ W (N ) ∩ {PH } to be *-isomorphic. Further, generalizing Lemma 5.3 of [17] ∗ ) is the minimal normal to operator tuples, one notes that N ∗ = (N1∗ , . . . , Nm ∗ ˜ and W ∗ (N ∗ ) ∩ {P J⊥ } are extension of S˜ in case S is pure; hence W (S) H ˜ are *-isomorphic. Thus S *-isomorphic. It follows that W ∗ (S) and W ∗ (S) ˜ = CIH⊥ if is irreducible if and only if W ∗ (S) = CIH if and only if W ∗ (S) ˜ and only if S is irreducible.

3. Invoking BDF- and K-Theory The observations in Section 3 of [4] leading to Proposition 3.1 there show that, for an appropriate positive measure μp (p ≥ m), Mp can be looked upon as a subnormal tuple of multiplications by the coordinate functions zi on the closure H 2 (μp ) of polynomials in L2 (μp ), with the minimal normal extension Np of Mp being the tuple (Np,z1 , . . . , Np,zm ) of multiplications by zi on L2 (μp ). (Classically, Mm and Mm+1 have been known to be subnormal with μm being the “surface area measure” on S2m−1 and μm+1 being the volumetric measure on B2m ). For p ≥ m + 2, the measure μp is a weighted volumetric measure supported on B2m with the weight factor, modulo a constant, being (1 − |z1 |2 − · · · − |zm |2 )p−m−1 . Using the Stone-Weierstrass Theorem, it is easy to see that, for any p ≥ m, C ∗ (Mp ) is the C*-algebra generated by the “Toeplitz” operators Tp,f with the symbols f being continuous functions on L2 (μp )

the support of μp (Tp,f g = PHp

f g, g ∈ Hp = H 2 (μp )).

Proposition 3.1. For any p, q ≥ m, there exists a unital *-isomorphism of C ∗ (Mp ) onto C ∗ (Mq ); in particular, for any p ≥ m, K0 (C ∗ (Mp )) = Z and K1 (C ∗ (Mp )) = 0. Proof. Let p ≥ m. As noted in Propositions 3.2 and 3.3 of [4], Mp is essentially normal and C ∗ (Mp ) is irreducible; also, the commutators (Mp,zi )∗ Mp,zi − Mp,zi (Mp,zi )∗ are non-zero. Hence C ∗ (Mp ) contains the full algebra of compact operators on Hp . Further, as observed in Lemma 3.5 of [4], for any p, q ≥ m, Mp and Mq are compalent under a unitary Vp,q from Hp onto Hq ηp,q ∗ so that the map A −→ Vp,q AVp,q , (A ∈ C ∗ (Mp )) sends the generators Mp,zi ∗ ∗ and IHp of C (Mp ) into C (Mq ) and also contains in its range the generators Mq,zi and IHq of C ∗ (Mq ); hence ηp,q is a (unital) *-isomorphism of C ∗ (Mp ) onto C ∗ (Mq ). It is well-known that K0 (C ∗ (Mm )) = Z and K1 (C ∗ (Mm )) = 0 (see [8], for example). Since K0 and K1 groups are preserved under a *-isomorphism of unital C*-algebras, we are done.

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As noted in Proposition 3.6 of [4], σ(Mp ) = B2m and σe (Mp ) = S2m−1 for any p ≥ m; in particular, Mp is Fredholm (and as per the definitions employed in [11], ind(Mp ) = −1 (refer to [14])). Clearly then the extension ι

φMp ◦π

0 −→ K(Hp ) −→ C ∗ (Mp ) −→ C(S2m−1 ) −→ 0

(∗)

obtains for any p ≥ m. As recorded in Theorem 1 of [9], C ∗ (Mm ) = {Tm,f + K : f ∈ C(S2m−1 ), K ∈ K(Hm )} and C ∗ (Mm+1 ) = {Tm+1,f + K : f ∈ C(B2m ), K ∈ K(Hm+1 )}; arguing exactly as there, one has that C ∗ (Mp ) = {Tp,f + K : f ∈ C(B2m ), K ∈ K(Hp )} for any p ≥ m + 2. ˜ p as follows from Since Mp (p ≥ m) is essentially normal, so is its dual M Remark 2 of [2]. Since Mp,zi are non-normal operators and Mp is irreducible, ˜ p ) is irreducible in the light of Proposition 2.1. Since Mp is pure so that C ∗ (M the commutators (Mp,zi )∗ Mp,zi − Mp,zi (Mp,zi )∗ are non-zero, so are the com˜ p,z − M ˜ p,z (M ˜ p,z )∗ as again follows by using the observa˜ p,z )∗ M mutators (M i i i i ˜ p ) contains the tions present in Remark 2 of [2]. Hence, for any p ≥ m, C ∗ (M ⊥ 2 ⊥ 2 full algebra of compact operators on Hp = H (μp ) = L (μp )H 2 (μp ). The ˜ p,z on H⊥ (and of (Np,z )∗ on L2 (μp )) is multiplication by z¯i and action of M p i i ˜ p ) is the C*-algebra generated by it is easy to see that, for any p ≥ m, C ∗ (M the “Toeplitz-like” operators Wp,f with the symbols f being continuous funcL2 (μ )

tions on the support of μp (Wp,f g = PH⊥ p f g, g ∈ Hp⊥ ). Also, arguing as in p ˜ m ) = {Wm,f + K : f ∈ C(S2m−1 ), K ∈ Theorem 1 of [9], one sees that C ∗ (M ⊥ ˜ p ) = {Wp,f + K : f ∈ C(B2m ), K ∈ K(Hp⊥ )} for any )} and C ∗ (M K(Hm p ≥ m + 1. Proposition 3.2. There exists a unital *-isomorphism of C ∗ (Mm ) onto ˜ m ); in particular, K0 (C ∗ (M ˜ m )) = Z and K1 (C ∗ (M ˜ m )) = 0. C ∗ (M ˜ m ) = S2m−1 Proof. As observed in part (a) of Examples 1 of [2], σe (M ˜ ˜ m) = (so that Mm is Fredholm). In view of Proposition 3 of [2], ind(M m+1 ind(Mm ) (see also [13]). (−1) ˜ m ) = ind(Mm ) = −1, and we consider the If m is odd, then ind(M extension EMm (which is (∗) with p = m) and the extension EM˜ m given by φ

◦π

˜m ι ⊥ ˜ m) M ) −→ C ∗ (M −→ C(S2m−1 ) −→ 0. 0 −→ K(Hm

As follows from the proof of Theorem 7 in [11], EMm and EM˜ m are equivalent. (Indeed, both EMm and EM˜ m can be identiﬁed with the same generator ˜ m are compalent. of Z = Ext(S2m−1 )). Thus Mm and M ˜ If m is even, then ind(Mm ) = −ind(Mm ) = 1. We note that the ˜ m,z )∗ , M ˜ m,z , . . . , M ˜ m,z ) is essentially normal; further, ˜m = ((M tuple M 1 2 m 2m−1 ˜ ˜ ˜ ) = (by Corollary 3.14 of [11]) and ind(M σe (Mm ) = σe (Mm ) = S m ˜ −ind(Mm ) = ind(Mm ) = −1 (by Proposition 9.1 of [11]). We now consider the extension EMm (which is (∗) with p = m) and the extension EM˜ given m by φ

˜

◦π

ι m ⊥ ˜ ) M 0 −→ K(Hm ) −→ C ∗ (M −→ C(S2m−1 ) −→ 0. m

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Again, it follows from the proof of Theorem 7 in [11] that EMm and EM˜ m are equivalent. (Here both EMm and EM˜ get identiﬁed with the same m ˜ are compalent. generator of Z = Ext(S2m−1 )). Thus Mm and M m ⊥ such that the In either case, one has a unitary U from Hm onto Hm η ∗ ∗ map A −→ U AU (A ∈ C (Mm )) sends the generators Mm,zi and IHm of ˜ m ) = C ∗ (M ˜ ) and also contains in its range a set of C ∗ (Mm ) into C ∗ (M m ∗ ˜ generators of C (Mm ); hence η is a (unital) *-isomorphism of C ∗ (Mm ) onto ˜ m ). C ∗ (M Proposition 3.3. For any p, q ≥ m + 1, there exists a unital *-isomorphism ˜ p ) onto C ∗ (M ˜ q ). of C ∗ (M Proof. Let p ≥ m + 1. If Np is the minimal normal extension of Mp , then σ(Np ) = B2m . Thus, for any (λ1 , . . . , λm ) ∈ B2m , the number 0 is in the m spectrum of the normal operator N = i=1 ((Np,zi )∗ − λi )(Np,zi − λi ) by Corollary 3.9 of [11]; if the tuple (Np,z1 − λ1 , . . . , Np,zm − λm ) were to be Fredholm, then 0 would not be in the essential spectrum of N (by Corollary 3.9 of [11] again) and hence would be an eigenvalue of N , which is easily seen not to be the case in the light of our description of μp and Np earlier. Thus (λ1 , . . . , λm ) ∈ σe (Np ) and one has σe (Np ) = B2m . Since σe (Mp ) is S2m−1 , it ˜ p ) = B2m . Thus, for p ≥ m + 1, one follows from Corollary 2 of [2] that σe (M has the extension ι

φM ˜ p ◦π

˜ p ) −→ C(B2m ) −→ 0. 0 −→ K(Hp⊥ ) −→ C ∗ (M Since B2m is contractible, Ext(B2m ) = 0 so that all the extensions EM˜ p (p ≥ ˜ q are ˜ p and M m + 1) are trivial and are equivalent to each other; thus M compalent for p, q ≥ m + 1, and that leads to the desired result. ˜ m+1 )) = Z ⊕ Z and K1 (C ∗ (M ˜ m+1 )) = 0; Proposition 3.4. One has K0 (C ∗ (M ˜ m ) and in particular, there is no homotopy equivalence between C ∗ (M ∗ ˜ C (Mm+1 ). Proof. In view of our observations in the proof of Proposition 3.3, the extension ι ⊥ ˜ m+1 ) ) −→ C ∗ (M 0 −→ K(Hm+1

φM ˜

m+1

−→

◦π

C(B2m ) −→ 0

obtains. That leads to the following six-term exact sequence of K-theory (with the symbols for group homomorphisms being suppressed): ⊥ ∗ ˜ 2m )) K0 (K(H ⏐ m+1 )) −→ K0 (C (Mm+1 )) −→ K0 (C(B ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⊥ ∗ ˜ K1 (K(Hm+1 )) ←− K1 (C (Mm+1 )) ←− K1 (C(B2m ))

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⊥ ⊥ )) = Z = K0 (C(B2m )) and K1 (K(Hm+1 )) = 0 = Since K0 (K(Hm+1 2m K1 (C(B )) (refer to [18]), one has in eﬀect the six-term exact sequence ∗ ˜ Z ⏐ −→ K0 (C (Mm+1 )) −→ Z ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ∗ ˜ 0 ←− K1 (C (Mm+1 )) ←− 0.

˜ m+1 )) = 0 and also the short exact sequence of groups That yields K1 (C ∗ (M ˜ m+1 )) −→ Z −→ 0. 0 −→ Z −→ K0 (C ∗ (M

(∗∗)

Since Z is a free group on the generator 1, it is easy to see that the sequence ˜ m+1 )) = Z ⊕ Z. Since (∗∗) is a split exact sequence so that K0 (C ∗ (M ∗ ˜ K0 (C (Mm )) = Z, and since K0 groups are preserved under a homotopy ˜ m) equivalence of unital C*-algebras (see Proposition 3.2.6 of [18]), C ∗ (M ˜ m+1 ) cannot be homotopy equivalent. and C ∗ (M Using ≈ to indicate the existence of a unital *-isomorphism and to indicate the lack of any homotopy equivalence between two C*-algebras, the results of Propositions 3.1, 3.2, 3.3 and 3.4 can be summarized as follows. ˜ m ); for any p ≥ m + 1, C ∗ (M ˜ p) ≈ For any p ≥ m, C ∗ (Mp ) ≈ C ∗ (M ˜ m+1 ); C ∗ (M ˜ m ) C ∗ (M ˜ m+1 ); also, for any p ≥ m, K0 (C ∗ (Mp )) = C ∗ (M ˜ m )) = Z; for any p ≥ m + 1, K0 (C ∗ (M ˜ p )) = Z ⊕ Z; and, for any K0 (C ∗ (M ∗ ∗ ˜ p ≥ m, K1 (C (Mp )) = K1 (C (Mp )) = 0.

References [1] Agler, J.: Hypercontractions and subnormality. J. Oper. Theory 13, 203–217 (1985) [2] Athavale, A.: On the duals of subnormal tuples. Integr. Equ. Oper. Theory 12, 305–323 (1989) [3] Athavale, A.: On the intertwining of joint isometries. J. Oper. Theory 23, 339– 350 (1990) [4] Athavale, A.: Model theory on the unit ball in Cm . J. Oper. Theory 27, 347– 358 (1992) [5] Athavale, A.: Unitary and spherical dilations: a hypercontractive perspective. Rev. Roumaine Math. Pures Appl. 38, 387–400 (1993) [6] Athavale, A.: Quasisimilarity-invariance of joint spectra for certain subnormal tuples. Bull. Lond. Math. Soc. 40, 759–769 (2008) [7] Bram, J.: Subnormal operators. Duke Math. J. 22, 75–94 (1955) [8] Cao, G.: Toeplitz algebras on strongly pseudoconvex domains. Nagoya Math. J. 185, 171–186 (2007) [9] Coburn, L.A.: Singular integral operators and Toeplitz operators on odd spheres. Indiana Univ. Math. J. 23, 433–439 (1973) [10] Conway, J.B.: The dual of a subnormal operator. J. Oper. Theory 5, 195–211 (1981)

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[11] Curto, R.E.: Fredholm and invertible n-tuples of operators. The deformation problem. Trans. Am. Math. Soc. 266, 129–159 (1981) [12] Davidson, K.R.: C*-Algebras by Example. Fields Institute Monographs. Amer. Math. Soc., Providence (1996) [13] Gleason, J.: On a question of Ameer Athavale. Irish Math. Soc. Bull. 48, 31–33 (2002) [14] Gleason, J., Richter, S., Sundberg, C.: On the index of invariant subspaces in spaces of analytic functions in several complex variables. J. Reine Angew. Math. 587, 49–76 (2005) [15] Ito, T.: On the commutative family of subnormal operators. J. Fac. Sci. Hokkaido Univ. 14, 1–15 (1958) [16] M¨ uller, V., Vasilescu, F.-H.: Standard models for some commuting multioperators. Proc. Am. Math. Soc. 117, 979–989 (1993) [17] Olin, R.F.: Functional relationships between a subnormal operator and its minimal normal extension. Pac. J. Math. 63, 221–229 (1976) [18] Rordam, M., Larsen, F., Laustsen, N.J.: An Introduction to K-Theory for C*-Algebras. Cambridge University Press, Cambridge, UK (2000) [19] Vasilescu, F.-H.: An operator-valued Poisson kernel. J. Funct. Anal. 110, 47–72 (1992) Ameer Athavale (B) Department of Mathematics Indian Institute of Technology Bombay Powai, Mumbai 400076, India e-mail: [email protected] Received: November 24, 2009. Revised: June 3, 2010.

Integr. Equ. Oper. Theory 68 (2010), 263–286 DOI 10.1007/s00020-010-1816-5 Published online July 13, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes Jordi Juan-Huguet Abstract. Let P be a linear partial differential operator with constant coefﬁcients. For a weight function ω and an open subset Ω of RN , the class EP,{ω} (Ω) of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P . Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradiﬀerentiable functions if and only if P is elliptic. These results remain true in the Beurling case EP,(ω) (Ω). Mathematics Subject Classification (2010). 35B65, 35H10, 46F05, 47F05. Keywords. Hypoelliptic polynomial, elliptic differential operator, non-quasianalytic classes, iterates of an operator.

1. Introduction In 1960, Komatsu [9], using tools introduced by H¨ ormander [7], characterized when a smooth function f ∈ C ∞ (Ω) in an open subset Ω ⊂ RN is real analytic in terms of the successive iterates of a elliptic partial differential operator P (D). In particular, given a elliptic differential operator P (D) of order m, a function f ∈ C ∞ (Ω) is real analytic if and only if for each compact subset K ⊂⊂ Ω there exists a constant C > 0 such that for each j ∈ N0 P j (D)f 2,K ≤ C j+1 (j!)m , where P j (D) is the jth iterate of P (D), i.e., P j (D) = P (D) ◦ · · · ◦ P (D). j

See also the theorem by Kotake–Narasimhan [11, Theorem 1]. In 1973, Newberger and Zielezny [19] treated this problem in the setting of the Gevrey classes: let G d (Ω) be the Gevrey class of exponent d > 1 and Jordi Juan-Huguet was partially supported by MEC and FEDER Project MTM2007-62643, Conselleria d’Educaci´ o de la GVA, Project GV/2010/040, grant F.P.U. AP-2006-04678 and the research net MTM2007-30904-E.

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let GPd (Ω) be the class of smooth functions in Ω such that for each K ⊂⊂ Ω there exists a constant C > 0 such that ∀j ∈ N0 , P j (D)f 2,K ≤ C j+1 (j!)d , then G d (Ω) = GPmd (Ω) whenever P is an elliptic polynomial with degree m. Moreover, in case P is a hypoelliptic polynomial and Q is an arbitrary polynomial, it is proved the equivalence between the inequality |Q(ξ)|2 ≤ dh (Ω). C(1 + |P (ξ)|2 )h , ∀ξ ∈ RN and the inclusion GPd (Ω) ⊂ GQ This research is continued by several authors like Bolley et al. [1], Zanghirati [22–24] and Bouzar and Chaili [3]. Langenbruch utilized generalized Gevrey classes in connection with different problems, like boundary values of zero solutions of hypoelliptic differential operators [12,13], diametral dimension of solution spaces [14] and isomorphic classiﬁcation [15]. The problem of the iterates consists in giving conditions on P in order to guarantee the equality G d (Ω) = GPmd (Ω). In this paper we extend the results of Komatsu [9] and Newberger and Zielezny [19] to the setting of non-quasianalytic classes in the sense of Braun–Meise–Taylor [4]. The precise definition of the spaces will be given in Sect. 2, in which we recall the definition of the non-quasianalytic classes of ultradifferentiable functions E(ω) (Ω) (Beurling type) and E{ω} (Ω) (Roumieu type) in the sense of Braun et al. [4] and the classes EP,(ω) (Ω) and EP,{ω} (Ω) of ultradifferentiable functions with respect to the iterates of P . In Sect. 3 we prove in Theorem 3.3 that EP,(ω) (Ω) and EP,{ω} (Ω), endowed with their natural topologies, are complete if and only if P is hypoelliptic. In case P is not hypoelliptic, a ﬁner topology on EP,∗ (Ω) can be deﬁned so that the space becomes complete. In spite of the importance of the completeness when dealing with functional analytic tools, as far as we know this is the ﬁrst time that the completeness of these spaces is discussed. In Sect. 4 we extend the results of Komatsu [9] and Newberger–Zielezny [19] for weight functions ω verifying a growth condition considered by Bonet et al. [2]. For this type of weight functions, we characterize in Theorem 4.12 when EP,{ω} (Ω) (respectively EP,(ω) (Ω)) coincides with the class of ultradifferen1 tiable functions E{σ} (Ω) (respectively E(σ) (Ω)), for σ(t) = ω(t m ), where m denotes the degree of the polynomial P.

2. Notation and Preliminaries 2.1. Non-Quasianalytic Classes in the Sense of Braun–Meise–Taylor We follow the point of view of Braun–Meise–Taylor (see [4]). A nonquasi-analytic weight function is an increasing continuous function ω : [0, ∞[→ [0, ∞[ with the following properties: (α) there exists L ≥ 0 with ω(2t) ≤ L(ω(t) + 1) for all t ≥ 0, ∞ (β) 1 ω(t) t2 dt < ∞,

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(γ) log(t) = o(ω(t)) as t tends to ∞, (δ) ϕ : t → ω(et ) is convex. For a weight function ω we deﬁne ω : CN → [0, +∞[ by ω(z) := ω(|z|) and again we denote this function by ω. The condition (β) is called non-quasianalytic condition and it implies ω(t) = o(t). Moreover, this condition implies the existence of functions with compact support in the class of ultradifferentiable functions. 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. Lemma 2.1. Given ϕ as above, we suppose that there exists L ≥ 0 such that ϕ(x + 1) ≤ L(1 + ϕ(x)) for all x ∈ [0, ∞[. Then, there exists y0 > 0 such that y −L ϕ∗ (y) − y ≥ Lϕ∗ L for all y ≥ y0 . Lemma 2.2. Given λ > 0 there exists a constant C > 0 (depending on λ) such that

x x+1 ∗ ∀x > 0. exp 2λϕ ≤ C exp λϕ∗ 2λ λ Proof. From the convexity of ϕ∗ we obtain

x 1 1 ∗ x 1 ∗ 1 ∗ + + ϕ ϕ ≤ ϕ . 2λ 2λ 2 λ 2 λ The conclusion follows with C = exp λϕ∗ λ1 .

Examples. The following functions are, after a change in some interval [0, M ], examples of weight functions: (i) (ii) (iii) (iv)

ω(t) = td for 0 < d < 1. s ω(t) = (log(1 + t)) , s > 1. ω(t) = t(log(e + t))−β , β > 1. ω(t) = exp(β(log(1 + t))α ), 0 < α < 1.

For a non-quasianalytic weight function ω, the spaces of ω-ultradifferentiable functions of Beurling and Roumieu case are deﬁned as follows. E(ω) (Ω) := {f ∈ C ∞ (Ω) : pK,λ (f ) < ∞, 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 pK,λ (f ) < ∞},

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where pK,λ (f ) := sup sup |f x∈K α∈NN 0

(α)

(x)|exp −λϕ

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∗

|α| λ

.

E(ω) (Ω) is a Fr´echet space, that is, a complete and metrizable locally convex space, while E{ω} (Ω) is a P LS-space, that is, a projective limit of a sequence En , where each En is an inductive limit of Banach spaces with compact linking maps. In the case that ω(t) := td (0 < d < 1), the corresponding Roumieu class is the Gevrey class with exponent 1/d. In the limit case d = 1, not included in our setting, the corresponding Roumieu class E{ω} (Ω) is the space of real analytic functions on Ω whereas the Beurling class E(ω) (RN ) gives the entire functions. The elements of E(ω) (Ω) (resp. E{ω} (Ω)) are called ultradifferentiable functions of Beurling type (resp. Roumieu type) in Ω. If a statement holds in the Beurling and the Roumieu case then we will use the notation E∗ (Ω). It means that in all cases * can be replaced either by (ω) or {ω}. The corresponding classes of test functions are deﬁned as follows: for a compact set K in RN , deﬁne D∗ (K) := {f ∈ E∗ (RN ) : suppf ⊂ K}, endowed with the induced topology. In [4, Remark 3.2 (1) and Corollary 3.6 (1)] it is shown that D∗ (K) = {0} is the strong dual of a nuclear Fr´echet space (i.e., it is a (DFN)-space). For an open set Ω in RN , deﬁne D∗ (Ω) := ind D∗ (K). −→ K⊂⊂Ω

According to [4](Proposition 4.7), the following inclusion D∗ (Ω) → E∗ (Ω) is continuous with dense range. The following lemma is well-known, but it is not easy to ﬁnd a precise reference. Lemma 2.3. The spaces E∗ (Ω) and D∗ (Ω) can be described with the L2 norm, i.e., we can replace pK,λ by the seminorms qK,λ (f ) := sup f (α) 2,K α∈NN 0 |α| ∗ , where exp −λϕ λ ⎛ f 2,K = ⎝

⎞ 12 |f |2 ⎠ .

K

Proof. We only need to prove that for each compact subset K ⊂⊂ Ω and λ > 0, there is other compact subset L ⊂⊂ Ω, μ > 0 and a constant D > 0 (depending only on K and λ) such that for all f ∈ E∗ (Ω), pK,λ (f ) ≤ DqL,μ (f ).

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We ﬁx K ⊂⊂ Ω and λ > 0. We take L ⊂⊂ Ω such that K ⊂ L ⊂ Ω. By the Sobolev Lemma, there exists a constant C > 0 such that |f (β) | ∀f ∈ C ∞ (Ω). sup |f (x)| ≤ C sup |β|≤N +1

x∈K

L

Then, sup |f

(α)

x∈K

(x)| ≤ C

sup |β|≤N +1

|f (α+β) |

L

|α|+|β| |α|+|β| | exp −λϕ∗ exp λϕ∗ λ λ |β|≤N +1 L

|α| + N + 1 ≤ C2 qL,λ (f ) exp λϕ∗ λ N +1

λ 2 |α| . ϕ∗ ≤ C3 qL,λ (f ) exp 2N +1 λ

=C

sup

|f

(α+β)

where we have applied that ϕ∗ is increasing and also Lemma 2.2 and H¨ older’s inequality. As a consequence, pK,

λ 2N +1

(f ) ≤ C3 qL,λ (f ).

So, given λ > 0 we take μ = 2N +1 λ.

2.2. The Classes EP,ω (Ω) We now consider smooth functions in an open set Ω verifying for each j ∈ N0

j j ∗ P (D)f 2,K ≤ C exp −λϕ , λ where K is a compact subset in Ω and P j (D) is the jth iterate of the partial · · ◦ P (D). If j = 0, then differential operator P (D), i.e, P j (D) = P (D) ◦ · j

P 0 (D)f = f . Given a polynomial P ∈ C[z1 , . . . , zN ] with degree m, P (z) = aα z α , |α|≤m

the partial differential operator P (D) is the following: P (D) =

|α|≤m

aα Dα , where Dα =

1 α ∂ . i

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Let ω be a weight function. Given a polynomial P , an open set Ω of RN , a compact subset K ⊂⊂ Ω and λ > 0, we deﬁne λ (K) = {f ∈ C ∞ (K) : f K,λ := sup P j (D)f 2,K EP,ω j∈N0

j exp −λϕ∗ < +∞}. λ

The spaces of ultradifferentiable functions with respect to the successive iterates of P are deﬁned as follows: Beurling case: EP,(ω) (Ω) = {f ∈ C ∞ (Ω) : f K,λ < +∞ for each K ⊂⊂ Ω and λ > 0}. It is endowed with the topology given by λ (K). EP,(ω) (Ω) := proj proj EP,ω ←− ←− K⊂⊂Ω λ>0

If {Kn }n∈N is a compact exhaustion of Ω we have k n (Kn ) = proj EP,ω (Kn ). EP,(ω) (Ω) = proj proj EP,ω ←− ←− ←− n∈N k∈N

n∈N

This metrizable locally convex topology is deﬁned by the fundamental system of seminorms { · Kn ,n }n∈N . Roumieu case: EP,{ω} (Ω) = {f ∈ C ∞ (Ω) : for each K ⊂⊂ Ω there is λ > 0 such that f K,λ < +∞}. Its topology is deﬁned by λ (K). EP,{ω} (Ω) := proj ind EP,ω ←− −→ λ>0 K⊂⊂Ω

3. Completeness of the Spaces EP,∗ (Ω) In order to extend the results of [19] we need to apply the Closed Graph Theorem and the Grothendieck’s Factorization Theorem. So, it is important to know whether the spaces EP,∗ (Ω) are complete or not. In this section we show that the spaces EP,∗ (Ω) are not necessarily complete spaces. In fact, completeness is characterized in terms of the hypoellipticity of the polynomial P . Moreover, in case completeness fails, a ﬁner topology on EP,∗ (Ω) is introduced so that the space becomes complete. This topology will be considered in Theorem 4.5. Proposition 3.1. Let Ω be an open subset of RN . If the space EP,∗ (Ω) is complete, then P is hypoelliptic. Proof. Proceeding by contradiction we assume that P is not hypoelliptic. We ﬁrst analyze the case that Ω = RN . Since P is not hypoelliptic, theorems [8, 11.1.5 and 10.1.25] imply the existence of a continuous function u ∈ C(RN )\C ∞ (RN ) such that P (D)u = 0.

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Beurling case. We take a regularizing sequence {ρn } with suppρn = B(0, n1 ) and we show that {u ∗ ρn } is a Cauchy sequence in EP,(ω) (RN ) which is not convergent. It is clear that u ∗ ρn ∈ C ∞ (RN ) for all n ∈ N. Moreover, P (D)(u ∗ ρn ) = P (D)u ∗ ρn = 0. As a consequence, P j (D)(u ∗ ρn ) = 0 if j = 0. Therefore, 1

u ∗ ρn K,λ ≤ (m(K)) 2 sup |u ∗ ρn (x)| < +∞, x∈K

for all K ⊂⊂ R and for all λ > 0, i.e, u ∗ ρn ∈ EP,(ω) (RN ) for each n ∈ N. In a similar way, N

1

u ∗ ρn − u ∗ ρl K,λ ≤ (m(K)) 2 sup |u ∗ (ρn − ρl )(x)| x∈K

which implies that {u ∗ ρn } is a Cauchy sequence in EP,(ω) (RN ). If {u ∗ ρn } converges to f ∈ EP,(ω) (RN ), then {u ∗ ρn } converges to f uniformly on the compact sets, hence f = u. This is a contradiction since u is not inﬁnitely differentiable. Roumieu case. The sequence {u ∗ ρn } constructed in the Beurling case is a Cauchy sequence in EP,{ω} (RN ) since the inclusion map EP,(ω) (RN ) → EP,{ω} (RN ) is continuous. We see that {u∗ρn } does not have limit in EP,{ω} (RN ). We call L2loc (RN ) = proj {f mesurable : f 2,K < ∞}, then the inclusion map ←− N K⊂⊂R

EP,{ω} (RN ) → L2loc (RN ) is continuous. If {u∗ρn } converges to f in EP,{ω} (RN ), then {u∗ρn } converges to f in L2loc (RN ). However, for each K ⊂⊂ RN 1

u ∗ ρn − u2,K ≤ (m(K)) 2 sup |u ∗ ρn (x) − u(x)| → 0 as n → +∞. x∈K

Then f = u, which is a contradiction since u is not C ∞ . In the case that Ω is an arbitrary open subset of RN , we can assume (after a suitable translation if necessary) that ∃u ∈ C(Ω + B(0, 1))\C ∞ (Ω) such that P (D)u = 0. Then the convolutions u ∗ ρn are deﬁned on Ω and we can proceed as above. In order to prove the converse of Proposition 3.1, we introduce the following spaces. Let ω be a weight function. Given a polynomial P , an open set Ω of RN , a compact subset K ⊂⊂ Ω and λ > 0. Deﬁne

j rK,λ (f ) := sup P j (D)f 2,K exp −λϕ∗ λ j∈N0 and LλP,ω (K) := {f ∈ L2 (K) : P j (D)f ∈ L2 (K) ∀j ∈ N0 , rK,λ (f ) < +∞}.

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Beurling case: LP,(ω) (Ω) = {f ∈ L2loc (Ω) : f ∈ LλP,ω (K) for each K ⊂⊂ Ω and λ > 0}. If {Kn }n∈N is a compact exhaustion of Ω this space is endowed with the topology given by LP,(ω) (Ω) = proj proj LkP,ω (Kn ) = proj LnP,ω (Kn ). ←− ←− ←− n∈N k∈N

n∈N

This metrizable locally convex topology is deﬁned by the fundamental system of seminorms {rKn ,n (·)}n∈N . Using standard arguments it follows that LP,(ω) (Ω) is a Fr´echet space. Roumieu case: LP,{ω} (Ω) = {f ∈ L2loc (Ω) : for each K ⊂⊂ Ω there is λ > 0 such that f ∈ LλP,ω (K)}. This space is endowed with the following locally convex topology: LP,{ω} (Ω) := proj ind LλP,ω (K). ←− −→ λ>0 K⊂⊂Ω

Proposition 3.2. The space of Roumieu type LP,{ω} (Ω) is complete. Proof. We ﬁx a compact subset K of Ω. If sufﬁces to prove that the countable inductive limit of Banach spaces X = ind LnP,ω (K) −→ n∈N

is complete. According to a Theorem of Mujica [18] (see also [20, Corollary 8.5.22 (ii)]), we only need to check that there is a Hausdorﬀ locally convex topology s on X such that the unit ball of each LnP,ω (K) is compact in (X, s). To do this it is enough to consider E := j∈N0 (L2 (Kn ), tn ), where tn denotes the weak topology, and deﬁne s as the topology induced on X by the injective map X → E, f → {P j (D)f }j∈N0 . Theorem 3.3. The space EP,∗ (Ω) is complete if and only if P is a hypoelliptic polynomial. Proof. By Theorem 3.2 it sufﬁces to prove that the spaces EP,∗ (Ω) and LP,∗ (Ω) coincide algebraically and topologically. Let m denote the degree of P , let Ω be an open subset of Ω and let f ∈ LP,∗ (Ω ) be given. We use condition H5 of [6, Theorem 6.36]. Since P (D)f ∈ L2loc (Ω ) = H0loc (Ω ), there loc exists δ > 0 such that f ∈ Hmδ (Ω ). Analogously, P j (D)f ∈ L2loc (Ω ) implies loc f ∈ Hjmδ (Ω ). As a consequence, f ∈ LP,∗ (Ω ) implies f ∈ s>0 Hsloc (Ω ) = C ∞ (Ω ) as a consequence of theorems [6, Theorems 6.7 and 6.13]. Denote by (Kj )j a fundamental sequence of compact sets in Ω such that each Kj is contained in the interior of Kj+1 . Our argument above shows that the restricλ tion maps LλP,ω (Kj+1 ) into EP,ω (Kj ) continuously, from where the conclusion follows.

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In order to construct a complete ﬁner locally convex topology on EP,∗ (Ω) in the general case, we ﬁx a compact exhaustion {Kn } of Ω and we consider the following system of seminorms on EP,(ω) (Ω) : { · n }n∈N ∪ {pn }n∈N where

f n := f Kn ,n

j = sup P (D)f 2,Kn exp −nϕ , n j∈N0 j

∗

and {pn }n∈N is a fundamental system of seminorms of E(Ω), i.e, pn (f ) := sup sup |f (α) (x)|. |α|≤n x∈Kn

Then, {max( · n , pn )}n∈N is a fundamental system of seminorms of a locally convex topology τ(ω),∞ on EP,(ω) (Ω). The proof of the following result is standard. Proposition 3.4. Let Ω be an open subset of RN . For a weight function ω and a polynomial P , the space (EP,(ω) (Ω), τ(ω),∞ ) is a Fr´echet space. In the Roumieu case, we consider the following topology: for n ∈ N and 1 n K ⊂⊂ Ω, we endow EP,ω (K) with the fundamental system of seminorms max · K, n1 , pm . m∈N

1

n It is easy to see that EP,ω (K) is a Fr´echet space. The topology τ{ω},∞ on EP,{ω} (Ω) is deﬁned by 1

n (EP,{ω} (Ω), τ{ω},∞ ) = proj ind EP,ω (K). ←− −→ n∈N

K⊂⊂Ω

1 n

The space ind EP,ω (K) is an (LF)-space, i.e, a countable inductive limit of −→ n∈N

Fr´echet spaces. Proposition 3.5. Let Ω be an open subset of RN . For a weight function ω and a polynomial P , the space (EP,{ω} (Ω), τ{ω},∞ ) is complete. The proof requires the following result for (LF)-spaces. Definition 3.6. Let X = ind Xn be an (LF)-space. X is called boundedly −→ n∈N

stable if on each set which is bounded in some Xn all but ﬁnitely many of the step topologies coincide. Next theorem due to Wengenroth follows from Theorem 6.4 (page 112) and Corollary 6.4 of [21, p. 113]:

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Theorem 3.7. Let X = ind Xn be an (LF)-space and { · n,m }m∈N a fun−→ n∈N

damental system of seminorms of Xn . If X is boundedly stable and satisfies the condition (P3*), i.e, ∀n ∃l ≥ n ∀m ≥ l ∃N ∈ N ∀M ∈ N ∃K ∈ N, S > 0 ∀x ∈ Xn xl,M ≤ S(xm,K + xn,N ), then X is complete. Proof of Proposition 3.5. It is enough to show that for each K ⊂⊂ Ω, the 1 1 n n (K) is complete. We denote Xn = EP,ω (K) and space X = ind EP,ω −→ n∈N

{ · n,m }m∈N = {max( · n1 , pm )}m∈N . To see that X is complete, we show that X veriﬁes the hypothesis of Theorem 3.7, i.e, X is boundedly stable and satisﬁes the condition (P3*). We apply Lemma 2.1 and the condition (α) of weight function to get a natural number L and y0 > 0 such that y − L for each y ≥ y0 . ϕ∗ (y) − y ≥ Lϕ∗ L That implies 1 ∗ exp − nL ϕ (jnL) 1 −→ 0 as j → +∞. (1) exp − n ϕ∗ (jn) To see that X is boundedly stable it sufﬁces to prove that any bounded set B in Xn is relatively compact in XnL . Since E(Ω) is a Montel space, we only need to show that if {fk }k∈N is a bounded sequence in Xn and converges to 0 in E(Ω), then 1 −→ 0 si k → +∞. {fk }k∈N converges to 0 in XnL , i.e, fk K, nL

Since {fk }k∈N is a bounded sequence in Xn , there exists a constant C > 0 such that for all k ∈ N, fk K, n1 ≤ C. Let η > 0, in view of (1) there exists j0 ∈ N0 such that

1 ∗ ϕ (jnL) P j (D)(fk )2,K exp − Ln

1 ≤ η sup P i (D)(fk )2,K exp − ϕ∗ (in) ≤ η C if j > j0 . n i∈N0 Therefore, 1 ≤ fk K, Ln

max

i=0,1,...,j0

1

ηC, P i (D)(fk )2,K e− Ln ϕ

∗

(iLn)

.

1 Since {fk }k∈N converges to 0 in E(Ω) we have fk K, Ln ≤ ηC if k is enough large. The property (P3*) is satisﬁed. Indeed, take l = n, N = 1, K = M and S = 1, then

∀n ∀m ≥ n ∀M ∈ N ∀x ∈ Xn xn,M ≤ xm,M + xn,1 ,

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since

x 1 ≤ x ≤ x n,1 m,M + xn,1 K, n xn,M = max xK, n1 , pM (x) = pM (x) ≤ xm,M ≤ xm,M + xn,1

4. Hypoelliptic and Elliptic Polynomials and the Growth of EP,∗ (Ω) The following result asserts that the class of ultradifferentiable functions Eω (Ω) is always contained in EP,σ (Ω) where σ depends on ω and the degree of P . Let ω be a weight function and m ≥ 1 the degree of P , it is easy to prove 1 that σ(t) := ω(t m ) is also a weight function. Moreover, ϕ∗σ (x) = ϕ∗ω (mx). We will denote ϕ∗ω simply by ϕ∗ . Theorem 4.1. Let Ω ⊂ RN be an open subset. For a weight function ω and a 1 (Ω) holds and polynomial P with degree m, the inclusion E∗(t) (Ω) ⊆ E P,∗(t m ) the inclusion map is continuous. Notation. E∗(t) (Ω) ⊆ E

1

P,∗(t m )

(Ω) means that both inclusions E(ω) (Ω) ⊆

EP,(σ) (Ω) and E{ω} (Ω) ⊆ EP,{σ} (Ω) hold. We need the following technical lemma. Lemma 4.2. Given a constant C ≥ 1 and a weight function ω, there exist two constants A and B (depending on C and ω) such that for all j ∈ N and λ > 0,

j λ ∗ B A j ∗ ϕ j . C exp λϕ ≤ exp λ exp λ B B λ Proof. From de definition of Young conjugate we get

∗ j exp λϕ ( ) = sup sj exp (−λω(s)) . λ s≥1

(2)

Choose l ∈ N such that 2l ≥ C. By condition (α), there exist A and B such that

t t l t ω(t) = ω C ≤ω 2 ≤ A + Bω . (3) C C C Then,

(2) j ∗ = sup(sC)j exp (−λω(s)) C exp λϕ λ s≥1

t j = sup t exp −λω C t≥C

(3) A ω(t) ≤ sup tj exp λ − B B t≥1 j

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A 1 j = exp λ sup t exp −λ ω(t) B t≥1 B

(2) A λ ∗ B = exp λ ϕ j . exp B B λ

Proof of Theorem 4.1. We set P (z) = |α|≤m aα z , P (D) = |α|≤m aα Dα and M := max{|aα | : |α| ≤ m}. Choose F large enough such that for each j ∈ N, M j (mj)N ≤ F j and take the constants A and B of Lema 4.2 such that for each j ∈ N,

j λ ∗ B A j ∗ ϕ j . (4) F exp λϕ ≤ exp λ exp λ B B λ To see E(ω(t)) (Ω) ⊆ E

1 (Ω), P, ω t m

α

take f ∈ E(ω(t)) (Ω) and we ﬁx K ⊂⊂

Ω, λ > 0 and j ≥ 1. We observe that the polynomial of the operator P j (D) · · · P has degree mj. Moreover, there exists C > 0 such that for is P j = P j

each α ∈ NN 0 ,

|α| f (α) 2,K ≤ C exp Bλϕ∗ . Bλ

We can choose C = pK,Bλ (f ). Hence, P (D)f 2,K ≤ M pK,Bλ (f ) j

j

(5)

∗

exp Bλϕ

|γ|≤mj

|γ| Bλ

mj ≤ pK,Bλ (f )M j (mj)N exp Bλϕ∗ Bλ

mj ≤ pK,Bλ (f )F j exp Bλϕ∗ Bλ

(4) mj ∗ ≤ pK,Bλ (f ) exp(λA) exp λϕ . λ Therefore,

P (D)f 2,K exp −λϕ j

∗

mj λ

≤ pK,Bλ (f ) exp(λA).

(6)

1 It has been proved E(ω(t)) (Ω) ⊆ E (Ω) and this inclusion is continuP,(ω(t m )) ous. This settles the Beurling case. In the Roumieu case, let f ∈ E{ω(t)} (Ω) be given. For each K ⊂⊂ Ω there λ

λ (K). Proceeding as above f ∈ E B exists λ > 0 such that f ∈ Eω(t)

1 (K) P,ω t m

E

1 (Ω). P, ω t m

⊂

Now, from (6) we get the continuity of the inclusion λ

λ (K) → E B Eω(t)

1 (K) P,ω t m

and the theorem follows.

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Proposition 4.3. E∗ (Ω) is a dense subspace of E

1 (Ω). P,∗ t m

Proof. First, we suppose Ω = RN . Beurling case. Applying Lema 2.1 there exist L ≥ 1 and y0 > 0 such that y − L for each y ≥ y0 . ϕ∗ (y) − y ≥ Lϕ∗ L This implies exp −λϕ∗ mj λ −→ 0 if j → +∞. (7) exp −λLϕ∗ mj λL Using Theorem 4.1 D(ω) (RN ) → E(ω) (RN ) → E

1 (RN ). P, ω t m

Let f ∈ EP,(ω) (RN ) be given. Proceeding as in [4, Lema 3.8] we take a regularizing sequence {ρn }n∈N in D(ω) (RN ) and prove that f ∗ ρn ∈ E(ω) (RN ). To get this aim we ﬁx a compact subset K of RN and λ > 0, then

|α|

(α) ∗ sup (f ∗ ρn ) exp −λϕ λ 2,K α∈NN 0

|α|

exp −λϕ∗ = sup f ∗ ρ(α) n λ 2,K N α∈N0

|α|

(α) f (· − y)ρn (y)dy exp −λϕ∗ = sup

λ N α∈N0

suppρn 2,K

|α|

(α) ∗ ≤ sup sup ρn (y) exp −λϕ |f (· − y)|dy

λ N y∈ supp ρ n α∈N0

suppρn 2,K

< ∞.

Now, it is enough to show that for each compact subset K of RN , λ > 0 and p∈N f − f ∗ ρn K,λ −→ 0

if n → ∞.

Given η > 0, using (7) and (6) there exists j0 ∈ N0 such that

mj P j (D)(f − f ∗ ρn )2,K exp −λϕ∗ λ

−λϕ∗ ( mj ) λ mi e i ∗ ≤ sup P (D)(f − f ∗ ρn )2,K exp −Lλϕ ∗ mj −Lλϕ ( Lλ ) Lλ i∈N0 e ≤ (f K,Lλ + pK,LλB (f ∗ ρn ))

∗ mj e−λϕ ( λ ) <η ∗ mj e−Lλϕ ( Lλ )

if j > j0 .

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As a consequence, f − f ∗ ρn K,λ ≤

max

i=0,1,...,j0

IEOT

∗ mi η, P i (D)(f − f ∗ ρn )2,K e−λϕ ( λ ) .

Hence f − f ∗ ρn K,λ ≤ η Roumieu case. Let f ∈ E

if n is large enough.

1 (R P, ω t m

N

). We take {ρn }n∈N ∈ D(ω) (RN ) as

above and ﬁx a compact subset K of RN . Since f ∈ E λ (K). Proceeding as exists λ > 0 such that f ∈ EP,ω λ

1 (RN ), there P, ω m λ above f ∗ ρn ∈ EP,ω (K)

L and f ∗ ρn converges to f in EP,ω (K). As a consequence, f ∗ ρn converges to f in E m1 (RN ).

P, ω

To ﬁnish, we suppose that Ω is an arbitrary open subset of RN . We ﬁx a compact subset K and λ > 0 and we choose a compact subset L such that ◦ K ⊂ L ⊂ L ⊂ Ω. Let f ∈ E 1 (Ω). We deﬁne f = f on L and f = 0 in P, ω t m

and according to [4, Proposition 6.4] ρn ∗ f other case. Then f ∈ is an ultradifferentiable function which coincides in f ∗ ρn on K if n is large enough. Then, given ε > 0 there exists n0 such that f −ρn0 ∗ fK,λ < ε.From this the conclusion follows. (RN ) D(ω)

1.

As a consequence of the former results we get: D∗ (Ω) is a dense subset of E m1 (Ω).

2.

The functions in of

P,∗ t 1 (Ω) with E P,∗ t m

compact support are a dense subset

E m1 (Ω). P,∗ t

According to a well known result of H¨ ormander (see [7, Theorem 3.2]), if P is an hypoelliptic polynomial and Q any polynomial, there are constants h > 0 and C > 0 such that |Q(ξ)|2 ≤ C(1 + |P (ξ)|2 )h ,

∀ξ ∈ RN .

Moreover, the smaller h is a rational number. For an hypoelliptic polynomial P and an arbitrary polynomial Q we show the equivalence between the inequality |Q(ξ)|2 ≤ C(1 + |P (ξ)|2 )h and 1 (Ω) for weight functions satisfying the folthe inclusion EP,∗(t) (Ω) ⊆ E Q,∗(t h ) lowing growth condition B-M-M: There exits a constant H ≥ 1 such that for all t ≥ 0 2ω(t) ≤ ω(Ht) + H.

(8)

This condition is considered by Bonet et al. [2] in order to characterize those weight functions ω such that there exists a sequence {Mp } with the property that the class of ultradifferentiable functions in the sense of Braun–Meise– Taylor associated to the weight ω coincides with the non-quasianalytic class in the sense of Komatsu (see [10]) deﬁned by the sequence {Mp }. Gevrey weights verify this condition.

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We will prove the following theorems. Theorem 4.4. Let P and Q be hypoelliptic polynomials with h > 0 and C > 0 such that |Q(ξ)|2 ≤ C(1 + |P (ξ)|2 )h , ∀ξ ∈ RN . Let Ω ⊂ RN be an open subset and ω a weight function, then there exits m0 such that if m ≥ m0 E

1 (Ω) P,∗ t m

⊆E

1 (Ω), Q,∗ t mh

and the inclusion map is continuous. Theorem 4.5. Let Ω ⊂ RN be an open subset, ω a weight function satisfying the condition B-M-M, cf. (8). Let P be an hypoelliptic polynomial and let Q be an arbitrary polynomial such that EP,∗(t) (Ω) ⊆ E h1 (Ω) for some Q,∗ t

h ≥ 1. Then |Q(ξ)|2 ≤ C(1 + |P (ξ)|2 )h ,

∀ξ ∈ RN .

In the proofs we need two technical lemmata. Lemma 4.6. Let ω be a weight function, m ≥ 1 and γ,μ > 0 such that γ ≤ μm. Then for each k ∈ N, i = 0, 1, . . . , k and λ > 0,

(k − i)μm kμm k iγ exp λϕ∗ ≤ exp (λω(k)) exp λϕ∗ . λ λ As a consequence, since ω(t) = o(t) there exists C > 0 (depending on ω) such that

(k − i)μm kμm iγ ∗ ∗ ≤ C exp(λCk) exp λϕ . k exp λϕ λ λ Proof.

(k − i)μm kμm ∗ k exp λϕ exp −λϕ λ λ

kμm (k − i)μm iγ ∗ ∗ = k exp −λ ϕ −ϕ . λ λ iγ

∗

Since ϕ∗ is a convex function, we have ϕ∗ (A) − ϕ∗ (B) ≥ ϕ∗ (A − B) if 0 ≤ B < A. Therefore,

(k − i)μm kμm iγ ∗ ∗ k exp λϕ exp −λϕ λ λ is less than or equal to

iμm iμms iμm ln(k) − sup − ω(es ) = exp λ k iμm exp −λϕ∗ λ λ λ s≥0 ≤ exp (λω(k)) taking s = ln(k) in the last inequality.

The next lemma is stated in [5, 1.4] without a proof. We include a proof here for the sake of completeness.

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Lemma 4.7. Let h > 0,λ > 0 be positive constants, then for all t ≥ 1

1 hj (1) sup tj exp −λϕ∗ ≤ exp λω t h λ j∈N0 and

1 hj 1 ∗ . (2) sup t exp −λϕ ≥ exp λω t h λ t j∈N0 j

Proof. Proof of (1):

hj hjs sup tj exp −λϕ∗ − ω(es ) = sup tj exp −λ sup λ λ s≥0 j∈N0 j∈N0 We take s = ln(t) h ≥ 0, therefore

1 1 hj = exp λω t h . sup tj exp −λϕ∗ ≤ sup tj t−j exp λω t h λ j∈N0 j∈N0 In order to show (2), having in mind j ≤ s < j + 1:

j hj hj j ∗ ∗ sup t exp −λϕ ln(t) − ϕ = sup exp λ λ λ λ j∈N0 j∈N0

s hs ∗ ln(t) − ϕ ≥ sup exp λ − ln(t) λ λ s≥0

ln(t) 1 = exp λϕ∗∗ t h

1 1 = exp λω t h . t Proof of Theorem 4.4. Using [7, Theorem 3.2] we can suppose h = μ, ν ∈ N. Then, for some constant C > 0, |Qν (ξ)|2 ≤ C(1 + |P μ (ξ)|2 )

μ ν,

where

ξ ∈ RN .

(9)

Let Ω be an open subset relatively compact in Ω. Given δ > 0, we set Ωδ := {x ∈ Ω : d(x, ∂Ω ) > δ} where ∂Ω is the boundary of Ω . The condition (9) and [7, Theorem 4.2] imply that there exist γ > 0 and C (which depends of P ,Q and the diameter of Ω ) such that for each s ≥ 0 and t > 0, γ ν γ μ sup τ Q (D)f 2,Ωs+τ ≤ C sup τ P (D)f 2,Ωs+τ + f 2,Ωs , 0<τ ≤t

0<τ ≤t

f ∈ C ∞ (Ω). where 0 0. Applying repeatedly the last inequality to s = ki ∈ N, kδ ≥ 1, δ(k−i)ν f for i = 0, 1, . . . , k we obtain (see [19, Theorem 1]) 1− k δ, t = k and Q k k k iγ Qkν (D)f 2,Ω2δ ≤ C k P (k−i)μ (D)f 2,Ω f ∈ C ∞ (Ω). (10) i δ i=0

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For k = 0 this inequality remains true. On the other hand, for i = 0, 1, . . . , ν − 1 we have |Qi (ξ)|2 ≤ 1 + |Qν (ξ)|2 ,

ξ ∈ RN .

Again, [7, Theorem 4.2] implies ∀i = 0, 1, . . . , ν − 1, Qi (D)f 2,Ω2δ ≤ C Qν (D)f 2,Ωδ + f 2,Ωδ ,

f ∈ C ∞ (Ω).

(11)

where C depends of δ. For j ∈ N0 , we put j = kν + l, k,l ∈ N0 , l ≤ ν − 1. Applying (11) to i = l and Qkν f , Qj (D)f 2,Ω2δ ≤ C Q(k+1)ν (D)f 2,Ωδ +Qkν (D)f 2,Ωδ , f ∈ C ∞ (Ω). (12) As a consequence, using (10) we obtain k+1 k + 1 k + 1 iγ P (k+1−i)μ (D)f 2,Ω Qj (D)f 2,Ω2δ ≤ C C k+1 i δ i=0 k iγ k k + C Ck P (k−i)μ (D)f 2,Ω , f ∈ C ∞ (Ω). i δ i=0 (13) We set m0 := 1b ≥ 1. Let Ω be an open subset of RN and let m ≥ m0 , f ∈ E m1 (Ω). We ﬁx K ⊂⊂ Ω and λ > 0. There exist Ω and δ > 0 P, ω t

such that K ⊂ Ω2δ ⊂ Ω ⊂⊂ Ω. We call E the constant of the Lemma 4.6 such that for each k ∈ N, i = 0, 1, . . . , k and λ > 0,

(k − i)μm kμm iγ ∗ ∗ k exp λϕ ≤ exp(λEk) exp λϕ . λ λ γ Denote F := 2C 1δ eλE . We can suppose F ≥ 1 and 1δ > 1. We take A and B the constants of Lemma 4.2 such that ∀k ∈ N and ∀λ > 0,

k λ ∗ B A k ∗ ϕ k . F exp λϕ ≤ exp λ exp λ B B λ We apply the inequality (13) and f ∈ E

1 (Ω) P, ω t m

to get

Qj (D)f 2,K ≤ Qj (D)2,Ω 2δ

iγ

k+1 k+1 k+1 (k + 1 − i)μm k+1 p ∗ ≤C C exp 2 Bλϕ i δ 2p Bλ i=0

k iγ k k (k − i)μm k ∗ exp Bλϕ +C C i δ Bλ i=0 when p is the ﬁrst entire after μm. Therefore, Qj (D)f 2,K

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is less than or equal to

k+1 k + 1 k + 1 iγ (k − i)μm + p exp 2p Bλϕ∗ C C k+1 i δ 2p Bλ i=0

k iγ k k (k − i)μm exp Bλϕ∗ + C Ck i δ Bλ i=0 since ϕ∗ is increasing. Now, using Lemmas 2.2 and 4.6 we have γ

k

1 kμm Qj (D)f 2,K ≤ D 2C eBλE exp Bλϕ∗ δ Bλ

kμm = DF k exp Bλϕ∗ . Bλ In view of Lemma 4.2,

kμm A Qj (D)f 2,K ≤ D exp Bλ exp λϕ∗ B λ

khνm = D exp(Aλ) exp λϕ∗ λ

jhm ≤ D exp(Aλ) exp λϕ∗ . λ

As a consequence for each j ∈ N,

jhm j ∗ Q (D)f 2,K exp −λϕ ≤ D exp(Aλ). λ Then, f ∈E

1 (Ω). Q, ω t mh

In order to see the inclusion map E

1 (Ω) P, ω t m

is continuous, let f ∈ E

→ E

1 (Ω) P, ω t m

1 (Ω) Q, ω t mh

and we ﬁx K ⊂⊂ Ω, λ > 0. Proceeding

as above:

jhm Qj (D)f 2,K exp −λϕ∗ λ

iγ

k+1 k+1 k+1 jhm exp −λϕ∗ ≤ C C k+1 P (k+1−i)μ (D)f 2,Ω i δ λ i=0

k iγ k k jhm k ∗ +C C exp −λϕ P (k−i)μ (D)f 2,Ω . i δ λ i=0

Note by Lemma 4.6, for each k ∈ N, i = 0, 1, . . . , k and λ > 0,

jhm (k − i)μm k iγ exp −λϕ∗ ≤ exp(λEk) exp −λϕ∗ . λ λ

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So, for each j ∈ N,

Q (D)f 2,K exp −λϕ j

∗

jhm λ

≤ DF sup P (D)f 2,Ω exp −λϕ k

l

∗

l∈N0

Hence,

281

lm λ

.

jhmB λ Qj (D)f 2,K exp − ϕ∗ B λ

lm A ≤ D exp λ sup P l (D)f 2,Ω exp −λϕ∗ . B l∈N0 λ

Proceeding as in Theorem 4.1, the Roumieu case is analogous.

Proof of Theorem 4.5. Roumieu Case. We ﬁx a compact subset K0 ⊂⊂ Ω. The following inclusions hold: EP,(ω(t)) (Ω) ⊆ EP,{ω(t)} (Ω) ⊆ E

1 (Ω) Q, ω t h

1

⊆ ind E n 1 (K0 ). −→ Q,ω t h n∈N

From Theorem 3.3 we get that EP,(ω(t)) (Ω) is a Fr´echet space. Now we con1

sider on E n

1 (K0 ) Q,ω t h

1

the topology of Theorem 3.5, so that ind E n 1 (K0 ) −→ Q,ω t h n∈N

is an (LF)-space. By Closed Graph Theorem and Grothendieck’s Factorization Theorem (see [16, Theorems 24.31 and 24.33]), there exists n0 ∈ N such that 1

EP,(ω(t)) (Ω) ⊆ E n0

1 (K0 ) Q,ω t h

with continuous inclusion. So, given any seminorm max pm , · Q,K0 , n1 , 0

1

of E n0

1 (K0 ), Q,ω t h

there exist C > 0, a compact K ⊂⊂ Ω, p ∈ N0 and λ > 0

such that for all f ∈ EP,(ω(t)) (Ω),

1 sup Qj (D)f 2,K0 exp − ϕ∗ (hjn0 ) ≤ max pm , f Q,K0 , n1 0 n0 j∈N0

j ≤ C sup P j (D)f 2,K exp −λϕ∗ . (14) λ j∈N0 If ξ ∈ RN , we denote fξ (x) = ei<x,ξ> and observe that Qj (D)ei<x,ξ> = Q(ξ)j ei<x,ξ>

and

fξα (x) = ξ α iα ei<x,ξ> .

Moreover, fξ ∈ EP,(ω) (Ω) since for a compact subset K ⊂⊂ Ω and λ > 0

j j j ∗ P (D)fξ 2,K = m(K)|P (ξ)| ≤ C exp λϕ λ by Lemma 4.2.

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Applying inequality (14) to fξ we get

j 1 ∗ j j ∗ sup |Q(ξ)| exp − ϕ (hjn0 ) ≤ C2 sup |P (ξ)| exp −λϕ . n0 λ j∈N0 j∈N0

(15)

For |Q(ξ)| and |P (ξ)| greater or equal than 1 we obtain from (15) and Lemma 4.7,

1 1 ω |Q(ξ)| h ≤ C2 exp (λω(|P (ξ)|)) . exp 2n0 Hence,

1 ω |Q(ξ)| h ≤ C3 + C4 ω(|P (ξ)|) ≤ C5 ω(|P (ξ)|)

whenever |P (ξ)|, |Q(ξ)| ≥ 1. On the other hand, condition B-M-M implies that for each k ∈ N there exists Hk such that 2k−1 ω(t) ≤ ω(Hk t) whenever t ≥ 1. Then, 1 ω |Q(ξ)| h ≤ ω(C6 |P (ξ)|) whenever |P (ξ)|, |Q(ξ)| ≥ 1. Having in mind ω vanishes on [0, 1] and it is increasing and |P | tends to +∞ if |ξ| tends to +∞ we ﬁnally conclude that there is C7 > 0 such that, 1

|Q(ξ)| h ≤ C7 |P (ξ)|

for every ξ ∈ RN .

The Beurling case is easier because EP,(ω) (Ω) is a metrizable space. ses E

Next we show that for any elliptic polynomial P of degree m, the clas 1 (Ω) and E ,∗(t) (Ω) are the same as sets and as topological vector m

P,∗ t

spaces. This is an extension of a result of Komatsu (see [9]). The converse in the Gevrey setting is due to M´etivier [17]. We need the following lemma due to Komatsu (see [9, Lemma 3]). Lemma 4.8. Let Ω be an open subset of RN . Suppose that P is an elliptic operator of order m and let ρ0 > 0 be given. Then, there exists a constant C > 0, which only depends on N, ρ0 and P, such that for each f ∈ C ∞ (Ω) and for each α ∈ N0 N verifying |α| ≤ m, f (α) 2,Ωρ+σ ≤ CP (D)f 2,Ωσ

|α| m

f 2,Ωσ

1− |α| m

+C

1 f 2,Ωσ ρ|α|

for every 0 < ρ ≤ ρ0 and σ > 0. Theorem 4.9. Let ω be a weight function and let Ω be an open subset of RN . For any elliptic polynomial P of degree m we have E

1 (Ω) P,∗ t m

⊆ E∗(t) (Ω)

and the inclusion map is continuous. Proof. Let Ω be an open subset relatively compact in Ω. We ﬁrst estimate the derivatives f (α) of order |α| = km, k ∈ N0 . We write α = p + q with

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|p| = m and |q| = (k − 1)m. Using Lemma 4.8 with ρ = kδ and σ = δ 1 − k1 , δ small enough, we get a constant C > 0 such that ∀f ∈ C ∞ (Ω ), (p) 2,Ωδ f (α) 2,Ωδ = f (q) m k (q) + ≤ C P (D)f (q) 2,Ω f . 2,Ω δ (1− 1 ) δ (1− 1 ) δ k k Applying this lemma k times as is (10) we obtain k im k k (α) k P (k−i) (D)f 2,Ω f 2,Ωδ ≤ C i δ i=0

(16)

whenever |α| = km. In the case that α ∈ N0 N is an arbitrary multi-index we write α = β + γ, where |β| = km and |γ| < m. We observe that xa y 1−a ≤ x + y holds if x, y ≥ 0 and 0 < a < 1. Therefore, P (D)f 2,Ωδ

|γ| m

f 2,Ωδ

1− |γ| m

≤ P (D)f 2,Ωδ + f 2,Ωδ .

Hence it follows from Lemma 4.8 that there is a constant C such that f (γ) 2,Ω2δ ≤ C (P (D)f 2,Ωδ + f 2,Ωδ )

∀f ∈ C ∞ (Ωδ ) y ∀|γ| < m. (17)

C depends of δ. We apply (17) to f (β) to obtain f (α) 2,Ω2δ ≤ C (P (D)f (β) 2,Ωδ + f (β) 2,Ωδ ).

(18)

Now the inequality (16) implies P (D)f (β) 2,Ωδ = (P (D)f )(β) 2,Ωδ k im k k k ≤C P (k+1−i) (D)f 2,Ω i δ i=0 k+1 k + 1 k + 1 im ≤ C k+1 P (k+1−i) (D)f 2,Ω . i δ i=0 From this inequality, (18) and (16) we conclude k+1 k + 1 k + 1 im f (α) 2,Ω2δ ≤ C C k+1 P (k+1−i) (D)f 2,Ω i δ i=0 k im k k + C C k P (k−i) (D)f 2,Ω . i δ i=0 As γ ≤ μm, we can use Lemma 4.7, proceeding as in Theorem 4.4 with 1 and γ = μb = m, to conclude that h = μν = m E

1 (Ω) P,∗ t m

with continuous inclusion.

⊆ E∗(t) (Ω)

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Corollary 4.10. Let ω be a weight function and let Ω be an open subset of RN and P an elliptic polynomial. Then the equality E m1 (Ω) = E∗(t) (Ω) P,∗ t

holds.

Proof. It is a consequence of Theorems 4.1 and 4.9. Finally, we characterize when E

1 (Ω) P,∗ t m

and E∗(t) (Ω) coincide for

weight functions verifying the growth condition B-M-M. Lemma 4.11. If E

1 (Ω) P,∗ t m

= E∗(t) (Ω) algebraically, then P is hypoelliptic

and the previous equality also holds in the topological sense. Proof. We will the proof in the Beurling case and for Ω = RN . We present 1 put σ(t) = ω t m . Since {f ∈ C ∞ (RN ) : P (D)f = 0} ⊂ EP,(σ) (RN ), our hypothesis implies that {f ∈ C ∞ (RN ) : P (D)f = 0} = {f ∈ E(ω) (RN ) : P (D)f = 0}. We ﬁx a compact subset K ⊂ RN and λ > 0. From the Open Mapping Theorem we deduce that there are a constant C > 0, m ∈ N and a compact set Q such that pK,λ (f ) ≤ C sup sup |f (α) (x)|

(19)

|α|≤m x∈Q

whenever f ∈ C ∞ (RN ) and P (D)f = 0. We now assume that P is not hypoelliptic. Then we can apply theorems [8, 11.1.5 and 10.1.25] to ﬁnd a function f ∈ C m (RN )\C ∞ (RN ) such that P (D) = 0. We take {ρn } a regularizing sequence. Then {f ∗ ρn } is a Cauchy sequence in C m (RN ) and from inequality (19) we conclude that also {f ∗ ρn } is a Cauchy sequence in E(ω) (RN ). This is a contradiction. Theorem 4.12. Let ω be a weight function verifying condition B-M-M. Suppose P is polynomial which degree is m. Then E m1 (Ω) = E∗(t) (Ω) P,∗ t

holds algebraically if and only if P is elliptic. In this case the equality E m1 (Ω) = E∗(t) (Ω) is also topological. P,∗ t

1 (Ω) = E∗(t) (Ω) by Theorem 4.9. In order Proof. If P is elliptic then E P,∗(t m ) to show the converse, we apply once again Theorem 4.9 to the elliptic poly2 nomial Q(ξ) = ξ12 + · · · ξN to get

E

1 (Ω) P,∗ t m

= E∗(t) (Ω) = E

1 (Ω). 2 ,∗ t 2 ξ12 +···+ξN

Moreover, according to Lemma 4.11, P is hypoelliptic and we can proceed as in Theorem 4.5 to deduce 1

1

2 2 ) ≤ C|P (ξ)| m (ξ12 + · · · + ξN

if |ξ| is large enough.

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That is, for some constant C > 0, 2 m ) ≤ C(1 + |P (ξ)|2 ). (ξ12 + · · · + ξN

Acknowledgments This article will be part of the PhD. Thesis of the author, which is supervised by Jos´e Bonet and Antonio Galbis. He wants to thank them, as well as Leonhard Frerick, for their suggestions and constant help.

References [1] Bolley, P., Camus, J., Rodino, L.: Analytic Gevrey hypoellipticity and iterates of operators. Rend. Sem. Mat. Univ. Politec. Torino. 45(3), 1–61 (1987) [2] Bonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways of deﬁne classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007) [3] Bouzar, C., Chaili, L.: A Gevrey microlocal analysis of multi-anisotropic differential operators. Rend. Semin. Mat. Univ. Politec. Torino 64(3), 305–317 (2006) [4] Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Result. Math. 17, 206–237 (1990) [5] Fern´ andez, C., Galbis, A., Jornet, D.: Pseudodifferential operators on nonquasianalytic classes of Beurling type. Studia Math. 167(2), 99–131 (2005) [6] Folland, G.B.: Introducction to Partial Differential Operators. Princeton University Press, Princeton (1995) [7] H¨ ormander, L.: On interior regularity of the solutions of partial differential equations. Comm. Pure Appl. Math. 11, 197–218 (1958) [8] H¨ ormander, L.: The Analysis of Linear Partial Differential Operators II. Springer-Verlag, Berlin (1990) [9] Komatsu, H.: A characterization of real analytic functions. Proc. Jpn. Acad. 36, 90–93 (1960) [10] Komatsu, H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Tokyo Sec. IA 20, 25–105 (1973) [11] Kotake, T., Narasimhan, M.S.: Regularitty theorems for fractional powers of a linear elliptic operator. Bull. Soc. Math. France 90, 449–471 (1962) [12] Langenbruch, M.: P -Funktionale und Randwerte zu hypoelliptischen Differentialoperatoren. Math. Ann. 239(1), 55–74 (1979) [13] Langenbruch, M.: Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partielle Differentialgleichungen. J. Reine Angew. Math. 311/312, 57–79 (1979) [14] Langenbruch, M.: On the functional dimension of solution spaces of hypoelliptic partial differential operators. Math. Ann. 272, 217–229 (1985) [15] Langenbruch, M.: Bases in solution sheaves of systems of partial differential equations. J. Reine Angew. Math. 373, 1–36 (1987) [16] Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)

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[17] M´etivier, G.: Propi´et´e des it´er´es et ellipticit´e. Comm. Partial Differ. Equ. 9(3), 827–876 (1978) [18] Mujica, J.: A completeness criterion for inductive limits of Banach spaces. In: Zapata, G.I. (ed.) Functional Analysis, Holomorphy and Approximation Theory II. North-Holland Mathematics Studies (1984) [19] Newberger, E., Zielezny, Z.: The growth of hypoelliptic polynomials and Gevrey clases. Proc. Am. Math. Soc. 39(3), 547–552 (1973) [20] P´erez-Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces. Noth-Holland Mathematics Studies, vol. 131 (1987) [21] Wengenroth, J.: Derived functors in funcional analysis. In: Lecture Notes in Mathematics, vol. 1810. Springer-Verlag, Berlin (2003) [22] Zanghirati, L.: Iterates of a class of hypoelliptic operators and generalized Gevrey clases. Suppl B.U.M.I 1, 177–195 (1980) [23] Zanghirati, L.: Iterates of quasielliptic operators and Gevrey classes. Boll. U.M.I. 18B, 411–428 (1981) [24] Zanghirati, L.: Iterati di operatori e regolarit` a Gevrey microlocale anisotropa. Rend. Sem. Mat. Univ. Padova 67 (1982) Jordi Juan-Huguet (B) Instituto de Matem´ atica Pura y Aplicada IUMPA Universidad Polit´ecnica de Valencia 46071 Valencia, Spain e-mail: [email protected] Received: December 6, 2009. Revised: June 16, 2010.

Integr. Equ. Oper. Theory 68 (2010), 287–299 DOI 10.1007/s00020-010-1817-4 Published online July 17, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Convolution Operators on Banach Lattices with Shift-Invariant Norms Nazar Miheisi Abstract. Let G be a locally compact abelian group and let µ be a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in Johansson (Syst Control Lett 57:105–111, 2008). We use Laplace transform methods to show that the norm of a convolution operator with symbol µ on such a space is bounded below by the L∞ norm of the Fourier–Stieltjes transform of µ. We also show that for any Banach lattice of locally integrable functions on G with a shift-invariant norm, the norm of a convolution operator with symbol µ is bounded above by the total variation of µ. Mathematics Subject Classification (2010). Primary 47A30, 47B38; Secondary 43A15. Keywords. Convolution operator, shift-invariant norm, laplace transform.

1. Introduction Let G be a locally compact abelian (LCA) group. Let M (G) denote the algebra of complex valued regular Borel measures on G of bounded variation. It is well known that for Lp (G), a convolution operator with symbol μ ∈ M (G) (i.e an operator f → f ∗ μ on Lp (G)) is bounded with norm less than or equal the dual group of G to |μ|(G), with equality when p = 1 or ∞. Denote by G (the group of continuous homomorphisms G → T, where T denotes the circle group). It is also known that the norm of a convolution operator on Lp (G) is μ(γ)|, where μ ˆ (= Fμ) denotes the greater than or equal to |ˆ μ|G = supγ∈G |ˆ Fourier–Stieltjes transform of μ. In this case, there is equality when p = 2. For a comprehensive treatment of harmonic analysis on LCA groups see [5] or [8] for example. In [6], a class of function spaces deﬁned on R+ —which we call Johansson spaces—was introduced and it was shown that the norms of convolution operators on these spaces has the same upper bound as for the Lp -spaces. We will describe these spaces in detail, and in the more general setting of

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LCA groups in Sect. 2. Indeed, this upper bound holds for any function space deﬁned on a discrete group provided that the norm is shift-invariant—that is, f (·) = f (· − y) for ﬁxed y ∈ G—as is easily veriﬁed. Given this, it was conjectured that it holds for any space of locally integrable functions deﬁned on R [6], and then subsequently for any LCA group. In Sect. 2.1, we introduce Johansson spaces deﬁned on LCA groups and give some basic results regarding the structure of these spaces. Then in Sect. 2.2 we will restrict the domain of definition to being the positive cone (i.e all x ≥ 0) of an Archimedean ordered group. This allows us to use Laplace transform methods to obtain a lower bound for the norms of convolution operators on these spaces, which is the same as the lower bound for the Lp -spaces. In Sect. 3 we consider a much larger class of Banach lattices, which include both the Lp -spaces and Johansson spaces. We will show that provided the norm is shift-invariant, the norms of convolution operators on these spaces have the same upper bound as the Lp -spaces, thus answering the earlier conjecture positively for Banach lattices. 1.1. Notation Throughout this paper G will denote a σ-compact LCA group with identity We write f for f dm. We denote 0, Haar measure m and dual group G. the characteristic function of E ⊂ G by χE . That is, 1 if x ∈ E χE (x) = 0 if x ∈ / E. M (G) denotes the algebra of complex valued regular Borel measures on G and L1loc (G) denotes the Fr´echet space (cf. Lemma 10) of complex valued measurable functions on G such that |f | < ∞ K

for every compact K ⊂ G. The topology on L1loc (G) is that of L1 -convergence on compact subsets. For f ∈ L1loc (G) and μ ∈ M (G) the convolution product f ∗ μ of f and μ is deﬁned by the formula (f ∗ μ)(x) = f (x − y) dμ(y) (x ∈ G). G

In addition, we deﬁne the shift operator Sy : L1loc (G) → L1loc (G) by Sy f (x) = f (x − y). Let X be a normed linear space. By X ∗ and X[1] we denote its topological dual and closed unit ball respectively. For x ∈ X and φ ∈ X ∗ , x, φ denotes their dual pairing (i.e φ(x)). We use the same notation for real or complex valued homomorphisms on G. For A ⊂ X, we write linA for the linear span of A. Finally, for any set Ω and any complex valued function f deﬁned on Ω, we write |f |Ω for supx∈Ω |f (x)|.

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2. Johansson Spaces 2.1. Johansson Spaces on LCA Groups Defintion 1. Let K ⊂ G be a compact neighbourhood. For 1 ≤ p < ∞, the Jop (G) consists of all functions f ∈ L1loc (G) such that the norm hansson space JK ⎛ ⎞ p1 f K = sup ⎝ |f (x − y)|p dm(y)⎠ x∈G

K

is ﬁnite. Proposition 2. Let K, K ⊂ G be compact neighbourhoods. Then · K and · K are equivalent norms. Proof. Without loss of generality we can assume both K and K to be neighbourhoods of the identity. Then K ∩ K contains an open set V , so χV ≤ χK . Since V contains the identity, the collection {V + a}a∈K gives an open cover of K . Since K is compact, there is a ﬁnite subcover {V + ai }ni=1 . Then we have n

χK ≤ χV +ai . i=1

Fix x ∈ G. Then it follows that n

p χK (y)|f (x − y)| dm(y) ≤ χV +ai (y)|f (x − y)|p dm(y) i=1 G

G

=

n

χV (y − ai )|f (x − y)|p dm(y)

i=1 G

≤

n

χK (y − ai )|f (x − y)|p dm(y)

i=1 G

=

n

χK (y)|f (x − y + ai )|p dm(y)

i=1 G

≤

n

i=1

f pK .

And hence f K ≤ n1/p f K . Swapping K and K in the preceding argument gives the result.

Given the previous result, throughout the remainder of this paper we will avoid reference to the set K and we will write J p (G) and f for p (G) and f K respectively. Proposition 2 also has the following interesting JK corollary. Corollary 3. (i) If G is compact then J p (G) = Lp (G). (ii) If G is discrete then J p (G) = ∞ (G).

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Proof. (i) Since all compact neighbourhoods K give equivalent norms, we can take K = G which gives Lp (G). (ii) If G is discrete, singleton sets are open so we can take K = {0} which gives ∞ (G). For completeness we now present some basic results regarding the structure of J p (G). Proposition 4 answers a question that was asked in [6]. Proposition 4. For 1 ≤ p < ∞, (i) J p (G) is a Banach space. (ii) If G is not compact, J p (G) is nonseparable. Proof. Let the norm on J p (G) be induced by the compact neighbourhood K. (i) Let (fn )n∈N ⊂ J p (G) be a Cauchy sequence. There is a subsequence (fnk )k∈N such that fnk+1 − fnk ≤ 2−k . For each m ∈ N, set gm =

m

|fnk+1 − fnk |,

g=

k=1

∞

|fnk+1 − fnk |.

k=1

For each gm we have gm ≤

m

fnk+1 − fnk < 1.

k=1

Fix y ∈ G. By Fatou’s lemma we see that p p lim inf gm ≤ lim inf gm ≤ lim inf gm p ≤ 1, y−K

m→∞

m→∞

m→∞

y−K

and hence g ≤ 1. Since G is σ-compact it can be covered by countably many translates of K. So if g < ∞ then g < ∞ a.e, and so the series f (x) = fn1 (x) +

∞

fnk+1 (x) − fnk (x)

k=1

converges absolutely for almost all x ∈ G. Since fn1 (x) +

k−1

fni+1 (x) − fni (x) = fnk ,

i=1

we see that f (x) = limk→∞ fnk (x) a.e. It now remains to show that f is the norm limit of the sequence (fn ). Choose ε > 0. There exists an N ∈ N such that fn − fm < ε whenever m, n ≥ N . For every m > N and x ∈ G, Fatou’s lemma shows that p |f − fm | ≤ lim inf |fnk − fm |p x−K

k→∞

x−K

≤ lim inf fnk − fm p k→∞

≤ εp .

So we conclude f ∈ J p (G) with fn → f in norm.

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(ii) If G is not compact, there is a sequence (xn )n∈N ⊂ G such that {xn − K} are pairwise disjoint. Let Ω ⊂ J p (G) be the collection of all f ∈ J p (G) such that f (G) ⊂ {0, 1}, f is constant on each xn − K and f (x) = 0 for almost all x ∈ / ∪∞ n=1 (xn − K). Clearly Ω is uncountable. For f, g ∈ Ω with f = g, there is an N ∈ N such that f = 1 − g on xN − K. From this it follows p

f − g ≥

p

|f − g| = xN −K

1 = m(K). xN −K

Thus J p (G) cannot be separable.

2.2. Johansson Spaces on Ordered Groups So far we have considered J P (G) for a general LCA group G. However, in this section we shall restrict G to be an Archimedean ordered group (for a treatment of these see [8]). In fact, we will consider Johansson spaces deﬁned only on the positive cone G+ of G, where we deﬁne G+ := {x ∈ G : x ≥ 0}. We deﬁne G− analogously. J p (G+ ) is deﬁned in precisely the same way as J p (G) but with G+ replacing G at each instance. We can consider this to be the closed subspace of J p (G) consisting of all f ∈ J p (G) with f (x) = 0 for almost all x ∈ G− \{0}. The case G = R is particularly important due to its potential applications in linear control. We will not discuss these here, but the interested reader is referred to [6]. In order to arrive at our main result about convolution operators on J p (G+ ) we will require the machinery of Laplace transforms. We now give a brief discussion of this. Let C• and R•+ denote the multiplicative groups of non-zero complex := Hom(G, C• )— numbers and positive real numbers respectively. Deﬁne G that is, G is the group of all continuous homomorphisms G → C• . Since C• = we have x, λ = x, β x, γ, R•+ ⊕ T, we see that for each x ∈ G and λ ∈ G and β ∈ Hom(G, R•+ ), and hence x, λ = x, γex,α , where where γ ∈ G α ∈ Hom(G, R) (here we are considering the additive group of real numbers). be the collection of all λ ∈ G with | x, λ| > 1 for all x ∈ Let Λ ⊂ G / G− . x,α So for λ ∈ Λ we have x, λ = x, γe with x, α > 0 for each x ∈ / G− . Choose a compact neighbourhood K ⊂ G and y ∈ G such that the interval [0, y] is contained in K. Let the norm on J p (G+ ) be induced by this set. For f ∈ J 1 (G+ ) and λ ∈ Λ,

|f (x)|| −x, λ| dm(x) =

G+

|f (x)|e−x,α dm(x)

G+

≤

∞

n=0ny−K

|f (x)|e−x,α dm(x)

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N. Miheisi ≤

∞

e−ny,α

n=0

≤

IEOT |f (x)| dm(x)

ny−K

f . 1 − e−y,α

(1)

Since J 1 (G+ ) ⊃ J 2 (G+ ) ⊃ . . . [6], we conclude that the integral (1) converges for all f ∈ J p (G+ ) and λ ∈ Λ. This allows us to make the following definition. Definition 5. For f ∈ J p (G+ ) (or μ ∈ M (G+ )) we deﬁne the Laplace transform L : J p (G+ ) → CΛ (resp. M (G+ ) → L∞ (Λ)) by Lf (λ) = fˆ(λ) = f (x) −x, λ dm(x) (λ ∈ Λ) G+

respectively,

Lμ(λ) = μ ˆ(λ) =

−x, λ dμ(x)

(λ ∈ Λ).

G+

It is straightforward to verify that the Laplace transform of μ ∈ M (G+ ) does indeed deﬁne a bounded function on Λ. For the case G = R, Λ corresponds to the right half plane C+ and this is just the ordinary Laplace transform as would be expected. The properties of the Laplace transform that we will be using are the following. Proposition 6. For f ∈ J p (G+ ), μ ∈ M (G+ ) and λ ∈ Λ, (i) L(f ∗ μ)(λ) = fˆ(λ)ˆ μ(λ). (ii) The linear functional f → fˆ(λ), J p (G+ ) → C is bounded for each λ ∈ Λ. The proof of Proposition 6(i) is identical to the one for the Fourier transform found in [8] and so we omit it here, and Proposition 6(ii) follows immediately from the previous calculation showing the convergence of the integral (1). Since the functional μ → μ ˆ(λ) is a character on the Banach algebra M (G+ ) (under convolution) for every λ ∈ Λ, we can regard Λ as a subset of Δ, where Δ denotes the maximal ideal space of M (G+ ) (for a comprehensive If we let Gμ treatment of Gelfand theory see [2]). The same is true for G. denote the Gelfand transform of μ ∈ M (G+ ), it follows that Lμ = Gμ|Λ and ˆ to denote both Lμ and Fμ. Fμ = Gμ|G , so there is no ambiguity in using μ μ|Λ . We will now show that |ˆ μ|G ≤ |ˆ μ|Λ . Proposition 7. For μ ∈ M (G), |ˆ μ|G ≤ |ˆ ⊂ Λ when considered as subsets of Δ (which Proof. First we will show that G is equipped with the relative weak∗ topology). and let α ∈Hom(G, R) be a ﬁxed homomorphism such that Take γ ∈ G

x, α > 0 for all x ∈ / G− . For each n ∈ N deﬁne λn ∈Hom(G, C• ) by 1

x, λn = x, γe n x,α .

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It is apparent that λn ∈ Λ for every n ∈ N. We claim that λn → γ in Δ. For each μ ∈ M (G+ ) we have 1 |ˆ μ(λn ) − μ ˆ(γ)| = −x, γ 1 − e− n x,α dμ(x) G 1 ≤ |1 − e− n x,α | d|μ|(x). G

Fix ε > 0. Since |μ| is ﬁnite there exists y ∈ G+ such that |μ|({x : x > y}) < ε 2 . Set ⎡ ⎤

y, α ⎥ N =⎢ ⎢ ⎥. ε ⎢ log 1 − 2|μ|([0,y]) ⎥ It is easily veriﬁed that for every n ≥ N we have 1 ε |1 − e− n x,α | < 2|μ|([0, y]) for each x ∈ [0, y]. Then for n ≥ N 1 ε |ˆ μ(λn ) − μ ˆ(γ)| < |1 − e− n x,α | d|μ|(x) + 2 [0,y]

< [0,y]

ε ε d|μ|(x) + 2|μ|([0, y]) 2

= ε. Hence λn → γ in Δ. μ|Λ = |ˆ μ|Λ , and Since Gμ ∈ C0 (Δ) ([2, page 203] for example) we have |ˆ ⊂ Λ, we conclude that |ˆ given that G μ|G ≤ |ˆ μ|Λ . In [1] it was shown that if μ m, then |ˆ μ|Λ = |ˆ μ|G . This is a consequence of the natural correspondence between the maximal ideal space However, if G is not discrete, the maximal ideal space of of L1 (G) and G. M (G) (or M (G+ )) is far more complicated and not easily characterised (see for example [9,10], or chapter 8 of [4]). We can now give the main result of this section. Theorem 8. For μ ∈ M (G+ ) define Tμ : J p (G+ ) → J p (G+ ) by Tμ f = f ∗ μ. μ|G . Then Tμ ≥ |ˆ Proof. From Proposition 6(ii) it follows that eλ : f → fˆ(λ) ∈ J p (G+ )∗ for each λ ∈ Λ. Given any f ∈ J p (G+ ) and λ ∈ Λ we have

f, Tμ∗ eλ = Tμ f, eλ = fˆ(λ)ˆ μ(λ) = μ ˆ(λ) f, eλ = f, μ ˆ(λ)eλ . Hence each μ ˆ(λ) is an eigenvalue of Tμ∗ and consequently Tμ = Tμ∗ ≥ |ˆ μ|Λ ≥ |ˆ μ|G .

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Theorem 8 was already stated in [6] for the case G = R. However, the proof contained an error: using the notation in [6], it was stated that the functions γ¯n → 0 as n → ∞, which is false. Consequently Theorem 8 corrects and extends the result. It is also worth noting that Theorem 8 is valid for any space of functions for which we can deﬁne a Laplace transform (or Fourier transform) and the functional eλ : f → fˆ(λ) is bounded for every λ.

3. Banach Function Lattices Definition 9. By a Banach function lattice (BFL) we mean a subspace X of L1loc (G) such that (X, · ) is a Banach lattice (with the obvious lattice operations) satisfying the following condition: (1) f ∗ ν ∈ X for each ν ∈ M (G) whenever f ∈ X. We do not assume that the norm topology on X is the same as the subspace topology inherited from L1loc (G). Note that Condition (1) above ensures that Sy f ∈ X for each y ∈ G whenever f ∈ X. This is because the shift operator Sy can be viewed as a convolution operator with symbol δy , where δy denotes the point mass at y. In order to prove our main result about convolution operators on BFLs, we ﬁrst need to establish some properties of the space L1loc (G). Lemma 10. L1loc (G) is a Polish space. Proof. Since G is σ-compact, there is a collection {Ki }i∈N of compact sets such that G = ∪i Ki . Let En = ∪n1 Ki . For each n ∈ N deﬁne the seminorm |f |. ρn (f ) = En

To see that

L1loc (G)

is completely metrisable, consider the metric d given by d(f, g) =

∞

2−n ρn (f − g) . 1 + ρn (f − g) n=1

It is straightforward to verify that this is a complete metric and that it induces the correct topology. Next we show separability. L1 (En ) is separable for each n ∈ N and so has a countable dense subset, An say. Let A = ∪∞ n=1 An . We claim that A is dense in L1loc (G). By the separability of L1 (En ), for each f ∈ L1loc (G) there exists a sequence (fn,k )k∈N ⊂ A such that ρn (fn,k − f ) < 1/k for every k ∈ N. Note that ρn (fn,k − f ) < 1/k implies ρ(fm,k − f ) < 1/k for all m ≤ n. Consider the sequence (fk,k )k∈N . For this sequence we have d(fk,k , f ) =

∞

2−n ρn (fk,k − f ) 1 + ρn (fk,k − f ) n=1

Vol. 68 (2010)

Convolution Operators on Banach Lattices

<

295

k ∞

2−n + 2−n k + 1 n=1 n=k+1

<

1 + k+1

∞

2−n

n=k+1

which clearly tends to 0 as k → ∞.

Lemma 11. The map y → Sy f , G → L1loc (G) is continuous for each f ∈ L1loc (G). Proof. Here it is sufﬁcient to show Syα f → f in L1loc (G) for every net (yα ) ⊂ G with yα → 0. Let U ⊂ G be a compact, symmetric neighbourhood of the the identity and K ⊂ G be an arbitrary compact set with m(K) > 0. Fix ε > 0 and f ∈ L1loc (G). Since f ∈ L1loc (G) and K + U is compact, there exists a compactly supported, continuous function g : G → C such that ε |f − g| < . 3 K+U

So for y ∈ U

|Sy f − Sy g| < K

ε . 3

Since g is compactly supported, there is a symmetric neighbourhood V of the identity such that |g(x) − g(x − w)| < ε/3m(K) whenever w ∈ V . So if y ∈ V ∩ U, ε ε = . |g − Sy g| < 3m(K) 3 K

K

From this it follows that |f − Sy f | = |f − g + g − Sy g + Sy g − Sy f | K

K

|f − g| +

≤ K

|g − Sy g| +

K

|Sy g − Sy f | < ε. K

We now require some results about BFLs. Lemma 12. Let X be a BFL. Then for K ⊂ G compact, the linear functional X → C, f → K f is bounded. Proof. For K ⊂ G deﬁne μ : X → C by μ(f ) = f. K

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Since μ is positive (i.e μ(f ) ≥ 0 whenever f ≥ 0) we have f ≥ g (equivalently f − g ≥ 0) implies μ(f − g) ≥ 0 and hence μ(f ) ≥ μ(g). We also have μ(|f |) ≥ |μ(f )|. Assume towards a contradiction that μ is not bounded. Then there exists a sequence (gn )n∈N ⊂ X such that |μ(gn )| ≥ 1 but gn < 1/n2 for 2 every n ∈ N. Let ∞fn = |gn | so fn ≥ 0, fn < 1/n and μ(fn ) ≥ |μ(gn )| ≥ 1. Let f = 1 fn which converges absolutely in X. For each N ∈ N, f≥

N

fn ,

1

and consequently μ(f ) ≥

N

μ(fn ) ≥ N,

1

which contradicts the fact that f is locally integrable.

Corollary 13. Every BFL is continuously included into L1loc (G). Proof. For a BFL X, take a sequence (fn )n∈N ⊂ X such that fn → 0. Then we have |fn | → 0 by continuity of | · |, and so by Lemma 12 |fn | → 0 K

for every compact K ⊂ G. Hence fn → 0 in L1loc (G).

Lemma 14. Let X be a separable BFL. Then the map y → Sy f , G → X is Borel measurable for every f ∈ X. Proof. Let i : X → L1loc (G) be the natural inclusion, which we know to be continuous from Corollary 13. Since i is injective, we can deﬁne an inverse on its image i−1 : i(X) → X, so for each f ∈ X we have the following commutative diagram: i

, i(X) X _?k = ?? { −1 { i ?? { { ? { { y →Sy f y →Sy f ?? {{ G Since the composition of Borel maps is Borel and y → Sy f , G → i(X) ⊂ L1loc (G) is continuous by Lemma 11, it only remains to show that i−1 is a Borel map. To show this we ﬁrst note that since X is a separable Banach space it is trivially a Polish space, as is L1loc (G) by Lemma 10. So for E ⊂ X Borel, we have (i−1 )−1 (E) = i(E) which is the continuous injective image of a Borel set between Polish spaces and thus is itself Borel [7, Theorem 15.1]. We can now prove our main result.

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Theorem 15. Let X be a BFL such that Sy f = f for each f ∈ X and y ∈ G. For μ ∈ M (G) define Tμ : X → X by Tμ f = f ∗ μ. Then Tμ is bounded and Tμ ≤ |μ|(G). Proof. First we show that Tμ is bounded. Deﬁne ⎫ ⎧ ⎬ ⎨ Γ = f → f : K ⊂ G compact ⊂ X ∗ . ⎭ ⎩ K

For each ψ ∈ Γ we have

f (x − y) dμ(y) dm(x) | Tμ f, ψ| = K G ≤ | Sy f, ψ| dμ G

≤ ψ|μ|(G)f . So ψ ◦ Tμ is bounded. The second inequality follows from the boundedness of Sy and ψ. Take a sequence (fn )n∈N ⊂ X such that fn → 0 and Tμ fn → g. By the boundedness of ψ ◦ Tμ and ψ it follows that Tμ fn , ψ → 0 and Tμ fn , ψ →

g, ψ respectively, and hence g, ψ = 0. Since Γ separates the points of X, g = 0. So by the closed graph theorem Tμ is bounded. Next we show that for X separable, Tμ ≤ |μ|(G). For f ∈ X deﬁne λf : G → X by λf (y) = Sy f which is Borel measurable by Lemma 14. For each φ ∈ X ∗ , the map y → λf (y), φ is also Borel measurable since it is the composition of Borel maps, and given that X is separable, Pettis’s measurabilty theorem [3, page 42] implies that λf is strongly μ-measurable. We also have λf dμ = f dμ = f |μ(G)| < ∞, G

G

so λf is μ-Bochner integrable. Deﬁne Rμ : X → X by Rμ f = λf dμ. G

For each f ∈ X we have Rμ f = λf dμ ≤ λf d|μ| = |μ|(G)f . G

G

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Hence Rμ ≤ |μ|(G). We claim that Rμ = Tμ . For any ψ ∈ Γ and f ∈ X we have

(Rμ − Tμ )f, ψ = λf dμ, ψ − f (x − y) dμ(y) dm(x) K G G f (x − y) dμ(y) dm(x) = λf (y), ψ dμ(y) − G

K G

= 0. The second equality followed directly from the fact that for φ ∈ X ∗ λf dμ, φ = λf , φ dμ, G

G

and the third equality followed from Fubini’s theorem. Since Γ separates the points of X we see that Rμ = Tμ and hence Tμ ≤ |μ|(G). Now we prove the general case. Let X be nonseparable. Choose f ∈ X and deﬁne M0 = lin{f } M1 = lin(M0 ∪ |M0 | ∪ Tμ M0 ) M2 = lin(M1 ∪ |M1 | ∪ Tμ M1 ) .. . where |Mk | = {|g| : g ∈ Mk } and Tμ Mk = {Tμ g : g ∈ Mk }. Set M = ∪∞ k=1 Mk . For each g ∈ M we have that g = limn gn where gn ∈ Mkn for some kn . Consequently, Tμ g = limn Tμ gn ∈ M and |g| = limn |gn | ∈ M since Tμ and | · | are continuous and M is closed. This shows that M is a Tμ -invariant Banach lattice. We claim that M is separable. Suppose that Mk is separable for some k ∈ N. Then |Mk | and Tμ Mk are separable by the continuity of | · | and Tμ , and so lin(M ∪ |Mk | ∪ Tμ Mk ) is separable. Since M0 is separable it follows that Mk is separable for every k ∈ N and hence M is separable. By considering the restriction of Tμ to M , the previous part of the proof can be applied. So for Tμ |M M : M → M we have Tμ g ≤ |μ|(G)g for each g ∈ M . In particular, Tμ f ≤ |μ|(G)f . Since f was arbitrary, this holds for every f ∈ X. Acknowledgments The author would like to gratefully acknowledge the ﬁnancial support of the UK Engineering and Physical Sciences Research Council (EPSRC), and the school of mathematics at the university of Leeds. He also wishes to thank Matthew Daws for many useful discussions and comments, as a result of which this work has been greatly improved, and Jonathan Partington for his ongoing guidance and support.

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References [1] Arens, R., Singer, I.M.: Generalised analytic functions. Trans. Am. Math. Soc. 81(2), 379–393 (1956) [2] Dales, H.G.: Banach Algebras and Automatic Continuity, London Mathematical Society. Oxford University Press, New York (2000) [3] Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977) [4] Graham, C.C., McGehee, O.C.: Essays in Commutative Harmonic Analysis. Springer, New York (1979) [5] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Springer, Berlin (1963) [6] Johansson, A.: Shift-invariant signal norms for fault detection and control. Syst. Control Lett. 57, 105–111 (2008) [7] Kechris, A.: Classical Descriptive Set Theory. Springer, New York (1995) [8] Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1962) [9] Taylor, J.L.: The structure of convolution measure algebras. Trans. Am. Math. Soc. 119(1), 150–166 (1965) [10] Taylor, J.L.: Measure Algebras. American Mathematical Society, Providence (1973) Nazar Miheisi (B) Department of Pure Mathematics University of Leeds Leeds LS2 9JT, UK e-mail: [email protected] Received: December 12, 2009. Revised: May 13, 2010.

Integr. Equ. Oper. Theory 68 (2010), 301–312 DOI 10.1007/s00020-010-1801-z Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Weighted Shift Operators Which are m-Isometries Teresa Berm´ udez, Antonio Martin´on and Emilio Negr´ın Dedicated to Professor Jos´e M. R. M´endez P´erez on occasion of his 60th birthday Abstract. We give a characterization of m-isometric operators on a separable Hilbert space. Moreover, we characterize the unilateral weighted shift operators which are m-isometries. Mathematics Subject Classification (2000). 47B37. Keywords. Isometry, m-isometry, unilateral weighted shift.

1. Introduction Agler obtained in [1] certain disconjugacy and Sturm–Lioville results for a subclass of the Toeplitz operators. These results were suggested by the study of operators T (called 2-isometry) which satisﬁes the equation I − 2T ∗ T + T ∗2 T 2 = 0. The class of m-isometric operators has been introduced in the same paper. A detailed study of m-isometries was developed by Agler and Stankus [2–4]. Throughout this paper H will denote a complex separable inﬁnite dimensional Hilbert space and (en )n≥1 an orthonormal basis of H. Let L(H) be the space of all bounded linear operators on H. Definition 1.1. Let m be a positive integer. An operator T ∈ L(H) is called an m-isometry if it satisﬁes the following identity: m m−k m (−1) (1.1) T ∗k T k = 0, k k=0

∗

where T denotes the adjoint operator of T .

302

T. Berm´ udez et al. A simple manipulation proves that (1.1) is equivalent to m m−k m (−1) T k x2 = 0, k

IEOT

(1.2)

k=0

for all x ∈ H. It is clear that 1-isometries are isometries and that any misometry is also an (m + 1)-isometry. The converse of the last property is not true (see Corollary 3.8, Remark 3.9 and [5, Proposition 8]). If (en )n≥1 is an orthonormal basis of H, then the unilateral weighted forward shift operator Sw and the unilateral weighted backward shift operator Bw on H, with weight sequence w = (wn )n≥1 ∈ ∞ , are deﬁned through the relations Sw en := wn en+1 , for n ≥ 1, and Bw en := wn en−1 , for n ≥ 2, and Bw e1 := 0. An excellent reference about the unilateral weighted shift operators is [8]. Notice that unilateral weighted backward shift operator can not be m-isometry for any positive integer number m, since Bw does not satisﬁes Eq. (1.2) for the vector e1 . Henceforth, we call unilateral weighted shift operator to the unilateral weighted forward shift operator. In this paper we give a characterization of m-isometries (Theorem 2.1). Moreover, we describe when a given unilateral weighted shift operator is an m-isometry in terms of the weights (Proposition 3.2 and Theorem 3.4); more precisely, the operator Sw is an m-isometry if and only if m−1 m−k−1 n... (n − k)...(n−m+1) |w0 · · · wk |2 k=0 (−1) k!(m−k−1)! 2 |wn | = , m−1 (n − 1 − k) (n−1)... ...(n−m) m−k−1 |w0 · · · wk |2 k=0 (−1) k!(m−k−1)! for n ≥ 1, where (n − k) denotes that the factor (n − k) is omitted. From this we obtain a result by Athavale [5]. The case m = 2 has been studied earlier by Patel [7]. Finally, we prove that the property of m-isometry is not inherit by the induced map b(T ), deﬁned on the hyperspace b(H) of all the bounded subsets of H (Example 4).

2. Preliminaries About m-Isometries For n = 0, 1, 2 . . . and k = 0, 1, 2 . . ., we denote

1 if n = 0 or k = 0, (k) n := n(n − 1)···(n − k + 1) otherwise. Given T ∈ L(H) and k = 0, 1, 2 . . ., we consider the operator βk (T ) :=

k k ∗j j 1 (−1)k−j T T . j k! j=0

Observe that if T is an m-isometry, then βk (T ) = 0, for every k ≥ m. From [2, page 388] we have that

Weighted Shift Operators Which are m-Isometries

Vol. 68 (2010)

∞

T ∗n T n =

303

n(k) βk (T )

k=0

for all n ≥ 0. Hence, if T is an m-isometry, then

T ∗n T n =

m−1

n(k) βk (T ),

k=0

for all n ≥ 0 and consequently

T n x2 =

m−1

n(k) βk (T )x, x

(2.1)

k=0

for all x ∈ H and all n ≥ 0, where ·, · denotes the inner product on H. The following property allows us to obtain an expression for the norm T n x of T n x, when T is an m-isometry, in terms of the norms of the vectors x, T x, . . . , T m−1 x. Theorem 2.1. Let T ∈ L(H). Then T is an m-isometry if and only if

n

2

T x =

m−1

m−k−1 n(n

(−1)

k=0

− 1) · · · (n − k) · · · (n − m + 1) k 2 T x , k!(m − k − 1)! (2.2)

for all n ≥ 0 and all x ∈ H. Therefore, the coefficient of T k x2 is a polynomial in n of degree m − 1, for k = 0, 1, . . . , m − 1. Proof. First, let us assume that T is an m-isometry. Notice that identity (2.2) is obvious for n < m. If n ≥ m, then by (2.1) we have that

T n x2 = =

m−1 j=0 m−1 j=0

n(j) βj (T )x, x n(j) j!

j

k=0

j−k

(−1)

j k 2 T x . k

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Developing and joining the terms T j x2 we obtain that

T n x2 = x2 + n T x2 − x2 + . . . +

m−1 2 n! T x − (m − 1)T m−2 x2 + (n−m+1)!(m − 1)!

+ (−1)m−1 x2

(−1)m−1 n(n − 1) . . . (n−m+2) n(n−1) +...+ = 1−n+ x2 + 2! (m−1)! n(n − 1) . . . (n − m + 2) + T m−1 x2 (m − 1)! =

n!m(−1)m−1 n!(m − 1)m(−1)m−1 x2 − T x2 + m!n(n − m)! m!(n − m)!(n − 1) +

=

n! T m−1 x2 (m − 1)!(n − m)(n − m + 1)

m−1 (−1)m−k−1 (m − k) n! T k x2 (n − m)! k!(m − k)!(n − k) k=0

=

m−1

m−k−1 n(n

(−1)

k=0

− 1) . . . (n − k) . . . (n − m + 1) k 2 T x . k!(m − k − 1)!

Hence the result is obtained for n ≥ m. Suppose that (2.2) holds for all n ≥ 0 and all x ∈ H. In particular, for n = m we have that equality (2.2) agrees with equality (1.2), that is m−1 (m − k) · · · 1 k 2 m(m − 1)) · · · m−k−1 T m x2 = T x (−1) k!(m − k − 1)! k=0 m−1 m−k−1 m = (−1) T k x2 . k k=0

So, the proof is completed.

3. The Unilateral Weighted Shift Operator In this section we give characterizations of the unilateral weighted shift operators which are m-isometries. Proposition 3.1. Let Sw be the unilateral weighted shift operator on H with weight sequence (wn )n≥1 . If Sw is an m-isometry, then wn = 0 for all n ≥ 1.

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Proof. Assume that there exists a positive integer n such that wn = 0. Since k en = 0 for all k ≥ 1, we obtain Sw m m k (−1)m−k en 2 = 1. Sw k k=0

This contradicts (1.2) since Sw is an m-isometry.

A simple manipulation of (1.2) gives a ﬁrst characterization of the unilateral weighted shift operators which are m-isometries. Proposition 3.2. Let Sw be the unilateral weighted shift operator on H with weight sequence (wn )n≥1 . Then Sw is an m-isometry if and only if m m (−1)m−k (3.1) |w0 · · · wn+k−1 |2 = 0 k k=0

for all n ≥ 1, where w0 := 1.

∞ Proof. Assume that Sw is an m-isometry. Let x := n=1 xn en ∈ H. Using Eq. (1.2) we have that m m−k m k (−1) x2 Sw k k=0 m ∞ m 2 m−k m 2 2 = (−1) x + (−1) |wn · · · wn+k−1 | |xn | k n=1 k=1 m ∞ m 2 m−k m 2 = (−1) x + (−1) |wn · · · wn+k−1 | |xn |2 = 0. k n=1 k=1

In particular, taking x = en and multiplying by |w0 · · · wn−1 |, we obtain m m 2 m−k m (−1) |w0 · · · wn−1 | + (−1) |w0 · · · wn+k−1 |2 = 0 k k=1

for all n ≥ 1. Thus m

m−k

(−1)

k=0

m |w0 · · · wn+k−1 |2 = 0 k

for all n ≥ 1. Suppose that equality (3.1) holds for every n ≥ 1. Taking into account the above equalities and that |w0 · · · wn−1 | = 0 by Proposition 3.1, it is clear that m m−k m k (−1) x2 = 0, Sw k k=0

hence Sw is an m-isometry.

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The following combinatorial lemma will be necessary in our proof of Theorem 3.4. We do not know if this result is published, so we have included the proof. Lemma 3.3. Let n and m be positive integers. Then ⎞ ⎛ m+1 m (n + k − 1) . . . (n + k − m) ⎠ ⎝ (n + m − h − j) (−1)k k!(m − k)!(n + k − 1 − h) j=1 k=0

= h(h − 1) . . . (h − m + 1), for all real number h. In particular, m k (n + k − 1) . . . (n + k − 1 − h) . . . (n + k − m) =0 (−1) k!(m − k)!

(3.2)

k=0

for h = 0, 1, . . . , m − 1. Proof. Denote P (h, n, m) :=

⎛

m k=0

⎞

⎜ k (n + k − 1) . . . (n + k − m) ⎝(−1) k!(m − k)!

m+1

⎟ (n + m − h − j)⎠

j=1 j=m−k+1

and Q(h, m) := h(h − 1) . . . (h − m + 1). It is clear that P (h, n, m) and Q(h, m) are polynomials in h of degree m. Moreover, for k = 0, 1 . . . , m, we have that P (n + k − 1, n, m) = Q(n + k − 1, m) = (n + k − 1) . . . (n + k − m). Hence P (h, n, m) − Q(h, m) is a polynomial in h of degree at most m with m + 1 zeros, so P (h, n, m) = Q(h, m). Fixed a sequence (wn )n≥1 , we denote, for m a positive integer number and n ≥ 0, m−1 (n − k) . . . (n − m + 1) n . . . |w0 · · · wk |2 , (3.3) Rn(m) := (−1)m−k−1 k!(m − k − 1)! k=0

where w0 := 1. Note that (m)

Rk

= |w0 · · · wk |2

(k = 0, 1 . . . , m − 1).

(3.4)

In the following result, we describe when a given unilateral weighted shift operator is an m-isometry. Theorem 3.4. Let Sw be the unilateral weighted shift operator on H with weight sequence (wn )n≥1 . Then Sw is an m-isometry if and only if |wn |2 =

(m)

Rn

(m)

Rn−1

>0

for n ≥ 1.

(3.5)

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307

Proof. Assume that Sw is an m-isometry. From Proposition 3.1 we obtain wn = 0 for any n ≥ 1. By Theorem 2.1 we have that, for every x ∈ H and for all n ≥ 0, m−1 (n − k) . . . (n − m + 1) k 2 n(n − 1) . . . n Sw x . Sw x2 = (−1)m−k−1 k!(m − k − 1)! k=0

In particular, for x = e1 we have that, |w1 · · · wn |2 =

(n − k) . . . (n − m + 1) n(n − 1) . . . |w1 · · · wk |k (−1)m−k−1 k!(m − k − 1)!

m−1 k=0

(m)

for all n ≥ 0. From equalities (3.3) and (3.4) we obtain Rn 0 and, consequently |wn |2 =

= |w0 · · · wn |2 =

(m)

Rn

(m)

Rn−1

(n ≥ 1).

Assume that (3.5) is veriﬁed. First note that (m)

|w0 · · · wn+k−1 |2 = Rn+k−1 (n ≥ 1, k ≥ 0). From Proposition 3.2, it is enough to prove that equality (3.1) is veriﬁed. For every n ≥ 1, we have that m m (m) 2 m−k m (−1) (−1) |w0 · · · wn+k−1 | = Rn+k−1 k k k=0 k=0 m m = (−1)m−k k k=0 ⎞ ⎛ m−1 (n+k − 1 − h) · · · (n+k − m) (n+k − 1) · · · m−h−1 2 (−1) |w0 · · · wh | ⎠ ×⎝ h!(m − h − 1)! m

m−k

h=0

=

m−1

(−1)m−h−1

h=0

⎛

×⎝

m k=0

=

m−1 h=0

⎛

×⎝

m−k

(−1)

⎞ m (n+k−1) · · · (n+k − 1 − h) · · · (n+k − m)⎠ |w0 · · · wh |2 h!(m − h − 1)! k

(−1)m−h−1 m! h!(m − h − 1)! m k=0

⎞ (n+k − 1−h) · · · (n+k − m) (n+k−1) · · · ⎠ |w0 · · · wh |2 =0, (−1)m−k k!(m − k)!

where the last equality is obtained from Lemma 3.3. So the result is proved.

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Remark 3.5. In the proof of Theorem 3.4 we have showed the following characterization: the unilateral weighted shift operator Sw with weight sequence (wn )n≥1 is an m-isometry if and only if Rn(m) = |w0 · · · wn |2 = 0, for every n ≥ 0. The following combinatorial lemmas are useful to give some particular cases of Theorem 3.4. Lemma 3.6. Let h be a real number and m be a positive integer. Then m−1 (h − k) · · · (h − m + 1) h · · · = 1. (3.6) (−1)m−k−1 k!(m − k − 1)! k=0

Proof. Denote R(h) be the left hand side of (3.6). Note that R(h) is a polynomial in h of degree at most m − 1. Moreover, for k ∈ {0, . . . , m − 1} we have that m−k−1 k · · · 1 0 (−1) · · · (−m + k + 1) = 1. R(k) = (−1) k!(m − k − 1)! Hence R(k) = 1 for m different values, so R(h) = 1.

Lemma 3.7. Let m be a positive integer. Then m

(−1)m−k

k=0

(m + k)! = 1. (m − k)! k!2

(3.7)

Proof. Recall that the hypergeometric function is deﬁned by F (a, b; c; z) :=

∞ (a)n (b)n n z (c)n n! n=0

where (a)0 := 1 and (a)n := a(a + 1) · · · (a + n − 1) for all positive integer n [6, page 238], and that the Legendre polynomials are given by the formula 1−z Pm (z) = F −m, m + 1; 1; 2 for m ≥ 0 [6, page 260]. Note that m

(−1)m−k

k=0

(m + k)! = (−1)m F (−m, m + 1; 1; 1) = (−1)m Pm (−1). (m − k)! k!2 (3.8) m

On the other hand, by [6, page 46], Pm (−1) = (−1) , and the conclusion is obtained. In the following corollary we obtain a result of Athavale proved in [5, Proposition 8]; that is, certain unilateral weighted shift operator associated

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to a diagonal positive kernel on the open unit disk is an (m + 1)-isometry, but not an n-isometry for every n ≤ m. Corollary 3.8. [5, Proposition 8] Let Sw (m) be the unilateral weighted shift operator on H with weight sequence n+m , (wn (m))n≥1 := n n≥1

for a given positive integer number m. Then Sw (m) is an (m + 1)-isometry, but is not an n-isometry for every n ≤ m. Proof. If m = 1, then it is easy to prove that Sw (1) is a 2-isometry and is not an isometry. Assume that m > 1. By Remark 3.5, the operator Sw (m) is an (m + 1)-isometry if and only if m m−k n · · · (n − k) · · · (n − m) |w0 · · · wk |2 = |w0 · · · wn |2 . (3.9) (−1) k!(m − k)! k=0

Taking into account Lemma 3.7 and |w0 · · · wj |2 =

j+m , j

(3.10)

we obtain equality (3.9). Moreover, if Sw (m) is an m-isometry, then m−1 m−k−1 n · · · (n − k) · · · (n − m + 1) |w0 · · · wk |2 = |w0 · · · wn |2 (−1) k!(m − k − 1)! k=0

(3.11) for all n ≥ 0. Let us prove that equality (3.11) is not true for n = m. Assume that n = m. It is clear that the left hand side of (3.11) is equal to 1 by = 1. So, the result is Lemma 3.7 and the right hand side of (3.11) is 2m m proved. Remark 3.9. Let Sw be the unilateral weighted shift with weight sequence (wn )n≥1 . 1.

From Theorem 3.4 we obtain the following characterizations: (a) Sw is an isometry if and only if |wn | = 1, for every n ≥ 1. (b) Sw is a 2-isometry if and only if |wn |2 =

n|w1 |2 − (n − 1) >0 (n − 1)|w1 |2 − (n − 2)

for n ≥ 1. Observe that n|w1 |2 − (n − 1) > 0 ⇐⇒ |w1 | ≥ 1. (n − 1)|w1 |2 − (n − 2) In particular, if w1 = 2, then Sw is the Dirichlet shift.

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Sw is a 3-isometry if and only if |wn |2 =

2.

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n(n−1) |w1 w2 |2 − n(n − 2)|w1 |2 + (n−1)(n−2) 2 2 (n−1)(n−2) 2 − (n − 1)(n − 3)|w |2 + (n−2)(n−3) |w w | 1 2 1 2 2

>0

for n ≥ 1. Theorem 3.4 assures that if Sw is an m-isometry, then the sequence of the absolute values |wn | is determined by the values |w1 |, . . . , |wm−1 |; that is, we have “certain degree of freedom” in the election of those values, but the |wn |, for n ≥ m, are determined completely. For examn|w1 |2 −(n−1) > 0 for n > 1, we ple, for any |w1 | > 1 and |wn |2 = (n−1)|w 2 1 | −(n−2) have that Sw is a 2-isometry which is not an isometry. Note that if (m) w1 = · · · = wm−1 = 1, from (3.3) and Lemma 3.6, we obtain Rn = 1, for every n ≥ 0, hence |wn | = 1 for every n ≥ 1. In particular, every operator Sw with w1 = · · · = wm−2 = 1 and wm−1 = 2, is an m-isometry by Theorem 3.4, but it is not an (m − 1)-isometry.

4. m-Isometries Do not Necessary Lift to M -Isometries on b(H) Now consider the hyperspace b(H) of all the nonempty bounded subsets of a Hilbert space H. In b(H) is deﬁned the Hausdorﬀ metric by h(C, D) := max{h (C, D), h (D, C)}

for C, D ∈ b(H),

which is a semidistance, where BH is the closed unit ball of H and h (C, D) := inf{ε > 0 : C ⊂ D + εBH } for C, D ∈ b(H). We can introduce the notion of “norm” of a set: C = sup x = h({0}, C) x∈C

for C ∈ b(H).

If T : H −→ H is a bounded and linear operator, we can deﬁne the map b(T ) : b(H) −→ b(H)C ∈ b(H) −→ b(T )C = T C. We have the following properties, for C, D ∈ b(H) and α ∈ C: b(T )(C + D) = b(T )C + b(T )D, b(T )(αC) = αb(T )C,

b(T )C ≤ T C.

That is, in certain sense, the map b(T ) is also linear and bounded. Therefore, it is natural to formulate the notion of m-isometry in this setting. We use a similar condition to the inequality (1.2) as the definition: the map b(T ) is called an m-isometry if it veriﬁes the equality m m (−1)m−k (4.1) b(T )k C2 = 0, k k=0

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for every C ∈ b(H). Note that if T is an isometry, then b(T ) is also an isometry. However, it is possible that T be a 2-isometry and b(T ) is not a 2-isometry, as we show in the next example. Example. Let T : H −→ H be the unilateral weighted shift operator deﬁned by 3n + 1 (n ≥ 1). wn := 3n − 2 The results of the previous section assures that T is a 2-isometry. If b(T ) is a 2-isometry, in the sense of the equality (4.1), then b(T )2 C2 − 2b(T )C2 + C2 = 0

(4.2)

6 10 e1

and y = e2 . Then for every C ∈ b(H). Take C = {x, y}, where x = 36 252 2 2 2 2 x2 = 100 , T x2 = 144 , T x = , y = 1, T y = 74 , T 2 y2 = 10 100 100 4 , hence 7 252 , C2 = 1, b(T )C2 = , b(T )2 C2 = 4 100 and 2 = 0. b(T )2 C2 − 2b(T )C2 + C2 = 100 That is, Eq. (4.2) is not veriﬁed and, consequently, b(T ) is not a 2-isometry. Acknowledgements The authors would like to thank the referee for helpful comments that improved the ﬁnal version of this paper.

References [1] Agler, J.: A disconjugacy theorem for Toeplitz operators. Am. J. Math. 112(1), 1–14 (1990) [2] Agler, J., Stankus, M.: m-Isometric transformations of Hilbert space. I. Integral Equ. Oper. Theory 21(4), 383–429 (1995) [3] Agler, J., Stankus, M.: m-Isometric transformations of Hilbert space. II. Integral Equ. Oper. Theory 23(1), 1–48 (1995) [4] Agler, J., Stankus, M.: m-Isometric transformations of Hilbert space. III. Integral Equ. Oper. Theory 24(4), 379–421 (1996) [5] Athavale, A.: Some operator theoretic calculus for positive definite kernels. Proc. Am. Math. Soc. 112(3), 701–708 (1991) [6] Lebedev, N.N.: Special Functions and their Applications. Dover, New York (1972) [7] Patel, S.M.: 2-Isometry operators. Glasnik Matematiˇcki 37(57), 143–147 (2002) [8] Shields A., Weighted shift operators and analytic function theory. In: Pearcy, C. (ed.) Topics in Operator Theory, pp. 49–128. Math. Surveys, vol. 13. American Mathematical Society, Providence (1974)

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Teresa Berm´ udez (B), Antonio Martin´ on and Emilio Negr´ın Departamento de An´ alisis Matem´ atico Universidad de La Laguna 38271 La Laguna (Tenerife), Spain e-mail: [email protected]; [email protected]; [email protected] Received: November 18, 2009. Revised: April 14, 2010.

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Integr. Equ. Oper. Theory 68 (2010), 313–335 DOI 10.1007/s00020-010-1830-7 Published online October 7, 2010 c The Author(s) This article is published with open access at Springerlink.com 2010

Integral Equations and Operator Theory

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The Pair of Operators T T and T T : J -Dilations and Canonical Forms Andr´e C. M. Ran and Michal Wojtylak Abstract. We describe a procedure of dilating an operator T in an inﬁnite dimensional Krein space, such that many of the spectral and algebraic [∗] [∗] properties of the operators T T and T T are preserved. We use the procedure to study canonical forms of those two operators in a ﬁnite dimensional Krein space. Mathematics Subject Classification (2010). Primary 47B50; Secondary 15B57. Keywords. Selfadjoint operator, Krein space, canonical form.

0. Introduction [∗]

[∗]

The problem of comparing the operators T T and T T in indefinite inner product spaces has already attracted some attention. One of the motivations was a result in [13] stating that a matrix T admits polar decomposition if [∗] [∗] and only if the canonical forms of T T and T T are the same. In the ﬁnite dimensional situation canonical forms of the matrices in question were considered in [9] for some special cases. Later on those results were generalized in Theorem 3.2 in [12] to provide a full description. A related result concerning an analogue of the singular value decomposition can be found in [3]. On the other hand, the inﬁnite dimensional case is far from being fully understood. For example, zero can be a singular critical point of one of the operators, while it is in the positive spectrum of the other operator. Further examples can be found in [15], where the notions of regular and singular critical point [∗] [∗] were studied for the pair T T and T T . In [14] the same pair of operators was studied in the context of local definitizability. The present paper treats both the inﬁnite and the ﬁnite dimensional case, since in its course we shall present an alternative proof of one of the main results of [12]. This will M. Wojtylak would like to express his gratitude for a research position at the Faculty of Sciences of the VU University Amsterdam from 2007 to 2009. The major part of the work has been carried out during that period of time.

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follow from more general considerations which hold in the inﬁnite dimensional situation. The main tool in this paper is a method of dilation (reduction) for the [∗] operator T , which is quite natural for the study of the properties of T T [∗] and T T . This construction, which we called J-dilation, has its origins in [8], while being also similar to a construction implicitly used in [12]. Despite its usefulness, so far this kind of dilation has not been studied systemati[∗] cally. Therefore, in the present paper we consider which properties of T T are preserved under the J-dilation procedure. The ﬁrst four sections are devoted to the general situation in inﬁnite dimensional Krein spaces. We prove that spectral properties at nonzero points in the complex plane, as well as definitizability and nilpotency are preserved. In Sect. 5 we present material that is valid in the general setting, but is on the other hand tailored to the study of the ﬁnite dimensional case. Our result here is a complete description of how Jordan chains corresponding to [∗] the zero eigenvalue of T T behave under the J-dilation. This leads to the proof of the quoted result in [12], which is presented in Sect. 6. In this light, in the subsequent section, we see the result on polar decomposition of [13] from a different angle. We conclude the paper with a concrete example.

1. J -Dilations and J -Restrictions We assume background knowledge on Krein spaces, see [1,6] for wide treatments of the subject. The indefinite inner product on a Krein space is always denoted by [ · , · ], even if there is more than one space in question. We use a Hilbert space structure on a Krein space only in a few examples. The theorems are formulated entirely in the Krein space language. By a subspace of a Krein space H we mean a closed linear space H0 ⊆ H with the indefinite inner product inherited from H. The space H0 is not necessarily a Krein space itself. If H is a Pontriagin space then the necessary and sufﬁcient condition for H0 being a Pontriagin space is its nondegeneracy. By E F we mean a direct sum of two subspaces, we will write E [] F if the spaces are additionally [ · , · ]-orthogonal. Let H and K be two Krein spaces and let T belong to the space B(H, K) [∗] of bounded linear operators from H to K. Then by T we mean the Krein space adjoint of the operator T . We deﬁne now the main object of the paper. Definition 1.1. Let H0 , K0 , H, K be Krein spaces. We say that an operator T ∈ B(H, K) is a J-dilation of T0 ∈ B(H0 , K0 ) (or conversely T0 is an J-restriction of T ) if the following three conditions are satisﬁed: (i) H0 is a subspace of H, K0 is a subspace of K. (ii) There exist subspaces Hi of H and Ki of K (i = 1, 2, 3) such that H = H0 [] H1 [] (H2 H3 ),

K = K0 [] K1 [] (K2 K3 ),

where H1 and K1 are Krein spaces, H2 and H3 (K2 and K3 ) are skewly linked neutral spaces such that H2 H3 (K2 K3 ) is a Krein space.

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The operator T has a following representation with respect to the above decomposition ⎞ ⎛ 0 T02 0 T0 ⎜ 0 0 0 0⎟ ⎟. (1.1) T =⎜ ⎝T20 0 T2 0⎠ 0 0 0 0 Note that (ii) implies that with xi , yi ∈ Hi (0 = 1, . . . , 3) we have

[x0 + x1 + x2 + x3 , y0 + y1 + y2 + y3 ] = [x0 , y0 ] + [x1 , y1 ] + [x2 , y3 ] + [x3 , y2 ]. (1.2) A similar formula holds for K as well. The J-dilation (J-restriction) will be called rigid if ker T = H1 H3 ,

im T = K0 K2 .

(1.3)

Note that in such case H3 = ker T ∩ ker T [⊥] ,

K2 = im T ∩ im T

[⊥]

.

(1.4)

Example 1. Let us analyze the following classical example (see e.g. [8,15] for extensions and modiﬁcations). Let H = K be the space L2 [0, 1] × C2 with the Π1 -inner product deﬁned by the fundamental symmetry J(f, x, y) = (f, y, x) for all f ∈ L2 [0, 1], x, y ∈ C. Consider the selfadjoint operator ⎞ ⎛ M√t 0 π(1) ⎟ ⎜ ⎟ ·, 1 0 0 T := ⎜ ⎠ ⎝ 0 0 0 where Mφ ∈ B(L2 [0, 1]) denotes the multiplication operator by a bounded measurable function φ, π(g) (where g ∈ L2 [0, 1]) maps x ∈ C to xg and 1 ∈ L2 [0, 1] is a function constantly equal one. Note that T , after interchanging the last two columns, is already in the form (1.1) with T0 = M√t . Zero is a singular critical point of the operator ⎛ √ ⎞ Mt 0 π( t) ⎜ √ ⎟ [∗] T T = T2 = ⎜ 0 ⎟ ⎝ ·, t 0 ⎠, 0 0 0 since ker T 2 = ker T is a degenerate space. On the other hand T02 does not have any critical points. And so we have discovered the ﬁrst property that is not being preserved by J-dilations. The next, a kind of obvious one, is the number of negative squares of the underlying space, see also Example 3. Although we have put the definitons of J-dilation and J-restriction in together, they are actually two different notions. The ﬁrst one requires ﬁnding an outer space, while the latter one says something about the inner structure of the operator. This splitting reﬂects also in the following three results. Proposition 1.2. Given T0 there always exists a rigid J-dilation T of T0 with im T closed.

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Proof. If both spaces H0 and K0 are trivial then the construction of a rigid J-dilation is obvious. Suppose now that at least one of the spaces H0 and K0 is nontrivial. We show ﬁrst that there exist operators A ∈ B(H0 ), B ∈ B(K0 , H0 ), C ∈ B(K0 ) such that the block operator matrix

A B ∈ B(H0 × K0 ) T0 C is boundedly invertible. Here we understand H0 × K0 as being equipped with the (unique) Banach space topology it has as a product of two Krein spaces (each of which has a unique Banach space topology induced by the Krein space structure). If the space H0 (K0 ) is trivial, then it is enough to set C (or A) as a boundedly invertible operator. If both spaces K0 and H0 are nontrivial, it is enough to choose A and C boundedly invertible and set B = 0. Then, by the Schur’s complement reasoning, zero is in the resolvent of the above block operator matrix. We set H = H0 × H1 × K0 × K0 and K = K0 ×K1 ×H0 ×H0 , where the spaces H1 and K1 can be chosen arbitrary. The indefinite inner product on H is given by ⎡⎛ ⎞ ⎛ ⎞⎤ y0 x0 ⎢⎜x1 ⎟ ⎜y1 ⎟⎥ ⎢⎜ ⎟ , ⎜ ⎟⎥ := [x0 , y0 ]H + [x1 , y1 ]H + [x2 , y3 ]K + [x3 , y2 ]K , 0 1 0 0 ⎣⎝x2 ⎠ ⎝y2 ⎠⎦ x3 y3 an analogous formula deﬁnes the inner product on K. We identify H0 and H1 (K0 and K1 ) with the ﬁrst and the second component of H (K) respectively. Moreover, we set H2 := {0} × {0} × K0 × {0}, H3 =: {0} × {0} × {0} × K0 , K2 := {0} × {0} × H0 × {0}, K3 =: {0} × {0} × {0} × H0 . Finally, we deﬁne ⎛ ⎞ T0 0 C 0 ⎜0 0 0 0⎟ ⎟ T =⎜ ⎝ A 0 B 0⎠ . 0 0 0 0 Example 2. We present1 an operator that does not have a rigid J-restriction. Let L be a closed, strictly positive but not uniformly positive subspace of a Krein space K and let G ∈ B(K) be a fundamental symmetry. The space K can be written as an · , · -orthogonal sum of L and its · , · -orthogonal complement L⊥ , where · , · stands for the Hilbert space inner product [G · , · ]. The operator T deﬁned as zero on L and identity on L⊥ is continuous. Suppose it has a rigid J-restriction T0 . Since ker T is non-degenerate, H3 equals {0}. Consequently, H1 = ker T . But ker T endowed with the original inner product inherited from K is not a Krein space, contradiction. On the other hand T has a nontrivial J-restriction. Indeed, let e ∈ L\ {0}. Then H1 = lin {e} and K1 = lin {Ge} are a Krein spaces (with the [⊥] [⊥] original inner product [ · , · ]). We set H0 = H1 , K0 = K1 , H2 = H3 = 1 We

thank the referee for his suggestions on this example.

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K2 = K3 = {0}. Note that

[Ge, L⊥ ] = e, L⊥ = 0, [⊥]

which leads to im T = L⊥ ⊆ K1

= K0 . Since T e = 0, the operator

T0 := T|H0 : H0 → L⊥ ⊆ K0 is a J-restriction of T . Proposition 1.3. Let H, K be nonzero Pontryagin spaces. Given T there exists a rigid J-restriction T0 of T . Proof. We apply Theorem IX.2.5 of [1] to the subspace ker T of H and to the subspace imT of K. As a consequence we get the decompositions H = H0 [] H1 [] (H2 H3 ),

K = K0 [] K1 [] (K2 K3 ),

satisfying (1.3), (1.4) and points (i) and (ii) of the definition of J-dilation. It is also apparent that T , with respect to the above decompositions, has the form (1.1). Hence, T0 appearing in (1.1) is a rigid J-restriction of T . We refer the reader to Thm. IX.2.5 of [1] for questions connected with uniqueness of this construction.

2. Further Properties of J -Dilations, The Adjoint Operator Treating H = H0 [] H1 [] (H2 H3 ) as a direct and orthogonal sum of three Krein spaces and likewise for K = K0 [] K1 [] (K2 K3 ) we see that [∗] the operator T has the following block operator form ⎛

[∗] ⎞ T20 [∗] 0 ⎟ ⎜ T0 0 ⎟ ⎜ [∗] ⎟ ⎜ 0 0 0 (2.1) T =⎜ ⎟. ⎜ [∗] ⎟

⎠ ⎝ [∗] T2 0 T02 0 0 0 0 A simple indefinite inner product argument shows that

[∗]

[∗]

[∗] 0 0 T20 T2 0 + T02 0 = , = 0 T20 = , + 0 0 0 T02 0

0 T2+

+ + ∈ B(K3 , H0 ), T02 ∈ B(K0 , H3 ) and T2+ ∈ B(K3 , H3 ), respecwith some T20 tively. Substituting this into (2.1) we obtain ⎛ [∗] ⎞ + T0 0 0 T20 ⎜ 0 [∗] 0 0 0 ⎟ ⎟. (2.2) T =⎜ ⎝ 0 0 0 0 ⎠ + 0 0 T2+ T02

Hence, if we interchange the roles of H2 and H3 and interchange the roles of [∗] [∗] K2 and K3 , we can see T as a J-dilation of T0 . This fact and the lemma [∗] below will allow us to interchange the roles of T and T in further reasonings.

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We deﬁne the operators Ti• ∈ B(H, Ki ) and T•j ∈ B(Hj , K) as respectively the ith row and jth column of the matrix T . Our main hero will be definitely the operator (2.3) T0• = T0 0 T02 0 , [∗]

[∗]

though some of the others will appear as well. Note that (T0• ) = (T )• 0 [∗] [∗] and (T• 0 ) = (T )0• . [∗]

Lemma 2.1. T is a rigid J-dilation of T0 if and only if T is a rigid J-dilation [∗] of T0 . Proof. Suppose that T is a rigid J-dilation of T0 . The inclusion K1 [] K2 ⊆ [∗] [∗] ker T is obvious. To see the opposite one takes y = y0 +y1 +y2 +y3 ∈ ker T , yi ∈ Ki (i = 0, 1, 2, 3). Then for every x ∈ H we have [T x, y] = 0. By (1.3) T maps H onto a dense subspace of K0 [] K2 . Consequently, by (1.2), y0 = 0, y3 = 0. Hence, y ∈ K1 [] K2 . [∗] [∗] Suppose now that imT K0 [] K3 . Then either im(T )0• K0 or [∗] im(T )3• K3 . In the ﬁrst case there exists a nonzero x0 ∈ K0 which satis[∗] ﬁes [T y, x0 ] = 0 for all y ∈ K. Consequently x0 ∈ ker T , which contradicts the rigidity of T . In the latter case there exists a nonzero x2 ∈ K2 such that [∗] [T y, x2 ] = 0 for all y ∈ K, which is again in contradiction with the rigidity of T . [∗]

[∗]

At this point we can derive formulas for the operators T T and T T : ⎛

[∗]

T0 T0 ⎜ 0 [∗] ⎜ T T =⎝ 0 + T0 T02

⎞ 0 0⎟ ⎟, 0⎠ 0

[∗]

T0 T02 0 0 + T02 T02

0 0 0 0

⎛

TT

[∗]

[∗]

T0 T0 ⎜ ⎜ 0 =⎜ ⎝T20 T0[∗] 0

⎞

0 0 0 0

0 0 0 0

+ T0 T20 ⎟ 0 ⎟ ⎟. +⎠ T20 T20 0

(2.4)

Using our row–column notation we can rewrite (2.4) as [∗]

[∗]

T T = (T0• ) T0• ,

[∗]

[∗]

T T = T• 0 (T• 0 ) .

(2.5) [∗]

Lemma 2.2. If T is a rigid J-dilation of T0 then the operators T• 0 and (T0• ) [∗] are injective and ker T T = ker T0• .

Proof. The operator T• 0 is clearly injective by (1.3). By Lemma 2.1 we have [∗] [∗] [∗] ker T = K1 [] K2 , hence (T )• 0 = (T0• ) is injective as well. The equality [∗] ker T T = ker T0• follows now from (2.5). [∗]

3. Powers of Operators T T , Annihilating Polynomials Starting from (2.5) one can easily prove by induction the following formulas [∗]

[∗]

[∗]

(T T )j = (T0• ) (T0 T0 )j−1 T0• , [∗]

[∗]

[∗]

(T T )j = T• 0 (T0 T0 )j−1 (T• 0 ) ,

j ≥ 1,

(3.1)

j ≥ 1.

(3.2)

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Observe that, denoting by P0 the projection P0 (x0 + · · · + x3 ) = x0 , (xi ∈ Hi , i = 1, . . . , 3), we get [∗]

[∗]

P0 (T T )j |H0 = (T0 T0 )j ,

j ≥ 0.

(3.3)

[∗]

A similar formula can be derived for (T T )j . Note also the following intertwining relations [∗]

[∗]

(T0 T0 )j T0• = T0• (T T )j , [∗]

[∗]

[∗]

j = 0, 1, . . . ,

[∗]

(T0 T0 )j (T• 0 ) = (T• 0 ) (T T )j , [∗]

[∗]

[∗]

j

j = 0, 1, . . . ,

[∗]

j

(3.4)

(T0• ) (T0 T0 ) = (T T ) (T0• ) ,

(3.5)

j = 0, 1, . . . .

(3.6) n

If an operator A is nilpotent then we put ν(A) := min {n ∈ N : A = 0}. [∗] [∗] It is an easy fact that T T is nilpotent if and only if T T is nilpotent, in [∗] [∗] such case |ν(T T ) − ν(T T )| ≤ 1 (see also [4]). Equation (3.1) implies the following proposition. Proposition 3.1. Let T be a J-dilation of T0 . If p(t) is an annihilating poly[∗] [∗] [∗] nomial for T0 T0 (for T T ) then tp(t) is an annihilating polynomial for T T [∗] [∗] (for T0 T0 , respectively). Consequently the operator T0 T0 is nilpotent if and [∗] only if T T is nilpotent and [∗]

[∗]

|ν(T0 T0 ) − ν(T T )| ≤ 1. [∗]

Proof. Let p(T0 T0 ) = 0. Then by (3.1) [∗]

[∗]

[∗]

[∗]

T T p(T T ) = (T0• ) p(T0 T0 )T0• = 0. [∗]

[∗]

On the other hand if p(T T ) = 0 then by (3.3) we have p(T0 T0 ) = 0 and consequently, [∗]

[∗]

[∗]

[∗]

T0 T0 p(T0 T0 ) = T0 p(T0 T0 )T0 = 0. Proposition 3.2. Let T be a closed range, rigid J-dilation of an operator T0 . Then the spaces [∗]

[∗]

ker(T T )j+1 / ker(T T )j ,

[∗]

[∗]

ker(T0 T0 )j / ker(T0 T0 )j−1

(3.7) [∗]

are linearly isomorphic for j ≥ 1. Consequently, if, in addition, T T is [∗] [∗] nilpotent then ν(T T ) = ν(T0 T0 ) + 1. Proof. Since T is a closed range rigid dilation of T0 , we get T0• surjective [∗] and (T0• ) injective (Lemma 2.2). Consequently, by (3.1) we have [∗]

[∗]

T0• (ker(T T )j ) = ker(T0 T0 )j−1 ,

j ≥ 1.

Therefore, for j ≥ 1 the mapping [∗]

Φj (x + ker(T T )j ) := T0• x + ker(T0 T0 )j−1 ,

[∗]

x ∈ ker(T T )j+1

is a well deﬁned linear isomorphism between the spaces listed in (3.7).

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As usually we deﬁne the Segre characteristic for the nilpotent operator A in a ﬁnite dimensional space as a decreasing sequence of sizes of Jordan blocks in the Jordan canonical form of A, extended with an inﬁnite number of zeros. Since, for j = 1, 2, . . ., the dimension of ker Aj / ker Aj−1 equals the number of Jordan chains in the Jordan canonical form of A of length larger or equal then j, we get the following. Corollary 3.3. Let H and K be finite dimensional. If T is a rigid dilation of [∗] T0 and T T is nilpotent with the Segre characteristic (nk )∞ k=1 , then the Segre [∗] characteristic of T0 T0 is (max {nk − 1, 0})∞ k=1 . Example 3. Let H0 = K0 be an inﬁnite dimensional Hilbert space and let T0 be the zero operator on H0 . Consider now an arbitrary closed range, rigid [∗] J-dilation T of T0 . According to Proposition 3.2 the space ker(T T )2 / [∗] ker(T T ) is inﬁnite dimensional. Hence, there are inﬁnitely many Jordan chains of length two corresponding to the zero eigenvalue. Therefore, neither H nor K is a Pontryagin space.

4. Spectral Properties and Definitizabilty [∗]

Proposition 4.1. Let T be a J-dilation of T0 with H0 = H. Then σ(T T ) = [∗] σ(T0 T0 ) ∪ {0}. Proof. A simple argument involving the Schur complement applied to the [∗] [∗] block operator matrix (2.4) shows that σ(T T ) = σ(T0 T0 ) ∪ {0}. By a well [∗] known result the latter set is equal to σ(T0 T0 ) ∪ {0}. Note also the following proposition, which shows, besides other things, [∗] [∗] that the nonzero point spectra of the operators T T and T0 T0 coincide. By the algebraic root space of an operator A we mean the space Sλ (A) := {f ∈ D(A) : ∃n ∈ N : (A − λ)n f = 0}. Proposition 4.2. Let T be a J-dilation of T0 and let λ ∈ C\ {0}. Then [∗]

[∗]

T0• maps Sλ (T T ) bijectively to Sλ (T0 T0 ); [∗] [∗] [∗] (T0• ) maps Sλ (T0 T0 ) bijectively to Sλ (T T ); [∗] [∗] Sλ (T T ) is finite dimensional if and only if Sλ (T0 T0 ) is finite dimensional; [∗] [∗] (iv) Sλ (T T ) is non-degenerate if and only if Sλ (T0 T0 ) is nondegenerate.

(i) (ii) (iii)

Proof. (i)&(ii) First note that by the intertwining relation (3.4) T0• maps [∗] [∗] [∗] [∗] Sλ (T T ) into Sλ (T0 T0 ). Similarly, by (3.6), (T0• ) maps Sλ (T0 T0 ) into [∗] [∗] [∗] Sλ (T T ). Since T0• (T0• ) = T0 T0 and the latter operator is clearly injective [∗] [∗] on Sλ (T0 T0 ), the operator (T0• ) | [∗] is injective as well. The mapping [∗]

[∗]

[∗]

Sλ (T0 T0 )

T T = (T0• ) T0• maps Sλ (T T ) bijectively onto itself. By injectivity of

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[∗]

[∗]

(T0• ) | and (T0• ) | is [∗] , each of the mappings T0• | [∗] [∗] Sλ (T T ) Sλ (T0 T0 ) Sλ (T0 T0 ) bijective. Point (iii) is now obvious. Point (iv) follows directly from bijectivity of [∗] [∗] [∗] and (T0 T0 )| T0• |S (T [∗] T ) , (T0• ) | [∗] , (T T )| [∗] . [∗] S (T T ) Sλ (T0 T0 )

λ

Sλ (T0 T0 )

λ

The reader might have already guessed that the case λ = 0 is much more difﬁcult, we will deal with it in the next section. For the notions of definitizability, deﬁnitizing polynomial and spectral function see e.g. [7,11]. Note, that in our setting all operators are bounded. We also take the usual definitions of the set of critical points c(A) and the positive and negative spectrum σ±± (A). We set R± := {x ∈ R : ±x > 0}. [∗]

[∗]

Proposition 4.3. The operator T T is definitizable if and only if T0 T0 is def[∗] [∗] initizable. If p(t) is a definitizing polynomial for T0 T0 (for T T ) then tp(t) [∗] [∗] is a definitizing polynomial for T T (for T0 T0 , respectively). Consequently, [∗] if T T is definitizable then [∗]

[∗]

[∗]

σ±± (T T ) ∩ R+ = σ±± (T0 T0 ) ∩ R+ ,

[∗]

σ±± (T T ) ∩ R− = σ∓∓ (T0 T0 ) ∩ R−

and [∗]

[∗]

c(T T ) ∪ {0} = c(T0 T0 ) ∪ {0}. [∗]

Proof. Let T0 T0 be definitizable with the deﬁnitizing polynomial p(t). By (3.1) we have [∗]

[∗]

[(T T )j x, y] = [(T0 T0 )j−1 T0• x, T0• y],

j ≥ 1, x, y ∈ H. [∗]

In consequence, tp(t) is a deﬁnitizing polynomial for T T . On the other hand [∗] if p(t) is a deﬁnitizing polynomial for T T then formula (3.3) shows that p(t) [∗] is a deﬁnitizing polynomial for T0 T0 as well. By [15, Theorem 3.1] tp(t) is [∗] deﬁnitizing for T0 T0 . The ‘consequently’ part is now obvious. By R0 we denote the semiring generated by ﬁnite intervals and their [∗] complements with endpoints not in c(T T ) ∪ {0} and by E and E0 we denote [∗] [∗] the spectral function of T T and T0 T0 respectively. [∗]

Theorem 4.4. Let T be a J-dilation of T0 and let T T be definitizable. Then [∗]

T0• E(σ) = E0 (σ)T0• ,

λ ∈ ρ(T T )\ {0}.

(4.1)

[∗]

Consequently, a spectral point λ ∈ c(T T )\ {0} is a singular critical point for [∗] [∗] T T if and only if it is a singular critical point for T0 T0 . Proof. The intertwining relation (3.4) implies [∗]

[∗]

T0• (T T − λ)−1 = (T0 T0 − λ)−1 T0• ,

[∗]

λ ∈ ρ(T T )\ {0},

which after integration over a suitable contour becomes (4.1) (cf. [15], proof of Theorem 4.1). [∗] Suppose that λ is a regular critical point of T T and let us take a [∗] bounded closed neighborhood τ of λ such that τ ∩ (c(T T ) ∪ {0}) = ∅.

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By the properties of the spectral function ([11, p.30], see also [15, Theorem 4.2] for usage of similar arguments as those below) there exists X ∈ B(K0 ) [∗] ˜ ) = E0 (τ ). Since λ is a regular critical point of T [∗] T , such that T0 T0 X E(τ there exists a constant c ≥ 0 such that E(σ) ≤ c,

σ ⊆ τ,

σ ∈ R0 .

(4.2)

Now for σ ⊆ τ such that λ ∈ σ ∈ R0 we get [∗] E0 (σ) = E0 (σ)E0 (τ ) = E0 (σ)T0 T0 XE0 (τ ) [∗] [∗] ≤ E0 (σ)T0• (T0• ) X · c ≤ T0• E(σ)(T0• ) X · c [∗] ≤ T0• · c · (T0• ) X · c. [∗]

Hence, λ is a regular critical point for T0 T0 . A similar argument shows the opposite implication.

5. Decomposing Spaces We say that a pair of subspaces H of H and K of K decomposes T if H and [∗] K are Krein spaces, T H ⊆ K and T K ⊆ H . Note that in such case the pair H[⊥] and K[⊥] decomposes T as well and T can be written in the form [∗] [∗] [∗] T = T [] T , while T = T [] T . For the definition of a canonical form of an H-symmetric matrix we refer to [5]. Since our paper contains both ﬁnite and inﬁnite dimensional situations we view matrices as operators. Let A be a selfadjoint operator in a ﬁnite dimensional Krein space E. A linear basis (ej )j of E will be called a canonical basis for the operator A if and only if the matrix representation of A in (ej )j is in a Jordan canonical form and the Gramm matrix ([ei , ej ])ij is of a special type as outlined in [5]. By the sign characteristic of A we understand the sign characteristic of the pair consisting of the matrix representation of A in a canonical bases (ej )j and the Gramm matrix ([ei , ej ])ij . Obviously this notion does not depend on the choice of a canonical bases. If e1 , . . . , ek belong to some canonical basis of A and form a full Jordan chain, then by the sign of the chain we mean as usually the number [e1 , ek ], which is either plus or minus one. From now on we concentrate on the zero eigenvalue. Note that the nonzero eigenvalues were analyzed in the previous section. One can easily apply the methods used in the proof of Proposition 4.2 to analyze the sign characteristic for nonzero eigenvalues in the ﬁnite dimensional case (see also Proposition 3 in [9]). Motivated by the ﬁnite dimensional situation we introduce the following definition. However, note that neither the assumption of the ﬁnite dimension[∗] ality of the space nor nilpotency of the operator T T is needed in this section. Definition 5.1. Let E, F be a pair of ﬁnite dimensional spaces that decomposes T . We say that it is of

Vol. 68 (2010) type (i)

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323

if dim E = dim F = 2k (with some k ≥ 1) and there exist linear bases g1 , . . . , g2k of E and h1 , . . . , h2k of F such that hj−1 : j = 2, . . . , 2k T gj = 0 : j=1 (5.1) [∗] gj−1 : j = 2, . . . , 2k T hj = 0 : j=1 and

[gi , gj ] = [hi , hj ] =

type (ii)

[∗]

0 : i + j = 2k + 1, ; 1 : i + j = 2k + 1

if dim E = dim F − 1 ≥ 1, there exist canonical bases e1 , . . . , ek [∗] [∗] for T T and f1 , . . . , fk+1 for T T such that [∗] ej−1 : j = 2, . . . , k + 1 T fj = 0 : j=1 T ej = fj , j = 1, . . . , k and

type (iii)

[ei , ej ] = εδi+j,k+1

i, j = 1, . . . , k,

[fi , fj ] = εδi+j,k+2

i, j = 1, . . . , k + 1

(5.2)

with some ε ∈ {−1, 1}; if dim F = dim E − 1 ≥ 1, there exist canonical bases e1 , . . . , ek+1 [∗] [∗] for T T and f1 , . . . , fk (k = dim F) for T T such that fj−1 : j = 2, . . . , k + 1 T ej = 0 : j=1 [∗]

T fj = ej ,

j = 1, . . . , k

and

type (iv) type (v)

[ei , ej ] = εδi+j,k+2 ,

i, j = 1, . . . , k + 1,

[fi , fj ] = εδi+j,k+1 ,

i, j = 1, . . . , k

(5.3)

with some ε ∈ {−1, 1}; if dim F = 1, dim E = 0; if dim E = 1, dim F = 0.

We will refer to the bases appearing in each point above as a corresponding (to a type) basis. [∗]

Remark 5.2. While in the definitions of types (ii)–(iii) the operators T T |E [∗] and T T |F are presented in their canonical bases, in the definition of type (i) this is not the case. Although the bases g1 , . . . , g2k and h1 , . . . , h2k are not canonical it is easy to transform them into canonical ones. In fact, the [∗] canonical basis of T T|E consists in this case of two Jordan chains 1 1 1 (g2k + g2k−1 ), (g2k−2 + g2k−3 ), . . . , (g2 + g1 ) 2 2 2

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and 1 1 1 (g2k − g2k−1 ), (g2k−2 − g2k−3 ), . . . , (g2 − g1 ) 2 2 2 [∗]

of opposite signs, the same concerns the canonical basis of T T |F with g everywhere replaced by h. Note that in each of the types zero is the only eigenvalue of the opera[∗] [∗] tors T T|E and T T |F , or one of the operators is trivial and the second one is zero. Hence, a necessary condition for an operator T to have a decomposing pair of spaces of one of the types (i)–(v) is that zero is an eigenvalue of at [∗] [∗] least one of the operators T T and T T . In the next section we will see that this condition is also sufﬁcient in the ﬁnite dimensional case. On the other hand Example 1 shows that in the inﬁnite dimensional Π1 -space zero can be [∗] [∗] an eigenvalue of both operators T T and T T , but there is no decomposing pair for T . ˜ F˜ be a pair of Proposition 5.3. Let T be a rigid J-dilation of T0 and let E, decomposing spaces for T0 of type (i) with the corresponding bases g˜1 , . . . , g˜2k ˜ 1, . . . , h ˜ 2k . Then there exists a pair of spaces E, F decomposing T and and h of type (i) with a corresponding bases g−1 , g0 , . . . , g2k and h−1 , h0 , . . . , h2k , such that ˜ j , (T• 0 )[∗] hj = g˜j , j = 1, . . . , 2k, (5.4) T0• gj = h Proof. First note that by surjectivity of T0• there exist g2k ∈ H and h2k ∈ K such that [∗] ˜ 2k , (T• 0 ) h2k = g˜2k . (5.5) T0• g2k = h We deﬁne now gj and hj by a recursive relation [∗]

gj−1 := T hj

hj−1 := T gj ,

j = 0, . . . , 2k.

(5.6)

[∗]

By Lemma 2.2 the operator (T0• ) is injective and hence [∗] [∗] [∗] ˜2 = g0 = T T g2 = (T0• ) T0• g2 = (T0• ) h 0.

For the same reasons g−1 , h0 , h−1 are nonzero. On the other hand [∗] [∗] [∗] ˜ 1 = T0 T [∗] h ˜ T g−1 = T T T g1 = T (T0• ) T0• g1 = T (T0• ) h 0 1 = 0.

(5.7)

Similarly [∗]

T h−1 = 0.

(5.8)

Hence, g−1 , g0 , . . . , g2k and h−1 , h0 , . . . , h2k are nonzero vectors satisfying hj−1 : j = 0, . . . , 2k T gj = 0 : j = −1 (5.9) [∗] gj−1 : j = 0, . . . , 2k , T hj = 0 : j = −1 cf. (5.1). Equations (5.4) are also satisﬁed by the exploited intertwining relations (3.4), (3.5). Deﬁne now the spaces E and F by

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The Pair of Operators T T and T T

E := lin {g−1 , . . . , g2k },

[∗]

325

F := lin {h−1 , . . . , h2k }.

[∗]

Clearly T E ⊆ F and T F ⊆ E, by (5.6),(5.7) and (5.8). We show now that [gi , gj ] = [hi , hj ] = δi+j,2k−1 ,

i, j = −1, . . . , 2k, i + j < 4k − 1. (5.10)

Without loss of generality we can assume that i ≤ j. Note that [∗]

[gi , gj ] = [T T gi , gj+2 ] = 0,

i, j = −1, 0.

For j > 0 and i ≤ 2k − 2 we have [∗]

[gi , gj ] = [T T gi+2 , gj ] = [T0• gi+2 , T0• gj ] ˜ i+2 , h ˜ j ] = δi+j+2,2k+1 = δi+j,2k−1 . = [h Next, ˜ 2k , h ˜ 2k ] = 0. [g2k−1 , g2k−1 ] = [T h2k , T h2k ] = [h The same calculations hold for g and h interchanged, which ﬁnishes the proof of (5.10). The cases i, j = 2k and i = 2k − 1, j = 2k of (5.10) are more difﬁcult. Suppose we replace g2k and h2k respectively by ˆ 2k := h2k + β0 h0 + β−1 h−1 gˆ2k := g2k + α0 g0 + α−1 g−1 , h with some αi , βi ∈ R (i = 0, −1) and we replace g2k−1 and h2k−1 respectively by [∗] ˆ 2k , gˆ2k−1 := T h

ˆ 2k−1 := T gˆ2k . h

In such case the modiﬁed systems g−1 , . . . , g2k−2 , gˆ2k−1 , gˆ2k ,

ˆ 2k−1 , h ˆ 2k h−1 , . . . , h2k−2 , h

[∗]

[∗]

[∗]

still satisfy (5.9) since gi ∈ ker T T = ker T0• and hi ∈ ker T T = ker(T• 0 ) for i = 0, −1. Note that (5.4) holds for any choice of αi , βi (i = 0, −1) as well. Observe that by (5.10) we have [ˆ g2k , gˆ2k ] = [g2k , g2k ] + α−1 , ˆ ˆ 2k ] = [h2k , h2k ] + β−1 [h2k , h

(5.11)

ˆ 2k−1 , h ˆ 2k ] = [g2k , g2k−1 ] + α0 + β0 . [ˆ g2k , gˆ2k−1 ] = [h

(5.13)

(5.12)

and Hence, it is easy to choose αi , βi (i = 0, −1) such that the inner products in (5.11), (5.12) and (5.13) are all zero. Consequently, E and F are non-degenerate spaces and hence they decompose T . From the construction it is obvious that the spaces E and F are of type (i) with g−1 , . . . , g2k−2 , gˆ2k−1 , gˆ2k , and ˆ 2k−1 , h ˆ 2k h−1 , . . . , h2k−2 , h as corresponding bases.

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˜ F˜ be a pair of Proposition 5.4. Let T be a rigid J-dilation of T0 and let E, decomposing spaces for T0 of type (ii) with the corresponding bases e˜1 , . . . , e˜k and f˜1 , . . . , f˜k+1 . Then there exists a pair of spaces E, F decomposing T and of type (iii) with a corresponding bases e0 , . . . , ek+1 and f0 , . . . , fk , such that T0• ej = f˜j , [∗]

(T• 0 ) fj = e˜j ,

j = 1, . . . , k + 1, (5.14)

j = 1, . . . , k.

Moreover, [fk , f0 ] = [f˜k , f˜1 ].

[ek+1 , e0 ] = [˜ ek+1 , e˜1 ],

(5.15)

Proof. Since the mapping T0• is onto, there exist ek+1 ∈ H such that T0• ek+1 = f˜k+1 . Now let ej and fj be deﬁned by fj := T ej+1 ,

[∗]

ej := T fj ,

j = 0, . . . , k.

(5.16) [∗]

[∗]

By definition, e0 , . . . , ek+1 and f0 , . . . , fk are Jordan chains for T T and T T respectively. By the intertwining relations (3.4) and (3.5) we have that (5.14) is satisﬁed. Furthermore, note that [∗] [∗] [∗] e0 = T T e1 = (T0• ) T0• e1 = (T0• ) f˜1 . [∗]

(5.17) [∗]

By Lemma 2.2 the operator (T0• ) is injective, thus e0 = 0. Since T f0 = [∗] T T e1 = e0 , we get f0 = 0 as well. On the other hand, [∗] [∗] T e0 = T (T0• ) f˜1 = T0 T0 f˜1 = 0. [∗]

[∗]

(5.18)

[∗]

Consequently, T T e0 = 0 and T T f0 = T T T e1 = T e0 = 0. Deﬁne now the spaces E and F by E := lin {e0 , . . . , ek+1 },

F := lin {f0 , . . . , fk }.

[∗]

Clearly T E ⊆ F and T F ⊆ E, by (5.16) and (5.18). What remains to prove is that [ei , ej ] = δi+j,k+1 [f˜k+1 , f˜1 ],

i, j = 0, . . . , k + 1

(5.19)

and [fi , fj ] = δi+j,k [f˜k+1 , f˜1 ] = δi+j,k+1 [˜ ek , e˜1 ],

i, j = 0, . . . , k.

(5.20)

Namely, if the two above formulas are satisﬁed then E and F are non-degenerate spaces and (5.15) holds as well. We proceed now like in the proof of the last proposition, the details are left to the reader. The cases i+j < 2k+2 of (5.19) can be proved directly. We replace ek+1 , if necessary, by eˆk+1 := ek+1 + αe0 , where α ∈ R is chosen in such way that [ˆ ek+1 , eˆk+1 ] = 0. Since [fi , fj ] = [T ei+1 , T ej+1 ] = [ei+1 , ej ], we have (5.20).

i, j = 0, . . . , k

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327

Proposition 5.5. Let T be a rigid J-dilation of T0 and let E˜ = {0}, F˜ be a pair of decomposing spaces for T0 of type (iv) and let f˜1 ∈ F˜ be such that [f˜1 , f˜1 ] = ±1. Then there exists a pair of spaces E, F decomposing T and of type (iii) with a corresponding canonical bases e0 , e1 and f0 , such that T0• e1 = f˜1 . Moreover, [e1 , e0 ] = [f0 , f0 ] = [f˜1 , f˜1 ].

(5.21) [∗]

The proof is similar to the proof of Proposition 5.4. Substituting T for T we get the analogues of Propositions 5.4 and 5.5 for types (iii) and (v).

6. Canonical Forms for the Finite Dimensional Case In this section we give an alternative proof of [12, Theorem 3.2], cf. also the main result in [3]. It is worth mentioning that the paper [12] contains a broad and interesting study of the problem of comparing the operators T T and T T also in the presence of different types of involutions. Nevertheless, we ﬁnd it important to present the proof below for two reasons. Firstly, our proof highlights the induction argument, which is only implicitly present in the proof of [12]. Secondly, our proof is shorter than the one in [12] and has a more geometrical nature thanks to the J-dilation procedure. We restrict ourselves to the nilpotent case, since the essential difﬁculty lies in the zero eigenvalue. From now on we assume that every space is ﬁnite dimensional. [∗]

Theorem 6.1. Let H and K be finite dimensional and let the operators T T [∗] and T T be nilpotent. Then there exists subspaces Ei of H and Fi of K (i = 1, . . . , n) such that H = E1 [] · · · [] En ,

K = F1 [] · · · [] Fn

and each pair Ei , Fi decomposes T and is of one of the types (i)–(v). Note that our decomposing pairs of type (i) are the same as blocks of type (2) in [12], decomposing pairs of type (ii) correspond to blocks of type (3) in [12], decomposing pairs of type (iii) correspond to blocks of type (4) in [12] and the sum of decomposing pairs of type (iv) and (v) constitutes the block of type (1) in [12]. Proof. We will prove the claim by induction with respect to [∗] [∗] N := max ν(T T ), ν(T T ) . [∗]

[∗]

Let us suppose ﬁrst that N = 1. Then T T = 0 and T T = 0. Hence, im T [∗] as well as im T are neutral spaces. We ﬁx a skewly linked companion K of [⊥] im T and let K := (im T K ) , so that K = (im T K ) [] K . [∗]

[∗]

Since K [⊥] im T , it is contained in the kernel of T . Also im T ⊆ ker T by [∗] [∗] T T = 0. On the other hand if x ∈ K then, by definition of K , [T x, z] =

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[∗]

[x, T z] = 1 for some z ∈ H. Hence, T x = 0 and consequently ker T im T K Now consider a similar decomposition

[∗]

=

[∗]

H = (im T H ) [] H . [∗]

Again, we have ker T = im T H . Moreover, [∗]

im T = T H ,

[∗]

im T = T K .

(6.1) [∗]

The pairs H , {0} and K , {0} decompose T , since T |H = 0, T |K = 0. Furthermore, T|H : H → {0} and T|{0} : {0} → K satisfy the theorem with respectively blocks of type (v) and (iv) only. To ﬁnish the proof of case N = 1 [∗] we need to show that T restricted to (im T H ) decomposes into blocks of types (i),(ii) and (iii). Obviously, type (ii) as well as (iii) are not possible, [∗] [∗] since T T and T T do not have Jordan chains longer then one. Fix a linear [∗] basis g21 , . . . , g2l of H and choose vectors g11 , . . . , g1l ∈ im T such that [g1i , g2j ] = δij

i, j = 1, . . . , l.

Set hi1 := T g2i , i = 1, . . . , l and let hi2 ∈ K for i = 1, . . . , l be such that [∗] T hi2 = g1i . Note that [hi1 , hj2 ] = [T g2i , hj2 ] = [g2i , T hj2 ] = [g2i , g1j ] = δij , [∗]

i, j = 1, . . . , l. with Ei := lin g1i , g2i , Fi :=

Hence the theorem holds for T |im T [∗] H lin hi1 , hi2 (i = 1, . . . , l) being of type (i). Now let us assume that the claim is true for N and consider the N + 1 case. Let T0 be a rigid J-restriction of T . By Proposition 3.2 we can apply [∗] the induction hypothesis to the operator T0 . Hence, H0 = E˜1 [] · · · [] E˜n , K0 = F˜1 [] · · · [] F˜n with each pair E˜i , F˜i (i = 1, . . . , n) decomposing T0 and being of one of the types (i)–(v). By multiple use of Propositions 5.3, 5.4 and 5.5 we get a system of spaces E1 , . . . , En , F1 , . . . , Fn . Since E˜i ∩ E˜j = {0} ˜i (i = 1, . . . , n), we get Ei ∩ Ej = {0} for i = j. for i = j, and T0• Ei = E Similarly, Fi ∩ Fj = {0}. Since each pair Ei , Fi decomposes T , the spaces E := E1 · · · En and F := F1 · · · Fn are non-degenerate and decompose T as well. Set S := T|E [⊥] and note that [∗]

[∗]

[∗]

T T = (T T )|E [] S S. [∗]

The Segre characteristic of (T T ) |E is given by Propositions 5.3–5.5. On [∗] the other hand the Segre characteristics of T T is determined up to Jordan [∗] chains of length one by the Segre characteristic of T0 T0 (see Corollary 3.3). [∗] Combining these two facts we conclude, that the operator S S has Jordan [∗] chains of length one only. The same property concerns SS . Hence, we can apply the ﬁrst induction step to S. What remains to show is that T|E satisﬁes the theorem. First note that [∗]

if i = j and x = T T y ∈ Ei , z ∈ Ej then [x, z] = 0.

(6.2)

Indeed, [x, z] = [T0• y, T0• z]. The latter equals zero, since T0• y ∈ E˜i , T0• z ∈ E˜j .

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Now let us consider the following construction. For each i, j = 1, . . . , n, i > j there exists a pair of spaces Eˆi , Fˆi , decomposing T , of the same type as Ei , Fi and such that E = E1 · · · Ei−1 Eˆi Ei+1 · · · En , F = F1 · · · Fi−1 Fˆi Fi+1 · · · Fn , and such that [Eˆi , Ej ] = {0},

(6.3)

and [Eˆi , Ep ] = [Ei , Ep ],

p = j.

(6.4)

By repeating recursively this procedure we get the desired decomposition of H and K. We will consider now several cases, corresponding to the types of pairs Ei , Fi and Ej , Fj . Without loss of generality we can assume that the type of the pair Ei , Fi increases with i. Moreover, note that there are neither pairs of type (iv) nor (v). 1. The ith and the jth pair are of type (iii). Let ei0 , . . . , eik+1 , f0i , . . . , fki (ej0 , . . . , ejl+1 , f0j , . . . , flj ) be bases corresponding to the type for Ei and Fi (Ej and Fj ) respectively. We modify the eik+1 vector only. Namely, we deﬁne eˆik+1 = eik+1 + αej0 , where α ∈ C is such that [ˆ eik+1 , eil+1 ] = 0. We set Eˆi := lin ei0 , . . . , eik , eˆik+1 . Note that by (6.2) and the above we have [Eˆi , Ej ] = {0}. We also set Fˆi := Fi . In this case we already have [Fi , Fj ] = {0}, since [fki , flj ] = [T eik+1 , T ejl+1 ] = [T T eik+1 , ejl+1 ], [∗]

which is zero by (6.2). Thanks to (6.2) we get (6.4) as well. Note that the systems ei0 , . . . , eik , eˆik+1 and f0i , . . . , fki are bases of Eˆi , Fi corresponding to type (i). 2. The ith pair is of type (iii) and the jth pair is of type (ii). Let j ei0 , . . . , eik+1 , f0i , . . . , fki (ej0 , . . . , ejl , f0j , . . . , fl+1 ) be bases corresponding to the type for Ei and Fi (Ej and Fj ) respectively. We deﬁne eˆik+1 = eik+1 + αej0 , where α ∈ C is such that [ˆ eik+1 , eil ] = 0. Moreover, we deﬁne fˆki := T (ˆ eik+1 ) = fki + αfk0 . Hence, j j ] = [ˆ eik+1 , T fl+1 ] = [ˆ eik+1 , ejl ] = 0. [fˆki , fl+1 [∗]

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i The spaces Eˆi := lin ei0 , . . . , eik , eˆik+1 , Fˆi := lin f0i , . . . , fk−1 , fˆki satisfy now the requirements. 3. The ith pair is of type (iii) and the jth pair is of type (i). Let j j , . . . , g2l , hj−1 , . . . , hj2l ) be bases corresponding to ei0 , . . . , eik+1 , f0i , . . . , fki (g−1 the type for Ei and Fi (Ej and Fj ) respectively. We set j eˆik+1 = eik+1 + α−1 g−1 + α0 g0j ,

where α−1 , α0 ∈ C are such that j j [ˆ eik+1 , g2l ] = [ˆ eik+1 , g2l−1 ] = 0.

Furthermore, we set fˆki := T eˆik+1 and proceed as before. 4. The ith and the jth pair are of type (ii). Similarly to 1., interchanging the roles of the spaces. 5. The ith pair is of type (ii) and the jth pair is of type (i). Similarly to 3., interchanging the roles of the spaces. 6. The ith and the jth pair are of type (i). We set j i i gˆ2k := g2k + α−1 g−1 + α0 g0j , [∗]

i gˆ2k−1 = T hi2k ,

ˆ i := hi + β−1 hj + β0 hj h 2k 2k −1 0

i ˆi h 2k−1 = T g2k

with αp , βq ∈ C (p, q = −1, 0) such that ˆ i , hj ] = 0, [ˆ gri , gsj ] = [h r s

r = 2k − 1, 2k, s = 2l − 1, 2l.

The pair of spaces i i i i i i i i Eˆi := lin g−1 , Fˆi := lin f−1 , . . . , g2k−2 , gˆ2k−1 , gˆ2k , . . . , f2k−2 , fˆ2k−1 , fˆ2k now satisﬁes the requirements. This completes the proof.

7. Polar Decomposition Revisited Let H and K be ﬁnite dimensional Krein spaces. We call an operator U ∈ [∗] [∗] B(H, K) unitary if U U = IK and U U = IH . We say that T ∈ B(H, K) admits a polar decomposition if there exists a unitary U ∈ B(H, K) and a selfadjoint A ∈ B(H) such that T = U A. Corollary 6 of [13] says the following. Theorem 7.1. The operator T ∈ B(H, K), where the Krein spaces H and K are finite dimensional, admits a polar decomposition if and only if the sign [∗] [∗] characteristics of T T and T T are the same. Note that the ‘only if’ part of the theorem is obvious. In the light of Theorem 6.1 we are now able to give a new proof of the ‘if’ part by showing an explicit construction of the unitary transformation U . We again restrict ourselves to the nilpotent case, the nonzero eigenvalues can be handled for example as in [13]. [∗]

Part of the proof. Suppose that T T is nilpotent and the sign characteristics [∗] [∗] of T T and T T are the same. We apply Theorem 6.1 and obtain a system of decomposing pairs Ei , Fi , i = 1, . . . , n. Since the sign characteristics of

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[∗]

T T and T T are the same, we can renumerate and group the decomposing spaces in the following way: H = E1 [] · · · [] Er [] Er+1 [] Eφ(r+1) [] · · · [] Er+q [] Eφ(r+q) K = F1 [] · · · [] Fr [] Fr+1 [] Fφ(r+1) [] · · · [] Fr+q [] Fφ(r+q) where • r, q ≥ 0 and φ is a bijection from the set {r + 1, . . . , r + q} to the set {r + q + 1, . . . , r + 2q}, • each decomposing pair Ei , Fi is of type (i) for i = 1, . . . , r, • either Er+i , Fr+i is of type (ii) and Eφ(r+i) , Fφ(r+i) is of type (iii) or Er+i , Fr+i is of type (iv) and Eφ(r+i) , Fφ(r+i) is of type (v), • dim Er+i = dim Fφ(r+i) , • if the four systems φ(r+i)

r+i r+i r+i er+i 1 , . . . , ek , f1 , . . . , fk+1 and e1

φ(r+i)

φ(r+i)

, . . . , ek+1 , f1

φ(r+i)

, . . . , fk

(k ≥ 0) are the corresponding bases for the pairs Er+i , Fr+i and Eφ(r+i) Fφ(r+i) respectively then φ(r+i)

r+i ] = [e1 [f1r+i , fk+1

φ(r+i)

, ek+1 ].

(7.1)

For i = 1, . . . , r the operator T |Ei is already in the polar decomposition form. Indeed, if g1 , . . . , g2k and h1 , . . . , h2k are corresponding bases for Ei and Fi respectively, then the mapping U : Ei → Fi deﬁned by U gj := hj (j = 1, . . . , k) is unitary and A := U −1 T is selfadjoint. We show now that each restriction T |Er+i [] Eφ(r+i) has a polar decomposition as well. Suppose that the pair Er+i , Fr+i is of type (ii) and Eφ(r+i) , Fφ(r+i) is of type (iii), the proof for the (iv)–(v)-case goes the same way. We deﬁne a mapping U : Er+i [] Eφ(r+i) → Fr+i [] Fφ(r+i) by φ(r+i)

= fj U er+i j

,

j = 1, . . . , k,

φ(r+i)

U ej

= fjr+i ,

j = 1, . . . , k + 1.

It is unitary by (5.2), (5.3) and (7.1). The reader can now easily check that the operator A := U −1 T is selfadjoint. We refer the reader to [2,3,10,16] for topics related to the polar decomposition.

8. Explicit Example [∗]

[∗]

Suppose that we want to generate a matrix T such that T T and T T are nilpotent and have given sign characteristics. Constructing the canonical forms and then using a random basis transformation seems to be the simplest method. However, the Jordan chains are unstable and while performing this numerically the Jordan structure may be destroyed. We describe now how to apply our dilation procedure to get desired examples. Our method consists of several steps. In step (j + 1) we construct spaces (j+1) = Cnj , K(j+1) = Cmj and a matrix T (j) ∈ Cnj ×mj . The indefinite H

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inner products on H(j+1) and K(j+1) will be given by invertible, selfadjoint matrices H (j) and K (j) respectively, which means [z, w] = K (j) z, w , z, w ∈ Kj , [x, y] = H (j) x, y , x, y ∈ Hj , where · , · denotes the standard inner product. Moreover, T (j+1) will always (j+1) be a rigid J-dilation of T (j) , i.e. T0 = T (j) . For simplicity we restrict ourselves to generating matrices with given Segre characteristic, since ﬁxing the signs is a minor problem here. Example 4. We construct a matrix T such that the Segre characteristics of [∗] [∗] T T and T T are (4, 3, 3, 2) and (3, 3, 3, 3, 1) respectively. Note that such a situation is possible according to Theorem 6.1. Namely, there is one pair of decomposing spaces of type (iii) with the lengths of Jordan blocks 4 and 3, one pair of decomposing spaces of type (i) with two Jordan blocks of length 3 [∗] [∗] for T T and two Jordan blocks of length 3 of T T , one pair of decomposing spaces of type (ii) with lengths of Jordan blocks 2 and 3 and ﬁnally one pair of decomposing spaces of type (iv). Step 0. We start with H0 = C,

K0 = {0} ,

and T (0) being the zero operator from H0 to K0 . The Segre characteristics [∗] [∗] for T (0) T (0) T (0) T (0) are (1) and (0) respectively. Step 1. In this step we want to ﬁnd a rigid J-dilation T (1) of T (0) such that [∗] [∗] T (1) T (1) and T (1) T (1) have the Segre characteristics (1, 1, 1, 1) and (2, 1, 1) [∗] respectively. Moreover, two blocks of length one for T (1) T (1) and two blocks [∗] of length one for T (1) T (1) has to form a decomposing pair of spaces of type (i), one decomposing pair of type (ii) and one decomposing pair of type (v). This means that the spaces H1 and K1 are both of dimension 4 and dim ker T = 2.

(8.1) H11 ,

A priori we have two possible choices for dimensions of the spaces H21 , 1 1 1 1 H3 and two possible choices for dimensions of K1 , K2 , K3 . Namely, we can have dim H01 = dim H0 = 1,

dim H11 := 3,

dim H21 = dim H31 := 0

(8.2)

dim H01 = dim H0 = 1,

dim H11 := 1,

dim H21 = dim H31 := 1,

(8.3)

dim K01 = dim K0 = 0,

dim K11 := 2,

dim K21 = dim K31 := 1

(8.4)

dim K01 = dim K0 = 0,

dim K11 := 0,

dim K21 = dim K31 := 2.

(8.5)

or analogously or However, if we set the dimensions according to (8.2) and (8.5) then the J-dilation will not be rigid, since the necessary condition that dim(H01 H21 ) = dim(K01 K21 ) is in such case violated. The same happens if we take (8.3)

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together with (8.4). What is left is (8.2) with (8.4) and (8.3) with (8.5). In the ﬁrst case we get the matrix ⎞ ⎛ 0 0 ⎟ ⎜ (1) T (1) = ⎝T20 0⎠ . 0 0 (1)

The J-dilation is rigid if and only if the 1×1 matrix T20 is invertible. Observe that dim ker T (1) = 3. Hence, by (8.1), we have to reject this case as well. Let us analyze the case (8.3) with (8.5). The matrix T (1) has then the block form (1) (1) T20 0 T2 0 (1) T = 0 0 0 0 (1) (1) is and the J-dilation is rigid if and only if the 2 × 2 matrix T20 T2 (1)

invertible. Keeping this constraint we can even pick the matrices T20 and (1) T2 at random. Note that in this case we have dim ker T (1) = 2, which agrees with (8.1). The inner products on H1 and K1 are given by the matrices ⎛ ⎛ ⎞ ⎞ 1 0 0 0 0 0 1 0 ⎜0 1 0 0 ⎟ ⎜0 0 0 1 ⎟ ⎟ ⎟ H1 := ⎜ K1 := ⎜ ⎝0 0 0 1 ⎠ ⎝1 0 0 0 ⎠ , 0 0 1 0 0 1 0 0 respectively.2 Step 2. Now we want to obtain Segre characteristics (3, 2, 2, 1) and (2, 2, 2, 2), hence dim H2 = dim K2 = 8. There are two possibilities of setting the dimensions, namely dim H12 := dim K12 := 0, dim H12 := dim K12 := 2,

dim H22 = dim H32 := dim K22 = dim K32 := 2. dim H22 = dim H32 := dim K22 = dim K32 := 1,

otherwise the necessary condition dim(H02 H22 ) = dim(K02 K22 ) for rigidity of the J-dilation is violated. However, we need to reject the latter case. Indeed, since dim kerT (1) = 2 we are not able to extend T (1) to an invertible (2) T (1) T20 matrix in a ﬁve-dimensional space. We set (2) (2) T02 T02 ⎞ ⎞ ⎛ ⎛ 0 0 0 0 H1 K1 0 I2 ⎠ 0 I2 ⎠ . K2 := ⎝ 0 H2 := ⎝ 0 0 I2 0 0 I2 0 2

Here is a chance to fix the signs as well.

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(2)

Again, the numbers for above blocks T20 , T2 , T02 of the T (2) matrices could (2) T (1) T20 be chosen arbitrary, with the only requirement that the matrix (2) (2) T02 T02 is invertible. Step 3. Finally, we construct operator T (4) such that the Segre characteristics [∗] [∗] of T (4) T (4) and T (4) T (4) are respectively (3, 3, 3, 3, 1) and (4, 3, 3, 2). As in the previous steps we determine the dimensions: dim H3 = 13 = 8 + 1 + 2 + 2, we set

⎛

H2 ⎜0 ⎜ H3 = ⎝ 0 0 and we choose is invertible.

(3) T20 ,

(3) T2

0 1 0 0

0 0 0 I2

and

(3) T02

⎞ 0 0⎟ ⎟, I2 ⎠ 0

dim K3 = 12 = 8 + 0 + 2 + 2, ⎛

K2 K3 = ⎝ 0 0

0 0 I2

⎞ 0 I2 ⎠ 0

in such way, that the matrix

T (2) (3)

T02

(3)

T20

(3)

T02

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.

References [1] Bogn´ ar, J.: Indefinite Inner Product Spaces. Springer, New York, Heidelberg (1974) [2] Bolshakov, Y., van der Mee, C.V.M., Ran, A.C.M., Reichstein, B., Rodman, L.: Polar decompositions in ﬁnite dimensional indefinite scalar product spaces: general theory. Linear Algebra Appl. 261, 91–147 (1997) [3] Bolshakov, Y., Reichstein, B.: Unitary equivalence in an indefinite scalar product: An analogue of singular value decomposition. Linear Algebra Appl. 222, 155–226 (1995) [4] Flanders, H.: Elementary divisors of AB and BA. Proc. AMS. 2, 871–874 (1951) [5] Gohberg, I., Lancaster, P., Rodman, L.: Indefinite Linear Algebra and Applications. Birkh¨ auser-Verlag, Basel (2005) [6] Iohvidov, I.S., Krein, M.G., Langer, H.: Introduction to Spectral Theory of Operators in Spaces with Indefinite Metric, Mathematical Research, vol. 9. Akademie-Verlag Berlin (1982) [7] Jonas, P.: On the functional calculus and the spectral function for definitizable operator in Krein space. Beitrage Anal. 16, 121–135 (1981) [8] Jonas, P., Langer, H., Textorius, B.: Models and unitary equivalence of cyclic selfadjoint operators in Pontrjagin spaces. Oper. Theory Adv. Appl. 59, 252– 284 (1992)

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[9] Kes, J.S., Ran, A.C.M.: On the relation between XX [∗] and X [∗] X in an indefinite inner product space. Oper. Matrices 1(2), 181–197 (2007) [10] Kintzel, U.: Procrustes problems in ﬁnite dimensional indefinite scalar product spaces. Linear Algebra Appl. 402, 1–28 (2005) [11] Langer, H.: Spectral functions of definitizable operators in Krein spaces. In: Proceedings of Graduate School “Functional Analysis”, Dubrovnik 1981. Lecture Notes in Mathematics, vol. 948, pp. 1–46. Springer, Berlin, (1982) [12] Mehl, C., Mehrmann, V., Xu, H.: Structured decompositions for matrix triples: SVD-like concepts for structured matrices. Oper. Matrices 3, 303–356 (2009) [13] Mehl, C., Ran, A.C.M., Rodman, L.: Polar decompositions of normal operators in indefinite inner product spaces, operator theory in Krein spaces and nonlinear Eigenvalue problems. Oper. Theory Adv. Appl. 162, 277–292 (2006) [14] Philipp, F., Ran, A.C.M., Wojtylak, M.: Local definitizability of T [∗] T and T T [∗] . arXiv:1004.1584v1 [15] Ran, A.C.M., Wojtylak, M.: Analysis of spectral points of the operators T [∗] T and T T [∗] in a Krein space. Integr. Equ. Oper. Theory 63, 263–280 (2009) [16] van der Mee, C.V.M., Ran, A.C.M., Rodman, L.: Polar decompositions and related classes of operators in spaces Πκ . Integr. Equ. Oper. Theory 44, 50–70 (2002) Andr´e C. M. Ran (B) Afdeling Wiskunde Faculteit der Exacte Wetenschappen Vrije Universiteit Amsterdam De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail: [email protected] Michal Wojtylak Instytut Matematyki Wydzial Matematyki i Informatyki Uniwersytet Jagiello´ nski ul. L ojasiewicza 6 30-348 Krak´ ow, Poland e-mail: [email protected] Received: November 27, 2009. Revised: September 9, 2010.

Integr. Equ. Oper. Theory 68 (2010), 337–355 DOI 10.1007/s00020-010-1802-y Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Duality, Tangential Interpolation, and T¨ oplitz Corona Problems Mrinal Raghupathi and Brett D. Wick Abstract. In this paper, we extend a method of Arveson (J Funct Anal 20(3):208–233, 1975) and McCullough (J Funct Anal 135(1):93–131, 1996) to prove a tangential interpolation theorem for subalgebras of oplitz corona theoH ∞ . This tangential interpolation result implies a T¨ rem. In particular, it is shown that the set of matrix positivity conditions is indexed by cyclic subspaces, which is analogous to the results obtained for the ball and the polydisk algebra by Trent and Wick (Complex Anal Oper Theory 3(3):729–738, 2009) and Douglas and Sarkar (Proc CRM, 2009). Mathematics Subject Classification (2010). Primary 47A57; Secondary 30H30, 30H50. Keywords. Distance formulae, Hilbert module, Nevanlinna-Pick interpolation, Toeplitz corona problem.

1. Introduction The classical corona problem asks whether the set of point evaluations, for points in the unit disk D, is dense in the maximal ideal space of H ∞ . A famous result of Carleson [9] shows that they are dense. Let A be an abelian Banach algebra, and let M be its maximal ideal space. A subset X of M is dense in M if and only if for any ﬁnite set of functions f1 , . . . , fn such that n 2 2 j=1 |fj (x)| ≥ δ > 0 for x ∈ X, there exists a set g1 , . . . , gn ∈ A such that f1 g1 + · · · + fn gn = 1. n 2 Arveson [6] studied a related problem replacing j=1 |fj (x)| ≥ δ 2 , by n the operator theoretic condition j=1 Tfj Tf∗j ≥ δ 2 I, where Tf is the T¨oplitz operator with symbol f acting on the Hardy space H 2 . He showed that under n this assumption there exists g1 , . . . , gn ∈ H ∞ such that j=1 fj gj = 1 and M. Raghupathi was supported in part by a National Science Foundation Young Investigator Award, Workshop in Analysis and Probability, Texas A&M University. B. D. Wick was supported by National Science Foundation CAREER Award DMS 0955432 and an Alexander von Humboldt Fellowship.

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2

|gj (z)| ≤ δ −2 . The constant δ −2 is optimal, as demonstrated by the choice f1 = 1. See Schubert [15] for the best possible constant. In general determining the best constants in the corona problem is considerably challenging. For the T¨ oplitz corona problem we do obtain the optimal constant. However, we make stronger assumptions. The objective of this paper is to show that a similar T¨ oplitz corona theorem holds for the case where A is weak∗ -closed subalgebra of H ∞ . Our result makes use of a modiﬁcation of the Arveson distance formula [6, Theorem 1], a reﬁnement of this due to McCullough [12, Theorem 1]. These modiﬁcations allow us to then demonstrate the ﬁrst main result of this paper, a tangential interpolation theorem for unital weak∗ -closed algebras A. j=1

1.1. Notation We denote by Lp the Lebesgue space on the unit circle with respect to normalized arc-length measure. The corresponding Hardy space will be denoted H p. Given a subset S of a Hilbert space H, we denote by [S] the smallest closed subspace that contains S. A function g ∈ H 2 is called outer if the closure of H ∞ g is H 2 . In this paper we adopt the following notation. Given a subalgebra A ⊆ H ∞ and an outer function g we let Kg be the reproducing kernel of the subspace [Ag], viewed as a subspace of H 2 . 1.2. Statement of Main Results This paper is concerned primarily with tangential interpolation theorems and their application to T¨ oplitz corona problems. We would like to give an overview of the main results. We begin by stating the tangential interpolation problem and our main result, which is Theorem 1.1. In Sect. 2, we will elaborate on the connections between the two problems. Let A be a unital, weak∗ -closed subalgebra of H ∞ . Let (x1 , . . . , xn ) be a sequence of points in the unit disk D, let (v1 , . . . , vn ) be a sequence of vectors in 2 and let (w1 , . . . , wn ) be a sequence of scalars. We identify a function F : D → 2 with a sequence of functions (fk )∞ k=1 in the usual way. Let F : D → 2 be such that fk ∈ A for all k ≥ 1. The function F induces an operator MF : H 2 → H 2 ⊗ 2 given by MF (h) = F h. Hence, MF can be identiﬁed with the column operator (Tf1 , . . .)t . Similarly, there is a map from ∞ 2 2 2 H ⊗ → H given by MF (hk ) = k=1 fk hk . In this case, the operator MF is identiﬁed with the row operator (Tf1 , . . .). We denote by C(A) the set of F such that fk ∈ A, for k ≥ 1, viewed as column operators. When viewed as row operators we use the notation R(A). In both instances the norm of MF , as an operator, coincides with supz∈D F (z)2 . We are concerned with the following extremal problem inf sup F (z)2 : vj , F (xj ) = wj for j = 1, . . . , n . z∈D

We say that a function F such that vj , F (xj ) = wj for j = 1, . . . , n is a solution to the tangential interpolation problem.

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Our main result is a characterization, in terms of matrix positivity conditions, for the existence of a solution F ∈ C(A) such that supz∈D F (z)2 ≤ α, where α is a prescribed constant. Theorem 1.1. Suppose that A be a unital weak∗ -closed subalgebra of H ∞ . Let (x1 , . . . , xn ) be a sequence of points from the unit disk D, let (v1 , . . . , vn ) be a sequence of 2 vectors, and let (w1 , . . . , wn ) be a sequence of scalars. Let Qg denote the matrix (1) Qg = (α2 vj , vi − wi wj )Kg (xi , xj ) . Then there exists a function F : D → 2 such that supz∈D F (z)2 ≤ α and vj , F (xj ) = wj if and only if Qg ≥ 0 for all outer functions g such that g2 = 1. A more careful examination of the proof of Theorem 1.1, which will be given in Sect. 4, shows that we need only consider a subset of the set of all outer functions. Before we state this corollary we introduce some notation. Given A we let L∞ (A) denote the smallest weak∗ -closed subalgebra of L∞ that contains A + A. The algebra L∞ (A) is the algebra of essentially-bounded measurable functions for some sub-sigma-algebra of the Lebesgue measurable sets on the circle. Therefore, there exists a sigma-algebra M consisting of Lebesgue measurable subsets of T such that L∞ (A) = L∞ (T, M, dm), where m is Lebesgue measure. We let Lp (A) denote the corresponding Lp space. If g is an outer function, then we denote Hg = [Ag]. Recall that the kernel function for this space is Kg . When g = 1 we denote Hg by H and the corresponding kernel is denoted K. Corollary 1.2. Retaining the notation of Theorem 1.1. 1. There exists a function F ∈ C(A) such that supz∈D F (z)2 ≤ α and vj , F (xj ) = wj for j = 1, . . . , n if and only if Qg ≥ 0 for all outer functions g such that |g| ∈ L2 (A) and g2 = 1. 2. For F ∈ R(A) let MF,g ∈ B(Hg ⊗ 2 , Hg ) be given by h → F h. If there ∗ ≥ δ 2 I for all outer functions g is a constant δ > 0 such that MF,g MF,g 2 such that |g| ∈ L (A) and g2 = 1, then there exists a function G in C(A) such that F G = 1 and supz∈D G(z) ≤ δ −1 . The difﬁculty with Theorem 1.1 is that the positivity condition is over a whole family of outer functions or kernels. In some applications we would rather have the condition over just a single kernel. We turn to establishing a tangential interpolation theorem where we replace the family of conditions Qg ≥ 0 for all outer functions g such that |g| ∈ L2 (A) by a single positivity condition. However, we can not guarantee a solution of optimal norm. This leads to the second main result of the paper. Theorem 1.3. Let A be a unital weak∗ -closed subalgebra of H ∞ , and let Hg be the subspace generated by Ag, where g is an outer function. Suppose that for each outer function g such that |g| ∈ L2 (A) there exists a similarity Sg : H → Hg . Also assume that there is a constant c, that is independent of g, such that Sg Sg−1 ≤ c for all such outer functions g.

340 1. 2.

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If [(α2 vj , vi − wi wj )K(xi , xj )] ≥ 0, then there exists F ∈ C(A) such that supz∈D F (z)2 ≤ αc and vj , F (xj ) = wj for j = 1, . . . , n. If MF MF∗ ≥ δ 2 on B(H), then there exists G ∈ C(A) such that F G = 1, and GC(A) ≤ cδ −1 .

The outline of the paper is as follows. In Sect. 2, we give background to the tangential interpolation problem, including standard background for reproducing kernel spaces. In Sect. 3, we provide an extension of Arveson’s Distance formula needed in our context. Section 4 puts the computations and ideas from the ﬁrst two sections together to prove Theorem 1.1. Finally, in Sect. 5 we prove Theorem 1.3, which essentially follows from Theorem 1.1, and then collect applications of Theorem 1.3 to the case of bounded analytic functions on Riemann surfaces. This application generalizes a result of Ball [7]. The corona problem and its variant the T¨ oplitz corona problem have been studied extensively in the past. The paper of Agler and McCarthy [3, Section 7] provides an excellent overview of the connection between matrix positivity conditions, families of kernels and corona problems. The connections between interpolation theory and T¨ oplitz corona problem for the bidisk and the Schur–Agler class are described in Agler and McCarthy [2] and Ball and Trent [8].

2. The Tangential Interpolation Problem In order to state our results and describe our setting we will need the terminology of reproducing kernel Hilbert spaces. We begin with a brief description. The reader should consult the text of Agler and McCarthy [1], or the paper of Aronszajn [5]. 2.1. Reproducing Kernel Hilbert Spaces Let X be a set and let C be a Hilbert space. We denote by F(X, C) the set of functions from X to C. A subset H ⊆ F(X, C) is called a C-valued reproducing kernel Hilbert space (RKHS) if H is a Hilbert space and for each x ∈ X, the evaluation map Ex : H → C deﬁned by f → f (x) is a bounded linear map on H. The kernel function of H is the map K : X × X → B(C) deﬁned by K(x, y) = Ex Ey∗ ∈ B(C). It is straightforward that the kernel function of H is a positive semidefinite function on X × X and that the span of {Ex ξ : x ∈ X, ξ ∈ C} is dense in H. Conversely, every B(C)-valued positive semidefinite function K on X × X gives rise to a C-valued RKHS H(K) in a canonical way and this correspondence is one-to-one [1]. We denote the Hilbert space associated to K by H(K). We suppress the kernel function, when the context is clear. For i = 1, 2, let Ki be a Ci -valued kernel function on X and let Hi = H(Ki ). Given a function F : X → F(C1 , C2 ) and a function g : X → C1 , let F g denote the pointwise product of F and g. We say that F : X → F(C1 , C2 ) is a multiplier from H1 to H2 if and only if F g ∈ H2 for all g ∈ H1 . We denote

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the set of multipliers from H1 to H2 by mult(H1 , H2 ). Since the space Hi is completely determined by its kernel function Ki we also use the notation mult(K1 , K2 ) to denote the space of multipliers. The closed graph theorem shows that the operator MF : H1 → H2 deﬁned by MF (g) = F g is bounded. The multiplier norm of a function F ∈ mult(K1 , K2 ) is deﬁned as F mult(K1 ,K2 ) := MF B(H1 ,H2 ) . If Ei,x denotes the evaluation map on Hi , and F ∈ mult(K1 , K2 ), then E2,x MF = F (x)E1,x

for all x ∈ X.

If H is a scalar-valued RKHS, then the evaluation map Ex is a linear functional and the unique element kx ∈ H such that f (x) = Ex (f ) = f, kx

for all f ∈ H is called the kernel function at the point x for H. In this case K(x, y) = Ex Ey∗ = ky , kx . Given two scalar-valued RKHSs Hi , for i = 1, 2, with kernel functions Ki , and f ∈ mult(H1 , H2 ), we have Mf∗ k2,x = f (x)k1,x , where ki,x denotes the kernel function for Hi at the point x. Given a scalar-valued RKHS H(K) we give the Hilbert space tensor product H ⊗ C the structure of a C-valued RKHS by deﬁning Ex (f ⊗ ξ) = f (x)ξ. A short calculation reveals that the kernel function for H ⊗ C is K(x, y)IC , where IC is the identity map on C, and that Ex∗ ξ = kx ⊗ ξ. If F ∈ mult(H, H ⊗ C), then, for each x ∈ X, F (x) ∈ B(C, C). We identify B(C, C) with C via the correspondence T → T (1). We have, MF∗ (kx ⊗ ξ), h = kx ⊗ ξ, F h = kx , h F (x)∗ ξ = ξ, F (x) kx , h . Therefore, MF∗ (kx ⊗ ξ) = (F (x)∗ ξ)kx = ξ, F (x) kx .

(2) ∗

Given F ∈ mult(H ⊗ C, H) we have F (x) ∈ B(C, C) and so F (x) ∈ B(C, C) = C, under our identiﬁcation. In this case we have, h ⊗ ξ, MF∗ kx = F (h ⊗ ξ), kx

= h(x)F (x)ξ = h, kx ξ, F (x)∗

= h ⊗ ξ, kx ⊗ F (x)∗ . Hence, MF∗ kx = kx ⊗ F (x)∗ .

(3)

In this paper we will be interested primarily in two special cases: the case where C1 = C2 = C, and the case where either C1 or C2 is 2 := 2 (N) and the other is C. We can view⎡ an⎤operator T ∈ B(H, H ⊗2 ) as a column operator matrix T1 ⎢T2 ⎥ of the form T = ⎣ ⎦. If A ⊆ B(H), then we denote by C(A) the set of col.. . umn operators [Ti ] such that Ti ∈ A for all i. There is a similar identiﬁcation of B(H ⊗ 2 , H) with the set of row operator matrices, and we denote the set of row operators with entries from A by R(A). It is easily checked that mult(H, H ⊗ 2 ) = C(mult(H)) and that mult(H ⊗ 2 , H) = R(mult(H)).

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2.2. Tangential Interpolation We now describe the tangential interpolation problem. Let X be a set and let K be a kernel on X. We will assume that K(x, x) = 0, for all x ∈ X. Given a ﬁnite sequence of points (x1 , . . . , xn ) ∈ X n , a sequence of vectors (v1 , . . . , vn ) ∈ C n , and a sequence of scalars (w1 , . . . , wn ). We say that a function F ∈ mult(H, H ⊗ C) is a solution to the associated tangential interpolation problem if and only if F (xj )∗ vj = wj for j = 1, . . . , n. Given a constant α we are interested in ﬁnding necessary and sufﬁcient conditions for the existence of a solution of norm at most α, that is, a multiplier F such that F mult(H,H⊗C) ≤ α and F (xj )∗ vj = wj

for j = 1, . . . , n.

As is the case with many complex interpolation problems of this type, there is a necessary matrix positivity condition, which we now derive. Let F be a solution to the above problem and let α ≥ F mult(H,H⊗C) . For x, y ∈ X and ξ, ζ ∈ C we have, MF MF∗ (ky ⊗ ζ), kx ⊗ ξ = MF∗ (ky ⊗ ζ), MF∗ (kx ⊗ ξ)

= (F (y)∗ ζ)ky , (F (x)∗ ξ)kx

= (F (y)∗ ζ)(F (x)∗ ξ)K(x, y)

(4)

Consider the restriction of MF∗ to the subspace K, that is the span nof the vectors {kx1 ⊗v1 , . . . , kxn ⊗vn }. An element of K is of the form k = j=1 cj kxj ⊗

vj . Since MF∗ has norm at most α we see that (α2 I − MF MF∗ )k, k ≥ 0. Substituting for k, using (4) and the fact that F (xj )∗ vj = wj , shows us that MF∗ |K ≤ α if and only if [(α2 vj , vi − wi wj )K(xi , xj )] ≥ 0.

(5)

If C = C, and vi = 1 ∈ C, then the above positivity condition is a necessary condition for the existence of a function f ∈ mult(K) such that f mult(K) ≤ α and f (xj ) = wj for j = 1, . . . , n. If K(z, w) = (1 − zw)−1 is the Szeg¨o kernel for the unit disk D, then the associated multiplier algebra is H ∞ (D) and the multiplier norm is the supremum norm. This is the setting of the classical Nevanlinna-Pick theorem. In this case, it is a well-known fact that the necessary matrix positivity condition is also sufﬁcient. In general, a single matrix positivity condition is not sufﬁcient to guarantee that there exist solutions of norm at most a given constant α. However, in certain situations we may be able to replace a single kernel function by a set of kernel functions {Kλ : λ ∈ Λ} such that mult(Kλ ) = mult(H). In addition, this collection of kernel functions has the property that the condition Qλ = [(α2 −wi wj )Kλ (xi xj )] ≥ 0 for all λ ∈ Λ is equivalent to the existence of a multiplier f ∈ mult(H) such that f mult(H) ≤ α and f (xj ) = wj . This is, in fact, the situation when we replace the algebra H ∞ (D) by a weak∗ -closed subalgebra A of H ∞ (D) [13]. To make this formal, we introduce the following definition.

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Definition 2.1. Let H be a scalar-valued RKHS on a set X. A set of kernels {Kλ : λ ∈ Λ} on X such that mult(Kλ ) ⊇ mult(H) has the tangential interpolation property for mult(H) if and only if for every ﬁnite sequence of points (x1 , . . . , xn ) from X, vectors (v1 , . . . , vn ) from C, and scalars (w1 , . . . , wn ) the condition Qλ := [(α2 vj , vi − wi wj )Kλ (xi , xj )]ni,j=1 ≥ 0

for all λ ∈ Λ

(6)

implies the existence of a multiplier F ∈ mult(H, H ⊗ C) such that F mult(H,H⊗C) ≤ α and F (xj )∗ vj = wj for j = 1, . . . , n. 2.3. Reformulation as a Distance Problem Suppose that we are given the data of tangential interpolation problem, that is, a sequence of vectors (v1 , . . . , vn ) in 2 , a sequence of points (x1 , . . . , xn ) in X and a sequence of scalars (w1 , . . . , wn ). Let us assume that there are solutions to this problem, that is, multipliers F ∈ mult(H, H ⊗ C) such that F (xj )∗ vj = wj for j = 1, . . . , n. Let J denote the set of functions G in mult(H, H ⊗ C) such that G(xj )∗ vj = 0 for j = 1, . . . , n. Given two solutions F1 , F2 to the tangential interpolation problem we see that the difference F1 − F2 ∈ J . Conversely, every solution must lie in F + J . Hence, the set of solutions to the interpolation problem is precisely F + J , where F is one particular solution. We will show in Sect. 4 that under the assumption K(x, x) = 0 for all x ∈ X, the multiplier algebra mult(K) is a unital, weak∗ -closed subalgebra of B(H(K)). We will also establish the fact that the evaluation map on mult(K), given by f → f (x), is weak∗ -continuous. Since the algebras we are interested in are weak∗ -closed the least norm of any solution to the interpolation problem is given by inf{F + Gmult(H,H⊗C) : G ∈ J }. This is the distance from F to J , that is, d(F, J ). The problem of determining necessary and sufﬁcient conditions for the existence of a solution of norm at most α is reduced to the problem of computing a formula for the distance d(F, J ). With this in mind we present a reﬁnement of a distance formula in [6,12] in the next section. We will show how to deduce a T¨oplitz corona theorem from a tangential interpolation theorem in Sect. 4.

3. A Distance Formula In this section, we recall some basic facts about the distance of an operator A ∈ B(H1 , H2 ) from a weak∗ -closed subspace. In the next section we will use this formula to compute the distance of a solution to the subspace J and thus obtain a tangential interpolation theorem. This a simple application of standard duality techniques.

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We begin by recalling some basic facts related to the dual Banach space structure of B(H). Let H be a separable Hilbert space. Given an operator T ∈ B(H), the trace of T is deﬁned by trace(T ) =

∞

T ej , ej

j=1

where {ej : j ∈ N} is an orthonormal basis. The sum in the definition of the trace does not depend on the choice of orthonormal basis. An operator is called a trace class operator if and only if trace(T ) is ﬁnite. The set of trace class operators is an ideal in B(H) and is a Banach space in the trace class norm T 1 = trace(|T |). We let T C(H) denote the space of trace class operators on H. It is well known that the dual of T C(H) can be identiﬁed naturally with B(H) and that the dual pairing is given by T, A = trace(T A),

where A ∈ B(H), T ∈ T C(H).

This pairing also identiﬁes T C(H) with the set of weak∗ -continuous linear functionals on B(H). An operator H ∈ B(H) is called a Hilbert-Schmidt ∞ operator if and only if HH ∗ ∈ T C(H), that is, j=1 Hej , Hej is ﬁnite. The set of all Hilbert-Schmidt operators will be denoted HS(H). This space is a Hilbert space when given the inner product H, K = trace(HK ∗ ). Given a trace class operator T there exist Hilbert-Schmidt operators H, K such that 1/2 T = HK ∗ and T 1 = H2 = K2 . Let A ∈ B(H), let T = HK ∗ ∈ T C(H), let hj = Hej and let kj = Kej . If h = ⊕hj and k = ⊕kj ∈ H ⊗ 2 , then, trace(AT ) = trace(AHK ∗ ) = trace(K ∗ AH) ∞ ∞ = AHej , Kej = Ahj , kj = (A ⊗ I)h, k . j=1

j=1

We also have 2

h =

∞ j=1

2

hj =

∞ j=1

2

Hej , Hej = trace(H ∗ H) = H2 .

Hence, h = H2 and k = K2 . Given an operator A ∈ B(H) and a weak∗ -closed subspace S ⊆ B(H), the distance from A to S is given by d(A, S) = inf{A + S : S ∈ S} = sup{|trace(AT )| : T ∈ S⊥ , T 1 = 1} where S⊥ denotes the preannihilator of S. If we write T = HK ∗ where 1/2 H, K ∈ HS(H) and T 1 = H2 = K2 , hj = Hej , and kj = Kej as before, then h = k = 1. Since T = HK ∗ ∈ S⊥ we have 0 = trace(SHK ∗ ) =

∞

Shj , kj = (S ⊗ I)h, k ,

j=1

for all S ∈ S. Hence, k ⊥ (S ⊗ I)h. Rewriting the distance formula we get, d(A, S) = sup{| (A ⊗ I)h, k | : h = k = 1, k ⊥ (S ⊗ I)h}.

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It will also prove useful to have such a formula when S ⊆ B(H1 , H2 ). Let H = H1 ⊕ H2 . We can identify B(H1 , H2 ) with a subspace of B(H) in the usual way, that is, A(h1 ⊕ h2 ) = 0 ⊕ Ah1 . Keeping the notation from (7) we can write h ∈ (H1 ⊕ H2 ) ⊗ 2 as h = h1 ⊕ h2 , where hi ∈ Hi ⊗ 2 for i = 1, 2. We have, (A ⊗ I)h, k = 0 ⊕ (A ⊗ I)h1 , k1 ⊕ k2 = (A ⊗ I)h1 , k2 . Since k ⊥ (S ⊗ I)h we see that k2 ⊥ (S ⊗ I)h1 . Hence, d(A, S) = sup{| (A ⊗ I)h1 , k2 | : h1 = 1, k2 = 1, k2 ⊥ (S ⊗ I)h1 }. These computations prove the following distance formula: Theorem 3.1. Let S be a weak∗ -closed subspace of B(H1 , H2 ) and let A ∈ B(H1 , H2 ). The distance of A from S is given by d(A, S) = sup{| (A ⊗ I)h1 , h2 | : hi ∈ Hi ⊗ 2 , hi = 1, h2 ⊥ (S ⊗ I)h1 }. (8)

4. A Tangential Interpolation Theorem for Subalgebras of H ∞ 4.1. T¨ oplitz Corona Problem We had made the claim at the end of Sect. 2 that the multiplier algebra was weak∗ -closed and that point evaluations are weak∗ -continuous. We now prove that claim. Lemma 4.1. Let K be a kernel on a set X such that K(x, x) = 0 for all x ∈ X. Then the algebra mult(H) is weak∗ -closed when viewed as a subalgebra of B(H(K)). In addition, the evaluation map on mult(K) given by f → f (x) is weak∗ -continuous. Proof. Let Mft be a net that converges to an operator T ∈ B(H) in the weak∗ -topology. We have, Mft h, k → T h, k for any pair of vectors h, k ∈ H. Let k = kx and h = ky . We have,

T ky , kx = lim Mft ky , kx = lim ky , Mf∗t kx = lim ft (x) ky , kx . t

t

t

If we choose x = y, and use the fact that K(x, x) = 0, then limt ft (x) = T kx ,kx K(x,x) . Hence, limt ft (x) exists. Let us denote the limit by f (x). We have h, T ∗ kx = f (x) h, kx for all h ∈ H and x ∈ X, and so T ∗ kx = f (x)kx . It follows that f is a multiplier of H and that T = Mf . The above argument also shows that if Mft → Mf in the weak∗ -topol ogy, then ft (x) → f (x). We now show how the hypothesis of the T¨ oplitz corona problem leads to a tangential interpolation problem. Suppose that F ∈ mult(H ⊗ C, H) and that MF MF∗ ≥ δ 2 I. We have, ∗ MF kx = kx ⊗ F (x)∗ , where we have identiﬁed B(C, C) with C. Hence, MF∗ ky , MF∗ kx = ky ⊗ F (y)∗ , kx ⊗ F (x)∗

= F (y)∗ , F (x)∗ K(x, y)

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Let Y ⊆ X and let KY be the span of {kx : x ∈ Y }. Since the space H is the closed linear span of the set of kernel functions kx for x ∈ X, the operator MF MF∗ − δ 2 I is positive if and only if (MF MF∗ − δ 2 I)|KY is positive for all ﬁnite sets Y ⊆ X. The latter condition is equivalent to the positivity of the matrix [( F (y)∗ , F (x)∗ − δ 2 )K(x, y)]x,y∈Y

(9)

for all ﬁnite sets Y ⊆ X. Proposition 4.2. Let H be an RKHS on X and let {Kλ : λ ∈ Λ} be a set of kernels with the tangential interpolation property for mult(H). Let Hλ = H(Kλ ), let F ∈ mult(H ⊗ C, H), and let MF,λ denote the operator of multiplication by F between the spaces Hλ ⊗ C and Hλ . Suppose that for each λ ∈ Λ, we ∗ ≥ δ 2 Iλ . Then there exists a multiplier G ∈ mult(H, H ⊗ C) have MF,λ MF,λ such that G ≤ δ −1 and F G = 1. ∗ ≥ δ 2 Iλ , given a ﬁnite set Y = {x1 , . . . , xn }, (9) shows Proof. Since MF,λ MF,λ that the matrix [( F (xj )∗ , F (xi )∗ − δ 2 )Kλ (xi , xj )] ≥ 0 for all λ ∈ Λ. This matrix is of the form Qλ in (6) for the case where the vectors are vj = F (xj )∗ , and the scalars wj = δ for j = 1, . . . , n. Since, Kλ has the tangential interpolation property, there is a contractive multiplier GY ∈ mult(H, H ⊗ C) such that δ = GY (xj )∗ F (xj )∗ = (F (xj )GY (xj ))∗ for j = 1, . . . , n. Since δ > 0 we get, F (xj )GY (xj ) = δ for j = 1, . . . , n. The net GY is contained in the unit ball of the space mult(H, H ⊗ C), which is weak∗ -compact subset of B(H, H ⊗ C). If F denotes the collection of all ﬁnite subsets of X, then there is a subnet {Ft } ⊆ F such that of {GFt } converges in the weak∗ -topology to a contractive multiplier G. Since point evaluations are weak∗ -continuous on mult(H, H⊗C) we see, for a ﬁxed x ∈ X, that F (x)G(x) = lim Ft F(x)GFt (x) = δ. Therefore F G = δ. It follows that F (δ −1 G) = 1 and δ −1 Gmult(H,H⊗C) ≤ δ −1 .

We have established that if {Kλ : λ ∈ Λ} is a set of kernels that have ∗ ≥ δ 2 Iλ the tangential interpolation property, then the condition MF,λ MF,λ implies the existence of a multiplier G ∈ mult(H, H ⊗C) such that G ≤ δ −1 and F G = 1. Our strategy for the proof of Theorem 1.1 is to exploit the distance formula and the existence of at least one solution to the tangential interpolation problem. 4.2. The Existence of Solutions Now let A be a unital weak∗ -closed subalgebra of the multiplier algebra of H(K). Let g ∈ H be a nonvanishing function. In particular, recall that an outer function g does not vanish at any point in the disk. We view mult(H) as a subalgebra of B(H). Let Hg be the closure of Ag in H, let Kg be the kernel of Hg , let kxg be the kernel function at the point x, and let Qg = [( vj , vi − wi wj )Kg (xi , xj )]. We have assumed that α = 1. However, since this amounts to a rescaling, there is no loss of generality in doing so. Since

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g does not vanish at any point x ∈ X, the kernel Kg (x, x) = 0 for any x. Therefore, the results of the previous section do apply. We will establish the fact the positivity of the matrix Qg implies the existence of a multiplier F ∈ C(A) such that F (xj )∗ vj = wj . We will then establish the fact that the closure of J g in Hg ⊗ 2 is the set of functions f ∈ Hg ⊗ 2 such that f (xj ), vj = 0 for j = 1, . . . , n. We say that the algebra A separates x and y if and only if there exists a function f ∈ A such that f (x) = f (y). We say that the algebra A separates a set of points Y if and only if A separates x and y for all x, y ∈ Y . Lemma 4.3. Every element of A is a multiplier of Hg . Proof. Let h ∈ Hg and let fn be a sequence in A such that fn g − hH → 0. Let f ∈ A. We have (f fn )g − f hH ≤ Mf fn g − hH → 0. Hence, f h ∈ Hg . Lemma 4.4. The algebra A separates x and y if and only if kxg and kyg are linearly independent. Proof. Suppose that A does separate x, y and that f ∈ A with f (x) = 1 and f (y) = 0. Note that f is a multiplier of Hg . Assume that αkxg + βkyg = 0. We have 0 = Mf∗ (αkxg + βkyg ) = αf (x)kxg + βf (y)kyg = αkxg . Since g is a nonvanishing function in Hg we know that kxg = 0 and so α = 0. A similar argument shows that β = 0. Conversely, suppose that A does not separate x and y. Let z ∈ X and let fn be a sequence in A such that fn g → kzg . We have kzg (x) = limn→∞ fn (x)g(x). On the other hand, fn (x) = fn (y) and so kzg (y) = lim fn (y)g(y) = lim fn (x)g(y) n→∞

n→∞

g(y) g(y) = lim fn (x)g(x) = kzg (x) . n→∞ g(x) g(x) Hence, g(x)Kg (y, z)−g(y)Kg (x, z) = 0 for all z ∈ X and so g(x)kyg −g(y)kxg = 0 with g(x), g(y) = 0. The relation x ∼ y if and only if f (x) = f (y) for all f ∈ A is an equivalence relation on X. Let {x1 , . . . , xn } be given. Let us reorder the points {x1 , . . . , xn } in such a way that there is a sequence n0 = 0 < n1 < · · · < np = n such that the sets Xk = {xi : nk−1 < i ≤ nk } are the equivalence classes for the above equivalence relation. Lemma 4.5. If Qg ≥ 0, then there exists a multiplier F ∈ C(A) such that F (xj )∗ vj = wj for j = 1, . . . , n. In addition, the subspace [J g] is precisely the set of functions in Hg ⊗ 2 such that f (xj ), vj = 0 for j = 1, . . . , n Proof. By Lemma 4.4 there exist functions e1 , . . . , ep such that ek |Xl (x) = δk,l for k, l = 1, . . . , p. To simplify notation let K = Kg , let Q = [qi,j ], where qi,j = ( vj , vi − wi wj )K(xi , xj ) and let Qk = [qi,j ]nk−1
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t = nk − nk−1 . We temporarily set Y = Xk and relabel the points of the set Xk as y1 , . . . , yt . We also relabel the corresponding vectors as v1 , . . . , vt and the scalars w1 , . . . , wt . Let e = ek . Since the algebra A fails to separate any two points of Y we see that there exists a sequence of nonzero scalars λ1 , . . . , λt such that kyi = λi ky1 . The matrix Qk is given by Qk = [( vj , vi − wi wj )K(y1 , y1 )λi λj ]. Since this is a square submatrix of Q we know that Qk ≥ 0. Since the λi are nonzero and K(y1 , y1 ) is nonzero we see that [ vj , vi ] ≥ [wi wj ]. Hence, the vector (w1 , . . . , wt )t is in the range ofthe matrix P = [ vj , vi ]. Therefore, t there are scalars α1 , . . . , αt such that j=1 αj vj , vi = wi for i = 1, . . . , t. t 2 Let ξ = j=1 αj vj ∈ . We let ξi denote the ith component of ξ. Consider the function F = (ξ1 e, ξ2 e, . . .)t , which belongs to C(A), because ξ ∈ 2 . We have that F (x) = ξ if x ∈ Y and is 0 if x ∈ {x1 , . . . , xn } \ Y . From the argument in the previous paragraph we see that for each k such that 1 ≤ k ≤ p we can ﬁnd ξk such that ξk , vi = wi for nk−1 < i ≤ nk . In addition, we can ﬁnd Fk such that Fk (x) = ξk for x ∈ Xk and Fk (x) = 0 if x ∈ {x1 , . . . , xn } \ Xk . Hence, wi if x ∈ Xk ∗ Fk (x) vi = . 0 if x ∈ {x1 , . . . , xn } \ Xk Hence, the function F = F1 + · · · + Fp has the property that F (xi )∗ vi = wi for i = 1, . . . , n. Let J be the set of functions F ∈ C(A) such that F (xj )∗ vj = 0 for j = 1, . . . , n. We claim that [J g] is the set of functions in Hg ⊗ 2 such that f (xj ), vj = 0. One inclusion is straightforward. If Fm g → h, then vj , h(xj ) = lim vj , Fm (xj )g(xj ) = g(xj ) lim Fm (xj )∗ vj = 0. m→∞

m→∞

The reverse inclusion is a little more involved. Let f ∈ [J g]. There exists a sequence Fm ∈ C(A) such that Fm g − f → 0. We need to modify Fm to a sequence F˜m such that F˜m g − f → 0 and F˜m ∈ J . Once again let X1 , . . . , Xp be the partition of the set {x1 , . . . , xn } given by the equivalence relation of point separation. Given a function a ∈ A we deﬁne a ⊗ ξ to be the function in C(A) given by (a ⊗ ξ)h = ah ⊗ ξ. Let ηk,m be the orthogonal projection of the vectors Fm (xnk ) onto the ﬁnite-dimensional subspace spanned by {vi : nk−1 < i ≤ nk }. Since, Fm g − f → 0, we have that Fm (xi )g(xi ) − f (xi ), vi = Fm (xi ), vi → 0 for i = 1, . . . , n. Hence, ηk,m → 0 as m → ∞. p Let Gm = k=1 ek ⊗ ηk,m , let 1 ≤ i ≤ n, and let l be such that nl−1 < i ≤ nl . We have, Gm (xi )∗ vi =

p

(ek ⊗ ηk,m )(xi )∗ vi =

k=1

p k=1

ek (xi ) vi , ηk,m .

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Since ek |Xl (x) = δk,l , the terms in the above sum for k = l are zero. Hence, the sum reduces to vi , ηl,m . Recall that ηl,m is the projection of F (xnl ) onto the span of the vectors {vi : nl−1 < i ≤ nl }. Hence, vi , ηl,m = vi , Fm (xnl ) = Fm (xnl )∗ vi . However, functions in A are constant on the sets Xk , which means Fm (xnl ) = Fm (xi ) and we get Gm (xi )∗ vi = Fm (xi )∗ vi for all i = 1, . . . , n. Hence, the function F˜m = Fm − Gm ∈ J . We have, p p p ek g ⊗ ηk,m = ek g ηk,m → 0 Gm g = (ek ⊗ Fm (xnk ))g ≤ k=1

k=1

k=1

as m → ∞. Finally, note that f ∈ Hg ⊗ 2 is orthogonal to kx ⊗ v if and only if f (x), v = 0. Hence, (Hg ⊗ 2 ) [J g] is the span of the vectors {kxi ⊗ vi : i = 1, . . . , n} = Kg . The distance formula, (8), in Theorem 3.1 shows that we must be able to classify the cyclic subspaces of the form (A ⊗ I)h, and to do so, we begin with a simple lemma. We will use the natural identiﬁcation between H ⊗ 2 and the 2 -direct sum of H, which we denote ⊕∞ j=1 H. Lemma 4.6. Let A ⊆ B(H) and let h ∈ H, then the cyclic subspace of H generated by C(A) and h is equal to [Ah] ⊗ 2 . Proof. The space C(A)h is generated by elements of the form ah ⊗ ej for j ∈ N. Hence, [Ah]⊗2 ⊆ [C(A)h]. On the other hand, an element of C(A)h is ∞ 2 2 of the form ⊕∞ j=1 aj h where j=1 aj h is ﬁnite and hence is in [Ah]⊗ . 4.3. Proof of Theorem 1.1 We will now prove our main theorem, Theorem 1.1, which is a tangential interpolation result for weak∗ -closed subalgebras of H ∞ . Let A ⊆ H ∞ be a unital weak∗ -closed subalgebra of H ∞ . So far, we have assumed no additional structure on the algebras. A function g ∈ H 2 is called outer if and only if H ∞ g is dense in H 2 . When A is subalgebra of H ∞ the space (C(A)⊗I)h, which is contained in (H 2 ⊗2 )⊗2 , can be identiﬁed with a subspace of H 2 ⊗2 of the form C(A)g for some outer function g. ∞ 2 a sequence in H 2 such that i=1 hi is finite. Lemma 4.7. Let {hi }∞ i=1 be ∞ 2 1 Then the function p(t) = i=1 |hi (t)| ∈ L (T) and there exists an outer 2 2 function g ∈ H such that p = |g| a.e. T. Proof. The fact that p ∈ L1 (T) is a straightforward argument. It is well-known that a non-negative function p ∈ L1 (T) is of the form 2 p = |g| for some outer function if and only if the function log p is summ 2 mable. Let pm = i=1 |hi | . If u1 denotes the outer part of h1 , then p1 = 2 2 |h1 | = |u1 | . Hence log p1 ∈ L1 . Since log p < p it follows, since p ∈ L1 , that + (log p) ∈ L1 . On the other hand we have log p ≥ log p1 and so (log p)− ≤ (log p1 )− . It follows that (log p)− ∈ L1 .

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2 2 let g be as in Lemma 4.7. If A Lemma 4.8. Let ⊕∞ i=1 hi ∈ H ⊗ and ∞ ∞ is a unital subalgebra of H , and h = j=1 hj ⊗ ej , then the map U : [(C(A) ⊗ I)h] → [C(A)g] defined by U [((MF ⊗ I)h)] = [MF g] is a unitary operator.

Proof. The map is clearly linear and surjective. We now show that U is isometric, which also proves that U is well-deﬁned. We have, ∞ 2 2 2 F g = |fi g| MF g = =

∞ i=1

=

∞

i=1

∞ ∞ ∞ 2 2 2 |fi | ( |hj | ) = |fi hj | j=1 2

j=1 i=1 2

MF hj = (MF ⊗ I)h .

j=1

We are now in a position to prove our main result, which is a tangential interpolation theorem for subalgebras of H ∞ . Given an outer function we will denote by Hg the closed subspace H 2 generated by elements of the form f g, where f ∈ A. We will denote by Kg the kernel function for this subspace. When g is the constant function g ≡ 1 we will suppress the subscript g. Proof of Theorem 1.1. We have seen that the existence of a solution F of norm at most α implies Qg ≥ 0 for all g. Hence, it is the converse that concerns us. Let us assume that there is at least one solution, say F0 , which exists by Lemma 4.5. We view A as a subalgebra of B(H). We deﬁne J to be the set of functions in C(A) such that F (xj )∗ vj = 0 for j = 1, . . . , n. Applying the distance formula (8), we get d(F0 , J ) = sup{| (MF0 ⊗ I)h, k |}, 2

where h ∈ H ⊗ , k ∈ (H ⊗ 2 ) ⊗ 2 , h = k = 1 and k ⊥ (J ⊗ I)h. By projecting onto the subspace [(C(A) ⊗ I)h] we can assume that k ∈ [(C(A) ⊗ I)h] [(J ⊗ I)h]. Let U be the unitary map from Lemma 4.8 and ∞ 2 2 let g be the outer function such that |g| = i=1 |hi | . We have, MF0 h, k = U MF0 h, U k = MF0 g, U k = MF0 g, k , where k = U k. This gives d(F0 , J ) = sup{| MF0 g, U k |} = sup{| MF0 g, k |}, where the supremum is over all outer functions g ∈ H 2 such that g = ∞ 2 2 1, |g| = j=1 |hi | , k ≤ 1, and k ∈ [C(A)g] [J g]. Lemma 4.5 shows that [J g] is the set of functions in f ∈ Hg ⊗ 2 such that f (xj ), vj = 0 for j = 1, . . . , n. Since f, kx ⊗ ξ = f (x), ξ , we see that Kg := [C(A)g] [J g] is the span of the vectors {kxg i ⊗ vi : i = 1, . . . , n}, where kxg is the kernel function for Hg at the point x.

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Therefore,

d(F0 , J ) ≤ sup g, MF∗0 k ≤ sup MF∗0 |Kg

where the supremum is taken over all outer functions g as above. If F ∈ J , then MF∗(kxg i ⊗ vi ) = F(xi )∗ vi kxg i = 0, and so MF∗ |Kg = 0. Hence, MF∗0 +F |Kg = MF∗0 |Kg from which it follows that d(F0 , J ) = F0 + F ≥ ∗ supg MF0 |Kg . The calculation leading to (5) shows that MF∗0 |Kg ≤ α if and only if the matrix Qg ≥ 0. Hence, the positivity of all the matrices Qg implies that d(F0 , J ) ≤ α. This in turn guarantees the existence of a solution to the tangential problem of norm at most α. The proof of Corollary 1.2 is a consequence of the following observation. If h ∈ H, then there exists a sequence fn ∈ A such that fn − h2 → 2 2 0. Hence, |fn | − |h| ≤ fn − h2 fn + h2 → 0 as n → ∞. There2

1

fore |h| ∈ L1 (A). If (hn ) ∈ H ⊗ 2 is a square summable sequence, then ∞ 2 ∈ L1 (A). Hence, the absolute value of the outer function g such n=1 |hn | 2 ∞ 2 that |g| = n=1 |hn | is an element of L2 (A). We have now established a tangential interpolation theorem for weak∗ closed subalgebras of H ∞ . Proposition 4.2 shows that the tangential interpolation result implies a T¨ oplitz corona theorem. This result can be viewed as analogous to the results obtained in Trent and Wick [16] and Douglas and Sarkar [10]. However, the method of proof is different. 4.4. Examples A better feeling for the result in Theorem 1.1 can be obtained by examining some special cases. We single out two classes of examples of subalgebras of H ∞. 1.

∞ Let B be an inner function and consider the algebra HB = C + BH ∞ . ∞ ∞ ∞ ∞ Note that B ∈ HB and so B ∈ L (HB ). Therefore, H = BBH ∞ is ∞ ∞ ) and we see that L∞ (HB ) = L∞ . contained in L∞ (HB We can also give a more explicit description of the subspaces Hg in this case. Let g be an outer function and let v = PH 2 BH 2 g. We claim that Hg = [v] ⊕ BH 2 . We have,

[(C + BH ∞ )g] = Cg + B[H ∞ g] = Cv ⊕ BH 2 . 2.

Let R be ﬁnite open Riemann surface. It is well-known that the universal covering space for R is the open unit disk D. Let p : D → R denote the covering map and let Γ denote the set of deck transformations, that is, automorphisms γ of the disk such that p ◦ γ = p. The automorphisms in the group Γ act on the disk and induce an action on the space H ∞ by composition. The set of ﬁxed points HΓ∞ is naturally identiﬁed with the space of bounded holomorphic functions on the Riemann surface. The automorphisms also act by bounded linear maps on the space H p and Lp and we use a subscript Γ to denote the associated set of ﬁxed points.

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In this case the algebra L∞ (HΓ∞ ) = L∞ Γ . The cyclic subspace Hg for an outer function |g| ∈ L2Γ can be described in terms of character automorphic functions. A character of Γ is a homomorphism from Γ into the circle group T and we denote the ˆ A function h ∈ H 2 is called character autospace of characters by Γ. ˆ such that h ◦ γ = σ(γ)h. The morphic if there exists a character σ ∈ Γ ∞ 2 2 closure of HΓ in H is the space HΓ . If g is an outer function such that ˆ such that g ◦ γ = σ(γ)g. |g| ∈ L2 (A), then there exists a character σ ∈ Γ In addition, the space Hg = Hσ2 := {f ∈ H 2 : f ◦ γ = σ(γ)f }. A proof of these facts can be found in [14]. Given a character σ we let K σ denote the reproducing kernel of Hσ2 . We get that the kernels of the character automorphic spaces Hσ2 , where ˆ have the tangential interpolation property for H ∞ . σ∈Γ Γ In Sect. 5, we will return to this example and show that we can replace this family of matrix positivity conditions by a single condition, at the expense of the optimal constant for the norm of a solution to the tangential interpolation problem. We point out that the tangential interpolation theorem gives us a new way to derive the Nevanlinna-Pick type interpolation results in [13,14] for the examples above.

5. Applications of Theorem 1.3: Similar Cyclic Modules Theorem 1.1 shows that the positivity of MF MF∗ ≥ δ 2 on a family of reproducing kernel Hilbert spaces is enough to guarantee the existence of a function G such that F G = 1 and MG ≤ δ −1 . This theorem is analogous to the results obtained in the work of [4,10,16]. If we drop the requirement that the function G have optimal norm, then in some cases we can replace the family of conditions by a single condition. Let Ix denote the ideal of functions in A such that f (x) = 0. We have already seen that [Ix g] is a codimension one subspace of Hg and that the orthogonal complement of Ix g is spanned by the kernel function kxg . Now let us return to the setting where A is a unital weak∗ -closed subalgebra of H ∞ and the function g is outer. We will establish a tangential interpolation theorem where we replace the family of conditions Qg ≥ 0 for all outer functions g such that |g| ∈ L2 (A) by a single positivity condition. However, we can not guarantee a solution of optimal norm. We will then apply our result to the case of ﬁnite open Riemann surfaces. Let g, h be two outer functions and let S : Hg → Hh be a bounded invertible operator that intertwines the action of A, that is, such that SMf = Mf S for all f ∈ A. By taking adjoints we see that Mf∗ S ∗ = S ∗ Mf∗ for all f ∈ A. If x ∈ D, then Mf∗ kxg = f (x)kxg and so we have Mf∗ S ∗ kxh = f (x)S ∗ kxh for all x ∈ D and f ∈ A. It follows that the vector S ∗ kxh is orthogonal to Ix g. Hence, S ∗ kxh = φ(x)kxg for all x ∈ D, where φ is a complex-valued function on the disk. In fact, φ is a multiplier from Hg → Hh and S = Mφ .

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Let H1 , H2 , K1 and K2 be n-dimensional Hilbert spaces. If A is a bounded operator from H1 to H2 and x1 , . . . , xn is a basis for the space H1 , then A ≤ α if and only if the matrix [α2 xj , xi − Axj , Axi ] is positive invertible transformation. semidefinite. Nowlet Si ∈ B(H i , Ki ) be a bounded If A ≤ α, then S2 AS1−1 ≤ α S2 S1−1 . Consider the special case where H1 = span{kxg 1 ⊗v1 , . . . , kxg n ⊗vn }, K1 = span{kxh1 ⊗ v1 , . . . , kxhn ⊗ vn }, H2 = span{kxg 1 , . . . , kxg n }, and K2 = span{kxh1 , . . . , kxhn }. Let A be the map A(kxg i ⊗vi ) = wi kxg i , let S = Mφ be the similarity between Hg and Hh described above, let S1 = S ∗ ⊗ I, and let S2 = S ∗ . ∗ ∗ Note that −1ifF ∈ C(A), with F (xi ) vi = wi , then MF |H1 is precisely A. If S S = c, then a straightforward computation shows that A ≤ α if and only if [(α2 vj , vi − wi wj )Kg (xi , xj )] ≥ 0. Hence the norm of S ∗ A((S ∗ )−1 ⊗ I) is at most cα which in turn implies that [(c2 α2 vj , vi − wi wj )Kh (xi , xj )] ≥ 0. We are now in a position to prove Theorem 1.3. Proof of Theorem 1.3. From the observations made above we see that the matrix positivity condition implies that [(α2 c2 vj , vi −wi wj )K g (xi , xj )] ≥ 0. The result in (1) now follows from Theorem 1.1. The proof of (2) follows, as before, from the tangential interpolation theorem established in (1). We now provide an example of a class of subalgebras of H ∞ to which the above theorem applies. Recall that if R is a ﬁnite open Riemann surface, and Γ is the associated group of deck transformations acting on the disk, then the ﬁxed-point algebra HΓ∞ is naturally identiﬁed with H ∞ (R). In this case the outer function g has the property that there is a character σ such that g ∈ Hσ2 , and Hg = Hσ2 . We will establish the existence of a similarity Sσ = Mφσ between HΓ2 and Hσ2 and show that there is a uniform bound on Sσ Sσ−1 . This result generalizes a theorem of Ball [7] from the setting of multiply-connected domains to Riemann surfaces. Proposition 5.1. Let Γ be a the group of deck transformations associated to a ˆ there exists a bounded invertible finite open Riemann surface. For each σ ∈ Γ, 2 2 function φσ such that φσ HΓ = Hσ . There is a constant β, independent of σ such that β −1 ≤ |φσ | ≤ β. We will need two results of Forelli [11]. Theorem 5.2 (Forelli). Let Γ be the group of deck transformations associated to a finite open Riemann surface R of genus m. Let γ1 , . . . , γm denote the generators of Γ. There exist m vectors v1 , . . . , vm ∈ L∞ Γ such that vi is 2 . In addition, v , . . . , v non-negative, and vi is orthogonal to HΓ2 ⊕ HΓ,0 1 m are linearly independent. ∗ If f is a real-valued function in L2 , then its conjugate function ∗ f is the 2 ∗ 2 unique real-valued function in L such that f + if ∈ H and f = 0.

Theorem 5.3 (Forelli). Let f ∈ L2Γ and let f ∗ denote the function conjugate to f , then f ∗ ◦ γi − f is constant, and the constant is given by f vi .

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Now we present the proof of Proposition 5.1. Proof of Proposition 5.1. Let γ1 , . . . , γm be a minimal set of generators of the group Γ. We let σk = σ(γk ). Since σ is a character, there exists θ1 , . . . , θm ∈ [0, 2π) such that σk = eiθk . Let vk,l = vl , vk for k, l = 1, . . . , m. Note that vl∗ ◦ γk = vl∗ + vk vl = vl∗ + vk , vl . Since v1 , . . . , vm are linearly independent the matrix V = [vk,l ] is invertible. Let c = (c1 , . . . , cm )t be the unique vector such that V c = (θ1 , . . . , θm )t . Since the entries of V are real we see that V −1 has real entries. m Hence, the vector m c ∈ R . Let f = k=1 ck vk . Note that f is a real-valued element of L∞ . Following the construction in [11] we let φσ = exp(f + if ∗ ). Since f is real-valued we see that |φσ | = exp(f ) and so φσ is bounded. Now, m cl vk,l φσ ◦ γk = exp f + if ∗ + i l=1

m = exp i cl vk,l φσ = exp(iθk )φσ = σ(γk )φσ . l=1

Hence, φσ ∈ Hσ∞ . m We have | k=1 ck vk | ≤ maxk=1,...,m vk ∞ c1 . Since θ1 , . . . , θm ∈ [0, 2π) there exists a constant K, that does not depend on σ, such that c1 ≤ K. Hence, there is a constant K such that e−K ≤ |φσ | ≤ eK for all ˆ σ ∈ Γ.

References [1] Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces. In: Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence (2002) [2] Agler, J., McCarthy, J.E.: Nevanlinna-Pick interpolation on the bidisk. J. Reine Angew. Math. 506, 191–204 (1999) [3] Agler, J., McCarthy, J.E.: What Hilbert spaces can tell us about bounded functions in the bidisk. http://www.arxiv.org/0901.0907 [4] Amar, E.: On the To¨eplitz corona problem. Publ. Math. 47(2), 489–496 (2003) [5] Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950) [6] Arveson, W.: Interpolation problems in nest algebras. J. Funct. Anal. 20(3), 208–233 (1975) [7] Ball, J.A.: Interpolation problems and Toeplitz operators on multiply connected domains. Integr. Equ. Oper. Theory 4(2), 172–184 (1981) [8] Ball, J.A., Trent, T.T.: Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables. J. Funct. Anal. 157(1), 1–61 (1998) [9] Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. (2) 76, 547–559 (1962)

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[10] Douglas, R.G., Sarkar, J.: Some remarks on the Toeplitz corona problem. Proc. CRM 51, 81–90 (2010) [11] Forelli, F.: Bounded holomorphic functions and projections. Ill. J. Math. 10, 367–380 (1966) [12] McCullough, S.: Nevanlinna-Pick type interpolation in a dual algebra. J. Funct. Anal. 135(1), 93–131 (1996) [13] Raghupathi, M.: Nevanlinna-Pick interpolation for C + BH ∞ . Integr. Equ. Oper. Theory 63(1), 103–125 (2009) [14] Raghupathi, M.: Abrahamse’s interpolation theorem and Fuchsian groups. J. Math. Anal. Appl. 355(1), 258–276 (2009) [15] Schubert, C.F.: The corona theorem as an operator theorem. Proc. Am. Math. Soc. 69(1), 73–76 (1978) [16] Trent, T.T., Wick, B.D.: Toeplitz corona theorems for the polydisk and the unit ball. Complex Anal. Oper. Theory 3(3), 729–738 (2009) Mrinal Raghupathi (B) Department of Mathematics Vanderbilt University Nashville, TN 37240, USA e-mail: [email protected] URL: http://www.math.vanderbilt.edu/∼mrinalr Brett D. Wick School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-1060, USA e-mail: [email protected] URL: http://people.math.gatech.edu/∼bwick6/ Received: December 11, 2009. Revised: March 18, 2010.

Integr. Equ. Oper. Theory 68 (2010), 357–381 DOI 10.1007/s00020-010-1793-8 Published online April 14, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On the Perturbations of Regular Linear Systems and Linear Systems with State and Output Delays Zhan-Dong Mei and Ji-Gen Peng Abstract. This paper is concerned with perturbation problems of regularity linear systems. Two types of perturbation results are proved: (i) the perturbed system (A + P, B, C) generates a regular linear system provided both (A, B, C) and (A, B, P ) generate regular linear systems; and (ii) the perturbed system ((A−1 + ΔA)|X , B, CΛA ) generates a regular linear system if both (A, B, C) and (A, ΔA, C) generate regular linear systems. These allow us to establish a new variation of constants formula of the control system (A + P, B). Moreover, these results are applied to the linear systems with state and output delays. The regularity and the mild expressibility is deduced, and a necessary and sufﬁcient condition for stabilizability of the delayed systems is proved. Mathematics Subject Classification (2010). Primary 47A55; Secondary 93C23. Keywords. Regular linear systems, perturbation, admissibility, C0 -semigroup, stabilizability.

1. Introduction The class of well-posed linear systems introduced by Salamon in [21] has become a well-understood class of systems (see, e.g., [24,28,30,31]). Many partial differential equations with boundary control and point observation can be formulated as well-posed linear systems [17,21,22]. In a well-posed linear system, the input and output functions are locally in Lp , and on any ﬁnite time-interval, the ﬁnal state and output function depend continuously on the initial state and the input function. A regular linear system is among the well-posed systems whose output function corresponding to a step input function and zero initial state is not very discontinuous at zero (in detail, This work was supported by the NSFC under the contact 60970149.

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see the definition in Sect. 2). Regular linear systems constitute a large subclass of well-posed linear systems, whose basic properties are rich enough to develop a parallel of the theory of control systems with bounded control and observation operators, as presented in Curtain and Pritchard [4]. Weiss showed in [28] that regular linear systems with unbounded control and observation operators allow nice generalizations of ﬁnite dimensional systems by admitting the differential representation x (t) = Ax(t) + Bu(t), y(t) = CΛ x(t) + Du(t), where CΛ is the Λ-extension of the observation operator C with respect to system operator A (see Sect. 2). Admissibility of control as well as observation operators are necessary for a linear system to be regular (see, e.g., [21,22,28,30,31]). However, unfortunately, it is hard to test the admissibility, not to mention the well-posedness and regularity of an inﬁnite-dimensional linear system with unbounded control and observation. There are many papers devoted to discussion on the admissibility of control and observation operators, most of which are interested in proving or disproving Weiss’ conjecture (see, e.g., [6,12–15,25,29, 34,35]). Here, we mention an important work due to Zwart [35]; he proved that the Weiss conjecture almost holds in Hilbert spaces and in the case that p = 2. Sometimes, if A generates a C0 -semigroup on the state space and the admissibility of the control operator (or observation operator) for A is easy to check, we can write the generator of semigroup of the system as a sum of two simple operators, (A−1 + ΔA)|X (or A + P ). Thus the perturbation method can be used to test the admissible for the considered generator of semigroup of the system (see [8,9]). Inspired by such perturbations of admissibility, we consider to use perturbation methods to test whether an inﬁnite dimensional linear system is regular or not. Clearly, output feedback can be regarded as an perturbation [31]. However, in practical applications, it is hard to check that whether an inﬁnite dimensional system is the feedback of another one or not; in contrast, it is easier to decompose the generator of semigroup of the system into a sum of two operators as in [8,9]. Motivated by this, in this paper, we consider the later scheme. More specifically, we shall prove in Sect. 3 two types of perturbation results. In detail, if both (A, B, C) and (A, B, P ) generate regular linear systems on appropriate Banach spaces, (A + P, B, C) generates a regular linear system; if both (A, B, C) and (A, ΔA, C) generate regular linear system on appropriate Banach spaces, ((A−1 + ΔA)|X , B, C) generates a regular linear system. Furthermore, in Sect. 4, we will see that such perturbation results allow us to establish a new variation of constants formula for linear control system with perturbation in generator of the system semigroup. An important application of our perturbation results is to deal with delay differential systems. It is well-known that such systems arise in the study of many problems with theoretical and practical importance. Semigroup theory was ﬁrst systematically applied to delay systems in the book [11] for ﬁnite dimensional state spaces. B´ atkai [1] generated such method to delay differential systems in general Banach space setting. Since then,

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there have many mathematicians followed the work (see e.g [2,7,9,10,18]). In Sect. 5, we apply our perturbation results to systems with state and output delays. Similar to [1], we establish in the sense of classical solution the equivalent relationship of delay system and the extended system without delay. Moreover, using the results developed in Sect. 4, we obtain a new variation of constants formula for functional differential equations in Lp -phase spaces in terms of the Λ-extension of the delay operator. In the ﬁnal section we consider by our perturbation results the stabilizability of delay differential systems. Such property is specially important for delay systems because it has been recognized that the delay presence in the state could induce bad performance and complicate controller design and system analysis (see [33]). We know that most authors focus on the stabilizability of state delay systems with a ﬁnite dimensional delay-free state space. It is shown by Olbrot [19] that the stabilizability of the state-input delay system x(t) ˙ = A0 x(t) + A1 x(t − 1) + P u(t) is equivalent to the condition Rank[Δ(λ), P ] = n,

(1.1)

for λ ∈ C with Reλ ≥ 0, where n is the dimension of the delay-free system and Δ(λ) : = λI − A0 − A1 e−λ . It is not difﬁcult to prove that (1.1) is equivalent to Range[Δ(λ), P ] = Cn×n .

(1.2)

Combining our perturbation results and [32], we will develop a necessary condition extending (1.2) to system with state delay and unbounded control term. Moreover, if the input space is of ﬁnite dimensions and the semigroup is immediately compact, the condition becomes sufﬁcient. Throughout this paper, we assume p ∈ (1, ∞) (see Definition 2.1). Let X be a Banach space, we denote by M |X = {x ∈ D(M ) : M x ∈ X} the part of M in X and IX by the identity operator on X. Assume that A generates a C0 -semigroup T := (T (t))t≥0 on X, without any additional statement. And X1 denotes the domain D(A) equipped with the graph norm. By definition in [5], the extrapolation space X−1 is the completion of X under the norm R(λ0 , A) · with R(λ0 , A) the resolvent of A at λ0 . Denote by {T−1 (t)}t≥0 the extrapolation semigroup of {T (t)}t≥0 on X−1 , with A−1 denoting its generator. It follows from [5] that A−1 is the continuously extension from D(A) to X. Deﬁne Pτ by (Pτ f )(t) = f (t) when 0 ≤ t ≤ τ and (Pτ f )(t) = 0 when t > τ . Denote R+ = [0, ∞). For u, v ∈ Lp (R+ , U ) and τ ≥ 0, the τ -concatenation of u and v, denoted by u♦τ v, is deﬁned by (u♦τ v)(t) =

u(t), t < τ, v(t − τ ), t ≥ τ.

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2. Preliminaries on Regular Linear Systems This section is to recall in a very sketchy way some concepts and results related to regular linear system in the sense of Salamon [21] and Weiss [30], those are to be used as the main tool throughout our work. Throughout this section X, U and Y are Banach spaces. Definition 2.1. Let Ω = Lp (R+ , U ) and Γ = Lp (R+ , Y ). A well-posed linear system on Ω, X, and Γ is a quadruple Σ = (T, Φ, Ψ, F ), where: (i) T = (T (t))t≥0 is a C0 -semigroup of bounded linear operators on X. (ii) Φ = (Φ(t))t≥0 is a family of bounded linear operators from Ω to X such that Φ(t + τ )(u♦τ v) = T (t)Φ(τ )u + Φ(t)v, (iii)

for any u, v ∈ Ω and any τ, t ≥ 0. Ψ = (Ψ(t))t≥0 is a family of bounded linear operators from X to Γ such that Ψ(t + τ )x = Ψ(τ )x♦τ Ψ(t)T (τ )x,

(2.1)

for any x ∈ X and any τ, t ≥ 0, and Ψ(0) = 0. (iv) F = (F (t))t≥0 is a family of bounded linear operators from Ω to Γ such that F (t + τ )(u♦τ v) = F (τ )u♦τ (Ψ(t)Φ(τ )u + F (t)v),

(2.2)

for any u, v ∈ Ω and any τ, t ≥ 0, and F (0) = 0. U is the input space of Σ, X is the state space of Σ, and Y is the output space of Σ. The operators Φ(τ ) are called input map. The operators Ψ(τ ) are called output map. The operators F (τ ) are called input/output map. By the representation theorem due to Weiss [26], there exists a unique operator B ∈ L(U, X−1 ), called admissible control operator for A, such that for any t ≥ 0 and u ∈ Lploc (R+ , U ), t T−1 (t − s)Bu(s)ds ∈ X

Φ(t)u = 0

where the integral exists in X−1 . In this case, we say that (A, B) generates an abstract linear control system on (X, U ) and denote Φ = ΦA,B for brief. By [27,28], there are unique operators Ψ(∞) : X → L2loc (R+ , Y ) and F (∞) : Lploc (R+ , U ) → Lploc (R+ , Y ) such that, for any τ ≥ 0, the operators Ψ(τ ) and F (τ ) are obtained by truncation: Ψ(τ ) = Pτ Ψ(∞), F (τ )(t) = F (∞)(t), for any t ≤ τ. We call Ψ(∞) the extended output map of Σ, and F (∞) the extended input/output map of Σ. It is not difﬁcult to obtain the following two equations related to the extended output map and the extended input/output map: Ψ(∞)x = Ψ(∞)x♦τ Ψ(∞)T (τ )x, ∀x ∈ X, ∀τ ≥ 0,

(2.3)

F (∞)(u♦τ v) = F (∞)u♦τ (Ψ(∞)Φ(τ )u + F (∞)v),

(2.4)

and

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for any u, v ∈ Lploc (R+ , U ) and τ ≥ 0. By the representation theorem in [27], there exists a unique operator C ∈ L(X1 , Y ), called admissible observation operator for A, such that for any t ≥ 0 and x ∈ D(A), CT (t)x = (Ψ∞ x)(t). In this case, we say that (A, C) generates an abstract linear observation system on (X, Y ) and denote Ψ = ΨA,C for brief. We say that the well-posed linear system Σ is regular if the limit t 1 lim (F∞ u0 )(s)ds = Dz (2.5) t→0 t 0

exists in Y for the constant input u0 (t) = z, z ∈ U, t ≥ 0. In this case, we also say that the quadruple Σ = (T, Φ, Ψ, F ) is a regular linear system on (X, U, Y ) generated by (A, B, C, D), and we denote ΣA,B,C,D = Σ and FA,B,C = F . Furthermore, we denote (A, B, C) = (A, B, C, 0) and ΣA,B,C = ΣA,B,C,0 for brief. In order to introduce the representation theorem of regular linear system, Weiss [30] introduced an extension of C, called Λ-extension with respect to A, which is deﬁned by CΛ x = lim CλR(λ, A)x λ→∞

(2.6)

with the domain D(CΛ ) = {x ∈ X : this above limit exists in Y }. It follows from [27, Theorem 4.5 and Proposition 4.7] that for any x ∈ X, y(t) = CΛ T (t)x a.e. in t ≥ 0 whenever C is admissible for A. With the above Λ-extension, for well-posed linear system ΣA,B,C , the regularity condition (2.5) is equivalent to each of the following two conditions: • Range(R(λ, A−1 )B) ⊂ D(CΛA ) holds for some (and hence for all) λ ∈ ρ(A). • For any u ∈ U , G(λ)u has a limits when λ → ∞, where G is the transfer function associated to F (∞). In this case, the transfer function G is given explicitly by G(λ) = CΛA R(λ, A−1 )B + D, Re(λ) > w0 (T ),

(2.7)

where w0 (T ) is the growth bound of the semigroup T , and we denote GA,B,C = G. In order to state the following theorem, we deﬁne Dp (M ) = {f (·) ∈ Lploc (R+ , X) : f ∈ D(M ) f or a.e. t ≥ 0, and M f (·) ∈ Lploc (R+ , X)}. Theorem 2.2. [28] Let Σ be a regular linear system with generating operator A, B, C and D on (X, U, Y ). Then, for given (x0 , u) ∈ X × Lp (R+ , U ), the state trajectory x(·) of Σ, given by x(t) := T (t)x0 + ΦA,B u, t ≥ 0, is a.e. differential in X−1 and x(t) ˙ = A−1 x(t) + Bu(t), x(0) = x0 f or a.e. t ≥ 0.

(2.8)

Furthermore, x(t) ∈ for a.e. t ≥ 0 and the output function y = ΨA,C (∞)x + FA,B,C,D (∞)u of Σ is given by D(CΛA )

y(t) = CΛA x(t) + Du(t) f or a.e. t ≥ 0.

(2.9)

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In particular, ΦA,B (·)u ∈ Dp (CΛA ) and the extended input–output map F (∞) is given by t (F (∞)u)(t) =

T−1 (t − s)Bu(s)ds + Du(t) f or a.e. t ≥ 0.

CΛA

(2.10)

0

Definition 2.3. [24,31] Let Σ be a regular linear system with input/output map F (t). An operator Γ ∈ L(Y, U ) is called an admissible feedback for Σ if IY − F (·)Γ has uniformly bounded inverse. Theorem 2.4. [24,31] Assume that (A, B, C) generates a regular linear system Σ = (T, Φ, Ψ, F ) on (U, X, Y ) with admissible feedback operator Γ. Then feedback system ΣΓ is also a regular system given by Γ T (·) ΦΓ (·) Γ Σ = ΨΓ (·) F Γ (·) T (·) + Φ(·)Γ(IY − F Γ)−1 Ψ Φ(·)(IU − ΓF (·))−1 = (IU − ΓF (·))−1 Ψ F (IU − ΓF (·))−1 with the generating (AΓ , B Γ , C Γ ): AΓ = (A−1 + BΓCΛ )|X D(AΓ ) := {z ∈ D((C)Λ ) : (A−1 + BΓCΛ )z ∈ X} B Γ = B, C Γ = CΛA . Γ

In addition, D(CΛA ) = D(CΛA ).

3. Perturbations of Regular Linear Systems In this section, we develop two perturbation theorems of regular linear system based on the feedback theory [31]. Such results will be used throughout this paper. To do this, we ﬁrst introduce some lemmas. Throughout this section X, U and Y are Banach spaces and we denote by I the identity operator on appropriate Banach space. Lemma 3.1. Let (A, P ) generate an abstract observation system on (X, X), then (A, IX , P ) generates a regular linear system with admissible feedback operator IX . Proof. By [7, Proposition 3.3], for any f ∈ Lploc (R+ , X), ΦA,I (·)f ∈ Dp (PΛ ) and PΛ ΦA,I (·)f Lp ([0,t],X) ≤ K(t)f Lp ([0,t],X) , where K(t) → 0 as t → 0. Let F (τ ) = Pτ PΛ Φ(·), it is not difﬁcult to verify that F satisﬁes (2.2). It is not hard to show that (A, IX , P ) generates a regular linear system. Moreover, since K(t) → 0 as t → 0, F (τ ) → 0 as τ → 0. Hence IX is an admissible feedback operator. Lemma 3.2. Assume that (A, B, P ) generates a regular linear system on (X, U, X). Then (A + P, B) generates an abstract linear control system. Furthermore, we have ΦA+P,B = ΦA+P,I FA,B,P + ΦA,B .

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˜ := Proof. Similar to the proof of [8, Theorem 3.3], consider the operators B P (I, B) : X × U → X−1 , C˜ = : X → X × U. 0 ˜ C) ˜ By the definition and Lemma 3.1, it is easy to prove that (A, B, generates a regular linear system given by ⎞ ⎛ T (ΦA,I , ΦA,B ) ⎟ ⎜ ΣA,B, ˜ C ˜ := ⎝ FA,I,P FA,B,P ⎠. ΨA,P 0 0 0 with admissible feedback operator IX×U . By Theorem 2.4, it follows that ˜ IX×U = B, ˜ B is admissible for A + P and AIX×U = A + P , B ΦA+P,B˜ IX×U = (ΦA+P,I , ΦA+P,B ) (I − FA,I,P )−1 = (ΦA,I , ΦA,B ) 0

(I − FA,I,P )−1 FA,B,P I

.

So we obtain ΦA+P,B = ΦA,I (I − FA,I,P )−1 FA,B,P +ΦA,B = ΦA+P,I FA,B,P + ΦA,B . (3.1) The following lemma is due to Hadd and Idrissi [9]. Lemma 3.3. Let (A, P ) and (A, C) be abstract linear observation systems. Then D(CΛA ) ∩ D(PΛA ) = D(CΛA+P ) ∩ D(PΛA ), and CΛA+P x = CΛA x, for any x ∈ D(CΛA ) ∩ D(PΛA ). Now we can prove the regularity invariance under some regularity perturbation. Theorem 3.4. Assume that (A, B, C) and (A, B, P ) generate regular linear systems on (X, U, Y ) and (X, U, X), respectively. Then (A + P, B, C) generates a regular linear system with FA+P,B,C = FA+P,I,C FA,B,P + FA,B,C . Proof. By assumption, (A, B, P ) and (A, B, C) generate regular linear systems, it follows from Theorem 2.2 that Range(ΦA,B (t)) ⊂ D(CΛA ) ∩ D(PΛA ) for a.e. t ≥ 0. By lemma 3.3, we obtain Range(ΦA,B (t)) ⊂ D(CΛA+P ) and CΛA+P ΦA,B (t) = CΛA ΦA,B (t), for a.e. t ≥ 0. Moreover, we derive from Lemma 3.1 that (A+P, I, C) generates a regular linear system, Range(ΦA+P,I (t))) ⊂ D(CΛA+P ) for a.e. t ≥ 0. Thus, by (3.1), ΦA+P,B (t)u = ΦA+P,I (t) FA,B,P (t)u + ΦA,B (t)u ∈ D(CΛA+P ) for any u ∈ Lploc (R+ , U ) and a.e. t ≥ 0. Hence we can deﬁne F : (F u)(t) = CΛA+P ΦA+P,B (t)u for any u ∈ p Lloc (R+ , U ) and a.e. t ≥ 0. By simple calculation, we obtain that F = FA+P,I,C (∞)FA,B,P (∞) + FA,B,C (∞). We can see from the definition of F that F0 = (Pt F )t≥0 satisﬁes (2.2). Taking Laplace transform on F , we obtain G(λ) = GA+P,I,C (λ)GA,B,P (λ) + GA,B,F (λ), thereby, for any z ∈ U , G(λ)z → 0 as λ → 0. The assertion follows immediately. In the above theorem, let Y = X and P = C, we obtain the following corollary:

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Corollary 3.5. Assume that (A, B, C) generates a regular linear system on (X, U, X). Then, (A + C, B, C) generates a regular linear system with the relation FA+C,B,C = FA+C,I,C FA,B,C + FA,B,C . Next, we will prove the second perturbation theorem. To do this, we begin with some important lemmas. Lemma 3.6. Assume that (A, ΔA, C) generates a regular linear system on (X, X, Y ). Then, ((A−1 + ΔA)|X , C) generates an abstract linear observa(A +ΔA)|X tion system. Furthermore, CΛ −1 = CΛA . Proof. Similar to the proof of [8,Theorem 3.3], consider the operators B := I (ΔA, 0) : X × Y → X−1 , C = : X → X × Y . Obversely, D(CΛA ) = C I A A D(CΛ ), CΛ = . CΛA It is not hard to prove that (A, B, C ) generates a regular linear system given by ⎞ ⎛ T (ΦA,ΔA , 0) ⎜ ⎟ ΣA,B,C := ⎝ ΨA,I FA,ΔA,I 0 ⎠ ΨA,C FA,ΔA,C 0 By Theorem 2.4, it follows that with admissible feedback operator IX×Y . I (A−1 +ΔA)|X (A +ΔA)|X IX×Y = (A−1 + ΔA)|X , CΛ = )= with D(CΛ −1 A CΛA (A−1 +ΔA)|X

D(CΛA ) and C is admissible for (A + ΔA)|X . Furthermore, CΛ CΛA .

=

Lemma 3.7. Let (A, ΔA) and (A, B) generate abstract linear control systems on (X, X) and (X, U ), respectively. Then ((A−1 + ΔA)|X , B) generates an abstract linear control system. Moreover, Φ(A−1 +ΔA)|X ,B = Φ(A−1 +ΔA)|X ,ΔA FA,B,I + ΦA,B . Proof. By [8, Theorem 3.3], it follows that ((A−1 + ΔA)|X , B) generates an abstract linear control system and (Φ(A−1 +ΔA)|X ,ΔA , Φ(A−1 +ΔA)|X ,B ) (I − FA,ΔA,I )−1 = (ΦA,ΔA , ΦA,B ) 0

(I − FA,ΔA,I )−1 FA,B,I I

.

Thus, Φ(A−1 +ΔA)|X ,B = ΦA,ΔA (I − FA,ΔA,I )−1 FA,B,I + ΦA,B = Φ(A−1 +ΔA)|X ,ΔA FA,B,I + ΦA,B . Theorem 3.8. Suppose that (A, B, C) and (A, ΔA, C) generate regular linear systems on (X, U, Y ) and (X, X, Y ), respectively. Then ((A−1 + ΔA)|X , B, CΛA ) generates a regular linear system with the relation

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F(A−1 +ΔA)|X ,B,CΛA = FA,ΔA,C (I − FA,ΔA,I )−1 FA,B,I + FA,B,C . (A

+ΔA)|

X Proof. By Lemma 3.6 and Lemma 3.7, it follows that CΛ −1 = CΛA −1 and Φ(A−1 +ΔA)|X ,B = ΦA,ΔA (I − FA,ΔA,I ) FA,B,I + ΦA,B . The combination of this and the assumption that (A, B, C) generates a regular linear (A +ΔA)|X system implies that Range(Φ(A−1 +ΔA)|X ,B (t)) ⊂ D(CΛ −1 ) for a.e.

(A

+ΔA)|

X Φ(A−1 +ΔA)|X ,B (t)u, t ≥ 0. Hence, we can deﬁne F : (F u)(t) = CΛ −1 p + for a.e. t ≥ 0 and any u ∈ Lloc (R , U ). Through a simple calculation, F = FA,ΔA,C (∞)(I −FA,ΔA,I (∞))−1 FA,B,I (∞)+FA,B,C (∞). We see from the definition of F that F1 = (Pt F )t≥0 satisﬁes (2.2). Moreover, the transfer function G of F satisﬁes G(λ) = GA,ΔA,C (λ)(I−GA,ΔA,I (λ))−1 GA,B,I (λ)+GA,B,C (λ), thereby, for any z ∈ U , G(λ)z → 0 as λ → 0. By the definition of regular linear systems, our assertion follows immediately.

In the above theorem, let U = X and ΔA = B, we can obtain the following corollary: Corollary 3.9. Let (A, B, C) generate a regular linear system on (X, X, Y ), then ((A−1 + B)|X , B, CΛA ) generates a regular linear system.

4. Variation of Constants Formula of Admissible Observation Perturbed Control System In this section, we will see that our perturbation results allow us to establish a new variation of constants formula for the mild solutions of (4.1) in terms of the initial semigroup T (t) and the Λ-extension PΛA of P with respect to A. To obtain the main theorem in this section, we ﬁrst introduce the following lemma due to Hadd [7]: Lemma 4.1. Assume X to be a Banach space. Let P ∈ L(X1 , X) be an admissible observation operator for A. Then, we have t T (t − s)PΛA TA+P (s)xds,

TA+P (t)x = T (t)x + 0

for any x ∈ X and t ≥ 0. Here T = (T (t))t≥0 and TA+P = (TA+P (t))t≥0 are the C0 -semigroups generated by A and A + P , respectively. Now we state our variation of constants formula as follows. Theorem 4.2. Let (A, B, P ) generate a regular linear system on the triple of Banach spaces (X, U, X). Then the mild solution of the following equation x(t) ˙ = (A + P )x(t) + Bu(t), t ≥ 0 with x(0) = x satisfies x(·) ∈ D

p

(PΛA )

(4.1)

and is given by

t T (t − s)PΛ x(s)ds + ΦA,B (t)u,

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

(4.2)

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A . for any x ∈ X and u ∈ Lp (R+ , U ). Moreover x(·) is a.e. differential in X−1

Proof. In (3.1), taking B = I, we obtain ΦA+P,I (t) = ΦA,I (t) + ΦA,I (t)(I − FA,I,P (t))−1 FA,I,P (t) = ΦA,I (t)(FA+P,I,P (t) + I). Multiplying the above equation by FA,B,P (t) from the right, we obtain ΦA+P,I (t)FA,B,P (t) = ΦA,I (t)(FA+P,I,P (t)FA,B,P (t) + FA,B,P (t)).

(4.3)

From (2.10), Corollary 3.3 and 3.5, we obtain that for u ∈ Lp (R+ , U ), ΦA+P,I (t)FA,B,P (t)u = ΦA,I (t)FA+P,B,P (t)u t T (t − s)PΛA+P ΦA+P,B (s)uds. =

(4.4)

0

On the other hand, it follows from [7] that TA+P (·)x ∈ Dp (PΛA ) for any x ∈ X. By the definition of state trajectory and (3.1), we have x(·) = TA+P (·)x + ΦA+P,B (t)u = TA+P (·)x + ΦA+P,I (·)FA,B,P (·)u + ΦA,B (·)u ∈ Dp (PΛA ). (4.5) Substituting (4.5) into (4.4), we obtain t T (t − s)PΛA+P (x(s) − TA+P (s)x)ds.

ΦA+P,I (t)FA,B,P (t)u = 0

t Observe that by Theorem 4.1, TA+P (t)x = T (t)x+ 0 T (t−s)PΛA TA+P (s)xds, so we obtain t x(t) = T (t)x + T (t − s)PΛA TA+P (s)xds 0

t T (t − s)PΛA+P (x(s) − TA+P (s)x)ds + ΦA,B (t)u

+ 0

t T (t − s)PΛ x(s)ds + ΦA,B (t)u,

= T (t)x + 0

which is just (4.2).

5. The Classical and Mild Solutions of Linear Systems with State and Output Delays As an application example of the results obtained in the last two sections, we consider in this section the following linear system with time delays in state

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and output: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

= A−1 x(t) + Lxt + Bu(t), t ≥ 0, x(0) = z, x0 = f, y(t) = Cx(t) + Dxt ,

d dt x(t)

(5.1)

where the state space X, input space U and output space Y are Banach spaces, A : D(A) ⊂ X → X be a generator of C0 -semigroup on X, L ∈ A ), z ∈ X, f ∈ Lp ([−1, 0], X), 1 < L (W 1,p ([−1, 0], X), X), B ∈ L (U, X−1 p < ∞, C ∈ L (D(A), Y ) where D(A) is equipped with the graph norm, and D ∈ L (W 1,p ([−1, 0], X), Y ). Similar to [1,10], we give the definition of the classical solution of system (5.1) as follows. 1,p (R+ , U ), we call x(·) to be the classical solution Definition 5.1. Let u ∈ Wloc of system (5.1) if

(i) x(·) ∈ C 1 ([0, ∞), X) ∩ C([−1, ∞), X), (ii) A−1 x(t) + Bu(t) ∈ X and xt ∈ W 1,p ([−1, 0], X) for any t ≥ 0, d x(t) = A−1 x(t) + Lxt + Bu(t), for any t ≥ 0. (iii) x(·) satisﬁes dt In order to investigate system (5.1), similar to [1], we convert it into a system without delay. To do this, we consider in this section the linear system with the state space E = X × Lp ([−1, 0], X), input space U and the output space Y : ⎧ d X (t) = (AL )−1 X (t) + Bu(t), ⎪ ⎪ ⎨ dt z (5.2) X (0) = , f ⎪ ⎪ ⎩ y(t) = CX (t), A L where AL = , whose domain is given by d 0 dσ D(AL ) = B=

B 0

x ∈ D(A) × W 1,p ([−1, 0], X) : f (0) = x , f

and C = (C, D), whose domain is D(C) × Lp ([−1, 0], X).

1,p Definition 5.2. For u ∈ Wloc (R+ , U ), we call X : R+ → E to be a classical solution of system (5.2), if (AL )−1 X (t) + Bu(t) ∈ E and X (t) ∈ C 1 (R+ , E ) for t ≥ 0.

Write AL as AL = A + ΔA, with A 0 0 L A= , ΔA = . d 0 0 0 dσ

(5.3)

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According to [1], A, with the domain D(A) = D(AL ), generates a C0 -semigroup on E given by T (t) 0 , (5.4) T (t) = SX (t) Tt where T (t) is the C0 -semigroup generated by A, Tt : X → Lr ([−1, 0], X) are operators deﬁned by T (t + θ)x, t + θ ≥ 0 (Tt x)(θ) = 0, if not and (SX (t))t≥0 is the left shift semigroup on Lp ([−1, 0], X) with generator QX deﬁned by d on D(QX ) = {f ∈ W 1,p ([−1, 0], X) : f (0) = 0}. dσ Moreover, it is easy to show that ΔA ∈ L (D(A), E ), where D(A) is equipped with the graph norm. Assume that φ = φ(t)t≥0 is a family of bounded linear operators from p L (R+ , X) to Lp ([−1, 0], X) deﬁned by f (t + θ), t + θ ≥ 0 (φ(t)f )(θ) = 0, if not. QX =

It follows from [30] that (SX , φ) generates an abstract linear control system. We denote such control operator by βX . In order to derive our main theorem in this section, we introduce in X the mass operator L (see [10]) as follows: Lx =

lim Leλ x,

λ→+∞

(5.5)

D(L) = {x ∈ X : lim Leλ x exists}, λ→+∞

where eλ : X → L ([−1, 0], X), eλ x = eλ· x. For mass operator, there exists an important proposition due to Said Hadd et al. [10] as follows. r

Lemma 5.3. Assume E to be a Banach space. Let K ∈ L(W 1,p ([−1, 0], X), E), then (QX , βX , K) is a regular triple if and only if D(K) = X. Here we say (A, B, C) to be a regular triple on (X, U, Y ) if, (i) (A, B) generates an abstract linear control system; (ii) (A, C) generates an abstract linear observation system; and (iii) Range(R(λ, A−1 )B) ⊂ D(CΛA ) holds for some (and hence for all) λ ∈ ρ(A). Obversely, if (A, B, C) generates a regular linear system, (A, B, C) is a regular triple. Using Lemma 5.3, we can obtain the following lemma. Lemma 5.4. Let K ∈ L(W 1,p ([−1, 0], X), E), where E is a Banach space. Let (QX , βX , K) generate a regular linear system. Then KΛQX = K − Kδ0 on W 1,p ([−1, 0], X). Here δ0 denotes the point evaluation at zero.

(5.6)

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Proof. It follows from the proof of [10, Proposition 3] that (5.6) holds on D(KΛQX )∩W 1,p ([−1, 0], X). So it is sufﬁcient to prove that W 1,p ([−1, 0], X) ⊂ D(KΛQX ). Let f ∈ W 1,p ([−1, 0], X), λ > 0, we have KλR(λ, QX )f = KλR(λ, QX )(f − e0 f (0)) + KλR(λ, QX )e0 f (0). (5.7) The ﬁrst term of the right hand of (5.7) has a limit K(f − e0 f (0)). Since (QX , βX , K) generates a regular linear system, by the combination of [10, (3.4)] and Theorem 2.2, e0 f (0) ∈ D(KΛQX ). Our assertion follows immediately. In order to make our computation convenient, we deﬁne a diagonal A 0 , with the domain D(A0 ) = D(A) × D(QX ). Oboperator by A0 = d 0 dσ T 0 versely, the semigroup generated by A0 is T = with the extrapo0 SX 0 T−1 . Now we can obtain the following lation semigroup T−1 = 0 (SX )−1 lemma, part of which can be found in [5, Exercise 3.8 (4)] without proof. 0 0 , where 1⊗Id : X → Lp ([−1, Lemma 5.5. Let K := −(A0 )−1 (1 ⊗ Id) 0 0], X) is defined by ((1 ⊗ Id)x)(θ) = x, for any x ∈ X and θ ∈ [−1, 0]. Then, 0 0 K is admissible for A0 and ΦA0 ,K = . φ(·) 0 Proof. For any (u, v)T ∈ Lp (R+ , E ) = Lp (R+ , X) × Lp (R+ , Lp ([−1, 0], X)) and t > 0, t u(s) 0 t ds = − . (5.8) T−1 (t−s)K (QX )−1 0 SX (t − s)(1 ⊗ u(s))ds v(s) 0

It is easy to check that

t 0

SX (t−s)(1⊗u(s))ds =

t max{t+·,0}

By definition, K is admissible for A0 and ΦA0 ,K (t)

u v

Moreover, we can compute ⎛ ⎞ t ⎝QX SX (t − s)(1 ⊗ u(s))ds⎠ (θ) = −u(t + θ), 0,

u(s)ds ∈ D(QX ). is equal to (5.8).

t + θ ≥ 0, t + θ < 0,

0

combining which with the definition of φ we obtain ΦA0 ,K (t) =

0 φ(t)

for any t ≥ 0.

0 0

By the perturbation method, we can obtain the following theorem. Theorem 5.6. Assume that (A, B, C) generates a regular linear system, both (QX , βX , L) and (QX , βX , D) generate regular linear systems. Then,

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A0 0 (A + ΔAA Λ , B, CΛ ) generates a regular linear system. If in addition, L = 0 and D = 0, then the system (5.2) is a regular linear system.

Proof. By Lemma 5.5, K is admissible for A0 . For any u ∈ Lp (R+ , U ) and t ≥ 0, we have

t T−1 (t − s)Bu(s)ds = 0

ΦA,B (t)u 0

So B is admissible for A0 and ΦA0 ,B (t) =

∈ E.

ΦA,B (t)

0

.

For any (x, f )T ∈ D(A0 ), p t0 t0 t0 CT(s) x ds ≤ 2p CT (s)xp ds + 2p DSX (s)f p ds. f 0

0

0

Obversely, (A, C) and (QX , D) generates abstract linear observation systems. QX ) So C is admissible for A0 , ΨA0 ,C = (ΨA,C , ΨQX ,D ) and CΛA0 = (CΛA , DΛ Q A0 X A with D(CΛ ) = D(CΛ ) × D(DΛ ). By the assumption that (QX , βX , L) generates aregular linear that ΔA system, it follows is admissible for A0 , QX 0 ΨQX ,L 0 LΛ 0 0 ΨA0 ,ΔA = and ΔAA with D(ΔAA Λ = Λ ) = X × 0 0 0 0 X D(LQ Λ ). By the assumption that (A, B, C) generates a regular linear system and Theorem 2.2, we obtain that Range(ΦA0 ,B (t)) ⊂ D(CΛA0 ) for a.e. t ≥ 0. Let F1 = CΛA0 ΦA0 ,B (·) and F (τ ) = Pτ F1 for τ ≥ 0. Then it is not difﬁcult to verify that F = FA,B,C , thus F satisﬁes (2.2). By the definition of FA,B,C , the Laplace transform of F1 is GA,B,C . Hence (A0 , B, C) generates a regular linear system and FA0 ,B,C = FA,B,C . Similarly, we can show that (A0 , K, C), (A0 , B, ΔA) and (A0 , K, ΔA) generate regular with the relations FA0 ,K,C = (FQX ,βX ,D , 0), linear systems FQX ,βX ,L 0 and FA0 ,K,ΔA = 0, respectively. FA0 ,B,ΔA = 0 0 0 By Theorem 3.8, both (A, B, CΛA0 ) and (A, B, ΔAA Λ ) generate regular A0 0 linear systems. It follows from Theorem 3.4 that (A + ΔAA Λ , B, CΛ ) generates a regular linear system. QX X = L − Lδ0 and DΛ = D − Dδ0 We can obtain by Lemma 5.4 that LQ Λ 1,p on W ([−1, 0], X). Thus under the assumption L = 0 and D = 0, we obtain QX X 0 = L and DΛ = D on W 1,p ([−1, 0], X). Hence A + ΔAA LQ Λ Λ = AL and A0 CΛ = C on D(A). This implies that (5.2) is a regular linear system.

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Remark 5.7. It is reasonable to consider the conditions L = 0 and D = 0. In fact, as is described in [10], the frequently-used delay operators given by 0 dμ(σ)z(σ), z ∈ C([−1, 0], X),

Rz := −1

where μ : [−1, 0] → L(X, Y ) is an operator valued function of bounded variation with no mass at 0, i.e., lim |μ|([− , 0]) = 0,

0

(5.9)

satisﬁes R = 0. In particular, if Y = X and μ := X[a,0] (·)IX for some a ∈ [−1, 0), then Rz = δa z := z(a). In this case μ satisﬁes (5.9), so we have R = 0. Now we prove the equivalence of system (5.1) and (5.2) under some assumptions. To do this, we ﬁrst introduce the following lemma: Lemma 5.8. [1] For λ ∈ C we have λ ∈ ρ(AL ) if and only if A + Leλ . Moreover, for λ ∈ ρ(AL ) the resolvent R(λ, AL ) is given by R(λ, A + Lλ )LR(λ, QX ) R(λ, A + Lλ ) , R(λ, AL ) = eλ R(λ, A + Lλ ) [eλ R(λ, A + Lλ )L + I]R(λ, QX ) where Lλ = Leλ . Theorem 5.9. Assume that (A, B) generates an abstract linear control system, (QX , βX , L) generates a regular linear system, and L = 0. Let u ∈ x(t) 1,p Wloc (R+ , U ). Denote by X (t) = the classical solution of system z(t) (5.2). Let x(t), t ≥ 0; m(t) = (5.10) f (t), − 1 ≤ t < 0. Then m(·) is classical solution of system (5.1) and z(t) = xt . Conversely, if x(t) is the classical solution of system (5.1), (x(t), xt )T is the classical solution of (5.2). x(t) Proof. Assume X (t) = to be the classical solution of system (5.2). z(t) By Theorem 3.4, Theorem 3.8, Lemma 5.5 and Theorem 5.6, we obtain FAL ,B,I = FA+ΔA,B,I = FA+ΔA,I,I FA,B,ΔA + FA,B,I = FA+ΔA,I,I F((A0 )−1 +K)|E ,B,ΔA + F((A0 )−1 +K)|E ,B,I = FA+ΔA,I,I [FA0 ,K,ΔA (I − FA0 ,K,I )−1 FA0 ,B,I + FA0 ,B,ΔA ] +FA0 ,K,I (I − FA0 ,K,I )−1 FA0 ,B,I + FA0 ,B,I .

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For λ big enough, taking Laplace transform on both sides of the above equation, we obtain the following equation of the corresponding transfer function B R(λ, (AL )−1 ) 0 QX R(λ, A−1 )B I 0 LΛ eλ 0 = R(λ, AL ) eλ I 0 0 0 R(λ, A−1 )B R(λ, A−1 )B 0 0 I 0 + + eλ 0 eλ I 0 0 I X = (5.11) [R(λ, A + Lλ )LQ Λ eλ + I]R(λ, A−1 )B. eλ Here the result R(λ, (QX )−1 )βX = eλ in [10] and Lemma 5.8 are applied. Since X is the classical solution, we obtain B (AL )−1 X (t) + u(t) ∈ E , for any t ≥ 0. (5.12) 0 Let λ ∈ ρ(AL ) be ﬁxed. From (5.12), we obtain B X (t) − R(λ, (AL )−1 ) u(t) ∈ D(AL ), 0 that is x(t) I X − [R(λ, A + Lλ )LQ Λ eλ + I]R(λ, A−1 )Bu(t) ∈ D(AL ). z(t) eλ This means z(t)(0) = x(t)

(5.13)

z(t) ∈ W 1,p ([−r, 0], X).

(5.14)

f = z(0) ∈ W 1,p ([−r, 0], X).

(5.15)

and

In particular,

By Lemma 5.8, we obtain R(λ, AL )X (t) R(λ, A + Lλ )x(t) + R(λ, A + Lλ )LR(λ, QX )z(t) = . eλ R(λ, A + Lλ )x(t) + [eλ R(λ, A + Lλ )L + Id]R(λ, QX )z(t) (5.16) On the other hand, multiplying (5.2) by R(λ, AL ) from the left, it follow that d R(λ, AL )X (t) = R(λ, (AL )−1 )(AL )−1 X (t) + R(λ, (AL )−1 )Bu(t) dt = − X (t)+λR(λ, AL )X (t) + R(λ, (AL )−1 )Bu(t). (5.17)

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Combining (5.16) and (5.17), we obtain d d x(t) + R(λ, A + Lλ )LR(λ, QX ) z(t) dt dt = −x(t) + λR(λ, A + Lλ )x(t) + λR(λ, A + Lλ )LR(λ, QX )z(t)

R(λ, A + Lλ )

X + [R(λ, A + Lλ )LQ Λ eλ + I]R(λ, A−1 )Bu(t)

(5.18)

and d d x(t) + [eλ R(λ, A + Lλ )L + Id]R(λ, QX ) z(t) dt dt = −z(t)+λeλ R(λ, A + Lλ )x(t) + λ[eλ R(λ, A + Lλ )L + Id]R(λ, QX )z(t)

eλ R(λ, A + Lλ )

X +eλ [R(λ, A + Lλ )LQ Λ eλ + I]R(λ, A−1 )Bu(t).

(5.19)

Subtracting eλ (5.18) from (5.19), it follows that R(λ, QX )

d z(t) = eλ x(t) − z(t) + λR(λ, QX )z(t). dt

(5.20)

d d Multiplying (5.20) by λ − QX from the left, we obtain dt z(t) = dσ z(t), for any t ≥ 0. Observe (5.13) and (5.14), we rewrite the relations of z(t) and x(t) as follows: ⎧ d d z(t) = dσ z(t), t ≥ 0 ⎪ ⎪ ⎨ dt (5.21) z(t)(0) = x(t), ⎪ ⎪ ⎩ z(0) = f.

Moreover, the combining of (5.15) and the assumption that X (·) is classical solution implies that x ∈ W 1,p ([−1, 0], X) ∩ C 1 (R+ , X). It follows from [10, 1,p ([−r, ∞), X). Hence from [1], we obtain Lemma 1] that x ∈ Wloc d d xt = xt . (5.22) dt dσ As in [1], we set y(t) = z(t) − xt . Then, the combining (5.21) and (5.22) implies that y(t) is the solution of the abstract Cauchy problem associated to QX on Lp ([−1, 0], X) with initial value y(0) = 0. Hence y(t) = 0 for any t ≥ 0. This implies that z(t) = xt . Multiplying (5.18) by λ − (A + Lλ ) from the left, and using (5.21), we obtain d x(t) = A−1 x(t) + Lxt + Bu(t) − LR(λ, A−1 )Bu(t). dt By the assumption that the mass operator L = 0, the above equation becomes d x(t) = A−1 x(t) + Lxt + Bu(t). dt Thus, x(t) is the classical solution of system (5.1). Now we consider the converse case. Assume x(t) to be the classical 1,p solution of system (5.1). Then, x(·) ∈ Wloc ([−r, ∞), X). It follows from [10, x(t) 1 + p Lemma 1] that x· ∈ C (R , L ([−r, 0], X)). Deﬁne X (t) = , using xt

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the converse procedure of the last part, it is not difﬁcult to verify that X (t) is the classical solution of system (5.2). Lemma 5.10. [23, Theorem 3.1] If (A , B) generates an abstract linear con1,p (R+ , F ) with trol system on Banach spaces E and F , then for any u ∈ Wloc ˙ = (A )−1 x(t) + Bu(t) has a classical A−1 x(0) + Bu(0) ∈ E, the system x(t) solution. By Theorem 5.9 and Lemma 5.10, (AL )−1 X (0) + Bu(0) ∈ E implies that system (5.1) has a classical solution under the same assumption as in Theorem 5.9. Furthermore, from the proof of Theorem 5.9, it follows that the combining of A−1 z + Bu(0) ∈ X, f ∈ W 1,p ([−1, 0], X) and f (0) = z implies the existence of mild solution of system (5.1). This can be specifically described by the following corollary. Corollary 5.11. Assume that (A, B) generates an abstract linear control system, (QX , βX , L) generates a regular linear system, and L = 0. Let u ∈ 1,p (R+ , U ), A−1 x + Bu(0) ∈ X, f ∈ W 1,p ([−1, 0], X) and f (0) = z. Then Wloc system (5.1) has a classical solution. With the equivalence of system (5.1) and system (5.2), it is reasonable to deﬁne the mild solution of (5.1) by the mild solution of (5.2) as in [1]. Moreover, from the proof of Lemma 5.10 (see [23]), we obtain that system (5.2) has a mild solution for any (z, f, u)T ∈ E × Lploc (R+ , U ). Hence we are able to introduce the following concept. Definition 5.12. Assume that (A, B) generates an abstract linear control system, (QX , βX , L) generates a regular linear system, and L = 0. For all x(t) (z, f, u)T ∈ E × Lploc (R+ , U ), denote by X (t) = the mild solution of z(t) system (5.2), we call m deﬁned by (5.10) the mild solution of system (5.1). Theorem 5.13. Assume that (A, B, C), (QX , βX , L) and (QX , βX , D) generate regular linear systems. Let L = 0 and D= 0. Then the mild solution x(t) X (t) of the system (5.2) is of the form and satisfies xt ∈ D(LΛ ) for xt a.e. t ≥ 0 and t x(t) = T (t)z + T (t − s)LΛ xs ds + ΦA,B u. (5.23) 0

Moreover, the output is given by QX y(t) = CΛA x(t) + DΛ xt ,

(5.24)

for a.e. t ≥ 0. Proof. Similar to the proof of [16, Proposition 2.5], the family (TL,U (t))t≥0 given by ⎞ ⎛ ⎞ ⎛ t x B x (T ) (t) + T (t − s) g(s)ds L 0 ⎠ f 0 TL,U (t) ⎝ f ⎠ = ⎝ L −1 S(t)g g

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with TL being the C0 -semigroup generated by AL generates a C0 -semigroup on E × Lp (R+ , U ). The generator of (TL,U (t))t≥0 is given by ⎞ ⎛ ⎞ ⎛ x B x (A ) + g(0) ⎠ 0 AL,U ⎝ f ⎠ = ⎝ L −1 f g Qg with the domain

⎧⎛ ⎞ ⎨ x x D(AL,U ) = ⎝ f ⎠ : ∈ E , g ∈ W 1,p (R+ , U ), f ⎩ g ⎫ ⎬ x B (AL )−1 g(0) ∈ E . + f 0 ⎭

⎛ ⎞ z Hence, for ⎝ f ⎠ ∈ D(AL,U ), X (t) is the classical solution of system u (5.2). By the density of D(AL,U ) in E × Lp (R+ , X) and the fact that X (t) is depend continuously on the initial values, the mild solution is of the form x(t) . xt It follows from Theorem 4.2 that t z X (t) = T (t) + T (t − s)ΔAA (5.25) Λ X (s)ds + ΦA,B (t)u. f 0

By Lemma 3.7, it follows that ΦA,B (t)u = ΦA0 +K,B (t)u = ΦA0 ,K (t)(I − FA0 ,K,I )−1 FA0 ,B,I u + ΦA0 ,B (t)u ΦA,B (t)u ΦA,B (·)u 0 0 + = φ(t) 0 0 0 ΦA,B (t)u = . φ(t)ΦA,B (·)u

(5.26)

Substituting (5.26) into (5.25) and, taking the ﬁrst component, we obtain (5.23). The formula (5.24) follows immediately from (2.9).

6. Stabilizability of Linear System with State and Output Delays This section discusses the stabilizability of system (5.1). Assume (M, N ) to generate a control linear system on the Banach spaces E and F. We say that the pair (M, N ) is stabilizable if there exists F ∈ L(D(M ), F), such that (i) (M, N, F ) generates a regular linear system Σ; (ii) the identity operator IF : F → F is an admissible feedback operator for Σ; (iii) the semigroup generated by M−1 + N FΛM is exponentially stable. In this case, we say that

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F stabilizes (M, N ). For this concept and for further comments we refer to [20,32]. Now we introduce a necessary condition for stabilizability. Lemma 6.1. [32] Assume (M, N ) to be stabilizable on the Banach spaces E and F, then there exists a constant δ > 0 such that Ran[λ − M−1 , N ] ⊃ E for all complex number λ with Reλ > −δ. Combining the above lemma and our perturbation results, we derive a necessary condition for stabilizability for delay equation (5.1). Theorem 6.2. Assume that (A, B) generates an abstract linear control system, (QX , βX , L) generates a regular linear system, and L = 0. If system (5.1) is stabilizable, X ⊂ (λ − A−1 − Lλ )X + BU

(6.1)

holds for all λ ∈ σ(AL ) with Reλ ≥ 0. Proof. Assume system (5.1) to be stabilizable. This means by Theorem 5.9 that system (5.2) is stabilizable. It follows from Lemma 6.1 that there exists a constant α > 0 such that Ran[λ − (AL )−1 , B] ⊃ E for any λ ∈ C with Reλ > −α. This implies, for any x ∈ X and f ∈ Lp ([−1, 0], X) there exist (x0 , f0 ) ∈ E and u ∈ U such that, for Reλ > −α, (x, f )T = (λ − (AL )−1 )(x0 , f0 )T + Bu. Let μ ∈ ρ(AL ) = ρ((AL )−1 ) be ﬁxed. By (5.11), it follows that x x0 I = (λ−(AL )−1 ) + [R(μ, A+Lμ )LΛ eμ + I]R(μ, A−1 )Bu f f0 eμ I + (μ − λ) [R(μ, A + Lμ )LΛ eμ + I]R(μ, A−1 )Bu ∈ E . eμ This indicates that x0 I + [R(μ, A + Lμ )LΛ eμ + I]R(μ, A−1 )Bu ∈ D(AL ) f0 eμ and x x0 I = (λ − AL ) + [R(μ, A+Lμ )LΛ eμ +I]R(μ, A−1 )Bu f f0 eμ

I eμ

+ (μ − λ) [R(μ, A + Lμ )LΛ eμ + I]R(μ, A−1 )Bu λx0 − A−1 X0 − Lf0 + Bu − LR(μ, A−1 )Bu = . d )f0 (λ − dσ

(6.2)

(6.3)

Substituting the assumption L = 0 into (6.2), we can derive x = λx0 − A−1 X0 − Lf0 + Bu

(6.4)

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f=

λ−

d dσ

377

f0 .

(6.5)

From (6.5), we obtain f0 = eλ x0 + R(λ, QX )f . Substituting f0 into (6.4) and putting f = 0, we obtain x = λx0 − A−1 x0 − Lλ x0 + Bu.

Hence the condition (6.1) holds.

In the rest of this section, we will give a necessary and sufﬁcient condition of stabilizability of system with ﬁnite-dimensional input space and impact original semigroup by combining our perturbation theorem and [3]. To do this, we ﬁrst introduce a lemma. Lemma 6.3. Assume that (A, B) generates an abstract linear control system and the semigroup (T (t))t≥0 generated by A is immediately compact, that is, t T (t) is compact for t > 0. Then t → (F (t) = 0 T−1 (s)Bds : U → X) is compact for t > 0. Proof. Let be small enough. we can compute t T−1 (s)Buds = T ( )F (t − )u.

(F (t) − F ( ))u =

By the assumption, T ( ) is compact, so F (t) − F ( ) is compact. From the admissibility of B for A, we obtain (F (t) − (F (t) − F ( )))u = F ( )u ≤ 1/p ΦA,B ( )u, for any u ∈ U. This implies that F (t) − (F (t) − F ( )) → 0, as → 0 in the uniform operator topology. Hence F (t) is compact for t > 0. Theorem 6.4. Assume that (A, B) generates an abstract linear control system, (QX , βX , L) generates a regular linear system, and L = 0. In addition, let T (t) be compact for t > 0 and the input space be finite dimensional. Then, system (5.1) is stabilizable if and only if X = {(λ − A−1 − Lλ )x + Bu : x ∈ X, u ∈ U, A−1 x − Bu ∈ X} holds for all λ ∈ σ(AL ) with Reλ ≥ 0. Proof. The necessity has been proved in Theorem 6.2. A−1 sufﬁciency. It is easy to see that the operator 0 A−1 B D = X × U generates a C0 -semigroup 0 0 by t T−1 (t) 0 T−1 (σ)Bdσ S (t) = . 0 IU

Now we prove the B with domain 0 on X−1 × U given

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Because of the admissibility of B for A, the restriction of S (t) to X × U is also a C0 -semigroup given by t T (t) 0 T−1 (σ)Bdσ , S0 (t) = 0 IU which is generated by

A=

A−1 0

B 0

with the domain

x ∈ X × U : A−1 x + Bu ∈ X . u t By Lemma 6.3, the operator t → ( 0 T−1 (t − σ)Bds : U → X) is immediately compact. Combining this with the compactness of IU and the the immediate compactness of T (·), we obtain that S0 (t) is immediately compact. Deﬁne L0 L= : W 1,p ([−r, 0], X × U ) 0 0 x and ξ = : [−r, ∞) → X × U and v = u˙ where u is assumed to be u smooth with u = 0 on [−1, 0]. Then the system (5.1) is converted into ˙ = Aξ(t) + Lξt + Bv(t), t ≥ 0 ξ(t) (6.6) ξ(t) = (f (t), 0), a.e. t ∈ [−1, 0], D(A) =

where B : U → X × U is deﬁned by Bv = (0, v)T . Obversely, B is bounded. By the analysis in Sect. 5 and [7], system (6.6) is equivalent to τ˙ (t) = ML τ (t) + Bv(t), t ≥ 0, (6.7) τ (0) = (z, u(0), φ, 0)T , A L ξ T where τ (t) = (ξ(t), ξt ) , ML = ∈ D(A)× with D(ML ) = d g 0 dσ W 1,p ([−1, 0], X × U ) : g(0) = ξ and B = (B, 0)T . By Lemma 5.8, λ ∈ σ(ML ) ⇔ λ ∈ σ(A + LEλ ) eλ 0 A−1 + Leλ B . with Eλ = and A + LEλ = 0 eλ 0 0 Since Leλ ∈ L(X), σ(A + LEλ ) = σ(A + Leλ ) ∪ {0} = σ(AL ) ∪ {0}. By [18], T is eventually compact, specifically, T (t) is compact for t > 1. Hence the set

σ + = {λ ∈ σ(ML ) : λ ≥ 0} is ﬁnite.

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Take any (x, u) ∈ X ×U and let λ ∈ σ + . By assumption, there exist x0 ∈ X and u0 ∈ U such that A−1 x0 −Bu0 ∈ X and (λ−A−1 −Leλ )x0 +Bu0 = x, which implies that x0 0 eλ x0 x (λ − A) + −L = . −u0 0 u + λu0 u Take any φ and ψ such that (x, u, φ, ψ)T ∈ X × U × Lp ([−1, 0], X) × Lp ([−1, 0], U ), and deﬁne φ0 = eλ x0 + R(λ, QX ). Then,

x0 0 φ0 (λ − A) + −L −u0 0 u + λu0 x φ = − LR(λ, QX×U ) . u ψ

The combination of the last two identities implies I −LR(λ, QX×U ) (x, u, φ, ψ)T , (λ − ML )(x0 , −u0 , φ0 , 0) + Bv0 = 0 I where v0 = u + λu0 . This is equivalent to Range(λ − ML ) + Range(B) = X × U × Lp ([−1, 0], X) × Lp ([−1, 0], U ) for any λ ∈ σ + . By [3, Theorem 1], system (6.7), or system (5.1) is stabilizable. Acknowledgements The authors wish to thank the anonymous reviewer for his/her helpful and insightful comments and suggestions.

References [1] B´ atkai, A., Piazzera, S.: Semigroup and linear partial differential equations with delay. J. Math. Anal. Appl. 264, 1–20 (2001) [2] B´ atkai, A., Piazzera, S.: Semigroups for Delay Equations, vol. 10. A K Peters, Ltd, Wellesley (2005) [3] Bhat, K.P.M., Wonham, W.M.: Stabilizability and detectability for evolution systems on Banach spaces. In: Proceedings of IEEE Conference Decision Control 15th Symposium Adaptive Processes, pp. 1240–1243 (1976) [4] Curtain, R.F., Pritchard, A.J.: Inﬁnite dimensional linear systems theory. In: Lecture Notes in Information Sciences, vol. 8. Springer-Verlag, Berlin (1978) [5] Engel, K.J., Nagel, R.: One Parameter Semigroups for Linear Evolutional Equations. Springer-Verlag, New York (2000)

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[6] Haak, B., Kunstmann, P.C.: Admissibility of unbounded operators and wellposedness of linear systems in Banach spaces. Integr. Equ. Oper. Theory 55(4), 497–533 (2006) [7] Hadd, S.: Unbounded perturbations of C0 -semigroups on Banach spaces and applications. Semigroup Forum 70, 451–465 (2005) [8] Hadd, S.: Exact controllability of inﬁnite dimensional systems persists under small perturbations. J. Evol. Equ. 5, 545–555 (2005) [9] Hadd, S., Idrissi, A.: On the admissibility of observation for perturbed C0 semigroups on Banach spaces. Syst. Control Lett. 55, 1–7 (2006) [10] Hadd, S., Idrissi, A., Rhandi, A.: The regular linear systems associated to the shift semigroups and application to control delay systems. Math. Control Signals Syst. 18, 272–291 (2006) [11] Hale, J.K.: Functional Differential Equations, Applied Mathematical Sciences, vol. 3. Springer-Verlag, Berlin (1971) [12] Jacob, B., Partington, J.R.: The Weiss conjecture on admissibility of observation operators for contraction semigroups. Integr. Equ. Oper. Theory 40(2), 231–241 (2001) [13] Jacob, B., Partington, J.R., Pott, S.: Conditions for admissibility of observation operators and boundedness of Hankel operators. Integr. Equ. Oper. Theory 47(3), 315–338 (2003) [14] Jacob, B., Zwart, H.: Counterexamples concerning observation operators for C0 semigroups. SIAM J. Control Optim. 43(1), 137–153 (2004) [15] Maci´ a, F., Zuazua, E.: On the lack of observability for wave equations: Gaussian beam approach. Asymptot. Anal. 32(1), 1–26 (2002) [16] Malinen, J., Staffans, O.J., Weiss, G.: When is a linear system conservative?. Quart. Appl. Math. 64, 61–91 (2006) [17] Malinen, J., Staffans, O.J.: Conservative boundary control systems. J. Differ. Equ. 231, 290–312 (2006) [18] M´ atrai, T.: On perturbations of eventually compact semigroups preserving eventual compactness. Semigroup Forum 69, 317–340 (2004) [19] Olbrot, A.W.: Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays. IEEE Trans. Automat. Control 23(5), 887–890 (1978) [20] Rebarber, R.: Conditions for the equivalence of internal and external stability for distributed parameter systems. IEEE Trans. Automat. Control 38, 994–998 (1993) [21] Salamon, D.: Inﬁnite-dimensional linear system with unbounded control and observation: a functional analytic approach. Trans. Am. Math. Soc. 300, 383–431 (1987) [22] Salamon, D.: Realization theory in Hilbert space. Math. Syst. Theory 21, 147–164 (1989) [23] Staffans, O.J., Weiss, G.: Transfer functions of regular linear systems. Part II: the system operator and the Lax-Phillips semigroup. Trans. Am. Math. Soc. 354, 3229–3262 (2002) [24] Staffans, O.J.: Well-Posed Linear Systems. Cambridge University Press, Cambridge (2005)

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[25] Tucsnak, M., Weiss, G.: Observation and Control for Operators Semigroups. Birkh¨ auser Verlag, Basel (2009) [26] Weiss, G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27, 527–545 (1989) [27] Weiss, G.: Admissible observation operators for linear semigroups. Isr. J. Math. 65, 17–43 (1989) [28] Weiss, G.: The representation of regular linear systems on Hilbert spaces. In: Kappel, F., Kunisch, K., Schappacher, W. (eds.) Control and Estimation of Distributed Parameter Systems (Proceedings Vorau 1988), pp. 401–416. Birkh¨ auser, Basel [29] Weiss, G.: Two conjectures on the admissibility of control operators. In: Desch, W., Kappel, F. (eds.) Estimation and Control of Distributed Parameter Systems, pp. 367–378. Birkh¨ auser, Basel (1991) [30] Weiss, G.: Transfer functions of regular linear systems. Part I: characterizations of regularity. Trans. Am. Math. Soc. 342(2), 827–854 (1994) [31] Weiss, G.: Regular linear systems with feedback. Math. Control Signals Syst. 7, 23–57 (1994) [32] Weiss, G., Rebarber, R.: Optimizability and estimatability for inﬁnite-dimensional linear systems. SIAM J. Control Optim. 39, 1204–1232 (2000) [33] Zhong, Q.C.: Robust Control of Time-Delay Systems. Springer-Verlag, London (2006) [34] Zwart, H., Jacob, B., Staffans, O.: Weak admissibility does not imply admissibility for analytic semigroups. Syst. Control Lett. 48(3), 341–350 (2003) [35] Zwart, H.: Sufﬁcient conditions for admissibility. Syst. Control Lett. 54, 973– 979 (2005) Zhan-Dong Mei (B) and Ji-Gen Peng Department of Mathematics Xi’an Jiaotong University Xi’an 710049, China e-mail: [email protected]; [email protected] Received: December 30, 2009. Revised: February 23, 2010.

Integr. Equ. Oper. Theory 68 (2010), 383–390 DOI 10.1007/s00020-010-1804-9 Published online June 8, 2010 c The Author(s) This article is published with open access at Springerlink.com 2010

Integral Equations and Operator Theory

Operator Hyperreflexivity of Subspace Lattices J. Braˇciˇc, K. Kli´s-Garlicka, V. M¨ uller and I. G. Todorov Abstract. We introduce and study the notion of operator hyperreflexivity of subspace lattices. This notion is a natural analogue of the operator reflexivity and is related to hyperreflexivity of subspace lattices introduced by Davidson and Harrison. Mathematics Subject Classfication (2010). Primary 47A15; Secondary 47L99. Keywords. Subspace lattice, operator hyperreflexivity, operator reflexivity.

1. Introduction Let H be a complex Hilbert space. By B(H) we denote the algebra of all bounded linear operators on H and by P(H) the lattice of all orthogonal projections in B(H). A subspace lattice is a lattice which contains the trivial projections 0 and I, and is closed in the strong operator topology. Note that every subspace lattice is complete, which means that it is closed under taking arbitrary infima and suprema. For a subspace lattice L ⊆ P(H), the reflexive hull of L is defined as RefL = {P ∈ P(H);

P x ∈ Lx,

for all x ∈ H}.

A subspace lattice L is said to be operator reflexive if RefL = L (see [11]). Recall that the classical notion of reflexivity of L means Lat Alg L = L, which is strictly stronger condition than operator reflexivity [11]. Note that not every subspace lattice is operator reflexive [5]. Here, for a family of operators S ⊆ B(H), we let Lat S = {P ∈ P(H); SP = P SP ∀ S ∈ S} be collection of orthogonal projections onto the subspaces invariant for S. For a subspace lattice L, we denote by Alg L the algebra of all operators A ∈ B(H) satisfying L ⊆ Lat {A}, i.e., operators that leave invariant the ranges of all projections in L. The research was partially supported by the following grants: MEB 090905, BI-CZ/09-10ˇ 005, BI-PL/08-09-016, IAA100190903 of GA AV CR.

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Let L ⊆ P(H) be a subspace lattice, P ∈ P(H), and let d(P, L) = inf{P − Q;

Q ∈ L} = inf

sup P x − Qx

Q∈L x≤1

denote the usual distance between P and L. In [4], Davidson and Harrison introduce, in analogy with the Arveson distance for algebras (see [1]), the following quantity for subspace lattices. Let L be a subspace lattice and P ∈ P(H). They set β(P, L) = sup P ⊥ AP ; A ∈ (Alg L)1 , where (Alg L)1 denotes the set of all contractions in Alg L. It is straightforward to see that β(P, L) ≤ 2d(P, L) for every P (see [4, p. 310]). A subspace lattice L is said to be hyperreflexive if there is a positive number κ such that d(P, L) ≤ κβ(P, L)

for all P ∈ P(H).

(1)

The infimum κ(L) of all positive numbers κ satisfying (1) is called the constant of hyperreflexivity for L. Every hyperreflexive subspace lattice is reflexive, however the converse does not hold in general. In this paper we introduce another quantity related to a subspace lattice which seems to be a more natural analog of the Arveson distance. Our idea is based on the definition of the Arveson distance for general spaces of operators. Let L be a subspace lattice and P ∈ P(H). Then we set α(P, L) = sup{d(P x, Lx);

x ≤ 1} = sup inf P x − Qx. x≤1 Q∈L

It is obvious from the definition that α(P, L) ≤ d(P, L). We say that a subspace lattice L is operator hyperreflexive if there exists a constant c > 0 such that d(P, L) ≤ cα(P, L),

for all P ∈ P(H).

(2)

The infimum c(L) of all positive numbers c satisfying (2) is called the constant of operator hyperreflexivity for L. It is clear that every operator hyperreflexive lattice is operator reflexive. The goal of this paper is to study operator hyperreflexivity for subspace lattices. In Sect. 2 we show that hyperreflexivity implies operator hyperreflexivity. The converse implication does not hold. We show in Sect. 3 that every finite subspace lattice is operator hyperreflexive. We also establish some basic properties of operator hyperreflexive subspace lattices. In the last section, we give an example of a subspace lattice which is operator reflexive but not operator hyperreflexive. The following diagram summarizes the relations among these properties of a subspace lattice: reflexivity =⇒ operator reflexivity ⇑ ⇑ hyperreflexivity =⇒ operator hyperreflexivity All the implications are strict.

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2. Hyperreflexivity Versus Operator Hyperreflexivity In this section we compare operator hyperreflexivity with hyperreflexivity of subspace lattices. Theorem 2.1. Every hyperreflexive subspace lattice is operator hyperreflexive. Moreover, if L is a hyperreflexive subspace lattice with constant of hyperreflexivity κ(L), then the constant of operator hyperreflexivity for L is at most 4κ(L). Proof. Let L be a subspace lattice and P ∈ P(H) be arbitrary. We claim that β(P, L) ≤ 4α(P, L). To see this, let A ∈ (Alg L)1 and x ∈ H, x ≤ 1, be arbitrary. Then, for every Q ∈ L, one has | P ⊥ AP x, x | = | (P ⊥ AP − Q⊥ AQ)x, x | ≤ | (P ⊥ − Q⊥ )AP x, x | + | Q⊥ A(P − Q)x, x | = | AP x, (P − Q)x | + | (P − Q)x, A∗ Q⊥ x | ≤ 2(P − Q)x. It follows that | P ⊥ AP x, x | ≤ 2 inf{(P − Q)x;

Q ∈ L} and consequently

⊥

sup{| P AP x, x |; x = 1} ≤ 2 sup{inf{(P − Q)x; Q ∈ L}; x = 1}. Note that the number on the left side of the last inequality is the numerical radius w(P ⊥ AP ) of the operator P ⊥ AP and that the number on the right hand side is 2α(P, L). By the Lumer’s formula, one has P ⊥ AP ≤ 2w(P ⊥ AP ), which gives P ⊥ AP ≤ 4α(P, L), and we may conclude that β(P, L) ≤ 4α(P, L). It is obvious now that for a hyperreflexive subspace lattice L one has c(L) ≤ 4κ(L), which in particular means that every hyperreflexive subspace lattice is operator hyperreflexive. In [4], several classes of subspace lattices were proved to be hyperreflexive. So we have the following immediate corollary of Theorem 2.1. Corollary 2.2. (i) Every nest N is operator hyperreflexive with constant of operator hyperreflexivity not exceeding 4. (ii) Let A be a hyperreflexive von Neumann algebra with hyperreflexivity constant a. Then the projection lattice L of A is operator hyperreflexive with operator hyperreflexivity constant not exceeding 4a. (iii) If L is a commutative subspace lattice, then it is operator hyperreflexive with operator hyperreflexivity constant not exceeding 20. Proof. By [4, Theorem 3.1], every nest is hyperreflexive with hyperreflexivity constant 1. Hence, by Theorem 2.1, (i) follows. Clauses (ii) and (iii) follow similarly by Theorem 4.1, respectively by Theorem 5.1, in [4]. As the following example shows, hyperreflexivity is a condition strictly stronger than operator hyperreflexivity. Example 2.3. Let H be a two-dimensional Hilbert space. Assume that P1 , P2 , P3 ∈ P(H) are of rank one and that (Pi H) ∩ (Pj H) = {0} and (Pi H) ∨ (Pi H) = H hold for all i, j = 1, 2, 3, i = j. Denote by L the lattice {0, P1 , P2 , P3 , I}. It is easy to see that Alg L is trivial, i.e., it consists only of scalar multiples of the identity operator. Thus, β(P, L) = 0 for every P ∈ P(H) which

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means that L is not hyperreflexive. On the other hand, it will be shown later, see Theorem 3.2, that every finite subspace lattice is operator hyperreflexive.

3. Basic Results We start this section by showing that every finite subspace lattice is operator hyperreflexive which is not the case for hyperreflexivity, see Example 2.3. We need the following lemma, cf. [9, Theorem 37.17]. Lemma 3.1. Let T1 , . . . , Tn ∈ B(H) be arbitrary and assume that n operators 2 α1 , . . . , αn are positive numbers such that i=1 αi < 1. Then there exists x ∈ H, x = 1, such that Ti x ≥ αi Ti , for every i = 1, . . . , n. Proof. Without loss of the generality we can assume that every operator Ti is non-zero. Choose ε > 0 such that αi2 < 1 − ε. For i = 1, . . . , n, α 2 i set αi = √1−ε . Then (αi ) < 1. For every i choose yi ∈ H, yi = 1, √ √ such that Ti∗ yi > 1 − εTi∗ = 1 − ε Ti . Set ui = Ti∗ yi −1 Ti∗ yi , so that ui = 1. By [2], there exists a vector x ∈ H of norm 1 such that . , n. Hence Ti x ≥ | Ti x, yi | = | x, Ti∗ yi | = | x, ui | ≥ αi , for all i = 1, . .√ ∗ ∗ | x, Ti yi ui | ≥ αi Ti yi ≥ 1 − εαi Ti = αi Ti . Theorem 3.2. Let L = {L1 , . . . , Ln } ⊂ P(H) √ be a finite subspace lattice. Then L is operator hyperreflexive and c(L) ≤ n. Proof. Let P ∈ P(H) and ε > 0. Consider the operators P − L1 , . . . , P − Ln . By Lemma 3.1, there exists x ∈ H with x = 1 and 1 (P − Lj )x ≥ √ − ε P − Lj n for all j = 1, . . . , n. So α(P, L) = sup min (P − Lj )y y=1 1≤j≤n

≥

√1 n

−ε

min P − Lj =

1≤j≤n

Since ε > 0 was arbitrary we have d(P, L) ≤

√1 n

− ε d(P, L).

√ n · α(P, L).

Proposition 3.3. Let M and L be subspace lattices with L ⊆ M. Suppose that M is operator hyperreflexive with constant a and that d(M, L) ≤ b α(M, L) holds for all M ∈ M. Then L is operator hyperreflexive with constant at most a + b + ab. Proof. Let P ∈ P(H). Then for every ε > 0 there is M0 ∈ M such that P − M0 ≤ d(P, M) + ε. Since L ⊂ M one has d(P x, Mx) ≤ d(P x, Lx), for every x ∈ H. Hence α(P, M) ≤ α(P, L). Note that for every L ∈ L and x ∈ H one has M0 x − Lx ≤ M0 x − P x + P x − Lx, which means that

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α(M0 , L) ≤ supx=1 M0 x−P x+α(P, L) = M0 −P +α(P, L). Therefore d(P, L) ≤ P − M0 + d(M0 , L) ≤ d(P, M) + ε + d(M0 , L) ≤ a α(P, M) + ε + b α(M0 , L) ≤ a α(P, L) + ε + b (M0 − P + α(P, L)) ≤ a α(P, L) + ε + b (d(P, M) + ε) + b α(P, L) ≤ (a + b)α(P, L) + ε + b (aα(P, M) + ε) ≤ (a + b + ab)α(P, L) + ε + bε. Hence L is operator hyperreflexive with constant at most a + b + ab.

Proposition 3.4. For each i ∈ N, let Li ⊆ P(Hi ) be an operator hyperreflexive subspace lattice with constant ai . If a = supi∈N ai < ∞, then L = ⊕Li is operator hyperreflexive with constant at most 16 + 17a. Conversely, if L = ⊕Li is operator hyperreflexive with constant a, then all Li are operator hyperreflexive with constant at most a. Proof. If P = ⊕Pi ∈ P(⊕Hi ), then d(P, L) = sup d(Pi , Li ) ≤ a sup α(Pi , Li ). i∈N

i∈N

Let x ˜i = (0, . . . , 0, xi , 0 . . . ) ∈ ⊕Hi . Then sup α(Pi , Li ) = sup sup d(Pi xi , Li xi ) = sup sup d(P x ˜i , L˜ xi ) ≤ α(P, L). i∈N

i∈N xi ≤1

i∈N ˜ xi ≤1

On the other hand, ⊕P(Hi ) is the projection lattice of the injective von Neumann algebra ⊕B(Hi ), which is hyperreflexive with constant at most 4, by [3] and [10]. By Corollary 2.2 (ii), ⊕P(Hi ) is operator hyperreflexive with constant at most 16. Now Proposition 3.3 gives that L is operator hyperreflexive with constant at most 16 + 17a. Assume now that L = ⊕Li is operator hyperreflexive with constant a and take a projection P = 0 ⊕ 0 · · · ⊕ Pi ⊕ · · · ⊕ 0, where Pi ∈ P(Hi ). It is easy to see that d(P, L) = d(Pi , Li ) and α(P, L) = α(Pi , Li ). Hence by hyp erreflexivity of L we have d(Pi , Li ) = d(P, L) ≤ aα(P, L) = aα(Pi , Li ).

4. Non Operator Hyperreflexive Lattice which is Operator Reflexive Let H be an infinite dimensional separable Hilbert space with an orthonormal basis e1 , e2 , . . .. For k ∈ N, let Hk = {e1 , . . . , ek }. Denote by SH the unit 1 and fix a sequence (tn )∞ sphere of H. Let 0 < ε < 64 n=1 ⊂ (0, 1) consisting of mutually distinct numbers. Lemma 4.1. There exist subspaces Mn ⊂ H(n ∈ N) such that: (i) (ii) (iii)

Mn ∩ Mm = {0} (m, n ∈ N, m = n); Mn ∨ Mm = H (m, n ∈ N, m = n); √ PMn ej < nε , for j = 2, . . . , n, and PMn e1 − PMs e1 > ε (s = n), where PM denotes the orthogonal projection on a subspace M ⊆ H;

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Mn can be written as Mn = Fn ⊕ ∨{e2j+1 + tn e2j+2 ; j ≥ 2n }, where Fn ⊂ H2n+1 is a 2n -dimensional subspace.

Proof. We construct the subspaces Mn by induction on n. Let n ∈ N and suppose that the subspaces M1 , . . . , Mn−1 satisfying (i)–(iv) have already been constructed. Let Es = Ms ∩ H2n+1 for s = 1, . . . , n − 1. By assumptions (i) and (iv), we have dim Es = 2n and Es ∩ Es = {0} for all s = s , 1 ≤ s, s ≤ n − 1.

2 2n Let un = (1 − ε)e1 + 2ε−ε j=2 e2n +j . Then un = 1. Let Ln ⊂ 2n −1 H2n+1 be the subspace spanned by the vectors un , e2n +2 , e2n +3 , . . . , e2n+1 . Clearly, dim Ln = 2n . By [5, Lemma 2], there exists a subspace Ln ⊂ H2n+1 such that PLn − P Ln < ε/n and Ln ∩ Es n= {0} for s = 1, . . . , n − 1. Define Mn = Ln ⊕ {e2j+1 + tn e2j+2 ; j ≥ 2 }. Suppose that the subspaces Mn (n ∈ N) have been constructed in the above described way. As in [5], conditions (i), (ii) and (iv) are satisfied. So it is sufficient to show (iii). For j ∈ {2, . . . , n}, one has ε ε PMn ej = PLn ej ≤ PLn ej + PLn − PLn < ej , un un + = . n n Finally, for s < n, we have PMn e1 − PMs e1 = PLn e1 − PLs e1 ≥ PLn e1 − PLs e1 − PLn − PLn − PLs − PLs ε ε ≥ e1 , un un − e1 , us us − − n s ≥ (1 − ε)un − us − 2ε √ = (1 − ε) 2(2ε − ε2 ) − 2ε > ε. 1 . Then there exists an operator reflexive lattice Corollary 4.2. Let 0 < ε < 64 1 such that the operator hyperreflexivity constant is greater than 2√ . ε

Proof. Fix ε > 0 and let Mn be the subspaces constructed in Lemma 4.1. Let L = {0, I, PMn ; n = 1, 2, . . .}. By conditions (i) and (ii) in Lemma 4.1, L is a lattice. Claim. For each x ∈ H the set {Lx; L ∈ L} is closed. Proof. For j ≥ 2 we have limn→∞ PMn ej = 0. Consequently, limn→∞ PMn y = 0 for each y ∈ {ej ; j ≥ 2}. Let x ∈ H, x = αe1 + y for some α ∈ C, y ∈ {ej ; j ≥ 2}. For α = 0 the statement was shown above, so assume that α = 0. By property (iii), we have √ √ PMn (αe1 ) − PMs (αe1 ) ≥ |α| · ε, for all n = s. So PMn x − PMs x ≥ |α|2 ε for all n = s large enough. Hence the set {Lx; L ∈ L} is closed. It follows from [11] that L is operator reflexive; in particular, it is strongly closed.

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Consider now the orthogonal projection Q ∈ P(H) onto the 1-dimensional subspace Ce1 . Clearly d(Q, L) = 1. For x ∈ Hn , x = 1 we have Qx − PM x = Qx − PL x ≤ Qx − PL x + PL − PL n

n

n

n

n

≤ x, e1 e1 − x, un un + n−1 ε = x, e1 (e1 −(1−ε)un )+ε √ √ ≤ e1 − (1 − ε)un + ε ≤ 2ε + 2ε ≤ 2 ε. √ Hence α(Q, L) ≤ 2 ε and the operator hyperreflexivity constant of L is 1 . greater or equal to 2√ ε Corollary 4.3. There exists an operator reflexive subspace lattice which is not operator hyperreflexive. Proof. Let (cn )∞ n=1 be a sequence of positive numbers tending to ∞. For each n find a Hilbert space Hn and an operator reflexive subspace lattice Ln in the operator hyperreflexivity constant of Ln is greater than P(Hn ) such that

∞

∞ H and L = L . Then L is operator reflexive cn . Let H = n n n=1 n=1 subspace lattice that is not operator hyperreflexive, by Proposition 3.4. Acknowledgement We would like to thank the referee for suggestions which helped to simplify the proof of Corollary 4.2. 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.

References [1] Arveson, N.T.: Ten Lectures on Operator Algebras, CBMS Regional Conference Series 55 Am. Math. Soc., Providence (1984) [2] Ball, K.M.: The complex plank problem. Bull. Lond. Math. Soc. 33, 433–442 (2001) [3] Christensen, E.: Perturbations of operator algebras II. Indiana Univ. Math. J. 26, 891–904 (1977) [4] Davidson, K.R., Harrison, K.J.: Distance formulae for subspace lattices. J. Math. Soc. (2) 39, 309–323 (1989) [5] Kli´s-Garlicka, K., M¨ uller, V.: Non-operator reflexive subspace lattice. Integr. Eq. Oper. Theory 62, 595–599 (2008) [6] Kraus, J., Larson, D.R.: Reflexivity and distance formulae. Proc. Lond. Math. Soc. 53, 340–356 (1986) [7] Larson, D.R.: Hyperreflexivity and a dual product construction. Trans. Am. Math. Soc. 294(1), 79–88 (1986) [8] Loginov, A.I., Shul’man, V.S.: Hereditary and intermediate reflexivity of W ∗ algebras, Izv. Akad. Nauk. SSSR, 39, 1260–1273 (1975); Math. USSR-Izv. 9, 1189–1201 (1975)

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[9] M¨ uller, V.: Spectral Theory of Linear Operators, 2nd edition, Operator Theory, Advances and Applications, vol. 139. Birkh¨ auser, Basel (2007) [10] Rosenoer, S.: Distance estimates for von Neumann algebras. Proc. Am. Math. Soc. 86(2), 248–252 (1982) [11] Shulman, V.S., Todorov, I.G.: On subspace lattices I. Closedness type properties and tensor products. Integr. Eq. Oper. Theory 52, 561–579 (2005) J. Braˇciˇc University of Ljubljana, IMFM Jadranska ul. 19 1000 Ljubljana, Slovenia e-mail: [email protected] K. Kli´s-Garlicka (B) Institute of Mathematics University of Agriculture ul. Balicka 253c 30-198 Krak´ ow, Poland e-mail: [email protected] V. M¨ uller Institute of Mathematics Academy of Sciences of the Czech Republic ˇ Zitna 25, 115 67, Praha 1, Czech Republic e-mail: [email protected] I. G. Todorov Department of Pure Mathematics Queens University Belfast Belfast BT7 1NN, UK e-mail: [email protected] Received: January 19, 2010. Revised: May 7, 2010.

Integr. Equ. Oper. Theory 68 (2010), 391–411 DOI 10.1007/s00020-010-1824-5 Published online July 21, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Multiplication Operators on the Lipschitz Space of a Tree Flavia Colonna and Glenn R. Easley Abstract. In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on C is Euclidean. After observing that the space L of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of L we call the little Lipschitz space. Mathematics Subject Classification (2000). Primary 47B38, 05C05. Keywords. Multiplication operators, trees, Lipschitz space.

1. Introduction Given a complex Banach space X consisting of functions deﬁned on a set Ω and a complex-valued function ψ deﬁned on Ω, the multiplication operator with symbol ψ is deﬁned as the operator Mψ f = ψf for each f ∈ X. A natural problem in operator theory is to tie the operator theoretic properties of Mψ , such as boundedness, compactness, or being an isometry, to the function theoretic properties of the symbol ψ. In this work, we carry out the study of the multiplication operators in the case when Ω is an inﬁnite tree T , thought of as a metric space under the distance that counts the number of edges linking two vertices, and X is the space L of the complex-valued Lipschitz functions on T . Such space can be regarded as a discrete analogue of a Banach space of analytic functions on the open unit disk D called the Bloch space, as it will be explained later.

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The Bloch space B(D) consists of the analytic functions f on D (called Bloch functions) whose derivative grows no faster than the density of the Poincar´e metric. That is, 1 |f (z)| = O , as |z| → 1. 1 − |z|2 The map f → supz∈D (1−|z|2 )|f (z)| is a semi-norm. The Bloch norm is given by f B = |f (0)| + sup(1 − |z|2 )|f (z)|. z∈D

The subspace B0 (D) consisting of the Bloch functions f such that 1 |f (z)| = o , as |z| → 1. 1 − |z|2 is the closure of the polynomials in B(D) and is called the little Bloch space. For a good reference on Bloch functions, see [6]. Multiplication operators on the Bloch space and on the little Bloch space of the open unit disk D have been studied in [2,7,8,26]. Characterizations for the boundedness of multiplication operators were obtained independently by Arazy [7], and by Brown and Shields [8]. In [26], Ohno and Zhao characterized the bounded and the compact weighted composition operators on the Bloch space and on the little Bloch space. As a corollary, they deduced that the only compact multiplication operators are those whose symbol is identically zero. In [2], Allen and the ﬁrst author gave estimates on the operator norm, determined the spectrum, and showed that the only isometric multiplication operators are those whose symbol is a constant of modulus one. In higher dimensions, Bloch functions have been introduced by Hahn [16] and the space of Bloch functions has been extensively studied by Timoney in [28,29] in the setting of bounded homogeneous domains. Krantz and Ma [19] studied Bloch functions on a strongly pseudoconvex domain. Multiplication operators on the Bloch space in higher dimensions have been studied by Zhu for the case of the unit ball [33] and by Allen and the ﬁrst author in the bounded homogeneous domain setting [3]. There is an extensive literature on the study of the weighted composition operators (of which the multiplication operators are a special case) between the Bloch space (as well as its generalizations known as the α-Bloch spaces) and other Banach spaces of analytic functions. References include [4,5,17,20–25,27,30,31]. Bloch functions on the unit disk (as well as in the several variable setting of a bounded homogeneous domain) are precisely the analytic functions f which are Lipschitz when regarded as maps between the metric spaces (D, ρ) and (C, dE ), where ρ is distance induced by the Bergman metric and dE is the Euclidean distance. That is, f is Bloch if and only if there is a constant C > 0 such that |f (z) − f (w)| ≤ Cρ(z, w),

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for all pairs of points z and w. Furthermore, the infimum of all such constants C is precisely the Bloch semi-norm (see [13] (Theorem 10) and [32] (Theorem 5.1.6) for the case of the unit disk, [33] (Theorem 3.6) for the case of the unit ball in Cn , and [1] (Theorem 3.1) for the general case of a bounded homogeneous domain). Inﬁnite trees are discrete structures which are widely regarded as discretizations of the hyperbolic disk in the complex plane [9]. The link between homogeneous trees (that is, trees whose vertices have the same number of neighbors) and the hyperbolic disk is made geometrically explicit in [11]. In recent times, many classical problems in harmonic analysis, potential theory, integral geometry and functional analysis have stimulated interest in formulating and studying corresponding problems in discrete structures such as trees, grids, or, more generally, graphs. The space L under consideration in this work consists of the complexvalued functions f deﬁned on the vertices of an inﬁnite tree T such that |f (v) − f (u)| ≤ C d(v, u),

(1)

for all v, u vertices of T , where d(v, u) is the distance between v and u, and C is a constant. The elements of L can also be characterized by the property that the difference between the values at two neighboring vertices (i.e. vertices which are connected by an edge) remains bounded throughout the tree. Indeed, condition (1) implies the boundedness of |f (v) − f (w)| over all pairs of neighboring vertices v and w. Conversely, if |f (v) − f (w)| is bounded above by some constant C independent of the neighboring vertices v and w, then for arbitrary vertices v and u, we have |f (v) − f (u)| ≤

n

|f (vj ) − f (vj−1 )|,

j=1

where n = d(v, u) and v = v0 , v1 , . . . , vn = u (vj neighbor of vj−1 , j = 1, . . . , n) are the vertices of the unique path from v to u. Thus, |f (v)−f (u)| ≤ n C, which yields (1). Fixing a vertex o as a root of the tree, the distance between the values of a function at a vertex and at its neighboring vertex closest to o can be interpreted as a derivative. The apparent analogy between Bloch functions on the disk and the Lipschitz functions on a tree and the recent works in operator theory based on the Bloch space provided the motivation for carrying out this investigation. In [10,12], Cohen and the ﬁrst author introduced the Bloch functions on a tree as the harmonic functions (i.e. satisfying the mean value property) which are Lipschitz as maps from (T, d) to (C, dE ). These articles, however, focused only on homogeneous trees (that is, trees whose vertices have the same number of neighbors). In the present work, besides dropping the homogeneity assumption, we do not restrict to harmonic functions because the multiplication operators that preserve harmonic functions must have constant symbol. Thus, the study carried out in this article does not deal with the probabilistic aspects that typically arise when studying the functional analysis and the harmonic analysis on trees.

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1.1. Organization of the Paper After giving some preliminary definitions and notation on trees, in Sect. 2 we prove that the correspondence f ∈ L → |f (o)| +

sup |f (v) − f (v − )|

v∈T,v=o

is a norm which endows L of a Banach space structure, where T is an inﬁnite tree rooted at o. Note that setting Df (o) = 0 and Df (v) = |f (v) − f (v − )|,

for v = o,

a Lipschitz function on a tree can then be thought of as a function f satisfying the condition Df (v) = O(1)

as d(v, o) → ∞.

We introduce the subspace L0 of L consisting of the functions f such that Df (v) = o(1)

as d(v, o) → ∞

and show that it closed and separable. In Sect. 3, we show that a multiplication operator on L is bounded if and only if it is bounded as an operator acting on L0 and characterize such an operator. In Sect. 4, we establish operator norm estimates. In Sect. 5, we determine the spectrum, the point spectrum, and the approximate point spectrum of the bounded multiplication operators acting on L or L0 , and in Sect. 6 we characterize those operators that are bounded below. In Sect. 7, we characterize the compact multiplication operators on L. Unlike the compact operators on the Bloch space in the disk, which reduce to the operator induced by the constant 0, in the tree case such operators constitute a large class. In Sect. 8, we provide estimates on the essential norm of the bounded multiplication operators on L. Finally, in Sect. 9, we show that the only isometric multiplication operators on L and L0 are those induced by a constant function of modulus one. 1.2. Preliminary Definitions and Notation By a tree T we mean a locally ﬁnite connected and simply connected graph, which, as a set, we identify with the collection of its vertices. Two vertices v and w are called neighbors if there is an edge [v, w] connecting them, and we use the notation v ∼ w. A vertex is called terminal if it has a unique neighbor. A path is a ﬁnite or inﬁnite sequence of vertices [v0 , v1 , . . .] such that vk ∼ vk+1 and vk−1 = vk+1 , for all k. An inﬁnite path is also called a ray. Given a tree T rooted at o and a vertex v ∈ T , a vertex w is called a descendant of v if v lies in the unique path from o to w. The vertex v is then called an ancestor of w. We call the parent of a vertex v = o the neighbor v − of v which is an ancestor of v. The vertex v is called a child of v − . For v ∈ T , the set Sv consisting of v and all its descendants is called the sector determined by v. Deﬁne the length of a ﬁnite path [v = v0 , v1 , . . . , w = vn ]

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(with vk ∼ vk+1 for k = 0, . . . , n − 1) to be the number n of edges connecting v to w. The distance, d(v, w), between vertices v and w is the length of the unique path connecting v to w. Deﬁne the length of a vertex v as |v| = d(o, v). A function on a tree is a complex-valued function on the set of its vertices. In this paper, we shall assume the tree to be without terminal vertices, (and hence inﬁnite), and rooted at a vertex o and shall denote by L∞ the space of the bounded functions f on the tree equipped with the supremum norm f ∞ = supv∈T |f (v)|.

2. The Lipschitz Space of a Tree By the remarks made in the introduction, the Lipschitz space of T can be thought of as the space L of the functions f on T such that Df ∈ L∞ . We shall denote by max|w|=n the maximum over the vertices w of length taken over all n. Similarly, we adopt the notations sup v=o for the supremum vertices v with v = o and |v|=n f (v), |v|n f (v), for the sum of f (v) taken over all vertices v of length n, less than n, and greater than n, respectively. For a Lipschitz function f on T , let f L = |f (o)| + Df ∞ . Theorem 2.1. The correspondence f ∈ L → f L is a norm which endows L of a complex Banach space structure. Proof. It is straightforward to see that L is a complex vector space and that the mapping f → Df ∞ is a semi-norm. Thus, to prove that L is a Banach space, it sufﬁces to show that f L = 0 if and only if f is identically 0 and that L is complete. It is clear that ||0||L = 0. Conversely, if f L = 0, then Df is identically 0, which implies that f is constant. But f (o) = 0, so f must vanish identically. To show that L is complete, let {fn }n∈N be a Cauchy sequence in L. We need to show that {fn } converges in norm to some function f ∈ L. We begin by showing that {fn } converges pointwise to some function on T arguing by induction on the distance from o. Since |fn (o)| ≤ fn L , the sequence {fn (o)} is Cauchy in C, and so it converges to some complex number we denote by f (o). Assume that for some N ∈ N, the sequence {fn (w)} converges to some complex number f (w) for all w ∈ T with |w| = N . Let v be a vertex of length N + 1. For n, m ∈ N, we have |fn (v)−fm (v)| ≤ D(fn −fm )∞ +|fn (v − )−fm (v − )|. By the inductive hypothesis, {fn (v − )} is convergent. Since D(fn −fm )∞ ≤ fn − fm L , we deduce that {fn (v)} is Cauchy in C and hence convergent to some complex number f (v). Therefore, by induction, {fn } converges pointwise to a function f . We now show that f is Lipschitz. If v is a vertex, v = o, and n ∈ N, then |f (v) − f (v − )| ≤ |f (v) − fn (v)| + |fn (v) − fn (v − )| + |fn (v − ) − f (v − )| = |f (v) − fn (v)| + Dfn (v) + |fn (v − ) − f (v − )|. (2)

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Since for each v ∈ T, Dfn (v) ≤ fn L and {fn } is Cauchy in L, and hence bounded, {Dfn (v)} is uniformly bounded by some constant C, and so (2) yields |f (v) − f (v − )| ≤ lim inf Dfn (v) ≤ C. n→∞

Hence f is Lipschitz and Df ≤ lim inf n→∞ Dfn . To conclude the proof of the completeness, we need to show that fn − f L → 0 as n → ∞. Since fn (o) → f (o), it sufﬁces to show that D(fn − f ) → 0 in L∞ as n → ∞. Arguing by contradiction, suppose there exist ε > 0 and a subsequence {fnj }j∈N such that D(fnj − f )∞ > ε for all j ∈ N. Then for each j ∈ N, pick two neighboring vertices vnj and wnj , with vnj child of wnj , such that |fnj (vnj ) − f (vnj ) − (fnj (wnj ) − f (wnj ))| ≥ ε. Since {fnj } is Cauchy in L, there exists a positive integer J such that for each j, h ≥ J, and v ∈ T , v = o, we have ε |fnj (v)−fnh (v)−(fnj (v − )−fnh (v − ))| ≤ D(fnj −fnh )∞ ≤ fnj −fnh L < . 2 In particular, for all h ≥ J, we have ε |fnJ (vnJ ) − fnh (vnJ ) − (fnJ (wnJ ) − fnh (wnJ ))| < . (3) 2 On the other hand, by the pointwise convergence of fnh to f , for all integers h sufﬁciently large ε (4) |fnh (vnJ ) − f (vnJ ) − (fnh (wnJ ) − f (wnJ ))| < . 2 Thus, by the triangle inequality, from (3) and (4) we deduce that |fnJ (vnJ ) − f (vnJ ) − (fnJ (wnJ ) − f (wnJ ))| < ε, contradicting the choice of vnJ and wnJ . The completeness of L is established. Definition 2.2. The little Lipschitz space is deﬁned as the subspace L0 of L consisting of all functions f on T such that lim Df (v) = 0.

|v|→∞

Recall that B0 (D) is the closure of the polynomials in B(D) [6]. The following result shows that there is a collection of functions in L0 which plays the role of the polynomials for the tree case. Theorem 2.3. The space L0 is the closure in L of the collection n P= ak pvk : n ∈ N, vk ∈ T, ak ∈ Q[i], k = 1, . . . , n , k=1

where pv is the characteristic function of the sector Sv determined by v. In particular, L0 is a closed separable subspace of L. Proof. Since Dpv equals the characteristic function χv of {v} for v = o, and Dpo is identically 0, it follows that pv ∈ L0 for all v ∈ T . Therefore by linearity, P is a subset of L0 . Thus, to prove that L0 is the closure of P in L, it

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sufﬁces to show that if f ∈ L0 , there exists a sequence {fn } in P, such that fn − f L → 0 as n → ∞. Fix f ∈ L0 and for n ∈ N, let f (v) if |v| ≤ n, gn (v) = f (vn ) if |v| > n, where vn denotes the ancestor of v of length n. Then, gn − f L = D(gn − f )∞ = sup |gn (v) − f (v) − (gn (v − ) − f (v − ))| v=o

= sup |f (v) − f (v − )| |v|>n

= sup Df (v) → 0

(5)

|v|>n

as n → ∞. Fix ε > 0 and n ∈ N. Observe that the set of vertices of a tree is countable. Since Q is dense in R, for each vertex v we may choose a complex number qv with rational real and imaginary parts such that |f (v)−qv | < ε/3. Deﬁne qv if |v| ≤ n, fn (v) = qvn if |v| > n, where vn is as above. Then, the functions fn and gn may be represented as q v χv + q v pv , fn = |v|
gn =

|v|
|v|=n

f (v)χv +

f (v)pv .

|v|=n

Observe that χv = pv − w− = v pw for each vertex v = o. Thus, fn is a ﬁnite linear combination of functions of the form pv with coefﬁcients in Q[i], so fn ∈ P. Observing that both fn and gn remain constant in each sector relative to the vertices of length n, we obtain fn − gn L ≤ |qo − f (o)| + 2 max |qv − f (v)| < ε. |v|≤n

(6)

Therefore, using (5) and (6), and passing to the limit as n → ∞, we obtain limn→∞ fn − f L ≤ ε. Since ε was arbitrary, limn→∞ fn − f L = 0. The proof is now complete. The following result will be used in Sect. 6 to determine estimates on the essential norm of the bounded multiplication operators. Proposition 2.4. Let {fn } be a sequence of functions in L0 converging to 0 pointwise in T and such that {fn L } is bounded. Then fn → 0 weakly in L0 . Proof. First suppose fn (o) = 0 for all n ∈ N so that fn L = Dfn ∞ . Then, the sequence {Dfn } converges to 0 pointwise. Observe that the subspace of L0 whose elements send o to 0 is isomorphic to the space c0 , consisting of the sequences indexed by T which vanish at inﬁnity, under the supremum

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norm via the correspondence f → Df . On the other hand, it is well known (e.g. [18]) that c0 has dual isomorphic to the space 1 of absolutely summable sequences via the correspondence g ∈ 1 → g˜ ∈ c∗0 , where for f ∈ c0 g˜(f ) = f (v)g(v). v∈T

Thus, under the identiﬁcation of L0 with c0 , if fn ∈ c0 converges pointwise to 0 and is bounded in c0 , then for any g ∈ 1 , we have |˜ g (fn )| = fn (v)g(v) ≤ |fn (v)||g(v)|. (7) v∈T

v∈T

Let c = supn∈N,v∈T |fn (v)|. Fixing any positive integer N , we may split the sum on the right-hand side of (7) into the two sums S1 (n, N ) = |fn (v)||g(v)| and S2 (n, N ) = |fn (v)||g(v)|. |v|≤N

|v|>N

Since fn → 0 uniformly on the set {v ∈ T : |v| ≤ N }, we see that S1 (n, N ) ≤ max |fn (v)|g1 → 0, |v|≤N

as n → ∞.

On the other hand, since g ∈ 1 , the tail end of the series v∈T |g(v)| approaches 0. Therefore, g (fn )| ≤ lim S1 (n, N ) + sup S2 (n, N ) ≤ c |g(v)|. lim |˜ n→∞

n→∞

n∈N

|v|>N

Letting N → ∞, we deduce that limn→∞ g˜(fn ) = 0. Hence, if fn (o) = 0, then fn converges to 0 weakly. In the general case, deﬁne Fn = fn −fn (o). By the previous part, Fn → 0 weakly. Since fn (o) → 0, we conclude that fn → 0 weakly as well.

3. Boundedness In this section, we characterize the boundedness of the multiplication operators on L and L0 . For v ∈ T , deﬁne ω(v) = sup {|f (v)| : f (o) = 0, f L ≤ 1} , f ∈L

ω0 (v) = sup {|f (v)| : f (o) = 0, f L ≤ 1} . f ∈L0

Lemma 3.1. For v ∈ T, ω(v) = ω0 (v) = |v|. Proof. Letting g(v) = |v|, for v ∈ T , we see that g ∈ L, g(o) = 0 and gL = Dg∞ = 1. Thus, for v ∈ T , we have |v| = |g(v)| ≤ ω(v). Conversely, let f be any function in L such that f (o) = 0 and f L ≤ 1. We are going to show that for v ∈ T , |f (v)| ≤ |v| arguing by induction on n = |v|. The assertion is trivial for n = 0. Let n ∈ N and assume |f (u)| ≤ |u|, for each vertex u of length less than n. Let v be a vertex of length n. Then |f (v)| ≤ |f (v) − f (v − )| + |f (v − )| ≤ Df ∞ + |v − | ≤ n, since Df ∞ ≤ 1 and |v − | = n − 1. Hence |v| = ω(v) for each v ∈ T .

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We now show that ω0 (v) = |v| for each v ∈ T . Since ω0 (v) ≤ ω(v) = |v|, it sufﬁces to show that |v| ≤ ω0 (v). This inequality holds trivially if v = o. So assume v is a ﬁxed vertex, v = o. We are going to construct a function fv ∈ L0 such that fv (o) = 0, fv L = 1, and fv (v) = |v|. For w ∈ T , deﬁne ⎧ ⎪ if |w| ≤ |v|, ⎨|w| fv (w) = 2|v| − |w| if |v| + 1 ≤ |w| ≤ 2|v| − 1, ⎪ ⎩ 0 if |w| ≥ 2|v|. Then fv ∈ L0 , fv (o) = 0, fv (v) = |v|, and fv L = 1. Hence, |v| ≤ ω0 (v).

Recall that a Banach space X of complex-valued functions on a set Ω is said to be a functional Banach space if for each ω ∈ Ω, the point evaluation functional eω : f ∈ X → f (ω) is bounded; that is, there exists a constant C > 0 such that |f (ω)| ≤ C, for each f ∈ X with f = 1. In [15], the authors proved following result. Lemma 3.2. [15, Lemma 11] Let X be a functional Banach space on the set Ω and let ψ be a complex-valued function on Ω such that Mψ maps X into itself. Then Mψ is bounded on X and |ψ(ω)| ≤ Mψ for all ω ∈ Ω. In particular, ψ is bounded. Proposition 3.3. L is a functional Banach space. Proof. Fix v ∈ T . Then for each f ∈ L with f L = 1, by Lemma 3.1 applied to the function g = f − f (o), we have |ev (f )| = |f (v)| ≤ |f (o)| + |g(v)| ≤ 1 + |v|. Therefore, the point evaluation functional ev on L is bounded.

Lemma 3.4. Let T be a tree and v ∈ T . (a)

If f ∈ L, then |f (v)| ≤ |f (o)| + |v|Df ∞ .

(b)

In particular, if f L ≤ 1, then |f (v)| ≤ |v| for each v ∈ T, v = o. If f ∈ L0 , then lim

|v|→∞

f (v) = 0. |v|

Proof. The conclusion in (a) is immediate if f is constant. Suppose f is a non-constant function in L. Then the function g deﬁned on T by g(v) =

1 (f (v) − f (o)) Df ∞

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maps o to 0 and gL = 1. So by Lemma 3.1, |g(v)| ≤ |v| for all v ∈ T . Thus, |f (v)| ≤ |f (o)| + |f (v) − f (o)| = |f (o)| + |g(v)| Df ∞ ≤ |f (o)| + |v|Df ∞ , for all v ∈ T . If f L ≤ 1, then |f (o)| ≤ 1 − Df ∞ , so for each v ∈ T, v = o, we have |f (v)| ≤ |f (o)| + |v|Df ∞ ≤ 1 + (|v| − 1)Df ∞ ≤ |v|. To prove (b), let f ∈ L0 , and assume ﬁrst that f (o) = 0. The assertion is clear if f is constant. So assume f is non-constant. Since lim|v|→∞ Df (v) = 0, corresponding to ε ∈ (0, Df ∞ ), there exists N ∈ N such that Df (v) < ε whenever |v| ≥ N . Thus, if w ∈ T with |w| = N , v is a descendant of w, and the path from w to v is [w = u0 , u1 , . . . , u|v|−|w| = v], with uk and uk−1 neighbors, k = 1, . . . , |v| − |w|, then |v|−|w|

|f (v)| ≤ |f (w)| + |f (v) − f (w)| ≤ |f (w)| +

|f (uk ) − f (uk−1 )|.

k=1

Thus, using part (a), we obtain |v|−|w|

|f (v)| ≤ |w|Df ∞ +

Df (uk )

k=1

≤ |w|Df ∞ + (|v| − |w|)ε = |w|(Df ∞ − ε) + |v|ε. Hence |f (v)| |w| < (Df ∞ − ε) + ε. |v| |v| As |v| → ∞, the right-hand side approaches ε. Therefore, letting ε → 0, we (v) = 0, proving the result when f (o) = 0. obtain lim|v|→∞ f|v| In the general case, let g(v) = f (v) − f (o) and observe that g ∈ L0 , and g(o) = 0. So by the previous case, f (v) f (o) g(v) = + →0 |v| |v| |v|

as |v| → ∞.

The proof of (b) is now complete.

Definition 3.5. For a function ψ on T , deﬁne σψ = sup |v|Dψ(v). v=o

Observe that, if σψ is ﬁnite, then ψ ∈ L0 . However, the converse is false, as shown by considering the function ψ(v) = |v|1/2 , for v ∈ T . Indeed, we have 1 Dψ(v) = |v|1/2 − (|v| − 1)1/2 = 1/2 → 0, |v| + (|v| − 1)1/2 as |v| → ∞, while |v|Dψ(v) → ∞.

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We now show that boundedness of the multiplication operator on L is equivalent to its boundedness on L0 and give a simple characterization in terms of the quantity σψ . Theorem 3.6. Let T be a tree and ψ a function on T . Then the following are equivalent statements. (a) Mψ is bounded on L. (b) Mψ is bounded on L0 . (c) ψ ∈ L∞ and σψ is finite. Proof. (c) =⇒ (a): Assume ψ ∈ L∞ with σψ < ∞, and let f ∈ L. We ﬁrst show that ψf ∈ L. Let v = o be given. Then, by the triangle inequality and the boundedness of ψ, we have |ψ(v)f (v) − ψ(v − )f (v − )| ≤ |ψ(v) − ψ(v − )||f (v)| + |ψ(v − )||f (v) − f (v − )| ≤ |f (v)|Dψ(v) + ψ∞ Df (v).

(8)

Using part (a) of Lemma 3.4, inequality (8) yields D(ψf )(v) ≤ (|f (o)| + |v|Df ∞ )Dψ(v) + ψ∞ Df ∞ ≤ |f (o)| Dψ∞ + (σψ + ψ∞ )Df ∞ .

(9)

Thus, ψf ∈ L. Hence, by Proposition 3.3 and Lemma 3.2, we conclude that Mψ is bounded on L. (a) =⇒ (c): Suppose Mψ is bounded on L. Then by Lemma 3.2, ψ is bounded with ψ∞ ≤ Mψ . To prove that σψ is ﬁnite, let f ∈ L. For each v ∈ T, v = o, we have |f (v)| ψ(v) − ψ(v − ) ≤ f (v)ψ(v) − f (v − )ψ(v − ) + ψ(v − ) f (v) − f (v − ) . Therefore |f (v)| Dψ(v) ≤ D(ψf )(v) + ψ∞ Df (v) ≤ Mψ f L + ψ∞ Df ∞ ≤ (Mψ + ψ∞ ) f L . Taking the supremum over all functions f ∈ L such that f (o) = 0 and f L ≤ 1, and using Lemma 3.1, we get |v|Dψ(v) ≤ Mψ + ψ∞ . Consequently, taking the supremum over all v ∈ T, v = o, we obtain σψ ≤ Mψ + ψ∞ . (c) =⇒ (b): Assume ψ ∈ L∞ and σψ < ∞. Then for f ∈ L0 and v ∈ T , v = o, by (8) and part (b) of Lemma 3.4, we have D(ψf )(v) ≤

|f (v)| |v|Dψ (v) + ψ∞ Df (v) → 0 |v|

as |v| → ∞. Thus, ψf ∈ L0 . The boundedness of Mψ on L0 follows from Lemma 3.2. (b) =⇒ (c): Arguing as in the proof of (a) =⇒ (c), if Mψ is bounded on L0 , then ψ ∈ L∞ and σψ ≤ Mψ + ψ∞ . Notice that neither condition in part (c) of Theorem 3.6 implies the other, as the following two examples show.

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Example 3.1. Let T be a tree containing a ray starting at o such that the set {w ∈ T \ρ : w ∼ v

for some v ∈ ρ}

is inﬁnite. Let ψ be the characteristic function of ρ. Then ψ is bounded on T , yet supv=o |v| Dψ(v) is inﬁnite. Example 3.2. Deﬁne ψ(o) = 0, and for v ∈ T , with |v| ≥ 1, let 1 ψ(v) = ψ(v − ) + . |v| |v| Observe that ψ(v) = k=1 k1 for all v ∈ T, v = o, so that ψ is unbounded. 1 On the other hand, Dψ(v) = |v| for each v ∈ T, v = o. In particular, supv=o |v|Dψ(v) = 1.

4. Operator Norm Estimates In this section, we provide estimates on the norm of the bounded multiplication operators on the Lipschitz space of a tree and show that they are sharp. Theorem 4.1. If Mψ is a bounded multiplication operator on L or L0 , then max{ψL , ψ∞ } ≤ Mψ ≤ ψ∞ + σψ .

(10)

Proof. We will prove the norm estimates for Mψ bounded on L. The argument for L0 is similar. First observe that by Lemma 3.2, we have the estimate ψ∞ ≤ Mψ . Furthermore, for f identically 1, Mψ f L = ψL . Therefore Mψ ≥ max{ψL , ψ∞ }. To prove the upper estimate, observe that ψL = |ψ(o)| + Dψ∞ ≤ ψ∞ + σψ .

(11)

Let f ∈ L. Apply the identity |f (o)| = f L − Df ∞ to (9) and use (11) to deduce Mψ f L ≤ |f (o)|ψL + (σψ + ψ∞ )Df ∞ ≤ (ψ∞ + σψ )f L .

The proof is now complete. By taking ψ to be a constant we see that Mψ f L = |ψ|f L = max{ψL , ψ∞ }f L .

Thus, the lower estimate in (10) is sharp. The following example shows that the upper estimate in (10) is sharp as well. Example 4.1. Let ψ = χo . Then ψ∞ = 1, σψ = 1, so that the upper bound on the norm of Mψ in Theorem 4.1 is 2. On the other hand, for f ∈ L, Mψ f L = |f (o)| + sup |ψ(v)f (v) − ψ(v − )f (v − )| = 2|f (o)|. v=o

In particular, if f is constant, then Mψ f L = 2f L . Thus, Mψ = 2 = ψ∞ + σψ .

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5. Spectrum In this section, we determine the spectrum, the point spectrum and the approximate point spectrum of a bounded multiplication operator on the Lipschitz space and little Lipschitz space of a tree. Recall that the spectrum of a bounded operator A on a Banach space X is deﬁned as σ(A) = {λ ∈ C : A − λI is not invertible} , where I is the identity operator. The spectrum of a bounded operator is a nonempty compact subset of C. The set of eigenvalues σp (A) of a bounded operator A is called the point spectrum of A, that is σp (A) = {λ ∈ C : ker(A − λI) = {0}} . The approximate point spectrum of A is deﬁned as the set σap (A) consisting of all λ ∈ C corresponding to which for each n ∈ N there exists xn ∈ X with xn = 1 such that (A − λI)xn → 0 as n → ∞. It is clear that σp (A) ⊆ σap (S) ⊆ σ(A).

(12)

Furthermore, it is well known [14, Proposition 6.7] that ∂σ(A) ⊆ σap (A).

(13)

Note that if λ ∈ C, then Mψ − λI = Mψ−λ . Thus, λ ∈ σ(Mψ ) if and only if Mψ−λ is not invertible. Theorem 5.1. Let Mψ be a bounded multiplication operator on L or L0 . Then (a) σp (Mψ ) = ψ(T ); (b) σ(Mψ ) = σap (Mψ ) = ψ(T ). Proof. Due to the equivalence between the boundedness of a multiplication operator on L and on L0 , it sufﬁces to prove the result in the case when Mψ is a bounded operator acting on L. To prove (a), assume λ ∈ σp (Mψ ). Then there is a nonzero function f ∈ L such that ψf = λf . Thus, there is a vertex v such that f (v) = 0 and ψ(v)f (v) = λf (v). This implies that ψ(v) = λ, so λ ∈ ψ(T ). Therefore σp (Mψ ) ⊆ ψ(T ). Conversely, if λ ∈ ψ(T ), then there exists v ∈ T such that ψ(v) = λ. Thus, the function χv is not identically 0, whereas (Mψ − λI)χv is identically 0. Therefore λ ∈ σp (Mψ ). We next show that σ(Mψ ) = ψ(T ). Using part (a) and recalling that the spectrum is closed, passing to the closure we obtain ψ(T ) ⊆ σ(Mψ ).

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Conversely, suppose λ ∈ ψ(T ). Then there exists c > 0 such that |ψ(v) − λ| ≥ c for all v ∈ T . Thus the function ϕλ deﬁned by ϕλ (v) = (ψ(v) − λ)−1 is bounded on T . Furthermore, by a straightforward calculation, we have sup |v|Dϕλ (v) ≤ sup v=o

v=o

1 1 |v|Dψ (v) ≤ 2 σψ . 2 c c

Therefore, σϕλ ≤ c12 σψ , which is ﬁnite and so, by Theorem 3.6, the operator Mϕλ is bounded on L. Hence Mψ−λ is invertible on L, which proves that λ∈ / σ(Mψ ). The proof of the equality σ(Mψ ) = ψ(T ) is now complete. The equality σap (T ) = ψ(T ) follows at once from the previous part and the inclusions (12) and (13). The proof of (b) is now established.

6. Boundedness from Below Recall that a bounded operator A on a Banach space X is bounded below if there exists a positive constant C such that Ax ≥ Cx for each x ∈ X. The following result relates the notion of approximate point spectrum of a bounded operator on a Banach space to the notion of an associated operator that is bounded below. Proposition 6.1. [14, Proposition 6.4] For a bounded operator A on a Banach space and for λ ∈ C, the following statements are equivalent. (a) λ ∈ / σap (A). (b) A − λI is injective and has closed range. (c) A − λI is bounded below. From Proposition 6.1 and Theorem 5.1, it follows that if Mψ is a bounded multiplication operator on L or L0 , then Mψ is bounded below if and only if 0∈ / ψ(T ). We deduce the following result. Theorem 6.2. The bounded operator Mψ on L or L0 is bounded below if and only if inf{|ψ(v)| : v ∈ T } > 0.

7. Compactness In this section we prove the equivalence of the compactness of a multiplication operator as an operator acting on L and on L0 and give a simple characterization of the compact multiplication operators on these spaces. Unlike the case of the multiplication operators on the Bloch space of the unit disk (as well as in higher dimensions), in which the class of such operators reduces to the one whose symbol is identically 0, there is a large class of compact operators on L and L0 . Lemma 7.1. A bounded multiplication operator Mψ on L (respectively, L0 ) is compact if and only if for every bounded sequence {fn } in L (respectively, L0 ) converging to 0 pointwise, the sequence ψfn L approaches 0 as n → ∞.

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Proof. We shall prove the result for Mψ bounded as an operator acting on L. The proof for Mψ acting on L0 is similar. Assume Mψ is compact on L and let {fn } be a bounded sequence in L converging to 0 pointwise. By rescaling the sequence, if necessary, we may assume fn L ≤ 1 for all n ∈ N. By the compactness of Mψ , {fn } has a subsequence {fnk } such that {ψfnk } converges in norm to some function f ∈ L. Observe that for v ∈ T , by part (a) of Lemma 3.4 applied to the function ψfnk − f , we have |ψ(v)fnk (v) − f (v)| ≤ |ψ(o)fnk (o) − f (o)| + |v|D(ψfnk − f )∞ ≤ (1 + |v|)ψfnk − f L . Therefore, ψfnk → f pointwise. Since by assumption, fn → 0 pointwise, it follows that f must be identically 0, whence ψfn L → 0. Since 0 is the only limit point in L of the sequence {ψfn }, it follows that ψfn L → 0 as n → ∞. Conversely, suppose that for every bounded sequence {fn } in L converging to 0 pointwise, the sequence ψfn L approaches 0 as n → ∞. Let {gn } be a sequence in L with gn L ≤ 1. Then |gn (o)| ≤ 1 and by part (a) of Lemma 3.4, for each v ∈ T , v = o, we have |gn (v)| ≤ |v|. Therefore, gn is uniformly bounded on ﬁnite subsets of T and so some subsequence, which for notational convenience we reindex as the original sequence, converges to some function g. Then, for v ∈ T , v = o, we have Dg(v) ≤ |g(v) − g(v − ) − (gn (v) − gn (v − ))| + Dgn (v). Fix ε > 0 and v ∈ T, v = o. Since gn → g pointwise, |g(o) − gn (o)| < ε/2, |gn (v) − g(v)| < ε/2 and |gn (v − ) − g(v − )| < ε/2 for all n sufﬁciently large. Therefore Dg(v) < ε + Dgn (v) for n sufﬁciently large, so g ∈ L. Therefore, the sequence {fn } deﬁned by fn = gn − g is bounded in L and converges to 0 pointwise, hence, by the hypothesis, ψfn L → 0 as n → ∞. We conclude that ψgn → ψg in norm, proving the compactness of Mψ . Theorem 7.2. Let Mψ bounded multiplication operator on L, or equivalently, L0 . Then the following statements are equivalent: (a) Mψ is compact on L. (b) Mψ is compact on L0 . (c) The following conditions hold: lim ψ(v) = 0,

(14)

lim |v|Dψ(v) = 0.

(15)

|v|→∞ |v|→∞

Proof. (a) =⇒ (c): Assume Mψ is compact on L. To prove (14) and (15), it sufﬁces to show that given a sequence {vn } in T such that 2 ≤ |vn | → ∞, we have limn→∞ ψ(vn ) = 0 and limn→∞ |vn |Dψ(vn ) = 0. Let {vn } be such a sequence and, for n ∈ N, let fn be the characteristic function of {vn }. Then fn → 0 pointwise and fn L = 1. Thus, by Lemma 7.1, it follows that

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|ψ(vn )| = ψfn L → 0 as n → ∞. Next, ⎧ ⎪ ⎨0 gn (v) = 2|v| − |vn | + 2 ⎪ ⎩ |vn |

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let if |v| < |v2n | , if |v2n | ≤ |v| < |vn |, if |v| ≥ |vn |.

Then gn → 0 pointwise, gn L = 2 and gn (vn− ) = gn (vn ) = |vn |. Since Mψ is compact, by Lemma 7.1 we obtain ψgn L → 0. On the other hand ψgn L ≥ |ψ(vn )gn (vn ) − ψ(vn− )gn (vn− )| = |vn ||ψ(vn ) − ψ(v − )| = |vn |Dψ(vn ).

Therefore limn→∞ |vn |Dψ(vn ) = 0. Conditions (14) and (15) are now established. (c) =⇒ (a): Assume (14) and (15) hold and that ψ is not the constant function 0, which clearly induces a compact multiplication operator. By Lemma 7.1, to prove that Mψ is compact, it sufﬁces to show that if {fn } is a sequence in L converging to 0 pointwise and such that s = supn∈N fn L < ∞, then ψfn L → 0 as n → ∞. Let {fn } be such a sequence and ﬁx a ε for all n sufﬁciently large, and there positive number ε. Then |fn (o)| < 3 ψ L exists M ∈ N such that ε ε |ψ(v)| < and |v|Dψ(v) < for |v| ≥ M. 3s 3s Furthermore, for each v ∈ T, v = o, we have D(ψfn )(v) ≤ |ψ(v) − ψ(v − )||fn (v)| + |ψ(v − )||fn (v) − fn (v − )| = Dψ(v)|fn (v)| + |ψ(v − )|Dfn (v). If |v| > M , then |v − | ≥ M , so that |ψ(v − )| < Lemma 3.4 and (16), we obtain

ε 3s .

(16)

Thus, by part (a) of

ε < ε. 3 Since fn → 0 uniformly on {v ∈ T : |v| ≤ M } as n → ∞, so does the sequence Dfn . Therefore, by (16), for n sufﬁciently large and for each v ∈ T, D(ψfn )(v) < ε. On the other hand, fn (o) → 0, and so ψfn L → 0, as n → ∞. The proof of the equivalence of (b) and (c) is analogous. D(ψfn )(v) ≤ (|fn (o)| + |v|Dfn ∞ )Dψ(v) +

From Theorem 7.2 it follows that, besides the multiplication operators whose symbol has ﬁnite support, compact operators include Mψ where ψ(v) = |v|1 p for p ≥ 1/2 and v = o.

8. Estimates on the essential norm In this section, we provide estimates on the essential norm of the bounded multiplication operators on L. The techniques we developed here were inspired by those of MacCluer and Zhao in [24], who obtained estimates

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on the essential norm of a weighted composition operator on the Bloch space of the unit disk. We recall that the essential norm Ae of an operator A on a Banach space X is the distance in the operator norm from the compact operators, that is, Ae = inf{A − K : K compact operator on X}. Definition 8.1. Given a bounded multiplication operator Mψ on L, deﬁne A(ψ) = lim sup |ψ(v)| n→∞ |v|≥ n

B(ψ) = lim sup |v|Dψ(v). n→∞ |v|≥ n

The quantities A(ψ) and B(ψ) are ﬁnite because ψ is bounded and σψ is ﬁnite by the boundedness of Mψ and Theorem 3.6. Theorem 8.2. Let Mψ be bounded on L or L0 . Then

1 Mψ e ≥ max A(ψ), B(ψ) . 2 Proof. For each n ∈ N, deﬁne fn = χ{v: |v|=n} . Then fn ∈ L0 , fn L = 1, and fn → 0 pointwise. Therefore, by Proposition 2.4, the sequence {fn } converges weakly to 0 in L0 . Since compact operators are completely continuous [14], it follows that limn→∞ Kfn L = 0 for any compact operator K on L0 . Therefore, if K is a compact operator on L0 , then Mψ − K ≥ lim sup (Mψ − K)fn L ≥ lim sup Mψ fn L . n→∞

n→∞

Thus, Mψ e ≥ inf{Mψ − K : K compact in L0 } ≥ lim sup Mψ fn L n→∞

= lim sup sup |ψ(v)fn (v) − ψ(v − )fn (v − )| n→∞ v=o

= lim sup |ψ(v)| = A(ψ). n→∞ |v|≥ n

Next we show that Mψ e ≥ 12 B(ψ). The result is trivial if B(ψ) = 0. So assume {vn } is a sequence of vertices of length greater than 2 such that |vn | is increasing unboundedly and lim |vn |Dψ(vn ) = B(ψ).

n→∞

For n ∈ N, let

⎧ ⎪ ⎨0 hn (v) =

(|v|+1)2 ⎪ |vn |

⎩ |vn |

if v = o, if 1 ≤ |v| < |vn |, if |v| ≥ |vn |.

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Then hn (o) = 0, hn (vn ) = hn (vn− ) = |vn |, and ⎧ 4 if |v| = 1, ⎪ ⎪ ⎨ |vn | Dhn (v) = 2|v|+1 if 1 < |v| < |vn |, |vn | ⎪ ⎪ ⎩ 0 if |v| ≥ |vn |. Thus, the supremum of Dhn is attained at the vertices v such that |v| = < 2. Letting gn = hhnn L , we see that |vn | − 1 and 1 ≤ hn L = 2|v|vnn|−1 | gn ∈ L0 , gn L = 1, and gn → 0 pointwise. Consequently, by Proposition 2.4, the sequence {gn } converges to 0 weakly. This implies that Kgn L → 0 as n → ∞ for any compact operator K on L0 . We deduce that for any such operator K Mψ − K ≥ lim sup (Mψ − K)gn L ≥ lim sup ψgn L . n→∞

n→∞

Therefore Mψ e ≥ inf{Mψ − K : K compact in L0 } ≥ lim sup D(ψgn )∞ .

(17)

n→∞

Next, observe that for each n ∈ N, gn (vn ) = gn (vn− ) =

|vn |2 2|vn |−1 ,

so

D(ψgn )(vn ) = |ψ(vn )gn (vn ) − ψ(vn− )gn (vn− )| |vn |2 . = Dψ(vn ) 2|vn | − 1

(18)

Therefore, from (17) and (18), we obtain Mψ e ≥ lim sup n→∞

|vn | 1 |vn |Dψ(vn ) = B(ψ), 2|vn | − 1 2

completing the proof. We now turn to the upper estimate. Theorem 8.3. If Mψ is bounded on L (or equivalently, L0 ), then Mψ e ≤ A(ψ) + B(ψ). Proof. Fix n ∈ N, deﬁne the operator Kn on L by f (v) if |v| ≤ n, Kn f (v) = f (vn ) if |v| > n,

where f ∈ L and vn is the ancestor of v of length n. In particular, Kn f ∈ L0 and Kn f (o) = f (o).

(19)

Furthermore, Kn f attains ﬁnitely many values, whose number does not exceed the number mn of vertices in the closed ball centered at o of radius n. Observe that if {gk } is a sequence in L with gk L ≤ 1 for each k ∈ N, then, a = supk∈N |gk (o)| ≤ 1 so that |Kn gk (o)| ≤ a. Furthermore, as a consequence of part (a) of Lemma 3.4, for each v ∈ T , v = o, and for each k ∈ N, we have

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|Kn gk (v)| ≤ n. Therefore, some subsequence of {Kn gk }k∈N must converge to a function g on T attaining constant values on the sectors determined by the vertices on the sphere centered at o of radius n. It follows that this subsequence converges to g in L as well. Thus, Kn is compact. Denote by I the identity operator and observe that the operator Mψ Kn is also compact. Furthermore, for v ∈ T , we have D[(I − Kn )f ](v) ≤ Df (v) ≤ f L .

(20)

On the other hand, by part (a) of Lemma 3.4, we see that |[(I − Kn )f ](v)| ≤ |v|f L .

(21)

We now use (21) and (20) to estimate ψ(I − Kn )f L : ψ(I − Kn )f L = sup |ψ(v)[(I − Kn )f ](v) − ψ(v − )[(I − Kn )f ](v − )| |v|>n

≤ sup

|v|>n

Dψ(v)|[(I − Kn )f ](v)| + |ψ(v − )|D[(I − Kn )f ](v)

≤ sup |v|Dψ(v) |v|>n

|[(I − Kn )f ](v)| |v|

+ sup |ψ(v − )|D[(I − Kn )f ](v) |v|>n

≤ sup |v|Dψ(v)f L + sup |ψ(v − )|f L . |v|>n

|v|>n

Therefore, using this estimate and taking the limit as n → ∞, we obtain Mψ e ≤ lim sup Mψ − Mψ Kn n→∞

= lim sup sup ψ(I − Kn )f L n→∞ f L =1

≤ B(ψ) + A(ψ),

completing the proof.

9. Isometries In this section we show that, in analogy to the classical case of the multiplication operators on the Bloch space of the unit disk, there are no nontrivial isometric multiplication operators. Theorem 9.1. The only isometric multiplication operators on L or L0 are the constant functions of modulus one. Proof. It is clear that the constant functions of modulus one are symbols of isometric multiplication operators on L and L0 . Thus, assume Mψ is an isometry on L or L0 so that, in particular, ψL = Mψ 1L = 1. For g =

1 2 χ{o} ,

we have gL = 2|g(o)| = 1 and thus 1 = ψgL = 2|ψ(o)g(o)| = |ψ(o)|.

(22)

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Hence, from (22) it follows that Dψ must be identically 0. Therefore, ψ is a constant function of modulus one.

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[20] Li, S.X.: Essential norm of the weighted composition operators on the Bloch spaces over the polydisk (Chinese, English and Chinese summary). Acta Math. Sinica (Chin. Ser.) 51(4), 655–662 (2008) [21] Li, S., Stevi´c, S.: Weighted composition operators from H ∞ to the Bloch space on the polydisc. Abstr. Appl. Anal. 2007, 13 (2007). (Article ID 48478) [22] Li, S., Stevi´c, S.: Weighted composition operators from α-Bloch spaces to H ∞ on the polydisk. Numer. Funct. Anal. Optim. 28(7), 911–925 (2007) [23] Li, S., Stevi´c, S.: Weighted composition operators between H ∞ and α-Bloch spaces in the unit ball. Taiwan. J. Math. 12, 1625–1639 (2008) [24] MacCluer, B.D., Zhao, R.: Essential norms of weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33(4), 1437–1458 (2003) [25] Ohno, S.: Weighted composition operators between H ∞ and the Bloch space. Taiwan. J. Math. 5(3), 555–563 (2001) [26] Ohno, S., Zhao, R.: Weighted composition operators on the Bloch space. Bull. Aust. Math. Soc. 63, 177–185 (2001) [27] Stevi´c, S.: Norm of weighted composition operators from Bloch space to H ∞ on the unit ball. Ars. Combin. 88, 125–127 (2008) [28] Timoney, R.M.: Bloch functions in several complex variables I. Bull. Lond. Math. Soc. 12, 241–267 (1980) [29] Timoney, R.M.: Bloch functions in several complex variables II. J. Reine Angew. Math. 319, 1–22 (1980) [30] Weighted composition operators from H ∞ into α-Bloch spaces on the unit ball. Acta Math. Sinica (English) 25, 265–278 (2009) [31] Zhou, Z., Chen, R.: Weighted composition operators from F (p, q, s) to Bloch type spaces on the unit ball. Int. J. Math. 19, 899–926 (2008) [32] Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990) [33] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005) Flavia Colonna (B) Department of Mathematical Sciences George Mason University Fairfax, VA 22030 USA e-mail: [email protected] Glenn R. Easley System Planning Corporation Arlington, VA 22201 USA e-mail: [email protected] Received: February 2, 2010. Revised: June 15, 2010.

Integr. Equ. Oper. Theory 68 (2010), 413–426 DOI 10.1007/s00020-010-1834-3 Published online October 19, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Stability Theory of General Contractions for Delay Equations Luis Barreira and Claudia Valls Abstract. We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations v = L(t)vt with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation v = (L(t) + M (t))vt , thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation v = L(t)vt + f (t, vt ). In addition, we consider general contractions e−λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time. Mathematics Subject Classification (2010). Primary 34K20. Keywords. Delay equations, nonuniform exponential stability.

1. Introduction Our main objective is to study the persistence of stability under perturbations of linear delay equations with a nonuniform exponential behavior. Namely, we consider nonuniform exponential contractions (see (1.2) and (1.6) for the definitions, respectively in the cases of continuous and discrete time), and we show that the (nonuniform) asymptotic stability of a given contraction persists under sufﬁciently small perturbations. We emphasize that in a certain sense our assumptions are the weakest possible under which it is possible to establish the persistence of the stability. We refer to [3] for a detailed related discussion in the case of ordinary differential equations. We also refer to the book [9] for a detailed discussion of the geometric theory in the inﬁnite-dimensional setting, with emphasis on delay differential equations. Partially supported by FCT through CAMGSD, Lisbon.

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More precisely, we consider a linear delay equation v = L(t)vt

(1.1) n

in the space C of continuous functions φ : [r, 0] → R , where r ≤ 0 is the delay (when r = 0 equation (1.1) is an ordinary differential equation). Here, L(t) : C → Rn are linear operators for each t ∈ R, and vt (θ) = v(t + θ) for θ ∈ [r, 0]. We refer to Sect. 2.1 for full details. Equation (1.1) is said to admit a nonuniform exponential contraction if there exist constants λ, D > 0 and ε ≥ 0 such that vt ≤ De−λ(t−s)+ε|s| vs ,

t≥s

(1.2)

for any solution t → vt of the equation. When ε > 0 this allows a “spoiling” of the uniform contraction e−λ(t−s) along each solution of (1.1) as the initial time s increases, corresponding to the extra exponential term eε|s| . Therefore, even though we have (nonuniform) exponential stability of solutions (since all Lyapunov exponents are negative), when ε > 0, to ensure that a solution of Eq. (1.1) is in a given neighborhood, the size of the initial condition must decay exponentially with s (while in the uniform case, i.e., when ε = 0, it can be chosen independently of s, and thus we have uniform exponential stability). We note that in Sect. 2.1 it is convenient to consider instead the notion of nonuniform exponential contraction in a larger space containing some discontinuous functions, although proceeding as detailed in [4], one can show that the two notions are equivalent. For clarity of the exposition, here we only use the space C of continuous functions. Assuming that equation (1.1) admits a nonuniform exponential contraction, we then consider the perturbed equation v = L(t)vt + f (t, vt ).

(1.3)

For a large class of sufﬁciently small perturbations f (either linear or nonlinear in the second variable), we show that all solutions of equation (1.3) also decay exponentially (see Theorems 2.3 and 2.4). In the case of linear perturbations, this corresponds to consider the perturbed linear equation v = (L(t) + M (t))vt .

(1.4)

Then our result on the persistence of stability asserts that if the perturbation M is sufﬁciently small (see Theorem 2.3), then equation (1.4) has a nonuniform exponential contraction. We note that in the case of nonlinear perturbations the initial conditions may need to be taken exponentially small in the initial time (although with a small exponential when compared to the negative Lyapunov exponents in the notion of nonuniform exponential contraction). We also obtain corresponding results in the case of discrete time. More precisely, we consider the delay difference equation x(m + 1) = Lm xm ,

m∈Z

(1.5)

for some linear operators Lm : X → Y , where X is the space of functions φ : [r, 0] → Y with values in some Banach space Y , for some integer r ≤ 0. The function xm ∈ X in (1.5) is deﬁned by xm (j) = x(m + j). Similarly,

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equation (1.5) is said to admit a nonuniform exponential contraction if there exist constants a, D > 0 and ε ≥ 0 such that xm ≤ De−λ(m−n)+ε|n| xn ,

m ≥ n.

(1.6)

Assuming that equation (1.5) admits a nonuniform exponential contraction, we show that for a large class of sufﬁciently small perturbations fm : X → Y (both linear and nonlinear) all solutions of the equation x(m + 1) = Lm xm + fm (xm ) also decay exponentially (see Theorems 3.1 and 3.2). As already noted, we consider the general notion of nonuniform exponential contraction, which includes as a very special case the notion of (uniform) exponential contraction. The latter is a particular case of the notion of (uniform) exponential dichotomy, essentially introduced by Perron in [14], which plays a central role in the stability theory of differential equations and dynamical systems. The theory of exponential dichotomies and its applications is widely developed. We refer to the books [7,8,11,12,15] for details and further references. On the other hand, the existence of an exponential dichotomy or of an exponential contraction is a strong requirement and, particularly in view of their central role, it is of interest to look for more general types of hyperbolic behavior. Here we consider the more general notion of nonuniform exponential contraction. In this respect, our work is also a contribution to the theory of nonuniformly hyperbolic dynamics (we refer to the book [1] for a detailed exposition of a large part of the theory). Using the so-called Lyapunov regularity theory, one can show that the nonuniform exponential behavior is very common when compared to what happens in the uniform setting. Essentially, the existence of nonzero Lyapunov exponents leads to a nonuniform exponential behavior. We refer to the books [1,3] for detailed discussions. Here we consider only the case of negative Lyapunov exponents, and of delay r = 0 in equation (1.5), with Y = Rp for some p ∈ N. In this case, each linear operator Lm is a p × p matrix Am , and if 1 lim sup log Am · · · A2 A1 x < 0 m m→+∞ for every vector x ∈ Rp \{0}, then equation (1.5) admits a nonuniform exponential contraction. Even more is true in the context of ergodic theory. Namely, the constant ε can be made arbitrary small although not necessarily zero. To formulate a rigorous statement we recall that a measurable transformation f : X → X is said to preserve a measure μ in X if μ(f −1 A) = μ(A) for every measurable set A ⊂ X. Let also Mp be the space of p × p matrices. Theorem 1.1. Let f : X → X and A : X → Mp be measurable transformations. If f preserves a finite measure μ in X such that log+ A ∈ L1 (X, μ), and 1 lim sup log A(f m (q)) · · · A(f (q))A(q))x < 0 m→+∞ m

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for μ-almost every q ∈ X and every x ∈ Rp \{0}, then for μ-almost every q ∈ X there is λ = λ(q) > 0 and for each ε > 0 there exists c = c(q, ε) > 0 such that A(f m−1 (q)) · · · A(f n (q)) ≤ ce−λ(m−n)+εn ,

m ≥ n.

(1.7)

Theorem 1.1 is a consequence of Oseledets’ multiplicative ergodic theorem in [13] (see for example [1] for a detailed discussion). In particular, it shows that in the context of ergodic theory the nonuniformity given by the constant ε in (1.7) can be made arbitrarily small for almost all trajectories. On the other hand, it follows from work of Barreira and Schmeling in [2] that for some classes of measure-preserving transformations, the set of points q ∈ X for which the constant ε in (1.7) cannot be made arbitrarily small is large from the point of view of Hausdorﬀ dimension. In addition, we consider the case of general nonuniform exponential contractions, in the sense that we consider a stable behavior with respect to an asymptotic rate ecρ(t) for some arbitrary function ρ (also in the case of discrete time). This includes as a very special case the usual exponential behavior with ρ(t) = t, and it allows considering situations when all Lyapunov exponents are inﬁnity.

2. The Case of Continuous Time 2.1. Setup Given r ≤ 0, let C = C([r, 0], Rn ) be the Banach space of continuous functions φ : [r, 0] → Rn with the norm φ = sup {φ(θ) : r ≤ θ ≤ 0} . Given linear maps L(t) : C → Rn for t ∈ R with (t, v) → L(t)v continuous, we consider the linear equation (1.1), where v denotes the right-hand derivative, and where vt (θ) = v(t + θ), θ ∈ [r, 0]. We assume that there exists κ > 0 such that t−r L(τ ) dτ ≤ κ t

for every t ∈ R. Then for each (s, φ) ∈ R × C there is a unique solution vt (·, s, φ), t ≥ s of equation (1.1) with vs (·, s, φ) = φ (see for example [10]). We deﬁne the evolution operator T (t, s) : C → C for t ≥ s by T (t, s)φ = vt (·, s, φ). We also extend the domain of T (t, s) to a larger space that contains some discontinuous functions. For this we write L(t) in the form 0 dθ [η(t, θ)]φ(θ),

L(t)φ = r

(2.1)

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for some n × n matrices η(t, θ) that are measurable in (t, θ) ∈ R × R and continuous from the left in θ. We also set mη (t) = Var η(t, ·), where Var denotes the total variation in [r, 0]. Note that L(t) = mη (t). Now let Cˆ be the space of functions φ : [r, 0] → Rn such that for each s ∈ [r, 0] there exist the limits lim φ(θ) and

θ→s−

lim φ(θ),

θ→s+

and φ is right-continuous at s, i.e., limθ→s+ φ(θ) = φ(s). We also consider ˆ Each operator L(t) can be extended to Cˆ using the supremum norm in C. the integral in (2.1), with the associated Riemann–Stieltjes sums taking the value [η(t, b) − η(t, a)]φ(b− ) for each subinterval [a, b] ⊂ [r, 0] (so that points at which both φ and η(t, ·) have discontinuities cause no problem). It can be shown that for each (s, φ) ∈ R × Cˆ there is a unique solution vt (·, s, φ) ∈ Cˆ for t ≥ s with vs (·, s, φ) = φ of the integral equation obtained from (1.1) (see [10] for a related discussion). The corresponding evolution operator Tˆ(t, s) : Cˆ → Cˆ is deﬁned for each t ≥ s by Tˆ(t, s)φ = vt (·, s, φ). Given an increasing differentiable function ρ : R → R with ρ(0) = 0, we say ˆ if that equation (1.1) admits a ρ-nonuniform exponential contraction (in C) for some constants λ, D > 0 and ε ≥ 0 we have Tˆ(t, s) ≤ De−λ(ρ(t)−ρ(s))+ε|ρ(s)| , t ≥ s. (2.2) The following examples of nonuniform exponential contractions are borrowed from [5]. The ﬁrst is an example of ordinary differential equation. Example 2.1. We recall that a linear ordinary differential equation v = A(t)v in Rn , with n × n matrices A(t) varying continuously with t ∈ R, can be seen as a linear delay equation. Namely, deﬁning linear operators L(t) : C → Rn by L(t)φ = A(t)φ(0), equation (1.1) becomes v = L(t)vt = A(t)vt (0) = A(t)v. Given ω > a > 0, it is shown in [5] that the scalar equation v = (−ω − at sin t)v

(2.3)

admits a nonuniform exponential contraction, and that the constant ε in (2.2) cannot be taken equal to zero. Example 2.2. Now we consider the scalar delay equation v = av + bv(t − τ )

(2.4)

for some constants a, b ∈ R and τ > 0. We assume that all roots in C of the characteristic equation λ = a + be−τ λ

(2.5)

have negative real part. Then it follows from the theory of representation of solutions of equation (2.4) by series of exponentials eλt with λ a root of (2.5)

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(see for example [6]) that equation (2.4) admits a nonuniform exponential contraction. We note that it is a uniform exponential contraction. We refer to [5] for further examples. 2.2. Robustness of Exponential Contractions We establish in this section the robustness of ρ-nonuniform exponential contractions. We consider the perturbed equation (1.4), where M (t) : C → Rn is a linear map such that (t, v) → M (t)v is continuous. We assume that t−r M (τ ) dτ ≤ κ t

for every t ∈ R, and we denote by T˜(t, s) : Cˆ → Cˆ the linear evolution operator associated to equation (1.4). Theorem 2.3. Assume that equation (1.4) admits a ρ-nonuniform exponential contraction, and that M (t) ≤ δe−ε|ρ(t)| ρ (t) for t ≥ 0 with δ < λ/D. Then equation (1.4) admits a ρ-nonuniform exponential contraction with T˜(t, s) ≤ De−(λ−δD)(ρ(t)−ρ(s))+ε|ρ(s)| ,

t ≥ s.

Proof. We ﬁrst note that by the variation-of-constants formula, t T˜(t, s) = T (t, s) +

Tˆ(t, τ )X0 M (τ )T˜(τ, s) dτ

(2.6)

s

for every t ≥ s, where (X0 u)(0) = u and (X0 u)(θ) = 0 for θ < 0. Now we consider the set J = {(t, s) ∈ R × R : t ≥ s}, and the Banach space B = U : J → Cˆ : U is continuous and U < ∞ with the norm U = sup U (t, s)e−ε|ρ(s)| : (t, s) ∈ J . We deﬁne an operator N in the space B by t Tˆ(t, τ )X0 M (τ )U (τ, s) dτ

(N U )(t, s) = T (t, s) + s

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for each U ∈ B. We have t Tˆ(t, τ ) · M (τ ) · U (τ, s) dτ

(N U )(t, s) ≤ T (t, s) +

s −λ(ρ(t)−ρ(s))+ε|ρ(s)|

≤ De

ε|ρ(s)|

+Dδe

t U

e−λ(ρ(t)−ρ(τ )) ρ (τ ) dτ

s −λ(ρ(t)−ρ(s))+ε|ρ(s)|

≤ De

+

Dδ ε|ρ(s)| e U , λ

(2.7)

and hence, Dδ U < ∞. λ Therefore, N (B) ⊂ B. Proceeding in a similar manner to that in (2.7), we can also show that Dδ N U1 − N U2 ≤ U1 − U2 λ for every U1 , U2 ∈ B. Since δ < λ/D, the operator L is a contraction, and there exists a unique function U ∈ B such that N U = U . By (2.6), we have U (t, s) = T˜(t, s). Now we estimate the norm of the operator T˜(t, s). The function x(t) = T˜(t, s) satisﬁes N U ≤ D +

−λ(ρ(t)−ρ(s))+ε|ρ(s)|

x(t) ≤ De

t + δD

e−λ(ρ(t)−ρ(τ )) ρ (τ )x(τ ) dτ

s

for every t ≥ s ≥ 0. Now we consider the continuous function Φ(t) satisfying −λ(ρ(t)−ρ(s))+ε|ρ(s)|

Φ(t) = De

t + δD

e−λ(ρ(t)−ρ(τ )) ρ (τ )Φ(τ ) dτ

(2.8)

s

for every t ≥ s ≥ 0. We can easily verify that Φ = (δD − λ)ρ (t)Φ. Furthermore, by (2.8) we have Φ(s) = Deε|ρ(s)| , and hence, Φ(t) = Deε|ρ(s)| e(δD−λ)(ρ(t)−ρ(s)) . By construction we have x(t) ≤ Φ(t) for every t ≥ s ≥ 0, and hence, x(t) ≤ De−(λ−δD)(ρ(t)−ρ(s))+ε|ρ(s)| , t ≥ s. Since U (t, s) = T˜(t, s), this completes the proof of the theorem.

2.3. Stability of Exponential Contractions We consider in this section the nonlinear equation (1.3), where f : R×Cˆ → Rn is a continuous function with f (t, 0) = 0 for every t ∈ R. Theorem 2.4. Assume that equation (1.1) admits a ρ-nonuniform exponential contraction, and that there exist constants δ > 0 and q > 0 such that f (t, u) − f (t, v) ≤ δρ (t)u − v(uq + vq )

(2.9)

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ˆ If qλ + ε < 0, then provided that δ is sufficiently for every t ∈ R and u, v ∈ C. small, the solution of equation (1.3) with initial condition (s, φ) ∈ R × C such that φ ≤ e−ε(1+2/q)|ρ(s)| satisfies vt ≤ 2De−λ(ρ(t)−ρ(s))+ε|ρ(s)| φ

for every

t ≥ s.

(2.10)

Proof. By the variation-of-constants formula we have t vt = T (t, s)φ + Tˆ(t, τ )X0 f (τ, vτ ) dτ, s

with X0 as in (2.6). We set t Tˆ(t, τ )X0 f (τ, vτ ) dτ

(Kv)t = T (t, s)φ + s

for each v in the space C = v : [s + r, +∞) → C continuous : vt ≤ 2Dφeγ(t,s) for t ≥ s , where γ(t, s) = −λ(ρ(t) − ρ(s)) + ε|ρ(s)|. We note that C is a complete metric space with the norm v = sup vt e−γ(t,s) : t ≥ s . Moreover, by (2.2) and (2.9) we have t Tˆ(t, τ )X0 f (τ, xτ ) dτ

(Kv)t ≤ T (t, s)φ +

s −λ(ρ(t)−ρ(s))+ε|ρ(s)|

≤ De

t +Dδ

φ

e−λ(ρ(t)−ρ(τ ))+ε|ρ(τ )| vτ q+1 ρ (τ ) dτ.

s

Now set β = ε(1 + 2/q). Since vt ≤ 2Dφeγ(t,s) and φ ≤ e−β|ρ(s)| , we obtain (Kv)t ≤ De−λ(ρ(t)−ρ(s))+ε|ρ(s)| φ q+1

+2

D

q+2

t δφ

e−λ(ρ(t)−ρ(τ ))+ε|ρ(τ )|−(q+1)λ(ρ(τ )−ρ(s))−2ε|ρ(s)| ρ (τ ) dτ

s

≤ De−λ(ρ(t)−ρ(s))+ε|ρ(s)| φ + 2q+1 Dq+2 δφe−λ(ρ(t)−ρ(s)) ≤ De−λ(ρ(t)−ρ(s))+ε|ρ(s)| φ +

t

e(−qλ+ε)(ρ(τ )−ρ(s)) ρ (τ ) dτ

s q+1

2

Dq+2 e−λ(ρ(t)−ρ(s)) δφ. |−qλ + ε|

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Therefore, Kv ≤ Dφ(1 + δμ) for some constant μ, and provided that δ is sufﬁciently small we have Kv ≤ 2Dφ. This shows that K(C) ⊂ C. Similarly, t Tˆ(t, τ )X0 (f (τ, uτ ) − f (τ, vτ )) dτ

(Ku)t − (Kv)t ≤ s

q+1

≤2

D

q+1

γ(t,s)

δe

t u − v

e(−qλ+ε)(ρ(τ )−ρ(s)) ρ (τ ) dτ

s

2q+1 Dq+1 γ(t,s) δe ≤ u − v, |−qλ + ε| and hence Ku − Kv ≤ δμu − v. Thus, provided that δ is sufﬁciently small the operator K is a contraction, and its unique ﬁxed point in C thus satisﬁes (2.10).

3. The case of Discrete Time 3.1. Setup For simplicity of the notation, we denote by [m, n], (−∞, n] and [n, +∞) respectively the sets [m, n] ∩ Z, (−∞, m] ∩ Z and [n, +∞) ∩ Z. Given a delay r ∈ Z− 0 and a Banach space Y , let X be the space of functions φ : [r, 0] → Y with the norm φ = max{|φ(j)| : j ∈ [r, 0]}, where |·| is the norm in Y . For any function x : (−∞, m] → Y and n ≤ m, we deﬁne xn ∈ X by xn (j) = x(n + j) for j ∈ [r, 0]. Given linear operators Lm : X → Y for m ∈ N, we consider the dynamics deﬁned by x(m + 1) = Lm xm .

(3.1)

k

For example, when Y = R each operator Lm can be written in the form Lm φ =

0

η(m, j)φ(j)

j=r

for some (uniquely determined) k × k matrices η(m, j) : Rk → Rk , and thus equation (3.1) can be written in the form x(m + 1) =

0

η(m, j)x(m + j).

j=r

For each n ∈ N and φ ∈ X, we obtain a unique function x : [n+r, +∞) → Y , denoted by x(·, n, φ), with xn = φ satisfying (3.1) for every m ≥ n. For each m ≥ n we deﬁne an operator T (m, n) on X by T (m, n)φ = xm (·, n, φ),

φ ∈ X.

(3.2)

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Clearly, T (m, n) is linear, T (m, m) = Id, and l ≥ m ≥ n.

T (l, m)T (m, n) = T (l, n),

Given an increasing function ρ : Z → Z satisfying ρ(−m) = ρ(m), we say that equation (3.1) admits a ρ-nonuniform exponential contraction if there exist constants λ, D > 0 and ε ≥ 0 such that T (m, n) ≤ De−λ(ρ(m)−ρ(n))+ε|ρ(n)| ,

m ≥ n.

(3.3)

3.2. Robustness of Exponential Contractions We establish in this section the robustness of nonuniform exponential contractions under linear perturbations. Namely, we consider the linear equation x(m + 1) = (Lm + Mm )xm .

(3.4)

Theorem 3.1. If equation (3.1) admits a ρ-nonuniform exponential contraction, and Mm ≤ δe−λ(ρ(m+1)−ρ(m))−ε|ρ(m+1)| ,

m ∈ N,

then for every sufficiently small δ > 0 equation (3.4) admits a ρ-nonuniform exponential contraction, with the constant a replaced by λ − log(1 + δD). Proof. Let T˜(m, n) be the evolution operator associated to (3.4), deﬁned in a similar manner to that in (3.2). We have Tˆ(m, n) = T (m, n) +

m−1

T (m, l + 1)Γ(Ml Tˆ(l, n)),

(3.5)

l=n

where Γ(0) = Id and Γ(l) = 0 for l ∈ [r, 0). Here the symbol Γ(Ml Tˆ(l, n)) denotes the function [r, 0] l → Γ(l)Ml Tˆ(l, n) in X. Setting αm = T˜(m, n), we ﬁnd that αm ≤ T (m, n) +

m−1

T (m, l + 1) · Ml αl

l=n

≤ De−λ(ρ(m)−ρ(n))+ε|ρ(n)| + δD

m−1

e−λ(ρ(m)−ρ(l)) αl .

l=n

Now we consider the sequence Φm with Φn = Deε|ρ(n)| deﬁned recursively by Φm = De−λ(ρ(m)−ρ(n))+ε|ρ(n)| + δD

m−1

e−λ(ρ(m)−ρ(l)) Φl .

l=n λ(ρ(m)−ρ(n))

Setting Γm = e

Φm we have

Γm = Deε|ρ(n)| + δD

m−1

Γl .

l=n

Clearly Γm+1 = (1 + δD)Γm , and thus, Γm = (1 + δD)m−n Γn = (1 + δD)m−n Deε|ρ(n)| = Delog(1+δD)(m−n)+ε|ρ(n)| .

(3.6)

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Since ρ is increasing we have ρ(m) − ρ(n) ≥ m − n, and it follows from (3.6) that Γm ≤ Delog(1+δD)(ρ(m)−ρ(n))+ε|ρ(n)| . This shows that Φm ≤ e−(λ−log(1+δD))(ρ(m)−ρ(n))+ε|ρ(n)| . Setting zm = αm − Φm , we obtain αm ≤ δD

m−1

e−λ(ρ(m)−ρ(l)) zl ,

m ≥ n.

l=n

Since the sequences (αm )m≥n and (Φm )m≥n are bounded, the supremum z = supm≥n αm is ﬁnite, and we obtain z ≤ δDz sup

m≥n

m−1

e−λ(ρ(m)−ρ(l)) ≤ δDz sup

m≥n

l=n

m−1

e−λ(m−l) .

l=n

Hence, z ≤ (δD/(1 − e−λ ))z. Provided that δ is sufﬁciently small, we obtain z ≤ 0, and thus αm ≤ Φm for every m ≥ n. Therefore, αm ≤ Φm ≤ e−(λ−log(1+δD))(ρ(m)−ρ(n))+ε|ρ(n)| .

This completes the proof of the theorem. 3.3. Stability of Exponential Contractions We consider in this section the delay difference equation x(m + 1) = Lm xm + fm (xm )

(3.7)

for some linear operators Lm as above, and some functions fm : X → Y for m ∈ N. We assume that fm (0) = 0 for each m ∈ N, and that there exist constants δ, q > 0 (independent of m) such that |fm (u) − fm (v)| ≤ δe−λ(ρ(m+1)−ρ(m+r))−β|ρ(m+1)| u − v,

(3.8)

for every m ∈ N and u, v ∈ X and with β = ε + σ, for some σ > 0 [with the same constants λ and ε as in (3.3)]. The following is our stability result. Theorem 3.2. If equation (3.1) admits a ρ-nonuniform exponential contraction and (3.8) holds, then for every sufficiently small δ > 0, the solution of equation (3.7) with initial condition (n, φ) ∈ N × X satisfies xm ≤ 2De−λ(ρ(m+r)−ρ(n))+ε|ρ(n)| φ

for every

m ≥ n.

(3.9)

Proof. In a similar manner to that in (3.5), the solution x = x(·, n, φ) of equation (3.7) satisﬁes the identities xm = T (m, n)φ +

m−1 j=n

T (m, j + 1)(Γfj (xj )),

m ≥ n.

(3.10)

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In view of (3.10), we consider the operator R deﬁned by (Rx)m = T (m, n)φ +

m−1

m≥n

T (m, k + 1)(Γfk (xk )),

(3.11)

k=n

in the space C = x : [n + r, +∞) → Y : xm ≤ 2Deγ(m,n) φ for m ≥ n , with γ(m, n) = −λ(ρ(m + r) − ρ(n)) + ε|ρ(n)|, equipped with the norm x = sup xm e−γ(m,n) /φ : m ≥ n . One can easily verify that C is a complete metric space. We note that T (m, n)φ(j) = T (m + j, n)φ(0)

for j ∈ [r, 0].

Therefore, for each j ∈ [r, 0] it follows from (3.11) that (Rx)(m + j) = [T (m + j, n)φ](0) +

m−1

T (m + j, k + 1)(Γfk (xk ))(0),

k=n

(3.12) and hence, |(Rx)(m + j)| ≤ T (m + j, n)φ +

m+j−1

T (m + j, k + 1) · |fk (xk )|.

k=n

By (3.3) and (3.8), since ρ is increasing we obtain |(Rx)(m + j)| ≤ De−λ(ρ(m+j)−ρ(n))+ε|ρ(n)| φ +

m−1

Dδe−λ(ρ(m+j)−ρ(k+r))−σ|ρ(k+1)| xk

k=n

≤ Deγ(m,n) φ +

m−1

Dδe−λ(ρ(m+r)−ρ(k+r))−σ|ρ(k+1)| xk .

k=n

Furthermore, since xk ≤ 2Deγ(k,n) φ we have |(Rx)(m + j)| ≤ Deγ(m,n) φ + 2D2 δeγ(m,n) φ

m−1

e−σ|ρ(k+1)| .

k=n

(3.13)

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Since ρ(k + 1) ≥ 1 + ρ(k) for each k ∈ Z, we have ρ(k + 1) ≥ k + ρ(1), and hence, m−1 γ(m,n) −σρ(1) −σk |(Rx)(m + j)| ≤ De φ 1 + 2Dδe e k=n

≤ Deγ(m,n) φ 1 +

−σρ(1)

2Dδe 1 − e−σ

.

(3.14)

Taking δ sufﬁciently small we obtain |(Rx)(m + j)| ≤ 2Deγ(m,n) φ. Therefore, R(C) ⊂ C. Now we show that R is a contraction. By (3.12) we have |(Rx)(m + j) − (Ry)(m + j)| ≤

m−1

T (m + j, k + 1) · |fk (xk ) − fk (yk )|

k=n

≤ Dδ

m−1

e−λ(ρ(m+j)−ρ(k+1))+ε|ρ(k+1)|

k=n

×e−λ(ρ(k+1)−ρ(k+r)) e−(ε+σ)|ρ(k+1)| xk − yk ≤ Dδ

m−1

e−λ(ρ(m+j)−ρ(k+r))−σ|ρ(k+1)| xk − yk

k=n

≤ Dδx − yeγ(m,n)

m−1

e−σ|ρ(k+1)| φ.

k=n

Proceeding as in (3.13) and (3.14) we obtain |(Rx)(m + j) − (Ry)(m + j)| ≤

Dδ eγ(m,n) φ · x − y 1 − e−σ

and hence, Dδ x − y. 1 − e−σ Taking δ sufﬁciently small, the operator R becomes a contraction in the complete metric space C. Hence, R has a unique ﬁxed point in C, thus satisfying (3.9). This completes the proof of the theorem. Rx − Ry ≤

References [1] Barreira, L., Pesin, Ya.: Nonuniform hyperbolicity. In: Encyclopedia of Mathematics and its Applications, vol. 115. Cambridge University Press, Cambridge (2007) [2] Barreira, L., Schmeling, J.: Sets of “non-typical” points have full topological entropy and full Hausdorﬀ dimension. Israel J. Math. 116, 29–70 (2000)

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[3] Barreira, L., Valls, C.: Stability of nonautonomous differential equations in Hilbert spaces. J. Differ. Equ. 217, 204–248 (2005) [4] Barreira, L., Valls, C.: Delay equations and nonuniform exponential stability. Discrete Contin. Dyn. Syst. S 1, 219–223 (2008) [5] Barreira, L., Valls, C.: Stability of delay equations via Lyapunov functions. J. Math. Anal. Appl. 365, 797–805 (2010) [6] Bellman, R., Cooke, K.: Differential-Difference Equations. Academic Press, Dublin (1963) [7] Chicone, C., Latushkin, Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence (1999) [8] Hale, J.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence (1988) [9] Hale, J., Magalh˜ aes, L., Oliva, W.: Dynamics in Inﬁnite Dimensions, Applied Mathematical Sciences, vol. 47. Springer, Berlin (2002) [10] Hale, J., Verduyn Lunel, S.: Introduction to Functional-Differential Equations, Applied Mathematical Sciences, vol. 99. Springer, Berlin (1993) [11] Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., vol. 840. Springer, Berlin (1981) [12] Massera, J., Sch¨ aﬀer, J.: Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, vol. 21. Academic Press, Dublin (1966) [13] Oseledets, V.: A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–221 (1968) [14] Perron, O.: Die Stabilit¨ atsfrage bei Differentialgleichungen. Math. Z. 32, 703– 728 (1930) [15] Sell, G., You, Y.: Dynamics of Evolutionary Equations, Applied Mathematical Sciences, vol. 143. Springer, Berlin (2002) Luis Barreira (B) and Claudia Valls Departamento de Matem´ atica Instituto Superior T´ecnico 1049-001 Lisboa Portugal e-mail: [email protected]; [email protected] Received: June 17, 2009. Revised: October 1, 2010.

Integr. Equ. Oper. Theory 68 (2010), 427–449 DOI 10.1007/s00020-010-1808-5 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Toeplitz Quantization and Asymptotic Expansions: Peter–Weyl Decomposition Miroslav Engliˇs and Harald Upmeier Abstract. We obtain the Peter–Weyl decomposition of the star product and star restriction associated to the Toeplitz calculus on complex and real symmetric domains, respectively, under the action of the maximal compact subgroup. Both the Berezin and the Berezin–Toeplitz cases are covered. Mathematics Subject Classification (2010). Primary 32M15; Secondary 47B35, 53C35, 53D55. Keywords. Bounded symmetric domain, real symmetric domain, star product, Toeplitz operator, Peter–Weyl decomposition.

1. Introduction This paper continues our study, started in [8] and [9], of Toeplitz quantization on complex and real bounded symmetric domains. In the complex case, the latter arise as the (spectral) open unit ball D in a Hermitian Jordan triple Z ∼ = Cd , and can also be written as the quotient D = G/K with G the semi-simple Lie group of all biholomorphic self-maps of D and K the automorphism group of the Jordan triple Z, which coincides with the stabilizer of the origin 0 ∈ D in G. For ν > p − 1, p being the genus of D, the holomorphic discrete series representations U (ν) = U act as (Ug f )(z) = f (g −1 z) · Jg−1 (z)ν/p ,

z ∈ D, g ∈ G,

on the space Hν = Hν (D) of all holomorphic functions in L2 (D, dμν ) with respect to a certain measure dμν on D; here Jg is the complex Jacobian of the holomorphic self-map g on D. The Toeplitz calculus, assigning to a function f on D the Toeplitz operator Tf : u → Pν (f u)

(1)

ˇ Grant no. 201/09/0473 and AV CR ˇ institutional research Research supported by GA CR plan AV0Z10190503.

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on Hν , where Pν : L2 (D, dμν ) → Hν is the orthogonal projection, enjoys the G-invariance property Ug∗ Tf Ug = Tf ◦g ,

∀g ∈ G.

(2)

The Berezin–Toeplitz star product ∗ is, formally, deﬁned by Tf ∗g := Tf Tg .

(3)

The rigorous interpretation is that f ∗ g :=

∞

Cj (f, g)ν −j ,

j=0

the right-hand side being the formal power series in ν1 whose coefﬁcients Cj are bidifferential operators deﬁned by the requirement that there be an asymptotic expansion Tf Tg −

N −1

ν −j TCj (f,g) = O(ν −N ),

N = 0, 1, 2, . . . ,

j=0

as ν1 0, in the sense of operator norms. This is the Berezin–Toeplitz quantization. By construction, the star product ∗ is G-invariant. The main point in the papers [8,9] is the idea that these concepts have a natural generalization to the case of real symmetric domains. The real bounded symmetric domain DR arises as a real form of a complex bounded symmetric domain, i.e. as the subset DR = {z ∈ DC : z # = z} ⊂ Z R ∼ = Rd , where DC = GC /K C ⊂ Z C ∼ = Cd is of the kind discussed in the previous # paragraph and z → z is a conjugate-linear involution of Z C which preserves DC . For the induced involution g # z := (g(z # ))# on the group GC we have DR = GR /K R with GR := {g ∈ GC : g # = g} (a reductive Lie group) and K R := GR ∩K C the stabilizer of 0 ∈ DR in GR . Again, K R also coincides with the automorphism group of the real Jordan triple Z R := {z ∈ Z C : z # = z}, which we will assume to be irreducible, and DR is the open unit ball of Z R . Corresponding to the Toeplitz calculus in the complex case, we now have the real Toeplitz calculus (or Toeplitz extension) T R assigning to a function f on DR a holomorphic function TfR in Hν (DC ), which has the GR -invariance property TfR◦g = Ug∗ TfR ,

∀g ∈ GR .

(4)

Explicitly, T R is given by the formula R Tf (z) = f (x)kx (z)dμR (x), DR

where kx is the “coherent state” family of unit vectors in Hν uniquely determined by k0 = 1,

kgx = Ug kx ,

g ∈ GR ,

x ∈ DR ,

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and dμR is the (unique) GR -invariant measure on DR normalized so that T1R (0) = 1. (Note that the real Toeplitz calculus again depends—just as the complex Toeplitz calculus—on the parameter ν > pC − 1, although this is not reﬂected in the notation.) The counterpart of the star product in the real case is the star restriction # : C ∞ (DC ) → C ∞ (DR ), formally deﬁned by R := TFC I, T#F

(5)

where T C is the Toeplitz calculus on the complex domain DC and I(z) is the holomorphic function on DC uniquely determined by the requirement Ug I = I

∀g ∈ GR ,

I(0) = 1.

(It follows that I = T1R .) On the rigorous level, #F is again deﬁned as a formal power series in ν1 , whose coefﬁcients are differential operators in a neighbourhood of DR in DC , such that (5) holds in the sense of asymptotic expansions as ν1 0. By construction, #(F ◦ g) = (#F ) ◦ g for all g ∈ GR , i.e. # is GR -invariant. Our main goal in this paper is to obtain an explicit Peter–Weyl decomposition of the star product (3) and the star restriction (5), under the action of the maximal compact subgroup K and K R , respectively. The polynomial algebra P(Z C ) over Z C has a Peter–Weyl decomposition P(Z C ) = Pm (Z C ) (6) m∈N+ (rC ) C

under the natural K -action. Here N+ (rC ) denotes the set of all integer partitions m1 ≥ · · · ≥ mrC ≥ 0,

m = (m1 , . . . , mrC ), C

of length rC , the rank of Z . Endowing P(Z C ) with the Fischer–Fock inner product

p, qF := q ∗ (∂)p(0),

q ∗ (z) := q(z),

we have the corresponding expansion ez,w = Km (z, w) m

of the Fischer–Fock kernel into the reproducing kernels Km (z, w) of the ﬁnitedimensional subspaces Pm (Z C ). Being K C -invariant polynomials on Z C × Z C (i.e. holomorphic in z, w), they give rise to K C -invariant constant coefﬁcient differential operators Km (∂, ∂) on Z C , acting via Km (∂, ∂)ez,v+u,z = Km (u, v)ez,v+u,z for any ﬁxed u, v ∈ Z C . Conversely, any K C -invariant differential operator on Z C is necessarily a linear combination of Km (∂, ∂), m ∈ N+ (rC ). This applies, in particular, to the “freezing at the origin” β L|0 := cαβ (0)∂ α ∂ α,β multiindices

430

M. Engliˇs and H. Upmeier

of any linear differential operator Lf (z) =

IEOT

β

cαβ (z)∂ α ∂ f (z)

α,β multiindices

on DC which is GC -invariant under the unweighted action: L(f ◦ g) = (Lf ) ◦ g, Any such L thus has the form L=

∀g ∈ GC ,

C cm Km ,

f ∈ C ∞ (DC ).

cm ∈ C,

(7)

m∈N+ (rC ) C where Km is the GC -invariant differential operator on DC uniquely determined by C Km f (0) =

Km (∂, ∂)f (0) , Km (e, e)

where e is a maximal tripotent in Z C . In order to describe the analogous decomposition for the real case, we start with a linear differential operator L on DR which is GR -invariant under the unweighted action. Freezing the coefﬁcients at the origin produces a K R -invariant constant coefﬁcient linear differential operator, which therefore must be of the form p(∇), where p is a K R -invariant polynomial on Z R and ∇ stands for the real differentiation (to distinguish it from the Wirtinger operators ∂, ∂). Being a polynomial, p extends uniquely to an element of P(Z C ), whose Peter–Weyl decomposition (6) contains nonzero components only for those n ∈ N+ (rC ) for which Pn (Z C ) contains a nonzero K R -invariant vector. Such signatures are known as spherical, or even; their complete description is available [14] and depends on the root system of the domain DR . For root system of type A, n is even if and only if its modulus |n| := n1 + · · · + nrC is an even integer, and we will not consider this case here. For the other root types (B, C, BC and D), all even signatures n = mC are obtained by “doubling” signatures m ∈ N+ (rR ) of length rR , the rank of the real triple Z R ; more specifically, mC = (m, m) := (m1 , m1 , m2 , m2 , . . . , mrR , mrR ) if rC = 2rR (types C and BC), and mC = 2m := (2m1 , 2m2 , . . . , 2mrR ) if rC = rR (types B and D). The corresponding subspace of K R -invariant elements in PmC (Z C ) is then always one-dimensional and consists of multiples of a single polynomial φm (x), normalized to be 1 at (any) maximal tripotent. Altogether, we thus see that any GR -invariant linear differential operator on DR has the form R cm Km , cm ∈ C, (8) L= m∈N+ (rR ) R where Km is the GR -invariant differential operator on DR uniquely determined by R f (0) = φm (∇)f (0). Km

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Peter–Weyl Decompositions

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Finally, consider the “mixed” case of a linear differential operator L : C ∞ (DC ) → C ∞ (DR ), i.e. of the form β LF (x) = cαβ (x)(∂ α ∂ F )(x), x ∈ DR , F ∈ C ∞ (DC ), α,β multiindices R

which is G -invariant in the sense that (LF )(gx) = L(F ◦ g)(x),

x ∈ DR ,

g ∈ GR .

(That is, these are exactly the kind of operators like #.) Suppose now that L has the additional “anti-holomorphic factorization” property L(HF ) = (ρH) · LF

(9)

for all holomorphic H on DC ; here ρ denotes the restriction operator from DC to DR . Then cαβ = 0 unless |β| = 0, and thus at the origin we have LF (0) = p(∂)F (0) R

for some K -invariant polynomial p on Z R . As in the previous paragraph, it follows that L must have the form ∂ cm Km , cm ∈ C, (10) L= m∈N+ (rR )

where is the G -invariant differential operator from DC into DR uniquely determined by ∂ Km

R

∂ F (0) := φm (∂)F (0), Km

F ∈ C ∞ (DC ).

(11)

There is a completely analogous characterization, in terms of ∂ F (0) := φm (∂)F (0) Km R

(12)

of G -invariant differential operators having instead of (9) the “holomorphic” factorization property L(HF ) = (ρH)LF , for all H holomorphic on DC . We call (7), (8) and (10) the Peter–Weyl decompositions of the corC R ∂ ∂ responding operators L, and cm Km , cm Km , cm Km and cm Km , respectively, their Peter–Weyl components (which are uniquely determined, since the components in the polynomial decomposition (6) are). In an obvious manner, these notions also extend to linear operators which are not differential but have the form of a formal power series whose coefﬁcients are differential operators. In particular, we have the Peter–Weyl decomposition (10) of the star restriction # : C ∞ (DC ) → C ∞ (DR ). This also includes the decomposition (10) of the star product ∗ : C ∞ (D) ⊗ C ∞ (D) ∼ = C ∞ (D × D) → C ∞ (D) for hermitian symmetric domains D, since the latter is obtained as a special case of the star restriction # if D is viewed as the real form of D ×D := {(z, w) : z, w ∈ D} with respect to the involution (z, w)# = (w, z). The polynomials φm (z) are then equal to Km (z, z)/Km (e, e), and (8) reduces to (7). Further details on the above material can be found in the ﬁrst paper [8] of this series, where there was also obtained the Peter–Weyl decomposition (8)

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M. Engliˇs and H. Upmeier

BR =

m∈N+ (rR )

R Km [[ν]]m

IEOT (13)

of the (real) Berezin transform B R on DR ; that is, of the GR -invariant integral transform on DR obtained as the composition B R = T R∗ T R of the real Toeplitz calculus T R with its adjoint T R∗ (with respect to the inner products in Hν and in L2 (DR , dμR )). Again, (13) also includes the Peter–Weyl decomposition (7) of the “complex” Berezin transform B C on DC , found by Arazy and Ørsted [1]. The generalized “Pochhammer symbols” [[ν]]m occuring in (13) can be computed explicitly, at least for root systems not of type A, cf. formula (19) below. The main result of this paper is, ﬁrst of all, the Peter–Weyl decomposition (10) of the star restriction # (and, hence, also of the star product ∗ in the complex case). Note that this is something which has hitherto not been known even for the most studied bounded symmetric domain, the unit disc in C. Second, we show that the decomposition (13) of the Berezin transform is related to the Peter–Weyl decomposition of another star restriction operator, namely the one which corresponds to the Berezin (rather than Berezin–Toeplitz) quantization in the complex case. This is done in Sects. 3 and 2, respectively. In Sect. 4 we explore in more detail the operators with the “factorization” property (9) and give a formula for # resembling the one of A. & J. Unterberger [12] for the Weyl calculus on the upper half-plane. The ﬁnal section, Sect. 5, treats brieﬂy the “ﬂat” case of DR = Rn or Cn , where an alternative treatment is available based on Fourier transform methods. Expansions of the same form, Am , #= [ν]m m∈N+ (rR )

with some coefﬁcients [ν]m and GR -invariant operators Am from DC into DR , have also been obtained in [9] using a geometric construction. These are of different kind from those in this paper, which come from representationtheoretic considerations (the operators Am do not reduce to a single Peter– Weyl component). All these concepts can also be studied in the equally important case of compact (symmetric) spaces. In a companion paper [10], we present a uniﬁed approach to obtain the decomposition (13) for the Berezin transform also for the case of symmetric spaces (real or Hermitian) of compact type.

2. Berezin’s Star Restriction In his original paper [6], Berezin in fact never considered the star product (3). Instead, he proceeded by assigning to an operator T on Hν (D) (for a complex bounded symmetric domain D) a certain function T on D, called its “symbol”. In our current terminology, T was exactly T ∗ T , the image of T

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433

under the adjoint T ∗ of the Toeplitz calculus T : f → Tf , viewed as a map from L2 (D, dμ) into the Hilbert-Schmidt operators on Hν . He then deﬁned his star-product by declaring that T ∗d S := TS. We denote this “Berezin” star product by ∗d , in order to distinguish it from the “Berezin–Toeplitz” star product * from (3) (A more systematic terminology would be “Wick” and “anti-Wick” calculus, referring to the normal ordering procedure in quantum ﬁeld theory). Initially, ∗d is only deﬁned on functions on D of the form T with T ∈ B(Hν ), the algebra of all bounded linear operators on Hν ; however (in the modern terminology), one again extends it to a genuine star product by interpreting it as formal power series in ν1 in the sense of asymptotic expansions as ν1 0 [7, Section 4]. Choosing, in particular, T = Tf and S = Tg for some functions f, g on D, one gets f ∗d T g = T T f Tg = Tf ∗g , or Bf ∗d Bg = B(f ∗ g),

(14)

showing that the two star products are “conjugate” by the Berezin transform. Our goal in this section is to construct an object corresponding to ∗d also in the real case, and establish its basic properties. Throughout, we will use various standard notions and facts about Jordan triple systems, bounded symmetric domains and their reproducing kernels; we refer to [8] for all unexplained notation and further details. Definition 1. The Berezin (or “weak ”) star restriction #d is the GR -invariant operator from functions on DC into functions on DR deﬁned formally by #d T C∗ A = T R∗ (AI), ∀A ∈ B(Hν ). On a rigorous level, #d is best deﬁned using the next proposition, which generalizes (14) to the real case. Proposition 2. #d B C = B R#d . Proof. Taking A = TFC , we get, by the definition of #, R #d B C F = #d T C∗ TFC = T R∗ (TFC I) = T R∗ T#F = B R #F. Recall that as ν → +∞, the real Berezin transform B R —and, hence, also the complex Berezin transform B C —have the asymptotic expansions ∞ ∞ ν −k RR BC = ν −k RC (15) BR = k, k, k=0

k=0

R C with some GR -invariant differential operators RR k on D and G -invariant difC C R on D , respectively. (In [8], R and R ferential operators RC k k k were denoted R C by Qk and Rk , respectively.) Furthermore, R0 and R0 are just the identity

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operators; thus B R and B C are invertible as formal power series in inverses ∞ ∞ C −1 ν −k QR , (B ) =: ν −k QC (B R )−1 =: k k, k=0

QR 0

IEOT 1 ν,

with

k=0

QC 0

and being the identity operators. Finally, by [8, Theorem 3], the star restriction # has a similar asymptotic expansion ∞ #= ν −k Lk k=0 R

for some G -invariant differential operators Lk : C ∞ (DC ) → C ∞ (DR ), with L0 the identity operator. The following corollary thus provides a rigorous definition of #d . Corollary 3. Berezin’s star restriction #d exists as a formal power series #d = B R #(B C )−1 in ν1 , whose coefficients are GR -invariant differential operators from DC into DR obtained upon multiplying out the three formal power series on the righthand side, i.e. ∞ C ν −n Mn , Mn := RR (16) #d = j Lk Ql . n=0

j+k+l=m

It was shown in [8, Proposition 6], that the coefﬁcients Lk in the asymptotic expansion of # have the anti-holomorphic factorization property (9). It turns out that the corresponding coefﬁcients Mk for #d have the holomorphic, rather than anti-holomorphic, factorization property. Proposition 4. For H holomorphic on DC , #d (HF ) = (ρH) · #d F.

(17)

Proof. From the definition (1) of the complex Toeplitz calculus and the fact that multiplication by a bounded holomorphic function preserves Hν , it is immediate that TFCH = TFC THC . Taking adjoints gives TFCH = THC TFC since (TFC )∗ = TFC . Using the same symbol H to denote also the operator of multiplication by H, we can write the last equality as T C H = THC T C . (We remark that this was also proved as the formula (95) in [8], by a different argument; cf. Remark 8 there.) Taking adjoints again, but this time with T C viewed as a map from functions F to operators TFC , yields HT C∗ = T C∗ THC ,

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where the last THC now stands for the operator of multiplication by THC from the left on B(Hν ). By the formula (94) in [8], we also have R THC TfR = T(ρH)f ,

or T R (ρH) = THC T R . Taking adjoints gives T R∗ THC = (ρH)T R∗ . Thus #d HT C∗ A = #d T C∗ (THC A) = T R∗ (THC AI) = (ρH)T R∗ (AI) = (ρH)T C∗ A. Theorem 5. The Peter–Weyl decomposition (10) of #d is #d =

m∈N+ (rR )

∂ Km , [[ν]]m

(18)

∂ where Km are the operators from (12), and [[ν]]m is as in (13). ∂ Proof. As both #d and Km have the holomorphic factorization property (17), it sufﬁces to prove that (18) holds on anti-holomorphic functions H. Since ∂ both #d and Km are GR -invariant, it is further enough to prove this at the origin. However, for H holomorphic, ∂H = ∇H, whence ∂ R H(0) = φm (∂)H(0) = φm (∇)H(0) = Km H(0). Km

By (13), the right-hand side of (18) thus coincides with B R H. On the other hand, it is well-known that the complex Berezin transform is given by B C F (z) = K(z, z)−1 F Kz , Kz L2 (DC ,dμν ) , where K(z, w) ≡ Kw (z) is the reproducing kernel of the space Hν . By the reproducing property, it follows that B C ﬁxes anti-holomorphic functions. Thus B C H = H. Furthermore, since # has the anti-holomorphic factorization property and #1 = 1, we have #H = H. Thus, by Proposition 2, #d H = #d B C H = B R #H = B R H, completing the proof.

On a heuristic level, the last theorem asserts that the Peter–Weyl components of #d can be obtained from those of B R upon “replacing ∇ by ∂”. This is parallel to the fact that # can similarly be obtained from the formal inverse (B R )−1 upon “replacing ∇ by ∂”, cf. [8, Remark 11]. We will have more to say on this in Sect. 4.

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An explicit formula for the coefﬁcients [[ν]]m in the formulas (18) and (13) was given (excepting domains of type A) in [8, Theorem 14]. Namely, νrC + dX − dY (2rR /rC )2|m| dX [[ν]]m = , (19) dm rR m 2rR m where rR (rR − 1) rR (rR − 1) aR , dY = rR cR + aR , 2

j−i+1 2 aR m −m mi − mj + j−i 2 2 aR

j−i−1 i j , = j−i a a + 1 R R 2 2 1≤i<j≤r m −m

dX = rR + dm

R

i

j

R

aR , cR being the “characteristic multiplicities” of D , and rR j−1 aR (ν)m = ν− 2 mj j=1

(20)

being the associated multi-Pochhammer symbol. The determination of the Peter–Weyl decomposition of B R for type A domains remains an open problem.

3. Berezin–Toeplitz Star Restriction For any Riemannian symmetric space M ∼ = G/K (Hermitian or not), the Iwasawa decomposition G = N AK gives rise to the “plane wave” (or “conical”) functions R

eλ (n · exp a · K) = eλ+ρ

,a

on M , parameterized by λ ∈ a∗C , the complexiﬁcation of the dual of the Lie algebra a of A. Here ρR is the half-sum of positive roots. The spherical functions (21) φλ (z) := eλ (kz)dk K

are obtained upon averaging over K (with respect to the normalized Haar measure). It is well-known that both eλ and φλ are joint eigenfunctions of any G-invariant differential operator L on M : Lφλ = L(λ)φ λ,

∀λ ∈ a∗C .

is an isomorphism of the algebra of all The Harish–Chandra map L → L G-invariant differential operators onto the algebra of all polynomials on a∗ invariant under the Weyl group W . All this remains in force also for some more general G-invariant operators than differential, in particular, for operators of convolution by K-invariant functions on M integrable with respect to the invariant measure; the only difference is that one needs to restrict to may be a more general W -invariant those λ for which Lφλ makes sense, and L function than a polynomial.

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The above applies, in particular, to our DC = GC /K C and DR = GR /K R and the Berezin transforms B C and B R , which are convolution operators with the integrable functions K(z, z)−1 and I(z)−1 , respectively. The corresponding eigenvalues

rC Γ ν − pC2−1 + λj Γ ν − pC2−1 − λj C

B (λ) = pC −1 Γ ν − pC2−1 + ρC − ρC j Γ ν − j 2 j=1 and R (λ) = B

rR Γ νR − pR2−1 + λj Γ νR −

Γ νR − pR2−1 + ρR j Γ νR − j=1

pR −1 2 pR −1 2

− λj

− ρR j

(22)

have been found in [13] and [14], respectively. (We remark that (22) agrees with the formula (59) in [8], in view of the identity ΓR (νR +δ − drXR ) = ΓR (νR ), where δj = 1+(j −1)aR , for the Gindikin Gamma-function associated to DR .) Here pC = (rC − 1)aC + bC + 2 C

C

is the genus of D , ρ and ρR are the half-sums of positive roots, and rC rC νR = ν, pR = pC . 2rR 2rR Theorem 6. For domains not of type A, the Peter–Weyl decomposition (10) of # is ∂ Km , (23) #= [[pC − ν]]m m∈N+ (rR )

where

∂ Km

are the operators from (11).

Proof. By [8, Proposition 6], # has the anti-holomorphic factorization prop∂ , it is enough to prove (23) on holomorphic functions erty (9); since so do Km ∂ are GR -invariant, it further sufﬁces to prove this H. Since both # and Km at the origin. Since ∂H = ∇H for holomorphic H, we can then replace, as ∂ R H(0) by Km H(0). Now by [8, Proposition 9], in the proof of Theorem 5, Km we have B R #H = ρH, or #H = (B R )−1 (ρH), as formal power series in ν1 . Altogether, we see that it is enough to show that the formal inverse (B R )−1 , as a GR -invariant operator on DR , has the Peter–Weyl decomposition (8) given by R Km . (24) (B R )−1 = [[pC − ν]]m m∈N+ (rR )

To this end, we start with the asymptotic expansion for the logarithm of the Gamma function [5, §1.18 (12)] ∞ 1 log(2π) (−1)n+1 Bn+1 (a) −n + z log Γ(z + a) ≈ z + a − log z − z + 2 2 n(n + 1) n=1

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as z → +∞, where Bn are the Bernoulli polynomials. Using (22), it follows that there is an asymptotic expansion R (λ) ≈ bν (λ) as ν → +∞, log B where bν (λ) is the formal power series in bν (λ) =

1 ν

∞ (−1)n+1 pR −n νR − n(n + 1) 2 n=0 r R 1 1 1 + λj + Bn+1 − λj − Bn+1 + ρR × Bn+1 j 2 2 2 j=0 1 − ρR −Bn+1 . j 2

Since Bn (1 − x) = (−1)n Bn (x), by [5, §1.13 (12)], we see that the terms with even n actually vanish, whence bν (λ) = −bpC −ν (λ).

(25) R

On the other hand, recalling the asymptotic expansion (15) for B , we get from (13) that ∞

R Km ≈ ν −j Rm j [[ν]]m j=0

as ν → +∞,

(26)

where Rm j denote the Peter–Weyl components RR Rm j = j m∈N+ (rR )

in the decomposition (8) of RR j . However, from the explicit formula (19) we see that [[ν]]m is a rational function of ν; thus (26) must hold, in fact, also for ν → −∞. Hence ∞ R Km ≈ (pC − ν)−j Rm as ν → +∞. j [[pC − ν]]m j=0 Thus

R Km [[pC −ν]]m

is the mth Peter–Weyl component of the asymptotic expansion ∞

(pC − ν)−j RR j.

(27)

j=0

However, by (15), (25) and the fact that the Harish–Chandra map is an isomorphism, (27) is the asymptotic expansion of (B R )−1 . Thus (24) holds. Since complex bounded symmetric domains are a special case of the real ones, we automatically obtain an analogous result for the complex case; for convenience, we state it as a separate corollary.

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Corollary 7. The Peter–Weyl decomposition (10) of the Berezin–Toeplitz star product (3) on a complex bounded symmetric domain D of rank r, genus p, and with characteristic multiplicity a, is given by Km (∂x , ∂y ) f (x)g(y) , (28) (f ∗ g)(0) = (p − ν)m x=y=0 m∈N+ (r)

where (ν)m is the multi-Pochhammer symbol (20) associated to D. Proof. Immediate from the formula [[ν]]m = (ν)m /Km (e, e) for [[ν]]m in the complex case. Note that even in the case of the simplest complex bounded symmetric domain, the unit disc, the formula in the last corollary, which then reads (f ∗ g)(0) =

∞

1 ∂mf ∂mg (0) (0), m!(2 − ν)m ∂z m ∂z m m=0

(29)

seems to be new. We emphasize again that all the formulas (23), (28), (29), etc., for # are just equalities of asymptotic expansions as ν → +∞. For instance, (29) means that for any N = 0, 1, 2, . . ., (f ∗ g)(0) −

N −1 j=0

1 ∂mf ∂mg (0) (0) = O(ν −N ) m!(2 − ν)m ∂z m ∂z m

as ν → +∞, but has no meaning for a ﬁxed value of ν [just as the starproduct (3)], such as ν = 2, 3, . . . when the denominators vanish. To put it 1 in (29) for m = 1 should be viewed as a in still another way: the term 2−ν formal power series 1 1 1 =− 2−ν ν 1−

2 ν

=−

∞ 2k , ν k+1

k=0

and similarly for the terms m = 2, 3, . . .. In this regard, the situation is much better for #d : since B R F really exists for, say, bounded continuous F on DC , as a genuine function on DR (unlike (B R )−1 F , which is just a formal power series in ν1 resulting from the asymptotic expansion of B R as ν → +∞), the quantity [[ν]]m in (13) and (18) is, for given ν and m, an honest number.

4. Liftings of Operators We have seen in the Introduction that to each GR -invariant differential operator L on DR one can construct a GR -invariant operator L∂ from DC to DR , having the anti-holomorphic factorization property, by stipulating that for the “freezing at the origin” L∂ |0 := p(∂)

if L|0 = p(∇).

(30)

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This “lifting” of operators L to L∂ was introduced in [8, Section 4] (where L∂ was denoted by LC ). It was further shown there that all GR -invariant differential operators from DC into DR having the anti-holomorphic factorization property arise as L∂ for some GR -invariant L on DR . In this section, we highlight some additional properties of L∂ , and then describe a different “lifting” construction which is more complicated but in some respects better behaved, and present some applications. Proposition 8. L∂ has the following properties. (i) L∂ is in fact the restriction to DR of an GR -invariant differential operator (still denoted L∂ ) on all of DC . (ii) For G, H holomorphic, ρ(L∂ HG) = (ρH) · L(ρG). (iii) L → L∂ is an algebra homomorphism, i.e. (LM )∂ = L∂ M ∂ . Proof. (i) For a ∈ DR , let φa ∈ GR be the (unique) geodesic symmetry interchanging a and the origin. It is known that φa x is a rational function of both x ∈ DC and a ∈ DR ; in particular, φa x as well as all its partial derivatives with respect to x extend to holomorphic functions of x, a ∈ DC . Now by the GR -invariance of L, we have for any a ∈ DR Lf (a) = (Lf )(φa 0) = L(f ◦ φa )(0). It therefore follows that Lf (a) =

cα (a)∇α f (a),

∀a ∈ DR ,

α

where cα are some holomorphic functions on DC . Deﬁne L∂ on DC by L∂ F (z) := cα (z)∂ α F (z), z ∈ DC . (31) α multiindex

If g ∈ GR , then from L(f ◦ g) = (Lf ) ◦ g we have cα (a) ∇α (f ◦ g)(a) = cα (ga)(∇α f )(ga) α ∞

α R

R

for all f ∈ C (D ) and a ∈ D . Since cα and g are holomorphic and ∇ coincides with ∂ on holomorphic functions, it follows that cα (z) ∂ α (F ◦ g)(z) = cα (gz) (∂ α F )(gz) α ∞

α C

C

∂ R C for all F ∈ C (D) and z ∈ D . Thus L is G α-invariant on D . Finally, at ∂ α the origin L |0 = α cα (0)∂ where α cα (0)∇ = L|0 . Thus the restriction of L∂ to DR coincides with the L∂ deﬁned by (30). (ii) Immediate from (31) and the fact that ∂(HG) = H∇G for G, H holomorphic. (iii) Let M f (a) = mβ (a)∇β f (a), a ∈ DR , β multiindex

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be another GR -invariant differential operator on DR , and deﬁne qγ by LM f (a) =: qγ (a)∇γ f (a). (32) γ

From

cα ∇α mβ ∇β =

qγ ∇γ

on DR

γ

α,β

and the facts that cα , mβ are holomorphic on DC and ∇ coincides with ∂ on holomorphic functions, it follows that also cα ∂ α mβ ∂ β = qγ ∂ γ on DC . γ

α,β

Thus (LM )∂ = L∂ M ∂ .

By the standard structure theorem for invariant differential operators on a general Riemannian symmetric space, any GR -invariant operator on DR comes as a function L = Φ(Δ1 , . . . , ΔrR )

(33)

(in the spectral-theoretic sense) of certain algebraically independent generators (“higher Laplacians”) Δ1 , . . . , ΔrR . As a corollary to (iii) of the last proposition, we see that L∂ = Φ(Δ∂1 , . . . , Δ∂rR ). This applies, in particular, to the Berezin transform B R and to its inverse (as formal power series) (B R )−1 . By [8, Proposition 9], or the proof of Theorem 6 above, we have # = ((B R )−1 )∂ . Thus we arrive at the following corollary. Corollary 9. Let B R = ΦR (Δ1 , . . . , ΔrR ) be the expression of B R in terms of the “higher Laplacians”. Then 1

(34) # = ρ R Δ∂1 , . . . , Δ∂rR . Φ Remark 10. For the unit disc, rR = 1 and Δ1 is the invariant Laplacian (1 − |z|2 )∂∂. The last result then resembles the formula for the Weyl calculus on the upper half-plane in A. & J. Unterberger [12, Th´eor`eme 5.1]. In a completely analogous manner, all the above applies to the “anti-holomorphic” (rather than “holomorphic”) lifting L∂ of L deﬁned by L∂ |0 := p(∂)

if L|0 = p(∇)

instead of (30); the only change is that the roles of the holomorphic and anti-holomorphic functions get interchanged. In particular, we have the analogue of Proposition 8, as well as the following analogue of Corollary 9. Corollary 11. Let ΦR be as in the last corollary. Then

#d = ρΦR Δ∂1 , . . . , Δ∂rR .

(35)

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From Proposition 8 (iii) we see that ρL∂ H = L(ρH) for H holomorphic, but not for general H. Similarly for L∂ (with H replaced by H). We now exhibit another “lifting” construction, which improves upon this point, however, at the cost of losing the holomorphic or anti-holomorphic factorization property. One motivation to study lifting procedures in a more general framework is the existence of other functional calculi, notably the Weyl calculus [3,4], which do not have factorization properties with respect to holomorphic or antiholomorphic functions. For x ∈ DR and φx ∈ GR the associated geodesic symmetry, let γx ∈ GR be the “transvection” γx (z) := φx (−z) sending 0 to x. Following [9], consider now a GR -invariant retraction π : DC → DR . As in [9], we will assume that the preimage π −1 (0) of 0 ∈ DR has the form π −1 (0) = DC ∩ Y = ΛDR for some real vector subspace Y ⊂ Z C and real-linear K R -invariant map Λ : Z R → Z C . By [9, Lemma 2.4], the mapping Φ : DR × (Y ∩ DC ) → DC deﬁned by Φ(x, y) = γx (y) is then a real-analytic diffeomorphism. Consider again a GR -invariant differential operator cα ∇α L= α R

on D , so that p(X) :=

cα (0)X α ,

X ∈ ZC

α

is a K R -invariant polynomial on Z R (and Z C ). For y ∈ DC ∩ Y = π −1 (0) and F ∈ C ∞ (DC ), set cα (0)(∂ + ∂)α F (y). (36) L∂+∂ F (y) := α

Since the right-hand side is K R -invariant, L∂+∂ extends by GR -invariance to a well-deﬁned GR -invariant operator (still denoted L∂+∂ ) on all of DC . Proposition 12. L∂+∂ has the following properties. (i) ρL∂+∂ = Lρ. (ii) L → L∂+∂ is an algebra homomorphism, i.e. (LM )∂+∂ = L∂+∂ M ∂+∂ . (iii) For G holomorphic, ρL∂+∂ G = ρL∂ G = L(ρG) and ρL∂+∂ G = ρL∂ G = L(ρG).

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Proof. (i) By GR -invariance, it is enough to check this at the origin, and there it follows by the construction (as ∂ + ∂ = ∇). (ii) Again, by GR -invariance it is enough to show that (LM )∂+∂ and L∂+∂ M ∂+∂ coincide on π −1 (0). Since (36) only involves cα (0), this reduces to showing that (LM )∂+∂ |0 = (L∂+∂ M ∂+∂ )|0 . But that is exactly (32), since ∂ + ∂ = ∇. (iii) Again, all three terms reduce just to α cα (0)∇α G(0) at the origin, and the equality at any other point of DR follows by GR -invariance. Similarly for the second assertion. We point out that, in general, it is no longer true that L∂+∂ has the holomorphic or anti-holomorphic factorization property, or that L∂+∂ G = L∂ G (or L∂+∂ G = L∂ G) away from DR for G holomorphic. The main advantage of L∂+∂ over L∂ or L∂ is the intertwining property ρL∂+∂ = Lρ. Together with (iii) of the last proposition, it implies that for any L as in (33), one has ). Lρ = ρΦ(Δ1∂+∂ , . . . , Δr∂+∂ R This has the following consequence for general star restrictions, which are deﬁned as follows: Let A be any operator calculus on DC , invariant in the sense of (2); that is, A is a mapping from a suitable class of functions on DC into operators on Hν (DC ) such that AF ◦g = Ug∗ AF Ug ,

∀g ∈ GC .

Similarly let B be any operator extension on DR , invariant in the sense of (4); that is, B is a mapping from a suitable class of functions on DR into Hν (DC ) such that Bf ◦g = Ug∗ Bf ,

∀g ∈ GR .

Generalizing (5), the star restriction #A,B with respect to the pair A, B is the mapping from functions on DC into functions on DR formally deﬁned by B#A,B F = AF I. Both A and B may further depend on some parameter ν 0, and #A,B is then deﬁned on the rigorous level in terms of the asymptotic expansions as ν → +∞, as before. For A = T C and B = T R , #A,B reduces to the Toeplitz star restriction # studied so far. As shown in [8, Section 7.3], the two star restrictions are related by #A,B = MB−1 # MA for some GC -invariant operator MA on DC and some GR -invariant operator MB on DR . (Under the Harish–Chandra isomorphism, MA and MB R (λ)2 = |B R (0)/T RR (0)|2 and M are given by M (0)|2 /B B (λ) = |BφR A (λ) = φλ φλ λ C (λ)2 = |A C 1(0)/T CC 1(0)|2 , where φR , φC are the spherical |A C 1(0)|2 /B φλ

R

φλ

φλ

functions (21) on D and DC , respectively. See [2, Theorem 4.9].) One thus gets the following corollary.

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Corollary 13. For A, B as above, ∂+∂

((B R )−1 )∂ MA #A,B = ρ MB−1 1 1 C = ρ B (Δ1∂+∂ , . . . , Δr∂+∂ ) R (Δ∂1 , . . . , Δ∂rR )ΦA (ΔC 1 , . . . , ΔrC ), R Φ Φ C where MB = ΦB (Δ1 , . . . , ΔrR ) and MA = ΦA (ΔC 1 , . . . , ΔrC ) are the expressions of MB and MA , respectively, in terms of the “higher Laplacians” on DR and DC .

Observe that, in particular, for MA = (B C )−1 and MB = (B R )−1 , #A,B reduces to #d , by Proposition 2. Thus the expression given by the last corollary in this case must coincide with (34). To verify this directly seems quite intriguing.

5. The Euclidean Case The results of the preceding sections extend also to the “ﬂat” situation when the bounded symmetric domain is replaced by the Euclidean n-space. In the real flat case of DR = Rn , with complexiﬁcation DC = Cn , the role of the “invariant” measure is played by the (unnormalized) Lebesgue measure dz, and the spaces Hν are the Segal–Bargmann–Fock spaces of entire functions ν n 2 Hν = L2hol (Cn , dμν ), dμν (z) := e−ν|z| dz, ν > 0, π with reproducing kernels K(z, w) = eνzw ,

n where, for the simplicity of notation, we write zw for z, wCn = j=1 zj wj . The complex and the real Toeplitz calculus described in the Introduction are given by C (37) TF H(z) = Pν (F H)(z) = F (w)H(w)eνzw dμν (w) Cn

for H ∈ Hν , F ∈ L∞ (Cn ), where Pν : L2 (Cn , dμν ) → Hν is the orthogonal projection, and ν n/2 2 TfR (z) = f (x)eνzx−νx /2 dx, (38) 2π Rn

respectively. (In the second formula, dx is the unnormalized Lebesgue mean sure on Rn , and x2 := xx = j=1 x2j .) It turns out that this case can be treated directly using Fourier transform methods [11]. Recall that the Bargmann transform ν n/4 2 2 f (x)eνzx−νx /2−νz /4 dx (39) βf (z) := π Rn

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is a unitary map of L2 (Rn ) onto Hν/2 , with inverse given by ν n/4 2 2 −1 β F (x) = F (z)eνxz−νx /2−νz /4 dμν/2 (z). π Cn

Comparing (38) and (39), we see that TfR (z) =

ν n/4 2 eνz /4 βf (z). 4π

The star restriction #, deﬁned by R T#F := TFC I,

I(z) = eνz

2

/2

,

can thus be recovered using the formula for β −1 . Proposition 14. In the real flat case, # = ρe−∂

2

/2ν

=ρ

ν −|α|

α multiindex

(−1)|α| ∂ 2α . α!2|α|

Proof. By the formulas above, n/4 4π −1 −νz 2 /4 C #F (x) = β e TF I (x) ν 2 2 2 2 ν 2n = n/2 2n eνxz−νx /2−νz /4−νz /4−ν|z| /2 2 π Cn 2 2 × eνw /2+νzw−ν|w| F (w)dwdz.

(40)

Cn

We thus need to show that the oscillatory integral 2 2 2 2 2 2 ν 2n eνxz−νx /2−νz /4−νz /4−ν|z| /2+νw /2+νzw−ν|w| dz n/2 2n 2 π

(41)

Cn

2

equals e−∂ /2ν δx (w) (in our sense of asymptotic expansions as ν → +∞). Since the integral kernel in (41) is translation invariant (it remains unchanged upon replacing x, z, w by x + y, z + y, w + y with any y ∈ Rn ), the coefﬁcients of the asymptotic expansion must be constant coefﬁcient differential operators; it therefore sufﬁces to evaluate (41) on the test functions F (w) = eνAw+νBw

(42)

for A, B ∈ Cn . Also, in view of the translation invariance, it is enough to consider x = 0. Then (40) gives, using the reproducing property of the kernels

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eνzw and writing z = a + bi, a, b ∈ Rn , #F (0)

2 2 ν 2n −νz 2 /4−νz 2 /4−ν|z|2 /2 e eνw /2+νzw−ν|w| eνAw+νBw dw n/2 2n 2 π Cn Cn n 2 2 2 2 ν = n/2 n e−νz /4−νz /4−ν|z| /2 eνA(z+B)+ν(z+B) /2 dz 2 π

=

Cn

n νAB+νB 2 /2

=

ν e

2n/2 π n

=

=

νn n/2 2 πn νn 2n/2 π n ν

n

2n/2 π n

= e−νA

2

/2

2

e−νa

+νAa+νAbi+νa2 /2−νb2 /2+νabi+νaB+νBbi

da db

Rn Rn

ν n eνAB+νB = 2n/2 π n =

2

2π ν 2π ν 2π ν

/2

2

e−νa

/2+νAa+νaB−ν(b−Ai−ai−Bi)2 /2−ν(A+a+B)2 /2

Rn

n/2 eνAB+νB

eνAB+νB

= (e−∂

2

/2

2

e−νa

/2+νAa+νaB−ν(a+A+B)2 /2

Rn

n/2 n/2

2

2

/2 −ν(A+B)2 /2

e

da

da

2

e−νa da

Rn n/2

2 2 π eνAB+νB /2−ν(A+B) /2 n/2 ν

/2ν

F )(0),

completing the proof.

In the ﬂat case, the Berezin transforms B R = T R∗ T R and B C = T C∗ T C are given by the heat solution operators B R = e∇

2

/2ν

,

B C = e∂∂/ν .

(43)

Therefore Proposition 2 immediately gives the corresponding result for the Berezin star restriction. Corollary 15. In the real flat case, #d = ρe∂

2

/2ν

.

Proof. Using the same test function (42), we have (ρF )(x) = eν(A+B)x and obtain, by (43), 2 2 2 2 #d F = e∇ /2ν ρe−∂ /2ν−∂∂/ν F = e∇ /2ν ρe−νA /2−νAB F

= e−νA = ρeνB

2

2

/2−νAB ∇2 /2ν ν(A+B)x

/2

e

F = ρe∂

e

2

/2ν

= e−νA

2

/2−νAB+ν(A+B)2 /2

ρF

F.

For completeness, we also review the results for the complex flat case of D = Cn , with complexiﬁcation D × D = {(z, w) : z, w ∈ Cn }. The space Hν (D × D) consists of all functions in L2 (Cn × Cn , dμν ), dμν (z, w) =

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2

( πν )2n e−ν|z| −ν|w| , which are holomorphic in z and w. The complex Toeplitz calculus is given by C F (x, y)H(x, y)eνzx+νyw dμν (x, y) TF H(z, w) = Cn Cn

for H ∈ H (D × D), F ∈ L∞ (D × D). The real Toeplitz calculus is obtained from (37) upon identifying operators T on Hν (D) with their integral kernels T(z, w) := T eν.w , eν.z Hν (D) (see [8, p. 5] for details); explicitly, ν n 2 f (x)eνxw+νzx−ν|x| dx. TfR (z, w) = π ν

Cn

The real and complex Berezin transforms, respectively, are again given by the heat solution operators B R = e∂∂/ν ,

B C = e(∂z ∂ z +∂w ∂ w )/ν .

(44)

We then have the following analogues of Proposition 14 and Corollary 15. Proposition 16. In the complex flat case, # = ρe−∂z ∂ w /ν . Proof. We again evaluate on the test function F (z, w) = eνzA+νBz+νwC+νDw for A, B, C, D ∈ Cn . For I(z, w) = eνzw we thus have, using the reproducing property, C F (x, y)I(x, y)eνzx+νyw dμν (x, y) TF I(z, w) = Cn Cn

eνzA+νBz+νwC+νDw+νxy+νzx+νyw dμν (x, y)

= Cn Cn

eν(B+z)(A+y)+νyC+νDy+νyw dμν (y)

= Cn

= eν(B+z+D)(w+C)+ν(B+z)A = eν(B+z+D)(w+C+A)−νDA . On the other hand, the restriction f (x) := (ρF )(x) = eνx(A+C)+ν(B+D)x satisﬁes

ν n

TfR (z, w) =

π

(45) 2

eνx(A+C)+ν(B+D)x eνxw+νzx−ν|x| dx

Cn

= eν(B+D+z)(A+C+w) . R , or Thus TFC I = e−νDA TρF

#F = e−νDA ρF = ρe−∂z ∂ w /ν F, completing the proof.

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Corollary 17. In the complex flat case, #d = ρe∂w ∂ z /ν . Proof. By (44), (45) and Proposition 2, #d F = e∂∂/ν ρe−∂z ∂ w /ν−(∂z ∂ z +∂w ∂ w )/ν F = e∂∂/ν e−νDA−νBA−νDC ρF = e−νDA−νBA−νDC eν(B+D)(A+C) ρF = eνBC ρF = ρe∂w ∂ z /ν F. We remark that in the last proof, we could alternatively have used the intertwining relations Lρ = ρL∂+∂ established in Proposition 12; indeed, for the G-invariant (G = Cn U (n)) operator L = e∂∂/ν on D = Cn , the corresponding operators L∂ , L∂ , L∂+∂ on D × D are given by L∂ = e∂z ∂ w /ν ,

L∂ = e∂w ∂ z /ν ,

L∂+∂ = e(∂z +∂w )(∂ z +∂ w )/ν .

Similarly in the real ﬂat case, for the GR -invariant (GR = Rn O(n)) oper2 ator L = e∇ /2ν on DR = Rn , the corresponding operators on DC = Cn are given by L∂ = e∂

2

/2ν

,

L∂ = e∂

2

/2ν

,

L∂ = e(∂+∂)

2

/2ν

,

respectively. Thus, for instance LρeνAw+νBw = eν(A+B)

2

/2 νAx+νBx

e

= ρL∂+∂ eνAw+νBw ,

and similarly in the complex case.

References [1] Arazy, J., Ørsted, B.: Asymptotic expansions of Berezin transforms. Indiana Univ. Math. J. 49, 7–30 (2000) [2] Arazy, J., Upmeier, H.: Covariant symbolic calculi on real symmetric domains, singular integral operators, factorization and applications. Oper. Theory Adv. Appl. 142, 1–27 (2003) [3] Arazy, J., Upmeier, H.: Weyl calculus for complex and real symmetric domains. Rend. Mat. Acc. Lincei 13, 165–181 (2002) [4] Arazy, J., Upmeier, H.: A one-parameter calculus for symmetric domains. Math. Nachr. 280, 939–961 (2007) [5] Bateman, H., Erd´elyi, A.: Higher transcendental functions, vol. I. McGraw-Hill, New York (1953) [6] Berezin, F.A.: Quantization. Math. USSR Izvestiya 8, 1109–1163 (1974) [7] Engliˇs, M.: Weighted Bergman kernels and quantization. Comm. Math. Phys. 227, 211–241 (2002) [8] Engliˇs, M., Upmeier, H.: Toeplitz quantization and asymptotic expansions on real symmetric domains, Math. Z. (to appear). Preprint available at http:// www.math.cas.cz/englis/70.pdf [9] Engliˇs, M., Upmeier, H.: Toeplitz quantization and asymptotic expansions: geometric construction. SIGMA Symmetry Integr. Geom. Methods Appl. 5, Paper 021, 30 pp (2009)

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[10] Engliˇs, M., Upmeier, H.: Real Berezin transform and asymptotic expansion for symmetric spaces of compact and non-compact type (submitted) [11] Folland, G.B.: Harmonic Analysis in Phase Space, Annals of Mathematics Studies 122. Princeton University Press, Princeton (1989) [12] Unterberger, A., Unterberger, J.: Quantification et analyse pseudo´ diﬀ´erentielle. Ann. Sci. Ecole Norm. Sup. (21) 17, 133–158 (1988) [13] Unterberger, A., Upmeier, H.: Berezin transform and invariant differential operators. Comm. Math. Phys. 164, 563–598 (1994) [14] Zhang, G.: Berezin transform on real bounded symmetric domains. Trans. Am. Math. Soc. 353, 3769–3787 (2001) Miroslav Engliˇs Mathematics Institute ˇ a 25 Zitn´ 11567 Prague 1 Czech Republic and Mathematics Institute Silesian University in Opava Na Rybn´ıˇcku 1 74601 Opava Czech Republic e-mail: [email protected] Harald Upmeier (B) Fachbereich Mathematik Universit¨ at Marburg 35032 Marburg Germany e-mail: [email protected] Received: February 12, 2010. Revised: April 23, 2010.

Integr. Equ. Oper. Theory 68 (2010), 451–472 DOI 10.1007/s00020-010-1838-z Published online November 13, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform H. Bounit, A. Driouich and O. El-Mennaoui Abstract. This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufﬁcient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon–Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56–63, 1999) on the validity of the inverse formula of the Laplace transform for C0 -semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26(5):1331–1341, 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957). Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00. Keywords. Class ﬁle, journal.

1. Introduction In this paper we consider the following abstract differential equation: . x (t) = Ax(t) + Bu(t), t ∈ [0, τ ), (1.1) x(0) = x0 , where 0 < τ ≤ ∞, (A, D(A)) is the generator of a C0 -semigroup T := (T(t))t≥0 on a Banach state space X of type ω(A), and the function u takes values in another Banach space U . We denote by ·, · the duality pairing for each Banach space. If B : U → X is a bounded linear operator and X and U are of ﬁnite dimension, then (1.1) is a setup of classical linear control systems theory, and there is a large literature (cf., e.g., [8, and references therein]) in the case where X and U are inﬁnite dimensional spaces. This This work was completed with the support of our TEX-pert.

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allows applications to partial differential equations, but in order to model, e.g., control from the boundary, one has to deal with unbounded operator B. Writing L(Z, W ) for the space of all bounded operators from a Banach space Z to a Banach space W and following the literature, e.g., [20], one only requires B ∈ L(U, X−1 ), where X−1 is the completion of X with respect to the norm x−1 = ||(λ0 − A)−1 x||X for a ﬁxed λ0 in the resolvent set ρ(A) (all those norms are equivalent; see [11]) with X → X−1 (i.e. with continuous injection). The semigroup T extends to a C0 -semigroup on X−1 , whose generator is an extension of A (see e.g., [11]). The extended semigroup is isomorphic to the initial one. We denote the extensions of T and A by the same symbols when it is clear from the context. Moreover, the resolvent sets of A and its extension are the same. The usual choice for spaces from which controls u are taken is L2 ([0, τ ), U ) (which is also the natural one if X and U are Hilbert spaces). A control operator B ∈ L(U, X−1 ) is called 2 finite-time τ L -admissible for A if, for some (and hence for all) τ ∈ (0, ∞) the integral 0 T(τ − σ)Bu(σ)dσ which exists as a Bochner integral in X−1 (see e.g. [3, Proposition 1.3.4]) and takes values in X for all u ∈ L2 ([0, τ ), U ) (see [29]). L2 -Admissibility of the control operator B is equivalent to the wellposedness on X of the control system (1.1) (see [12]). In the case where the rank one operator B, deﬁned by Bu(t) = bu(t) for b ∈ X−1 is (ﬁnite-time) L2 -admissible, we say that b is an L2 -admissible input element. An operator B ∈ L(U, X−1 ) is called finite-time weakly L2 -admissible control operator for A (see [28]) if bv := Bv is an L2 -admissible input element for A for every v ∈ U . This notion of ﬁnite-time L2 -admissibility is invariant under scalings e−α· T. So, to investigate L2 -admissibility of control operators, we may assume that the semigroup T is exponentially stable (i.e. ω(T) < 0). It follows from the closed graph theorem that if B is ﬁnite-time L2 admissible control operator for A, then for each τ > 0, there exists Mτ > 0 such that τ T(τ − σ)Bu(σ)dσ

≤ Mτ uL2 ([0,τ ),U )

u ∈ L2 ([0, τ ), U ) .

X

0

Let A generate a C0 -semigroup and let B be a ﬁnite-time L2 -admissible control operator for A. Then the inhomogeneous linear partial differential equation (1.1) has, for every initial condition and every locally square integrable input, a unique (weak) solution which depends continuously on the initial condition and the input. For exponentially stable semigroup T, ﬁnite time L2 -admissibility is equivalent to inﬁnite-time L2 -admissibility, henceforth called L2 -admissibility for short (see [13]). Inﬁnite time L2 -admissibility is equivalent to the existence of a constant M > 0 satisfying ∞ T(σ)Bu(σ)dσ 0

X

≤ M uL2 ([0,∞),U )

u ∈ L2 ([0, ∞), U ) .

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Weiss [28] observed that L2 -admissibility of B implies the following resolvent condition which is often easier to test: there exists M > 0 such that (λI − A)−1 B ≤

M (Re(λ))

1/2

,

λ ∈ C,

Re(λ) > 0.

(1.2)

Then it is shown in [28] that the converse does not hold in the general Banach space context and it is conjectured that the converse would hold if X and U were Hilbert spaces. However, the Weiss conjecture has been shown to hold for certain special cases, such as right invertible semigroups [28], normal and analytic semigroups [16], analytic contraction semigroups [22]. Moreover, [22] showed that the Weiss conjecture holds for a bounded analytic semigroup if and only if the fractional power (−A)1/2 is inﬁnite-time L2 -admissible for A. For a direct approach of this result we refer to [4]; an analogous result has been recently proved in [5] for the weighted admissibility case. This paper deals with the following question: under what conditions is a control operator, B, L2 -admissible for A? What distinguishes the present study from earlier contributions is the attention which we pay not only to the relationship between the operator A, the control operator B and the semigroup T(t), but also to the geometry of the underlying control space. The restriction that the control space must be a UMD-space is essential will be seen from an example. This result brings out the intimate relationship between the control theory of inﬁnite dimensional systems and the inverse Laplace transform. However, this characterization (Proposition 3.2) of L2 -admissibility of control operators is useful, and yields a proof of the result by [9] on the validity of the (complex) inverse Laplace transform for C0 -semigroups in UMD-Banach spaces. In the functional analysis area, by using our approach to the characterization of admissibility, we retrieve some well-known results on incomplete and complete inverse Laplace transforms of the resolvent of a semigroup in general Banach spaces and in UMD spaces, respectively. The paper is organized as follows: in Sect. 2 we recall some basic facts on UMD-spaces and some results on the validity of the (complex) inverse Laplace transform for C0 -semigroups. The quoted results are due to [9] with exceptions to Lemma 2.2. Section 3 contains our main result on the characterization of L2 -admissibility of unbounded control operators in UMD control spaces. There is a dual notion of L2 -admissibility for observation operators; the results of this paper can be translated into these terms, as well. Based on this, we show that the weak L2 -admissibility is linked to the convergence of the (complex) inverse Laplace transforms of the semigroups on general Banach spaces.

2. UMD-Spaces and the Inverse Laplace Transform We start with a brief overview on the (complex) inverse Laplace transform. Let X denote a Banach space and A the generator of a C0 -semigroup T in X with domain D(A). It is well known that the resolvent R(λ, A) := (λI − A)−1

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exists and is given by the Laplace transform of the semigroup T +∞ R(λ, A)x = e−λt T(t)xdt

(x ∈ X)

0

for Re(λ) > w(A), where w(A) denotes the growth bound of A (see e.g., [2,23,24]). Conversely, let α > w(A). It is well known (see [17, p. 349], or α+iN [24, pp. 28–29]) that the integral α−iN eλt R(λ, A)xdλ converges as N → ∞ uniformly for t in compact intervals of (0, +∞), and the following holds 1 lim 2πi N →∞

α+iN

λt

e R(λ, A)xdλ =

α−iN

T(t)x for x for 2

t>0 t = 0,

(2.1)

for all x ∈ D(A). This inversion of the Laplace transform for C0 - semigroup in a Hilbert space H has been generalized to all x ∈ H in [27]. Now, let us recall some facts on the Hilbert transform. Let p ∈ (0, ∞) and ε > 0. Deﬁne the operator Hε on Lp (R, X) by f (s) 1 ds (t ∈ R). Hε f (t) := π t−s |t−s|≥ε

A Banach space is called UMD-space (or said to have the UMD property) if for some (and hence all) p ∈ (1, ∞) (see e.g., [1,6]) Hf := limε0 Hε f exists in Lp (R, X) and deﬁnes a bounded operator H on Lp (R, X). The operator H is called the Hilbert transform on Lp (R, X). Every UMD-space is reﬂexive and its dual is also a UMD-space. Typical examples of UMD-spaces are s for p, q ∈ (1, ∞) Lp (Ω)-spaces, Sobolev spaces Wps (Ω) or Besov spaces Bp,q and their closed subspaces. We refer to [1] for more details about UMD-spaces and the Hilbert transform. For a general Banach space X, we also use the bilateral vector-valued Fourier transform FX deﬁned for f ∈ L1 (R, X) and for almost every λ ∈ R by −1/2

∞

FX (f )(λ) := (2π)

e−iλs f (s)ds.

−∞

Recall that if X is a Hilbert space, then the vector-valued Fourier transform FX extends to an isometry from L2 (R, X) to itself (this is the vector-valued version of Planchrel’s theorem) due to [14]). In [9] it has been proved that the (complex) inversion of the Laplace transform for strongly continuous semigroups is valid in UMD spaces, namely they proved: Theorem 2.1. Let A be the generator of a C0 -semigroup T in a Banach space, X, and let α > w(A) and τ > 0. Assume that X is a U M D-space; then the

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convergence (2.1) holds for all x ∈ X uniformly for t ∈ ]τ, τ1 [ and we have T(t)x =

1 2πi

α+i∞

eλt R(λ, A)xdλ.

(2.2)

α−i∞

Note that the authors in [9] showed that the UMD-property is essential. In the sequel, and without loss of generality, we assume that the operator A generates an exponentially stable C0 -semigroup in X denoted by T, that is, w(A) < 0. So, we begin by deﬁning the following operators which will be used constantly in the rest of the paper: 1 TN (t) = 2π

+N

eiλt R(iλ, A)dλ

(N ≥ 0).

(2.3)

−N

Thanks to the above theorem, the operators TN (t) converge strongly as N → ∞ uniformly for t ∈ ]τ, 1/τ [ to T(t) (here α = 0). The following lemma is implicitly contained in [9] and will be improved in Corollary 2.4. Lemma 2.2. Let A generate an exponentially stable C0 -semigroup in X. Then b for b ≥ a > 0, the integral a TN (t)xdt converges in X for x ∈ X as N → +∞. Further, if X is an UMD space, then the convergence takes place in D(A). Proof. Let x ∈ X; then x1 = A−1 x ∈ D(A). The resolvent identity yields: b a

1 TN (s)xds = 2π 1 = 2π 1 = 2π 1 = 2π

b +N eiλs R(iλ, A)xdλds a −N

+N −N

+N −N

+N

−N

1 − 2π

eibλ − eiλa R(iλ, A)xdλ iλ eibλ − eiλa R(iλ, A)Ax1 dλ iλ

eibλ − eiλa dλ x1 iλ

+N (eibλ − eiλa )R(iλ, A)x1 dλ.

−N

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Let us study the ﬁrst integral on the right hand-side of the above equality. If we denote by χE the characteristic function of the set E ⊂ R, then we ﬁnd 1 √ 2π

+N ibλ e − eiλa 1 dλx1 = √ F(χ[a,b] )(−λ)χ[−N,N ] (−λ)dλx1 iλ 2π

−N

R

1 = √ 2π =

2 π

b F(χ[−N,N ] )(λ)dλx1 a

bN aN

sin(λ) dλx1 , λ

which converges to zero as N → ∞. Since x1 ∈ D(A), according to (2.1) the second integral also converges in X. Moreover, this integral can be rewritten as follows +N +N 1 ibλ iλa −1 1 (e − e )R(iλ, A)x1 dλ = A (eibλ − eiλa )R(iλ, A)xdλ. 2π 2π −N

−N

Now, if X is a UMD-space, then by virtue of Theorem 2.1 the integral +N ibλ (e − eiλa )R(iλ, A)xdλ converges in X, which implies that the inte−N b gral a TN (t)xdt converges in D(A), and we have exactly b lim

N →∞

TN (t)xdt = T(b)A−1 x − T(a)A−1 x ∈ D(A).

a

We now take a look at the situation where A generates an analytic semigroup on X. As a second result, for a UMD-Banach space X, the convergence stated (2.1) takes place on the entire space X. Proposition 2.3. Let A be the generator of an analytic C0 -semigroup in X. Then the following assertions hold: N (i) For x ∈ X, the integral −N R(iλ, A)xdλ converges in X as N → +∞, and hence in D(A) for all x ∈ D(A). N (ii) If X is a UMD space and τ ≥ 0, then for all x ∈ X, −N eiλτ R(iλ, A)xdλ converges in X as N → ∞, and hence in D(A) for all x ∈ D(A). Proof. Part (i) Assume ﬁrst that x ∈ D(A). Using the resolvent identity, we obtain x R(z, A)Ax for z ∈ ρ(A)\{0}. (2.4) R(z, A)x = + z z + + Let ∂CN be the boundary of the half-disc CN deﬁned by π

+ . = λ ∈ C : |λ| ≤ N, |argλ| ≤ CN 2

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Then Cauchy’s theorem yields N R(iλ, A)xdλ = −i R(z, A)xdz −N

+ ∂CN

π/2 =

N eiθ R(N eiθ , A)xdθ.

−π/2

Since A generates an analytic semigroup on X, the set {N eiθ R(N eiθ , A)} is N uniformly bounded (see [24]). It follows that −N R(iλ, A)xdλ is uniformly N bounded, that is, −N R(iλ, A)xdλ ≤ αx for all x ∈ X and some α > 0. Moreover, using the identity (2.4) we obtain dz R(z, A) x+ Axdz R(z, A)xdz = z z + ∂CN

+ ∂CN

+ ∂CN

π/2 = πix +

iR(N eiθ , A)Axdθ

for all x ∈ D(A).

−π/2

Once again, since A generates an analytic semigroup, the integral on the right-hand side of the above equality converges to zero as N → ∞. Using the last equality, the density of D(A) in X and the fact that the integral N R(iλ, A)xdλ is uniformly bounded, and an elementary equicontinuity −N argument guarantees the convergence in X for every x ∈ X. Using this together with the fact that x ∈ D(A) and τ ≥ 0, we get +N +N eiλτ R(iλ, A)xdλ = A−1 eiλτ R(iλ, A)ydλ, −N

N

for some

y ∈ X.

(2.5)

−N

Therefore −N R(iλ, A)xdλ converges in D(A) for all x ∈ D(A). Part (ii). Let X be a UMD-space. By virtue of Theorem 2.1, the convergence holds in X for all τ > 0, and for τ = 0 the convergence in X is guaranteed by the ﬁrst assertion in Part (i). For x ∈ D(A), the convergence in D(A) takes place accordingly to (2.5), Theorem 2.1, and the second assertion in Part(i) above. The following corollary is immediate: Corollary 2.4. Let A be the generator of an analytic C0 -semigroup on a Banach space X. Then the assertions in Lemma 2.2 are still true for b > a ≥ 0.

3. A Characterization of Admissibility Now we come to the main result characterizing L2 -admissibility in UMDspaces. We begin by introducing some notions and recalling some results on

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equivalent conditions for L2 -admissibility which will be used in the sequel. In Sect. 1, we have noted that the definition of L2 -admissibility is independent on the parameter τ , according the semigroups properties. Moreover, this definition is unaffected by a reﬂection of the control u about the ordinate, so we may integrate over the product TBu where convenient, instead of calculating the convolution. We summarize these results as follows (see e.g. [13,18,28]). Theorem 3.1. Let X and U be Banach spaces and B ∈ L(U, X−1 ). Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

for A; B is a finite-time L2 -admissible control operator τ For some τ > 0 and any u ∈ L2 ([0, τ ], U ), 0 T(τ − σ)Bu(σ)dσ ∈ X; 2π For any u ∈ L2 ([0, 2π], U ), 0 T(2π − σ)Bu(σ)dσ ∈ X; 2π For any u ∈ L2 ([0, 2π], U ), 0 T(σ)Bu(σ)dσ ∈ X; There exists M > 0 such that for any u ∈ L2 ([0, 2π], U ), 2π T(σ)Bu(σ)dσ ≤ M uL2 ([0,2π],U ) . X

0

If in addition ω(A) < 0, then all the above items are equivalent to the (i.e., the infinite-time L2 -admissibility): following ∞ (v) 0 T(σ)Bu(σ)dσ ∈ X for any u ∈ L2 ([0, ∞), U ). In order to formulate the problem precisely, we have to introduce the following notations and assumptions. Let T be an exponentially stable C0 -semigroup on X with generator (A, D(A)), and B ∈ L(U, X−1 ). Note that the restriction imposed to the semigroup T (i.e. ω(A) < 0) is unessential and hence all the results obtained in this paper are still true. For the time being, we assume these conditions are met. For any input u ∈ L22π (U ) := L2 ([0, 2π], U ) the X−1 -valued function σ → T(σ)Bu(σ) and the X-valued function σ → TN (σ)Bu(σ) are Bochner integrable on [0, 2π] (see [17] Cor1 after Thm. 3.5.3 and Thm. 3.7.4, and [10] Chap. II) where TN is given by (2.3). We then set 2π

2π T(σ)Bu(σ)dσ

Φ(u) = 0

and

TN (σ)Bu(σ)dσ,

ΦN (u) =

N ∈ R+ .

0

We may now formulate and prove the main theorem of this section, giving a necessary and sufﬁcient condition for the L2 -admissibility of B. The necessary condition here is essentially based on a geometric property of the control space U , that is, the UMD-property. The result encompasses Hilbert control spaces, but the proposition below yields the criterion’s necessity. Proposition 3.2. Let X and U be Banach spaces. Let A be the generator of an exponentially stable semigroup T on X and B ∈ L(U, X−1 ). Then the following assertions hold: (i) If ΦN is uniformly bounded on L(L22π (U ), X), then B is L2 -admissible for A.

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(ii) If U is a UMD-space and B is L2 -admissible for A, then ΦN is uniformly bounded on L(L22π (U ), X). Proof. Part (i). Notice that for any μ ∈ ρ(A) and constant input u0 ∈ U , we have μ2 R2 (μ, A)Bu0 ∈ D(A). By the inverse Laplace transform on general Banach space (see e.g., [2,23,24]), we obtain X − lim TN (τ )μ2 R2 (μ, A)Bu0 = T(τ )μ2 R2 (μ, A)Bu0 . N →∞

(3.1)

To prove that B is L2 -admissible for A, it sufﬁces to prove that for any step function u : [0, 2π] → U with compact support that does not contain zero, we have the following uniform estimate: 2π T(τ )Bu(τ )dτ ≤ M uL2 (U ) , for some M > 0. 2π X

0

So, consider a step function u with compact support that does not contain zero. Then there exists ε > 0 such that 2π ΦN (u) = TN (σ)Bu(σ)dσ 0

2π = TN (σ)Bu(σ)dσ. ε

Let MHY be the Hille–Yoshida constant associated to the semigroup T(t). Since ΦN is uniformly bounded with ΦN ≤ MΦ for some MΦ > 0 and for all N ∈ N, then μ2 R2 (μ, A)ΦN for μ > 0 is uniformly bounded, too, and we have 2π 2 2 2 2 μ R (μ, A)ΦN (u) = TN (σ)μ R (μ, A)Bu(σ)dσ ε

≤ MΦ MHY uL22π (U ) .

(3.2)

Invoking (3.1), an elementary equicontinuity argument and vector-valued dominated convergence theorem (in X), respectively, we obtain 2π X − lim

N →∞

2π TN (τ )μ R (μ, A)Bu(τ )dτ = 2

T(τ )μ2 R2 (μ, A)Bu(τ )dτ,

2

ε

ε

which implies via (3.2) that for all μ > 0 2π 2π T(τ )μ2 R2 (μ, A)Bu(τ )dτ ≤ sup TN (τ )μ2 R2 (μ, A)Bu(τ )dτ N ε

X

ε

≤ MΦ MHY uL22π (U ) .

X

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On the other hand, we have 2π T(τ )μ2 R2 (μ, A)Bu(τ )dτ ε

X

IEOT

2π 2 2 = μ R (μ, A) T(τ )Bu(τ )dτ ε

X

2π

and ε T(τ )Bu(τ )dτ takes its value in X due to the fact that the considered input u(t) is a step function. Passing to the limit (i.e., μ → +∞) in the above inequality, we deduce 2π T(τ )Bu(τ )dτ ≤ MΦ MHY uL2 (U ) . (3.3) 2π ε

X

The same argument also shows that ε T(τ )Bu(τ )dτ ≤ M MHY uχ[ε,ε ] L2 (U ) . 2π ε

X

2π

Thus, the sequence ( ε T(τ )Bu(τ )dτ )ε ) is a Cauchy sequence in X. 2π 2π Moreover, X−1 −limε→0 ε T(τ )Bu(τ )dτ = 0 T(τ )Bu(τ )dτ ∈ X. By 2π virtue of X → X−1 , and the fact that ε T(τ )Bu(τ )dτ satisﬁes (3.3), we ﬁnd 2π T(τ )Bu(τ )dτ ≤ M MHY uL2 (U ) , 2π X

0

for any step function u : [0, 2π] → U with compact support that does not contain zero. This shows that B is L2 -admissible for A by a density argument. Part (ii). Assumes that B is L2 -admissible for A. By definition of ΦN , for any u ∈ L22π (U ), we have 2π ΦN (u) = TN (σ)Bu(σ)dσ 0

1 = 2π 1 = 2π

2π N

eiλσ R(iλ, A)Bu(σ)dλdσ

0 −N

N e

iλσ

−N

2π R(iλ, A)Bu(σ)dσdλ 0

N 2π∞ 1 = e−iλt T(t)eiλσ Bu(σ)dtdσdλ 2π −N 0

0

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Admissibility of Control Operators in UMD Spaces 1 = 2π

2π∞ N 0

=

1 2π

461

eiλ(σ−t) T(t)Bu(σ)dλdtdσ

0 −N

∞ 2π 0

0

e−iN (σ−t) − eiN (σ−t) T(t)Bu(σ)dσdt. i(t − σ)

The use of Fubini’s theorem in this chain of equalities is justiﬁed by the fact that the maps (λ, σ) → eiλσ R(iλ, A)Bu(σ), and (t, λ, σ) → eiλ(σ−t) T(t)Bu(σ) belong to L1 ([0, 2π]×[−N, N ], X), and to L1 ([0, ∞)×[0, 2π]×[−N, N ], X−1 ), respectively. On the other hand, we have for t-a.e 1 ΦN (u) = 2πi ⎡

∞

=

−

∞ 0

1 2i

⎞

ε0 |t−σ|≥ε

∞

e−iN (σ−t) − eiN (σ−t) ⎟ Bu(σ)χ[0,2π] (σ)dσdt⎠ , (t − σ)

⎛

⎜ eiN t T(t)B ⎝ lim

⎞ −iN σ

e

ε0 |t−σ|≥ε

0

1 2πi

1 2i

⎜ T(t) ⎝ lim

0

⎢ 1 =⎣ 2πi

−

⎛

∞

⎛

⎜ e−iN t T(t)B ⎝U − lim

u(σ)χ[0,2π] (σ) ⎟ dσdt⎠ , t−σ ⎞⎤ iN σ

e

ε0 |t−σ|≥ε

0

u(σ)χ[0,2π] (σ) ⎟⎥ dσdt⎠⎦ , t−σ

T(t)BeiN t H(e−iN · u(·)χ[0,2π] (·))(t)dt

∞

T(t)Be−iN t H(eiN · u(·)χ[0,2π] (·))(t)dt,

0

where the limits as ε 0 in the ﬁrst and the second inequalities are taking in X−1 and U , respectively. The limits as ε 0 in the second equality exist in Lp (R+ , U )(1 < p < ∞) since U is a UMD-space and B ∈ L(U, X−1 ). Since the Hilbert transform H is bounded on L2 (R, U ), the inputs: u1 (·) = e−iN · H(eiN · u(·)χ[0,2π] (·))

and u2 (·) = eiN · H(e−iN · u(·)χ[0,2π] (·))

are in L2 (R+ , U ). Moreover, we have u1 L2 (R+ ,U ) ≤ C(2)uL22π (U ) and u2 L2 (R+ ,U ) ≤ C(2)uL22π (U ) where C(2) is the norm of the Hilbert transform on L2 (R, U ). Finally, using the L2 -admissibility (see Theorem 3.1) of B, we ﬁnd ΦN (u)X ≤ Thus the proof is done.

M C(2) uL22π (U ) . 2

Note that the extension of this result to the Lp -admissibility with p ∈ (1, ∞) is possible due to the boundedness of the Hilbert transform on Lp (R, U ).

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Lemma 3.3. Assume that A satisfies (H). The the sequence TN given by (2.3) obeys the following evolution property:

t

DN (t + t − τ )T(τ )dτ,

TN (t)T(t ) = TN (t + t ) −

(3.4)

0

where DN is the Dirichlet kernel given by sin(N t) πt Proof. By definition of TN , we have

(t ∈ R).

DN (t) =

1 TN (t)T(t ) = 2π

1 = 2π

=

1 2π

1 = 2π

+N

(3.5)

eiλt R(iλ, A)T(t )dλ

−N

+N e

iλt

−N

∞

e−iλτ T(t + τ )dτ dλdτ

0

+N∞

eiλ(t−τ ) T(t + τ )dτ dλ

−N 0

+N e

iλ(t+τ )

∞

e−iλτ T(τ )dτ dλ

t

−N

1 = TN (t + t ) − 2π

+Nt

eiλ(t+t −τ ) T(τ )dτ dλ.

−N 0

Fubini’s theorem yields

+Nt

e

iλ(t+t −τ )

t T(τ )dτ dλ =

−N 0

N T(τ )(

eiλ(t+t −τ ) dλ)dτ

−N

0

t sin(N (t + t − τ )) T(τ )dτ . =2 t + t − τ 0

Thus the formula (3.4) is proved.

The following proposition shows that the UMD-property in Proposition 3.2 is essential. In other words, Proposition 3.2 does not hold in general Banach control spaces. This sharpens the example given in [9], where it was shown that the UMD-property in this setting is essential. Proposition 3.4. Let T(t) be the shift group on L1 (R) and B = IdL1 (R) (i.e. U = L1 (R)). Then, there exists u ∈ L2 (R+ , L1 (R)) such that ΦN (u) tends to infinity as N → ∞.

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Proof. The left shift group (T(t))t∈R is deﬁned by (T(t)f )(x) := f (t + x) (f ∈ L1 (R), x ∈ R, t ∈ R). 1,1 (R). Since T(t)t∈R Its generator A is given by Af := df ds for f ∈ D(A) = W is a bounded C0 -group, the spectrum of A is a subset of iR. Now let us consider the input u∗ (t) = T(2π − t)f χ[1,π] , for f ∈ L1 (R). Then, u∗ ∈ L2 (R+ , X), and relation (3.4) we obtain

2π ΦN (u ) = TN (τ )u∗ (τ )dτ ∗

0

π TN (τ )T(2π − τ )f dτ

= 1

π = (π − 1)TN (2π)f − 1

⎛ 2π ⎞ ⎝ DN (τ )T(2π − τ )f dτ ⎠ dt. t

2π By virtue of Riemann–Lebesgue’s theorem, we have limN →∞ t DN (τ )T 2π (2π − τ )f dτ = 0 for all t ∈ [1, π]. Moreover ( t DN (τ )T(2π − τ )f dτ )N is uniformly bounded on [1, π]. So, by dominated convergence theorem we obtain ⎛ ⎞ π 2π ⎝ DN (τ )T(2π − τ )f dτ ⎠ dt = 0. lim N →∞

1

t

In Prop. 6 in [9], it has been proved that there exists a function f ∈ L1 (R) such that TN (2π)f does not converge as N → ∞. This completes the proof. Remark 3.5. (i) Note that for U = X(:= L∞ (R)), and for all B ∈ L(X), ΦN (u) converges for any u ∈ L22π (U ) (since in L∞ (R) every C0 -semigroup is generated by a bounded operator (see [24, p. 54]), and Banach–Steinhaus’s theorem implies that ΦN is uniformly bounded on the space L(L2 ([0, 2π], L∞ (R)), L∞ (R)), but L∞ (R) is not a UMDspace. (ii) From the Proposition 3.2, we rediscover the result of [9] on the validity of the (complex) inverse Laplace transform for C0 -semigroups in UMD-Banach spaces. Indeed, from Proposition 3.2, we deduce that in UMD-space X, TN (t) is always uniformly bounded. From [17], we known that TN (t)x converges in D(A). Thus, for all x ∈ X, we have that TN (t)x converges from a bounded subinterval of (0, ∞) and its limit is exactly T(t)x. This can be obtained by considering the control operator B = IdX , which is L2 -admissible for A, and hence ΦN is uniformly bounded on L22π (X). Indeed, for t0 > 0 and x ∈ X, consider the input u0 (t) := T(t0 − t)xχ[t0 /2,t0 ] . Regarding carefully the proof of Proposition 3.4, we get ΦN (u0 ) = t20 TN (t0 )x + L N (t0 )x, where L N (t0 ) constitute a family of bounded linear operators on X, and L N (t0 )x converges

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to zero as N tends to ∞; hence it is uniformly bounded accordingly to Banach–Steinhaus’s theorem. Moreover, ΦN (u0 ) converges in X to t0 2 T(t0 )x accordingly to Corollary 4.2, which implies that TN (t0 )x converges to T(t0 )x. (iii) Maybe it is possible to replace L1 (R) by L2 (R, Y ) in Proposition 3.4 whenever Y is a non-UMD Banach space, so the validity of the equivalence in Proposition 3.2 would somehow characterize UMD. Remark 3.6. Let A generate a bounded C0 -semigroup (not necessarily exponentially stable) T(t). Let Y be another Banach space and let C : D(A) → Y be a linear operator deﬁned on the domain of A. Assume that C is continuous with respect to the norm x1 = (I − A)−1 x on D(A). By definition, C is a ﬁnite time L2 -admissible observation operator if, for some (and hence every) τ > 0, there is a constant Mτ > 0 such that τ CT(t)x2Y dt ≤ Mτ x2X

x ∈ D(A).

0

Again, this notion is invariant under scaling e−α· T of the semigroup T and, for T exponentially stable, it is equivalent to inﬁnite-time L2 -admissibility (henceforth called L2 -admissibility for short), i.e., to the existence of M > 0 satisfying ∞ CT(t)x2Y dt ≤ M x2X

x ∈ D(A).

(3.6)

0

The problem of whether an observation operator C is L2 -admissible for A has received much attention recently. For wide information on this topic, we refer the reader to the excellent survey [18] and the references therein. If X is reﬂexive, the notion of ﬁnite-time L2 -admissible observation operators is, to a great extent, dual to the notion of L2 -admissible control operators, see e.g., [30] or [25]. As UMD-spaces are reﬂexive [6], and if the state space X is also reﬂexive (but not necessarily a UMD-space), then all the above results have dual versions. For Hilbert control spaces, and without recourse to the Hilbert transform explicitly, the following result constitutes a short proof of the Proposition 3.2. Proposition 3.7. Let X be a reflexive Banach space and U be a Hilbert space. Then B is L2 -admissible for A if (and only if ) ΦN is uniformly bounded on L(L22π (U ), X). Proof. By definition of ΦN , we have 2π TN (σ)Bu(σ)dσ . sup ΦN = u L2 (U ) =1 2π 0

X

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Let u ∈ L22π (U ) and ϕ ∈ X ∗ , then we have 2π 2π TN (σ)Bu(σ)dσ, ϕ = TN (σ)Bu(σ), ϕ dσ 0

0

2π =

∗ u(σ), (TN (σ)B) ϕ dσ

0

1 = √ 2π

2π

u(σ), FU ∗ [(R(i·, A)B)∗ χ[−N,N ] ϕ](σ) dσ.

0

It follows that ΦN =

2π sup TN (σ)Bu(σ)dσ, ϕ =1,ϕX ∗ =1

uL2 (U )) 2π

1 = √ 2π

0

sup

2π u(σ), FU ∗ [(R(i·, A)B)∗ ϕχ[−N,N ] (σ)] dσ

ϕX ∗ =1,uL2 (U ) =1 2π 0

1 = √ sup 2π ϕX ∗ =1 1 sup = √ 2π ϕX ∗ =1

2π FU ∗ [(R(i·, A)B)∗ ϕχ[−N,N ] ](σ)2 dσ, (3) 0

N

(R(iλ, A)B)∗ ϕ2 dλ,

(4)

−N

where in the passage from equality (3) to equality (4) we have used the vector-valued Fourier–Plancherel’s theorem for the Hilbert space-valued Fourier transform. On the other hand, it is well-known that if B is an L2 -admissible control operator for A, then its adjoint B ∗ is an L2 -admissible observation operator for A∗ (see e.g., [30]), which implies, once ∞ again, via vector-valued Fourier–Plancherel’s theorem and (3.6), that −∞ (R(iλ, A)B)∗ ϕ2 dλ is ﬁnite for all ϕ ∈ X ∗ . This ﬁnishes the proof.

4. From the Weak Admissibility to the Inverse Laplace Transform The action of the operator families ΦN on a certain dense subspace of L22π (U ) has certain properties we will now exploit. We will, therefore, try to relate the weak L2 -admissibility of control operators and the inverse Laplace transform for the semigroups on general Banach spaces. Proposition 4.1. Let X, U be Banach spaces and τ ∈ [0, 2π] and v ∈ U . Then ΦN (vχ[0,τ ] ) converges in X if B is weakly L2 -admissible.

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Proof. Applying ΦN to the input u = vχ[0,τ ] , and using Fubini’s theorem as in Proposition 3.2, we obtain 1 ΦN (u) = 2π

τ N

eiλσ R(iλ, A)Bvdλdσ

0 −N

1 = 2π 1 = 2π

τ N ∞ 0 −N 0

∞

1 π

∞ T(t)B

R

0

∞ =

τ

T(t)B 0

=

eiλ(σ−t) T(t)Bvdtdλdσ

0

eiN λ(σ−t) − e−iN λ(σ−t) vdσdt i(σ − t)

sin(N (σ − t)) vχ[0,τ ] (σ)dσdt σ−t

T(t)BvuτN (t)dt,

0

where the new scalar input uτN is given by sin(N (σ − t)) uτN (t) := χ[0,τ ] (σ)dσ π(σ − t) R

= (DN ∗ χ[0,τ ] )(t), where (∗) denotes the convolution product and DN denotes the Dirichlet kernel in (3.5). The scalar control sequence (uτN ) deﬁned above is, of course, in L2 [0, 2π]. Due to the weak L2 -admissibility of B (see Theorem 3.1 (v)), it therefore sufﬁces to show that uτN converges in L2 [0, 2π]. More generally, for f ∈ L2 (R) it is well known that limN →∞ (DN ∗ f ) = f in L2 (R). This follows from (DN ∗ f )(t) = (F(χ[−N,N ] ) ∗ f )(t), combined with the Plancherel’s theorem. Thus ΦN (u) converges in X as N → ∞, and we have τ lim ΦN (vχ[0,τ ] ) = T(t)Bvdt ∈ X. N →∞

0

Motivated by this result, one wonders if the convergence in Proposition 4.1 is sufﬁcient to determine the weak L2 -admissibility of a control operator. This converse result fails, as is established in Corollary 4.7. Combining Proposition 3.2 and Corollary 4.1, we get: Corollary 4.2. Let U be a U M D-space. Then B is L2 -admissible for A if (and only if) ΦN (u) converges in X for any u ∈ L22π (U ) and its limit is Φ(u).

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Proof. The ‘if ’ part. By definition, we have ΦN ∈ L(L22π (U ), X). Since ΦN (u) converges in X for all u ∈ L2 ([0, 2π], U ), Banach–Steinhaus’s theorem shows that ΦN is uniformly bounded on L(L22π (U ), X). According to Proposition 3.2 (i), B is L2 -admissible for A. The ‘only if’ part. Assume that B is L2 -admissible for A. By virtue of Proposition 3.2 (ii) ΦN is uniformly bounded on L(L22π (U ), X). Since the step functions are dense in L22π (U ), it sufﬁces to show that ΦN (u) converges in X for any step functions on [0, 2π] given by: u = u1 χ[t0 ,t1 ] +

n

ul χ[tl ,tl+1 ] ,

l=1

where ul ∈ U and for 0 = t0 < t1 < t2 · · · < tl · · · < tn < 2π. By linearity of ΦN and the fact that for 0 < τ1 < τ2 we have ΦN (vχ[τ1 ,τ2 ] ) = ΦN (vχ[0,τ1 ] ) − ΦN (vχ[0,τ2 ] ). Therefore it sufﬁces to show that ΦN (vχ[0,τ ] ) converges in X as N → ∞ for all τ > 0. Note that if X is a UMD space, then by Lemma 2.2 ΦN (vχ[τ1 ,τ2 ] ) converges in X for any 0 < τ1 < τ2 and v ∈ U , while the convergence of ΦN (vχ[0,τ2 ] ) is not guaranteed at this time. But this is true, since B is weakly L2 -admissible according to Proposition 4.1. Now let us show that its limit is exactly Φ(u). Uniform boundedness theorem implies that there is K > 0 such that ΦN L(L2 ([0,2π],U ),X) ≤ K. Let u ∈ L2 ([0, 2π], U ) and K = max(K, M ). Given ε > 0, we ﬁrst choose δ > 0 sufﬁciently small such that ΦN (uχ[0,δ] )X ≤ K uχ[0,δ] L2 ([0,2π],U ) ≤ ε/3 Φ(uχ[0,δ] )X−1 ≤ K uχ[0,δ] L2 ([0,2π],U ) ≤ ε/3. where χE is the characteristic function of the set E. The fact that ΦN (u) = ΦN (uχ[0,δ] ) + ΦN (uχ[δ,2π] ) for every δ > 0, we obtain ΦN (u) − Φ(u)X−1 = ΦN (uχ[δ,2π] ) − Φ(uχ[δ,2π] ) + ΦN (uχ[0,δ] ) − Φ(uχ[0,δ] )X−1 ≤ ΦN (uχ[δ,2π] ) − Φ(uχ[δ,2π] )X−1 + ΦN (uχ[0,δ] ) − Φ(uχ[0,δ] )X−1 ≤ ΦN (uχ[δ,2π] ) − Φ(uχ[δ,2π] )X−1 + ΦN (uχ[0,δ] )X−1 + Φ(uχ[0,δ] )X−1 The dominated convergence theorem shows that we can choose δ sufﬁciently small that the ﬁrst norm on the right of the above inequality is less than ε/3 and the above inequality gives ΦN (u) → Φ(u) as N → ∞ in X−1 . By assumption, we have ΦN (u) converges in X as N → ∞ for any u ∈ L2 ([0, 2π], U ). By virtue of X ⊂ X−1 with continuous injection and uniqueness of the limit, we deduce that ΦN (u) converges to Φ(u) in X as N → ∞ for any u ∈ L2 ([0, 2π], U ). This completes the proof. Corollary 4.3. Let B ∈ L(U, X−1 ). Then the operator B is weakly L2 admissible if (and only if ) ΦN (ζ · u) converges in X for any ζ ∈ L2 [0, 2π] and u ∈ U.

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Proof. Thanks to Corollary 4.2, the proof follows immediately by considering the input elements bu := Bu ∈ X−1 for u ∈ U , and the fact that the associated control space (i.e. U = C) is a UMD space. For the convergence of ΦN (vχ[0,τ ] ), we have the following equivalent condition in terms of the resolvent of the generator of the semigroup T. Proposition 4.4. Let X and U be Banach spaces and τ ∈ [0, 2π]. Then ΦN (vχ[0,τ ] ) converges in X for all v ∈ U if (and only if ) the integral N iλτ (e − 1)R(iλ, A)A−1 Bvdλ converges in X. −N Proof. Let v ∈ U and τ ∈ [0, 2π]. Fubini’s theorem yields 1 ΦN (vχ[0,τ ] ) = 2π 1 = 2π

τ N

eiλσ R(iλ, A)Bvdλdσ

0 −N

N −N

eiλτ − 1 R(iλ, A)Bvdλ. iλ

By the resolvent equation, we have R(iλ, A)Bv = −A−1 Bv + iλR(iλ, A)A−1 Bv,

λ ∈ R.

It follows that 1 ΦN (vχ[0,τ ] ) = − 2π 1 + 2π

N

eiλτ − 1 −1 A Bvdλ iλ

−N

N

(eiλτ − 1)R(iλ, A)A−1 Bvdλ.

(4.1)

−N

Now, we are going to study the convergence of the two integrals on the lefthand side of the above equality separately. For the ﬁrst integral we have 1 2π

N −N

eiλτ − 1 −1 1 A Bv = iλ 2π =

1 2π

F(χ[0,τ ] )(λ)χ[−N,N ] (λ)dλA−1 Bv

R

τ

F((χ[−N,N ] )(λ)dλA−1 Bv

0

= ∞

1 π

N τ

sin(λ) dλA−1 Bv. λ

0

Since A−1 Bv ∈ X and 0 sin(λ) λ dλ converges, then the ﬁrst integral converges in X. Finally we conclude that ΦN (vχ[0,τ ] ) converges in X if and only N if −N (eiλτ − 1)R(iλ, A)A−1 Bvdλ converges in X.

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Corollary 4.5. Let X and U be Banach spaces, τ ≥ 0 and v ∈ U . Then the following assertions hold: N (a) If −N eiλτ R(iλ, A)A−1 Bvdλ converges in X then ΦN (vχ[0,τ ] ) converges in X. (b) Let X be a UMD-space and ΦN (vχ[0,τ ] ) converges in X then N iλτ e R(iλ, A)A−1 Bvdλ converges in X. −N N Proof. Part (a). Assume that −N eiλτ R(iλ, A)A−1 Bvdλ converges in X N for all τ ≥ 0 ( in particular for τ = 0), it follows that −N (eiλτ − 1) R(iλ, A)A−1 Bvdλ converges in X (N → ∞). Finally, by virtue of Proposition 4.4, ΦN (vχ[0,τ ] ) converges in X. Part (b). Assume that ΦN (vχ[0,τ ] ) converges in X. Proposition 4.4 N implies that −N (eiλτ − 1)R(iλ, A)A−1 Bvdλ converges in X. By the fact N that X is a UMD-space, Theorem 2.1 shows that −N eiλτ R(iλ, A)A−1 Bvdλ always converges in X for all τ > 0 (Range(A−1 B) ⊂ X). Combining the N above facts, we conclude that −N R(iλ, A)A−1 Bvdλ converges in X, which N implies that −N eiλτ R(iλ, A)A−1 Bvdλ converges in X for all τ ≥ 0. As promised, we are now in a position to show that the convergence of ΦN (vχ[0,τ ] ) is not sufﬁcient for the weak L2 -admissibility of the control operator B. Before we give the proof, let us state the following remark. Remark 4.6. Let A be a generator of a C0 -semigroup on a Hilbert space X. Using real interpolation (see, e.g., [7] or [26]), we deﬁne X− 12 ,∞ = [X, X−1 ] 12 ,∞ In [26], the space X− 12 ,∞ was characterized as follows: K −1 X− 12 ,∞ = x ∈ X−1 , s.t. ∃K, α ≥ 0, ≥ 0, (λI − A) x ≤ √ , ∀λ > α . λ Furthermore, a necessary condition for B to be weakly L2 -admissible is that the input element bu = Bu satisﬁes the resolvent estimate (1.2) for all u ∈ U , which implies that Range(B) ⊆ X− 12 ,∞ .

(4.2)

Moreover, it has been proved in [28] that we have equality in (4.2) if the semigroup is normal and analytic, and we do not have equality in general. An analogous result has been recently proved in Proposition 4.2 in [15] in some general cases. Corollary 4.7. Let X be a Hilbert space and A be the generator of an analytic C0 -semigroup on X. Then they are B ∈ L(U, X−1 ) which are not weaklyL2 -admissible, and for which ΦN (uχ[0,τ ] ) converges in X for any τ ≥ 0 and u ∈ U. Proof. There exists B ∈ L(U, X−1 ) which is not weakly-L2 -admissible for A; this follows from Remark 4.6 and the fact that X− 12 ,∞ ⊂ X−1 is strict.

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Now, letting A generate an analytic C0 -semigroup on X, appealing to Corollary 2.3 (ii) and Proposition 4.4,and the fact that Hilbert spaces are UMD, τ we deduce that ΦN (vχ[0,τ ] ) = 0 TN (t)Bvdt converges in X (notice that Range(A−1 B) ⊂ X) for all v ∈ U . As a curious result, we gain a relationship between the notion of the weak L2 -admissibility and the validity of the inverse Laplace transform in general Banach spaces. In particular, Proposition 4.8 is an extension of the [17] result stated in the beginning of Sect. 2. Proposition 4.8. Let A be a generator of a C0 -semigroup T on X. Let U be a another Banach space and B ∈ L(U, X−1 ) be a weakly-L2 -admissible operator for A. Then for all τ ≥ 0 and x ∈ Range(A−1 B), the integral N iλτ e R(iλ, A)xdλ converges in X, and we have −N 1 N →∞ 2π

N

X − lim

eiλτ R(iλ, A)xdλ =

−N

T(τ )x x 2

for for

τ >0 τ = 0.

Proof. Let τ ≥ 0 and v ∈ U . Since B is a weakly-L2 -admissible control operN ator, Propositions 4.1 and 4.4 imply that −N (eiλτ − 1)R(iλ, A)A−1 Bvdλ converges in X. Thus, to complete the proof, we must prove only that N R(iλ, A)A−1 Bvdλ converges in X. Thanks to Corollary 4.3, ΦN (e·iα −N χ[0,τ ] v) converges in X for all α ∈ R. By virtue of Proposition 4.4 and making a straightforward computation, this is equivalent to the converN gence of −N (ei(λ+α)τ − 1)R(iλ, A)R(−iα, A)Bvdλ in X. Now, if α is chosen N in such a way that ατ = (2k + 1)π, then we deduce that −N (−eiλτ − N 1)R(iλ, A)R(−iα, A)Bvdλ converges in X. This implies that −N (−eiλτ − 1)R(iλ, A)A−1 Bvdλ also converges in X according to resolvent identity, and N hence −N R(iλ, A)A−1 Bvdλ converges in X (N → ∞). N iλτ 1 e R(iλ, A)xdλ. From identity Putting I(λ) := X − limN →∞ 2π −N τ A−1 Bv (4.1), we obtain 0 T(t)Bvdτ = − 2 + I(τ ) − I(0), for all τ > 0. Comt bining the famous semigroup identity T(t)x − x = 0 T(s)Axds for x ∈ D(A) −1 with the above equality yields: I(τ ) − T(τ )A−1 Bv = I(0) − A 2 Bv . Hence, it −1 remains only to show that I(0) = A 2 Bv . But for I(0), we have 1 I(0) = X − lim N →∞ 2π

N

R(iλ, A)A−1 Bvdλ

−N

1 = X−1 − lim N →∞ 2π

N

R(iλ, A)A−1 Bvdλ,

(since A−1 Bv ∈ X).

−N

N Thanks to (2.1), the integral −N R(iλ, A)A−1 Bvdλ valued in X−1 converges N in X−1 to πA−1 Bv. As −N R(iλ, A)A−1 Bvdλ converges also in X and as X → X−1 , we obtain I(0) =

A−1 Bv . 2

Thus the proof is done.

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Acknowledgements Our special thanks to the referees for their suggestions and comments for improvements. The referees suggested Remark 3.5 (iii).

References [1] Amann, H.: Linear and quasilinear parabolic problems. In: Abstract Linear Theory, vol. 1. Birkh¨ auser, Basel (1995) [2] Arendt, W.: Vector-valued Laplace transform and Cauchy problems. Isr. J. Math. 59, 327–352 (1987) [3] Arendt, W., Batty, C., Hieber, C., Neubrander, F.: Vector valued Laplace transforms and Cauchy problems. In: Monographs in Mathematics, vol. 96. Birkh¨ auser, Basel (2001) [4] Bounit, H., Driouich, A., El-Mennaoui, O.: A direct approach to the Weiss conjecture for analytic semigroups. Czechoslovak Math. J. 60(2), 527–539 (2010) [5] Bounit, H., Driouich, A., El-Mennaoui, O.: A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups. Semigroup Forum (under revision) [6] Burkholder, D.L.: Martingales and singular intergrals in Banach spaces. In: Handbook of the Geometry of Banach Spaces, vol. I, pp. 233–269. NorthHolland, Amsterdam (2001) [7] Butzer, P.L., Berens, H.: Semigroups of operators and approximations. Die Grundlehren der math. Wiss. 145. Springer-Verlag, York (1967) [8] Curtain, R.F., Zwart, H.: An introduction to inﬁnite-dimensional linear systems theory. In: Texts in Applied Mathematics, vol. 21. Springer-Verlag, New York (1995) [9] Driouich, A., El-Mennaoui, O.: On the inverse Laplace transform for C0 -semigroups in UMD-spaces. Arch. Math. 72, 56–63 (1999) [10] Diestel, J., Uhl, J.J.: Vector Measures. Math. Surveys, no. 15. Amer. Math. Soc., Providence (1977) [11] Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, New York (2000) [12] Emirsajlow, Z., Townley, S.: From PDE with a boundary control to the abstract state equation with unbounded control operator: a tutorial. Eur. J. Control 7(1), 1–23 (2000) [13] Grabowski, P.: Admissibility of observation functionals. Int. J. Control 62, 1163–1173 (1995) [14] Greiner, G., Nagel, R.: On the stability of strongly continuous semigroups of positive operators on L2 (μ). Annali Della Scuola Normale Superiore di Pisa Classe di Scienze Sr. 4 10(2), 257–262 (1983) [15] Haak, B.H., Haase, M., Kunstmann, P.C.: Perturbation, interpolation, and maximal regularity. Adv. Differ. Equ. 11(2), 201–240 (2006) [16] Hansen, S., Weiss, G.: The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on L2 . Syst. Control Lett. 16, 219–227 (1991) [17] Hille, E., Philllips, R.S.: Functional Analysis and Semigroups. Amer. Math. Soc. Transl. Ser. 2 (1957)

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[18] Jacob, B., Partington, J.R.: Admissibility of control and observation operators for semigroups: a survey. In: Ball, J.A., Helton, J.W., Klaus, M., Rodman, L. (eds.) Current Trends in Operator Theory and its Applications. Proceedings of the IWOTA 2002. Operator Theory: Advances and Applications, vol. 149, pp. 199–221. Birkh¨ auser Verlag [19] Jacob, B., Partington, J.R.: The Weiss conjecture on admissibility of observation operators for contraction semigroups. Integr. Equ. Oper. Theory 40, 231– 243 (2001) [20] Jacob, B., Partington, J.R.: Admissibility of control and observation operators for semigroups: a survey. In: Current Trends in Operator Theory and its Applications, vol. 149 of Oper. Theory Adv. Appl., pp. 199–221. Birkh¨ auser, Basel (2004) [21] Jacob, B., Zwart, H.: Counterexamples concerning observation operators for C0 -semigroups. SIAM J. Control Optim. 43(1), 137–153 (2004) [22] Le Merdy, C.: The Weiss conjecture for bounded analytic semigroups. J. Lond. Math. Soc. (2) 67(3), 715–738 (2003) [23] Nagel, R.: One-Parameter Semigroups of Positive Operators. LNM 1184, Berlin-Heidelberg-New York (1985) [24] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin (1983) [25] Staffans, O.J.: Well-posed linear systems. In: Encyclopedia of Mathematics and its Applications, vol. 103. Cambridge University Press, Cambridge (2005) [26] Triebel, H.: Interpolation Theory, Functions Spaces, Differential Operators. North-Holland, Amsterdam (1978) [27] Yao, P.F.: On the inversion of the Laplace transform of C0 -semigroups and its applications. SIAM J. Math. Anal. 26(5), 1331–1341 (1995) [28] Weiss, G.: Two conjectures on the admissibility of control operators. In: Desch, W., Kappel, F. (eds.) Estimation and Control of Distributed Parameter Systems, pp. 367–378. Birkh¨ auser Verlag (1991) [29] Weiss, G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27, 527–545 (1989) [30] Weiss, G.: Admissibile observation operators for linear semigroups. Isr. J. Math. 65, 17–43 (1989) H. Bounit (B), A. Driouich and O. El-Mennaoui D´epartement de Math´ematiques Facult´e des Sciences Universit´e Ibnou Zohr BP 8106 Hay Dakhla 80000 Agadir Morocco e-mail: [email protected]; [email protected]; [email protected] Received: November 21, 2009. Revised: October 22, 2010.

Integr. Equ. Oper. Theory 68 (2010), 473–485 DOI 10.1007/s00020-010-1818-3 Published online July 13, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

A Reflexivity Result Concerning Banach Space Operators with a Multiply Connected Spectrum Onur Yavuz Abstract. We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λj with radius rj for j = 1, . . . , n from the unit disk. We assume that T is a bounded linear operator on a separable reﬂexive Banach space whose spectrum contains ∂Ω and does not contain the points λ1 , λ2 , . . . , λn , and the operators T and rj (T − λj I)−1 are polynomially bounded. Then either T has a nontrivial hyperinvariant subspace or the WOT-closure of the algebra {f (T ) : f is a rational function with poles oﬀ Ω} is reﬂexive. Mathematics Subject Classification (2010). Primary 47A15; Secondary 47A60. Keywords. Invariant subspaces, reﬂexive operator algebras, polynomially bounded operators, functional calculus.

1. Introduction In 1978 Scott Brown proved that every subnormal operator deﬁned on a Hilbert space has a nontrivial invariant subspace [3]. Later, using the Scott Brown technique, Brown, Chevreau, and Pearcy showed that a Hilbert space contraction whose spectrum contains the unit circle has a nontrivial invariant subspace [5]. Ambrozie and M¨ uller generalized this result to Banach space operators in the sense that the adjoint of a polynomially bounded Banach space operator whose spectrum contains the unit circle has a nontrivial invariant subspace [1]. When the corresponding Banach space is reﬂexive the operator itself has a nontrivial invariant subspace. In [12] this result is extended to operators with a multiply connected spectrum. Once the existence of a nontrivial invariant subspace for a certain operator is proved, it is natural to ask how rich the invariant subspace lattice of the operator is, or stated formally, whether the operator is reﬂexive or not. Recall that an algebra of operators, A is said to be reﬂexive if A = Alg Lat(A) where Alg Lat(A) is the set of operators S such that Lat(T ) ⊂ Lat(S) for

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every T ∈ A. An operator T is said to be reﬂexive if the closure of the algebra generated by T in the weak operator topology is reﬂexive. In 1980, Olin and Thompson proved that every subnormal operator is reﬂexive [9]. In [6] it was proved that every Hilbert space contraction whose spectrum contains the unit circle is either reﬂexive or has a hyperinvariant subspace. The main technique used in the proofs mentioned above is showing the existence of full analytic invariant subspaces, a concept which was introduced in [9] and generalized in [4], where the authors show that every contraction with isometric calculus is reﬂexive. Recently, Rejasse has proved that if T is a polynomially bounded operator deﬁned on a reﬂexive Banach space, then either it has a nontrivial hyperinvariant subspace or it is reﬂexive [11], which is the reﬂexivity result associated with [1]. In this paper we will extend this result to operators with a multiply connected spectrum. Let X be a separable, reﬂexive complex Banach space. Let L(X) denote the algebra of all bounded linear operators on X. We will write D for the unit disk and T for the unit circle. We consider a multiply connected domain deﬁned as follows. Let λ1 , λ2 , . . . , λn be points in D and r1 , r2 , . . . , rn be positive numbers such that the closures of the open disks centered at λj with radius rj , which we will denote by B(λj , rj ) are contained in D and are pairwise disjoint. Let Cj be the boundary of B(λj , rj ) and Uj be the unbounded component of C \ Cj for j = 1, 2, . . . , n. We will write C0 for T, U0 for D, and λ0 for 0 when convenient. The multiply connected domain bounded by the circles (Cj )nj=0 will be called Ω. Let us write βj (λ) = rj (λj − λ)−1 for j = 1, 2, . . . , n and β0 (λ) = λ. Assume that T is a bounded linear operator on X such that the operators βj (T ) are polynomially bounded for j = 0, 1, 2, . . . , n. Recall that a bounded linear operator T deﬁned on a complex Banach space X, is said to be polynomially bounded if there exists a constant K > 0 such that p(T ) ≤ K sup{|p(λ)| : |λ| ≤ 1} for all polynomials p, and the smallest such constant is called the polynomial bound of T . Assume also that σ(T ) contains ∂Ω and does not contain the points λj for j = 0, 1, . . . , n. We will denote by R(Ω) the algebra of all rational functions with poles oﬀ Ω, and by WTΩ the closure of the algebra {f (T ) : f ∈ R(Ω)} in the weak operator topology. We will say a closed subspace M of X is a rationally invariant subspace for T if M is invariant for the algebra WΩ T . In [12] it was proved that there exists a nontrivial rationally invariant subspace for T . The purpose of this paper is to prove the following theorem: Theorem 1.1. Let T be a bounded linear operator on X whose spectrum contains ∂Ω and does not contain the points λ1 , λ2 , . . . , λn , and such that the operators T and βj (T ) for j = 1, 2, . . . , n are polynomially bounded. Then either the operator T has a hyperinvariant subspace or the operator algebra WΩ T is reflexive. In the following section we summarize some relevant results from [12]. In Sect. 3, we give some factorization results. In Sect. 4 we discuss full analytic rationally invariant subspaces in relevance to reﬂexivity in our context, and in the last section we prove our main result.

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2. Preliminaries In this section we will review some results from [12] which will be used in the proof of our main theorem. The following definition which is a modiﬁcation of Apostol set [1] was given in [12]. Definition. A subset Λ of Ω is called an Ω-Apostol set if all points on ∂Ω but countably many are radial limits of Λ. By applying the original result [2, Lemma 2.1] by Apostol to the operators βj (T ) for j = 0, 1, . . . , n, we obtain the following result. Theorem 2.1. Let T ∈ L(X) be an operator whose spectrum contains ∂Ω and does not contain the points λ1 , λ2 , . . . , λn , and such that the operators T and rj (T − λj I)−1 for j = 1, 2, . . . , n are polynomially bounded. Suppose that for some ε > 0 and k ≥ 1, the set Λε,k := {λ ∈ Ω : ∀˜ε > ε ∃ u ∈ X with u = 1 and T u − λu < ε˜(dist(λ, ∂Ω))k } is not an Ω-Apostol set. Then T has nontrivial hyperinvariant subspaces. Let us denote by A(Ω) the Banach algebra of continuous functions in Ω which are analytic on Ω with sup norm and by H ∞ (Ω) the Banach algebra of bounded analytic functions on Ω. It is well-known that H ∞ (Ω) can be identiﬁed with the corresponding Hardy space H ∞ (∂Ω) and we will use this identiﬁcation freely throughout the paper. We denote by L1 (∂Ω) the Banach space of all complex integrable functions with respect to arc length measure. From now on we will write H ∞ for H ∞ (∂Ω) or H ∞ (Ω) depending on the context and L1 for L1 (∂Ω). Note that the algebra H ∞ can be viewed as the dual of L1 /⊥ H ∞ with the following action: [f ], h = f (ζ)h(ζ)|dζ| ∂Ω ∞

1

for h ∈ H and f ∈ L . As in [12] we may assume that the functional calculus f → f (T ) extends to H ∞ and the extended functional calculus is isometric. We consider the bilinear form on H ∞ deﬁned by (x, x∗ ) → x ⊗T x∗ where x ⊗T x∗ , h = h(T )x, x∗ . By a similar discussion in [12] we can show that if for some x ∈ X and x∗ ∈ X ∗ the functional x ⊗∗T x∗ is not weak∗ continuous then T has a hyperinvariant subspace. So we may assume that these functionals are weak∗ continuous, in other words T has a weak∗ continuous functional calculus. Similarly we may assume that T ∗ has a weak∗ continuous functional calculus. If we deﬁne x ⊗T ∗ x∗ , f = x, f (T ∗ )x∗ for x ∈ X and x∗ ∈ X ∗ , then we see that x ⊗T x∗ , f = x ⊗T ∗ x∗ , f for every f ∈ H ∞ . We recall here the Ahlfors function which played a big role in [12]. Ahlfors function is an inner function deﬁned on Ω whose derivative does not vanish on the boundary. The Ahlfors function satisﬁes the properties that are stated in the following theorem. We refer the reader to [8] for the definition of the Ahlfors function and the proof of the theorem.

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Theorem 2.2. Let u be the Ahlfors function associated with Ω and p ∈ Ω. Then 1. u maps Ω onto D exactly n + 1 times, 2. u extends analytically across each Cj , and maps each Cj homeomorphically onto T, 3. u is not zero on any Cj for j = 0, 1, 2, . . . , n. We may assume that u(T )n x → 0 for every x ∈ X. Indeed, for the hyperinvariant spaces M = {x : u(T )n x → 0} for T and M∗ = {x∗ : u(T )∗n x → 0} for T ∗ , if neither M = X nor M∗ = X ∗ holds, it follows from Theorem 3.2 [A2] that u(T )∗ , and so T ∗ has hyperinvariant subspaces. Since X is reﬂexive this would also imply that T has hyperinvariant subspaces. As a consequence of the above discussions it will be enough to prove the following theorem to obtain our main result. Theorem 2.3. Let T be a bounded linear operator on X with a weak∗ continuous isometric H ∞ functional calculus which maps 1(z) ≡ z to T . Assume that the set Λk,ε is an Ω-Apostol set for every ε > 0 and k ≥ 1 and u(T )n converges to 0 in the strong operator topology. Then either the operator T has a hyperinvariant subspace or the algebra WTΩ is reflexive.

3. Factorization Let X, Y, Z be complex Banach spaces, and B : X × Y → Z a continuous bilinear application. The following definition was given in [11]. l Definition. The bilinear application B is said to have the EC,M,M property if there exist C ∈ (0, 1) and two positive constants M and M that satisfy the following: For every L ∈ Z, x ∈ X, w1 , w2 , . . . , wp ∈ Y , and δ > 0, there exits u ∈ X and v ∈ Y such that

B(u + x, v) − L ≤ CL B(u, wj ) < δ u ≤ M L1/2 v ≤ M L1/2 . The following result was proved in [11]. l Proposition 3.1. Suppose that B verifies the EC,M,M property. Let L1 , . . . , LN ∈ Z, x ∈ X, y1 , . . . , yN ∈ Y , and ρj > 0 such that

B(x, yj ) − Lj < ρj

j = 1, . . . , N.

Then there exist u ∈ X, v1 , . . . , vN ∈ Y such that N √ M k=1 ρk √ and u − x ≤ 1− α √ M ρj yj − vj ≤ √ , 1− γ B(u, vj ) = Lj for all j = 1, . . . , N .

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The proposition below is an analogue of Proposition 2.3 in [11]. Proposition 3.2. Let C, M , and M be three positive constants with C < 1. l Assume that the bilinear application B verifies the EC,M,M property. Let N be a dense subset of Z. Consider an element x ∈ X, two sequences of real numbers ( k )k≥1 and (δk )k≥1 such that ( k ) strictly decreases to 0 and √ j=0,1,...,n j=0,1,...,n in Y and (Lj,k )k≥1 in k≥1 k δk < ∞, and sequences (yj,k )k≥1 Z with Lj,k < δk for every k ≥ 1 and every j = 0, 1, . . . , n. Then there j=0,1,...,n j=0,1,...,n exist u ∈ X, two families of sequences (vj,k )k≥1 and (zj,k )k≥1 in Y , and constants C1 , C2 , C3 such that Lj,k = B(u, vj,k ) δk vj,k ≤ C1 k≥1

u − x ≤ C2

k δk

k≥1

B(u, zj,k ) ∈ N zj,k − yj,k < C3 k for every k ≥ 1 and j = 0, 1, . . . , n.

∞ Proof. Choose a sequence of positive numbers (an ) such that n=1 an ≤ 1 and for all n put αn = min{δn , a1 ε1 , . . . , an εn }. Since N is dense in Z, we can ﬁnd K1j ∈ N for every j = 0, 1, . . . , n such that B(x, y1j ) − K1j < α1 B(x, 0) −

Lj1

and we obviously have

< δ1 .

By applying the previous proposition to (K1j ), (Lj1 ), (y1j ), and qj where qj = 0 0 1 n , v11 , . . . , v11 and for all j, we can ﬁnd x1 ∈ X and sequences in Y v11 0 n z11 , . . . , z11 such that √ 2M (n + 1) δ1 x1 − x ≤ √ 1− γ √ M ε1 a1 j y1j − z11 ≤ √ 1− γ √ M δ1 j ≤ v11 √ 1− γ j ) = Lj1 B(x1 , v11

j B(x1 , z11 ) = K1j

for j = 0 . . . , n

By induction we will prove that for every k ≥ 1, there exist K1j , . . . , Kkj ∈ N, xk−1 , xk ∈ X and vil , zil ∈ Y (i = k − 1, k and l = 1, . . . , i) such that 1. B(xi , vilj ) = Ljl for i = k − 1, k and l = 1, . . . , i 2. B(xi , zilj ) = Klj for i = k√− 1, k and l = 1, . . . , i (n+1) δk √ 3. xk − xk−1 ≤ 2kM1− γ

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√ M εl ak √ for i = k − 1, k and l = 1, . . . , i 1− γ √ M εk ak j j yk − zk,k ≤ 1−√γ √ √δk vk,l − vk−1,l ≤ M 1− γ for i = k − 1, k and l = 1, . . . , i √ ∗j √δk vk,k ≤ M 1− γ

j j − zk−1,l ≤ 4. zkl

5. 6. 7.

We have already seen that the above statements hold for k = 1. Assume that they hold for k. Since N is dense in Z, there exist Kk+1 ∈ N such that j j ) − Kk+1 ≤ αk+1 , B(xk , zk,k

B(xk , 0) − Ljk+1 ≤ δk+1 . Using (1) and (2) we also have j B(xk , vk,l ) − Ljk ≤ αk+1

for l = 1, . . . , k

j ) B(xk , zk,l

for l = 1, . . . , k.

−

Kkj

≤ αk+1

and

Applying the previous proposition once more, we can see that properties (1)–(7) hold for k + 1. Using these properties one can show that the j j sequences (xk ), (zk,l ), (vk,l ) are Cauchy for every l ≥ 1 and k > l. Let j j u = limk→∞ xk , zj,l = limk→∞ zk,l , and vj,l = limk→∞ vk,l . In what follows we will give some factorization results which are relevant in our context. We start with the following proposition given in [12]. The constants τ and c3 are some speciﬁc constants described in [12] with 0 < c3 < 1. Proposition 3.3. Assume that the hypothesis of Theorem 2.3 is satisfied. Fix a nonnegative function f ∈ L1 (∂Ω) with f 1 = 1 and y ∗ ∈ X ∗ . Then for m sufficiently large, there exist x ∈ X, x∗ ∈ X ∗ such that x ≤ τ, x∗ ≤ 1, m and x ⊗T (u(T )∗ x∗ + y ∗ ) − Mf < c3 . Proposition 3.4. Assume that the hypothesis of Theorem 2.3 is satisfied. Let y1 , . . . , yp ∈ X, y ∗ ∈ X ∗ , ε > 0, and a nonnegative function f ∈ L1 . Then there exist w ∈ X and w∗ ∈ X ∗ such that 1. w ⊗T (w∗ + y ∗ ) − [f ] ≤ c3 f 1 , ∗ < ε, for every 2. yj ⊗T w j = 1, . . . , p 3. w ≤ τ f 1 , w∗ ≤ f 1 . Proof. Choose n so that u(T )n yj ≤

ε

. By applying the previous prop osition to the function f /f 1 , and to the functional y ∗ / f 1 , we get ∗ v ⊗T u(T )∗ n v ∗ + y − Mf / f 1 < c3 1/2 f 1

f 2

1/2

for some v and v ∗ with v ≤ η and v ∗ ≤ 1. Let w = vf 1 1/2 n f 1 u(T )∗ v ∗ .

and w∗ =

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Then w ⊗T (w∗ + y ∗ ) − Mf < c3 f 1 , n y ⊗T w∗ = y ⊗T f 1 u(T )∗ v ∗ = u(T )n y ⊗T f 1 v ∗ 1/2

≤ u(T )n yv ∗ f 1 < ε, w ≤ η f 1 , and w∗ ≤ f 1 . Note that in the previous proposition we assumed f to be nonnegative. We remove that condition in the following proposition. Fix an integer N such that c3 + πN −1 < 1, and a positive constant c satisfying 1 − N −1 (1 − c3 − πN −1 ) < c < 1. Proposition 3.5. Assume that the hypothesis of Theorem 2.3 is satisfied. Given y1 , . . . , yp ∈ X, y ∗ ∈ X ∗ , ε > 0, and f ∈ L1 , there exist x ∈ X and x∗ ∈ X ∗ such that 1. x ⊗T (x∗ + y ∗ ) − [f ] ≤ cf 1 , ∗ < ε for every 2. yj ⊗T x j = 1, . . . , p, 3. x ≤ τ f 1 , x∗ ≤ f 1 . In other words, the bilinear application which maps (x, x∗ ) to x ⊗T x∗ l has the Ec,τ,1 property. Proof. Without loss of generality we may assume f 1 = 0. For j = 0, 1, . . . , N − 1, let Bj be the set of all complex numbers that are of the form reit with r > 0 and −π/N ≤ t − 2πj/N < π/N . Fix a representative of f and N −1 deﬁne Aj = f −1 (Bj ) for j = 0, 1, 2, . . . , N − 1. Then f 1 = j=0 f χAj 1 . Fix 0 ≤ j0 ≤ N − 1 such that f χAj0 1 ≥ N −1 h1 , and set v = e2πj0 i/N . For each ζ ∈ Aj0 , we have f (ζ) ≤ |f (ζ)|πN −1 . |v|f (ζ)| − f (ζ)| = |f (ζ)| v − |f (ζ)| So v|f |χAj0 − f χAj0 1 ≤ πN −1 f χAj0 1 . By the previous proposition there exist vectors w ∈ X and w∗ ∈ X ∗ such that w ⊗T (w∗ + y ∗ ) − [|f |χAj0 ] ≤ c3 f 1 , yj ⊗T w∗ < ε for j = 1, . . . , p, w ≤ τ f 1 , and w∗ ≤ f 1 . Set x = vw and x∗ = w∗ . Properties (2) and (3) are automatically satisﬁed. Furthermore, vw ⊗T (w∗ + y ∗ ) − [f ] ≤ v(w ⊗T (w∗ + y ∗ ) − [|f |χAj0 ]) + v[|f |χAj0 ] − [f ] ≤ c3 f χAj0 1 + v|f |χAj0 − f χAj0 + f χAj0 1 j =j0

≤ (c3 + πN −1 )f χAj0 1 + f 1 − f χAj0 1

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≤ f 1 − f χAj0 1 (1 − c3 − πN −1 ) ≤ f 1 (1 − N −1 (1 − c3 − πN −1 ) ≤ cf 1 .

4. Analytic Rationally Invariant Subspaces Analytic rationally invariant subspaces for Hilbert space operators are deﬁned in [7]. We adapt it in the Banach space context. Definition. A rationally invariant subspace M of T is called an analytic rationally invariant subspace for T if there exists an analytic function e from Ω to M ∗ with e(λ) = eλ such that

(T |M − λI)∗ eλ = 0

for

λ ∈ Ω.

∗

Moreover, if λ∈Ω eλ = M , then the subspace M is called a full analytic rationally invariant subspace. Remark 1. Note that if M is an invariant subspace for T and S = T |M , then S ∗ has also a weak∗ functional calculus and h(S ∗ )(x∗ |M ) = h(T ∗ )(x∗ )|M. ∗

So we have x ⊗T x∗ = x ⊗S (x∗ |M ∗ ). Note that we also have h((T |M ) )eλ = h(λ)eλ for λ ∈ Ω and h ∈ H ∞ . Remark 2. When M ⊂ X is an analytic rationally invariant subspace for T , it is easy to check that the following hold. Recall that βj (λ) = rj (λj − λ)−1 ∗ ∗ for j = 1, 2, . . . , n and β0 (λ) = λ. There exist vectors y00 and {yj,k }∞ k=1 for ∗ j = 0, 1, . . . , n in M such that ∗ ∗ + βjk (λ)yj,k for λ ∈ Ω (1) e(λ) = y0,0 0≤j≤n k≥1 n ∗ ∗ T ∗ y0,0 + rj yj,1 =0 j=1 ∗ ∗ = y0,k−1 for k ≥ T ∗ y0,k ∗ ∗ ∗ βj (T )yj,k+1 = yj,k for k ≥ 1 and ∗ lim sup yj,k 1/k

(2) 1

(3) j = 1, . . . , n

=1

(5) ∗

dim ker((βj (T )|M − βj (λ)) ≥ 1 ∗ eλ = yj,n λ∈Ω

(4) (6) (7)

0≤j≤n k≥1

∗ ∗ Conversely, if there exist vectors y00 and {yj,k }∞ k=1 for j = 0, 1, . . . , n in M which satisfy properties (2)-(7) then the analytic map e deﬁned in (1) turns M into an analytic rationally invariant subspace for T . ∗

The following proposition is an analogue of Proposition 5.3 in [11]. Since the proof is a slight modiﬁcation of the original one, we omit it.

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Proposition 4.1. Assume that the hypothesis of Theorem 2.3 is satisfied. If X is full analytic rationally invariant for T , then WΩ T is reflexive. Let λ0 be an element in Ω. For x ∈ X we will denote by Mx the closure ¯ f (λ0 ) = 0}. Clearly Mx is rationally of the subspace {f (T )x : f ∈ A(Ω), invariant for T . We may assume that (T − λ0 I)x = 0 for every x ∈ X, since otherwise ker(T − λ0 I) would be a hyperinvariant subspace for T . So in other words we may assume that Mx = {0} for every x ∈ X. We will denote by CF (T ) the set of all x ∈ X such that Mx is full analytic. The following proposition is analogous to Proposition 5.4 in [11] and we will omit the proof. Proposition 4.2. Assume that the hypothesis of Theorem 2.3 is satisfied. If the set CF (T ) is dense in X, then the operator algebra WΩ T is reflexive.

5. Main Result It is well known that every h ∈ H ∞ (Ω) can be expressed uniquely as h = h0 + h1 + h2 + · · · + hn where hj ∈ H ∞ (Uj ) and hj (∞) = 0 for every j = 0, 1, 2, . . . , n. For a proof see [10] for instance. This identiﬁcation agrees with the identiﬁcation H ∞ (Uj )“ = ”H ∞ (Cj ). This will lead to a generalized power series expansion for f ∈ H ∞ (∂Ω) as follows (See [7]): f (ζ) = fˆ0 (0) + fˆj (k)βjk (ζ) 0≤j≤n k≥1

¯ k (ζ)|dζ| for j = 0, 1, . . . , n. with fˆj (k) = Cj fj (ζ)B j We consider the multiplication operator Mz which acts on H ∞ (Ω) such that f (z) → zf (z). By some simple calculations we get the following: ˆ ˆ Remark 3. 1. (M z (f ))0 (0) + r1 f1 (1) + · · · + rn fn (1) = 0 ˆ 2. (M z (f ))0 (k) = f0 (k − 1) for k ≥ 1 and j = 1, . . . , n. ˆ 3. (β j .f ) (k + 1) = fj (k) for k ≥ 1 and j = 1, . . . , n. j

Consider the functionals deﬁned on H ∞ by f → fˆj (k) for j = 0, 1, . . . , n and k ≥ 1. Since these functionals are weak∗ continuous, they can be represented by some elements in L1 /⊥ H ∞ which we will denote by [Ejk ] for j = 0, 1, . . . , n and k ≥ 1. Similarly [E0,0 ] will denote the element of L1 /⊥ H ∞ which corresponds with the functional that maps f to fˆ0 (0). Let K > 0 be the greatest among the polynomial bounds of the operators βj (T ) for j = 0, 1, . . . , n. Note that [Ej,k ] ≤ 2πK for every j = 0, 1, . . . , n and k ≥ 1, and [E00 ] ≤ 2πK. Lemma 5.1. Let N denote the linear span of the functionals [Ejk ] for j = 0, 1, . . . , n and k ≥ 1, and [E0,0 ]. Then N is dense in L1 /⊥ H ∞ . Proof. Assume that N is not dense in L1 /⊥ H ∞ . Then there exists a function in H ∞ (Ω) which lies in the intersection of kernels of these functionals.

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However, such a function does not exist as we have a representation of all functions in terms of coefﬁcients fˆj (k). In what follows we will prove an analogue of Proposition 6.1. of [11]. Proposition 5.2. Assume that the hypothesis of Theorem 2.3 is satisfied. Assume that there exist a vector x ∈ X and functionals x∗0,0 and (x∗j,k )j=1,...,n k≥1 such that x ⊗T x∗jk = [Ejk ]

for j = 0, 1, . . . , n x ⊗T x∗0,0

lim sup x∗j,k 1/k

k≥1

and

= [E00 ]

≤1

for j = 0, 1, . . . , n

Then Mx is an analytic rationally invariant subspace for T . Proof. Let us write M for Mx . We note that M ∈ Lat WΩ T . By the previous remark we have x ⊗T x∗ = x ⊗(T |M )∗ (x∗ |M ). We claim that the function deﬁned from M ∗ to L1 / ⊥ H ∞ by x∗ −→ x ⊗(T |M )∗ (x∗ |M ) is one-to-one. To prove the claim assume that x ⊗(T |M )∗ ¯ such that (x∗ |M ) = 0. Let y ∈ M . Then there exists a sequence (fk ) in A(Ω) fk (λ0 ) = 0 and fk (T )x converges to y as k −→ ∞. So we have y, x∗ = lim fk (T )x, x∗

k−→∞

= lim x ⊗T x∗ , fk . k−→∞

By the previous remark the last limit is equal to limk−→∞ x ⊗(T |M )∗ (x∗ |M ), fk which is 0. Since y is an arbitrary element of M it follows that the function deﬁned above is one-to-one. Now we will show that the vectors j=0,1,...,n and x∗0,0 satisfy the properties (2)–(7) of Remark 2. (x∗j,k )k≥1 Note that for f ∈ H ∞ , we have x ⊗T T ∗ x∗0,0 , f = x ⊗T x∗0,0 , Mz (f ) . Then by (1) of Remark 2, we have ⎞

⎛ n n ∗ ⎠ ⎝x ⊗T T ∗ x∗0,0 + rj xj,1 , f = (Mz (f ))0 (0) + rj fˆj (1) = 0. j=1

j=1

This proves that (2) of Remark 2 holds. Now by (2) of Remark 3, we have x ⊗T x∗0,k , f = x ⊗T x∗0,k , Mz (f ) = (M z (f ))0 (k) = fˆ0 (k − 1) = x ⊗T x∗ , f

0,k

So this gives us (2) of Remark 2. Now by (3) of Remark 3 we get x ⊗T βj (T ∗ )x∗j,k+1 , f = x ⊗T x∗j,k+1 , βj .f

ˆ = (β j .f )j (k + 1) = fj (k) = x ⊗T x∗j,k , f

for k ≥ 1 and j = 0, 1 . . . , n.

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Thus, property (3) of Remark 2 holds. Properties (4)–(7) of Remark 2 obviously hold. So M is an analytic rationally invariant subspace for T . The following proposition is an analogue of Proposition 6.2. in [11]. Proposition 5.3. Assume that the hypothesis of Theorem 2.3 is satisfied. Assume that there exist a vector x ∈ X and functionals x∗00 , (x∗j,k )j=1,...,n , k≥1 ∗ j=1,...,n and (yj,k )k≥1 such that x ⊗T x∗ij = [Eij ]

for j = 0, 1, . . . , n x ⊗T x∗0,0

k≥1

= [E0,0 ]

lim sup x∗j,k 1/k ≤ 1 for j = ∗ ∈ N for j = 0, 1, . . . , n x ⊗T yj,k

and

0, 1, . . . , n and

k≥1

∗ yj,n = X ∗.

0≤j≤n k≥1

Then Mx is full analytic rationally invariant subspace for T . Proof. By the above proposition we know that Mx is an analytic rationally invariant subspace for T . For all k ≥ 1 and j = 0, 1, . . . , n, the functional ∗ can be written as a linear combination of [E0,0 ] and [Ej,k ] for k ≥ 1 x ⊗T yj,k and j = 0, 1, . . . , n. It follows that

∗ yj,k |M ⊂

0≤j≤n k≥1

x∗j,k |M.

0≤j≤n k≥1

Thus,

x∗j,k |M = M ∗

0≤j≤n k≥1

We prove below the main result of this paper. Proof of Theorem 2.3. Since X is reﬂexive and separable, X ∗ is separable as well. Let (x∗l ) be such that {x∗l : l ≥ 0} = X ∗ . Note that we can write (x∗l ) in the form (x∗j,k ) where l = (k − 1)(n + 1) + j for k ≥ 1 and j = 0, 1 . . . , n. Let x be a vector in X, δ > 0, and ( k ) be a sequence of real numbers which is strictly decreasing to 0. For k ≥ 1 and j = 0, 1, . . . , n, deﬁne [hj,k ] =

[Ej,k ]δ 2 K2πk 6

and

[h0,0 ] =

[E0,0 ]δ 2 . K2πk 6

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δ ∗ ∗ Let δk = (n+1)k 8 . By Proposition 3.2., there exist y ∈ X, yj,k , zj,k , and three constants C1 , C2 , C3 such that ∗ [hj,k ] = y ⊗T yj,k ∗ yj,k ≤ C1 δ

y − x ≤ C2 δ ∗ y ⊗T zj,k ∈N

∗ zj,k − x∗j,k ≤ C3 k

If we let t∗j,k =

∗ (n + 1)k 8 yj,k , δ2

∗ we see that t∗j,k , y, and zj,k satisfy the hypothesis of the previous proposition. Thus My is full analytic for T . Since y − x ≤ C1 δ, the set CF (T ) is dense in X. Therefore, by Proposition 4.2, the operator algebra WΩ T is reﬂexive.

References [1] Ambrozie, C., M¨ uller, V.: Invariant subspaces for polynomially bounded operators. J. Funct. Anal. 213, 321–345 (2004) [2] Apostol, C.: Utraweakly closed operator algebras. J. Oper. Theory 2, 49–61 (1979) [3] Brown, S.: Some invariant subspaces for subnormal operators. Integral Equ. Oper. Theory 1, 310–333 (1978) [4] Brown, S., Chevreau, B.: Toute contraction ` a calcul fonctionnel isom´etrique est r´eﬂexive. C.R. Acad. Sci. Paris 307, 185–188 (1998) [5] Brown, S., Chevreau, B., Pearcy, C.: On the structure of contraction operators, II. J. Funct. Anal. 76, 30–55 (1988) [6] Chevreau, B., Exner, G., Pearcy, C.: On the structure of contraction operators, III. Michigan Math. J. 36, 29–62 (1989) [7] Chevreau, B., Li, W.S.: On certain representations of H ∞ (G) and the reﬂexivity of associated algebras. J. Funct. Anal. 128, 341–373 (1995) [8] Fischer, S.D.: Function Theory on Planar Domains. Wiley, New York (1983) [9] Olin, R., Thompson, J.: Algebras of subnormal operators. J. Funct. Anal. 37, 271–301 (1980) [10] Paulsen, V.I.: Completely Bounded Maps and Dilations. Longman Scientiﬁc & Technical, Harlow (1986) [11] Rejasse, O.: Factorization and reﬂexivity results for polynomially bounded operators. J. Oper. Theory 60, 219–238 (2008) [12] Yavuz, O.: Invariant subspaces for Banach space operators with a multiply connected spectrum. Integral Equ. Oper. Theory 58, 433–446 (2007)

Vol. 68 (2010)

Banach Space Operators

Onur Yavuz (B) Faculty of Engineering and Natural Sciences Sabanci University Tuzla, 34956 Istanbul, Turkey e-mail: [email protected] Received: January 30, 2010. Revised: June 6, 2010.

485

Integr. Equ. Oper. Theory 68 (2010), 487–502 DOI 10.1007/s00020-010-1805-8 Published online June 22, 2010 c The Author(s) 2010. This article is published with open access at Springerlink.com

Integral Equations and Operator Theory

Stability Analysis in Continuous and Discrete Time, using the Cayley Transform Niels Besseling and Hans Zwart Abstract. For semigroups and for bounded operators we introduce the new notion of Bergman distance. Systems with a finite Bergman distance share the same stability properties, and the Bergman distance is preserved under the Cayley transform. This way, we get stability results in continuous and discrete time. As an example, we show that bounded perturbations lead to pairs of semigroups with finite Bergman distance. This is extended to a class of Desch–Schappacher perturbations. Mathematics Subject Classification (2010). Primary 47D60; Secondary 93D05. Keywords. C0 -semigroups, Cayley transform, continuous time, discrete time, stability.

1. Introduction Consider the linear differential equation x(t) ˙ = Ax(t),

x(0) = x0 ,

(1.1)

with the state x in the separable Hilbert space X and A the inﬁnitesimal generator of the strongly continuous semigroup (eAt )t≥0 . A standard way of numerically solving this differential equation is the Crank–Nicolson method [7]. In this method the differential equation (1.1) is replaced by the difference equation −1 ΔA ΔA xd (n), xd (0) = x0 , (1.2) I− xd (n + 1) = I + 2 2 where Δ is the time step. Since we look at the stability properties of the semigroup, we can choose Δ freely. For symplicity we take Δ = 2. The operator (I + A)(I − A)−1 is known as the Cayley transform of A, and we denote it by Ad . A natural question is whether the solution xd (n) = And x0 of (1.2) is a good approximation of the solution eAt x0 of (1.1). We will not consider this

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question, but concentrate on the stability properties of both equations. If X is ﬁnite-dimensional, and thus A a matrix, then it is well-known that both equations share the same stability properties, i.e. A has all his eigenvalues in the open/closed left-half plane if and only if Ad has all its eigenvalues in the open/closed unit circle. This property on the eigenvalues hold for the operators A and Ad as well. However, for inﬁnite dimensional spaces this tells little about the stability of the solutions. The central question in this paper is the following. If we know that the semigroup is strongly stable, so for all x0 ∈ X, eAt x0 → 0, as t → ∞, what can be said about the solutions of the difference equation (1.2), and hence about And x0 for n → ∞? It is well known that if (eAt )t≥0 is a contraction semigroup, that is At e ≤ 1, then Ad ≤ 1 and thus And ≤ 1, for all n ≥ 0, for a detailed proof see e.g. [8, Theorem 3.4.9], although the result is much older. If (eAt )t≥0 is a bounded analytic semigroup, then And ≤ M2 , for all n ≥ 0, see [5]. Thus, in these cases the solutions of (1.2) are bounded. If additionally, the semigroup is strongly stable, then (And )n≥0 is strongly stable as well, see [5]. We extend the class of semigroups which behave nicely with respect to the Cayley transform, by introducing the new notion of Bergman distance. ˜ We say that two semigroups, (eAt )t≥0 and (eAt )t≥0 , have a finite Bergman distance if the following two inequalities are satisﬁed for all x0 ∈ X: ∞

1 ˜ (eAt − eAt )x0 2 dt < ∞, t

(1.3)

∗ 1 ˜∗ (eA t − eA t )x0 2 dt < ∞. t

(1.4)

0

∞ 0

Note that the measure t−1 dt is the invariant measure for the multiplication group R+ . The space L21 (R+ ) with this measure is isometrically isomorphic to the unweighted Bergman space A2 (Π+ ), see [3, Theorem 1]. Thus two semi∗ ˜ ˜∗ groups have ﬁnite Bergman distance, if (eAt − eAt )x0 and (eA t − eA t )x0 are in the Bergman space for all x0 ∈ X. In Sect. 6, we investigate which pair of generators have ﬁnite Bergman distance. Among others, we show that if A and A˜ generate exponentially stable semigroups, and if A − A˜ is bounded, then they have a ﬁnite Bergman distance. For the sequences of bounded operators, (And )n≥0 and (A˜nd )n≥0 , we say that they have a finite Bergman distance if the following two inequalities are satisﬁed for all x0 ∈ X: ∞ 2 1 k Ad x0 − A˜kd x0 < ∞, k

k=1

∞ 2 1 ∗k x Ad x0 − A˜∗k 0 < ∞. d k

k=1

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One of our main results is, that the Cayley transform conserves the Bergman distance. That is, the following equality holds for all x0 ∈ X: ∞ ∞ 2 1 ˜ k At At 21 (e − e )x0 dt = (1.5) Ad − A˜kd x0 . t k k=1

0

We prove this equality in Sect. 4. Furthermore, operators with ﬁnite Bergman distance have similar stability properties. In Sect. 2, we show this for the continuous-time case and in Sect. 3 we examine the discrete-time case. If (And )n≥0 and (A˜nd )n≥0 have a ﬁnite Bergman distance and (And )n≥0 is strongly stable, i.e. And x → 0 as n → ∞, then also (A˜nd )n≥0 is strongly stable. Combining this with Eq. (1.5), leads to the following theorem: ˜

Theorem 1.1. Let (eAt )t≥0 and (eAt )t≥0 have a finite Bergman distance. Then (And )n≥0 is strongly stable if and only if (A˜nd )n≥0 is strongly stable. Furthermore, the other implication also holds. Thus, if (And )n≥0 and ˜ have a ﬁnite Bergman distance, then (eAt )t≥0 and (eAt )t≥0 have similar stability properties. We prove this in Sect. 5. (A˜nd )n≥0

2. Stability in Continuous Time The ﬁnite Bergman distance devides semigroups into classes. In this section we show that within these classes of semigroups the stability properties are the same. First, we deﬁne what we mean by stability of semigroups. Definition 2.1. The C0 -semigroup (eAt )t≥0 is bounded if there exists a constant M ≥ 1 such that eAt ≤ M,

for all t ≥ 0.

At

The C0 -semigroup (e )t≥0 is exponentially stable if there exist constants M ≥ 1 and ω > 0 such that eAt ≤ M e−ωt ,

for all t ≥ 0.

The C0 -semigroup (eAt )t≥0 is strongly stable if for all x0 ∈ X, eAt x0 → 0,

as t → ∞.

Van Casteren, [1], gave the following characterisation of bounded and strongly stable semigroups. Lemma 2.2. The semigroup (eAt )t≥0 is bounded if and only if there exists a M such that for all t ≥ 0, and all x0 ∈ X, 1 t

t

eAs x0 2 ds ≤ M x0 2

0

with M independent of t and x0 .

and

1 t

t 0

∗

eA s x0 2 ds ≤ M x0 2 (2.1)

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Furthermore, if (eAt )t≥0 is bounded and for all x0 1 lim t→∞ t

t

eAs x0 2 ds = 0,

(2.2)

0

then (eAt )t≥0 is strongly stable. With Lemma 2.2, we can show that two semigroups with a ﬁnite Bergman distance, have the same stability properties. ˜

Theorem 2.3. Let (eAt )t≥0 and (eAt )t≥0 have a finite Bergman distance. Then 1. 2. 3.

˜

(eAt )t≥0 is bounded if and only if (eAt )t≥0 is bounded, ˜ (eAt )t≥0 is exponentially stable if and only if (eAt )t≥0 is exponentially stable, ˜ (eAt )t≥0 is strongly stable if and only if (eAt )t≥0 is strongly stable.

Proof. We prove the boundedness or stability of (eAt )t≥0 , given the bound˜ edness or stability of (eAt )t≥0 . By symmetry, the other implication then also holds. We begin with item 1. 1.

For all t > 0 and x0 ∈ X, the following inequalities hold: 1 t

t e

As

0

1 x0 ds ≤ t 2

t As

2e 0

t ≤2 0

˜ As

x0 − e

1 x0 ds + t 2

t

˜

2eAs x0 2 ds

0

1 As ˜ ˜ e x0 − eAs x0 2 ds + 2 sup eAt 2 x0 2 s t

≤ M1 x0 2 , ˜

2.

where we have used (1.3) and the boundedness of (eAt )t≥0 . Similarly, we obtain the dual result. Hence by Lemma 2.2, we conclude that (eAt )t≥0 is bounded. For t > 1, we have for all x0 ∈ X t 1

1 As e x0 2 ds ≤ 2 s

t 1

1 As ˜ e x0 − eAs x0 2 ds + 2 s 2

≤ M2 x0 ,

t 1

1 As ˜ e x0 2 ds s (2.3)

where we have used the ﬁnite Bergman distance and the exponential ˜ stability of (eAt )t≥0 .

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The exponential stability of (eAt )t≥0 trivially implies that (eAt )t≥0 is bounded. By item 1., we have that (eAt )t≥0 is bounded as well. Combining this with (2.3), we ﬁnd t

2

ln(t)e x0 = At

1

t ≤ 1

1 At e x0 2 ds s 1 A(t−s) 2 As e e x0 2 ds s t

≤ M1 1

1 As e x0 2 ds ≤ M1 M2 x0 2 . s

So for t > 1 we have that eAt 2 ≤

M1 M2 . ln(t)

Since for large t this will be less one, we have that (eAt )t≥0 is exponentially stable. ∞ ˜ 3. Since 0 1s eAs x0 − eAs x0 2 ds < ∞, for every ε > 0, there exists a tε ∞ 1 As ˜ such that tε s e x0 − eAs x0 2 ds < ε. For x0 ∈ X, there holds 1 lim t→∞ t

t e

As

0

1 x0 ds ≤ lim t→∞ t 2

t

˜

2eAs x0 − eAs x0 2 ds

0

1 + lim t→∞ t

t

˜

2eAs x0 2 ds.

0

Using (2.2), we have that 1 lim t→∞ t

t e

As

0

1 x0 ds ≤ lim t→∞ t 2

tε

˜

2eAs x0 − eAs x0 2 ds

0

t

1 + lim t→∞ t

˜

2eAs x0 − eAs x0 2 + 0

tε

≤ lim

t→∞

tε t

tε 0

t + lim

t→∞ tε

2 As ˜ e x0 − eAs x0 2 ds s 2 As ˜ e x0 − eAs x0 2 ≤ 0 + 2ε. s

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Since this holds for all ε > 0, we have shown that t 1 eAs x0 2 ds = 0, lim t→∞ t 0

At

and so (e )t≥0 is strongly stable by Lemma 2.2.

3. Stability in Discrete Time The discrete-time case is similar to the continuous-time case, the ﬁnite Bergman distance also creates classes of sequences of bounded operators. Elements within a class share the same stability properties. First, we deﬁne what we mean by stability in discrete time. Definition 3.1. The operator sequence (And )n≥0 is bounded if there exists a constant M ≥ 1 such that And ≤ M, (And )n≥0

The operator sequence and γ ∈ (0, 1) such that

is power stable if there exist constants M ≥ 1

And ≤ M γ n , The operator sequence

for all n ≥ 0.

for all n ≥ 0.

(And )n≥0 is strongly stable if And x0 → 0, as n → ∞.

for all x0 ∈ X,

Now, we recall a result by Van Casteren [1], and next we show the stability properties are preserved by the ﬁnite Bergman distance. Lemma 3.2. The operator sequence (And )n≥0 is power stable, if and only if there exists a M such that N N ∗k 2 1 Akd x0 2 ≤ M x0 2 and 1 Ad x0 ≤ M x0 2 . (3.1) N N k=1

k=1

with M independent of N and x0 . Furthermore, if (And )n≥0 is power stable, then (And )n≥0 is strongly stable if and only if N 1 Ad x0 2 = 0, N →∞ N

lim

(3.2)

k=1

for all x0 ∈ X. Theorem 3.3. Let (And )n≥0 and (A˜nd )n≥0 have finite Bergman distance. Then the following assertions hold: 1. (And )n≥0 is bounded if and only if (A˜nd )n≥0 is bounded, 2. (And )n≥0 is power stable if and only if (A˜nd )n≥0 is power stable, 3. (And )n≥0 is strongly stable if and only if (A˜nd )n≥0 is strongly stable. Proof. We prove the boundedness or stability of (And )n≥0 , given the boundedness or stability of (A˜nd )n≥0 . By symmetry, the other implication then also holds. The proofs are similar to the ones in the continuous time.

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Using Eq. (3.1) and the power stability of (A˜nd )n≥0 , we ﬁnd for all x0 ∈ X N N N 2 2 1 1 Akd x0 2 ≤ 1 2 Akd x0 − A˜kd x0 + 2 A˜kd x0 N N N k=1

k=1

≤2

k=1

2 2 1 k Ad x0 − A˜kd x0 + 2 sup A˜kd x0 k k

N k=1

≤ M1 x0 2 . Similarly, we obtain the dual result. By Lemma 3.2, (And )n≥0 is power stable. 2. We have N N N 2 1 1 1 k ˜k 2 Akd x0 2 ≤ 2 Ad x0 − A˜kd x0 + 2 Ad x0 k k k

k=1

k=1

k=1

≤ M2 x0 2 ,

(3.3)

where we have used the ﬁnite Bergman distance and the power stability of (A˜nd )n≥0 . The power stability of (A˜nd )n≥0 implies that (A˜nd )n≥0 is bounded, so by item 1. (And )n≥0 is bounded as well. Combining this with equation (3.3): ln(n + 1)And x0 2 ≤ ≤

n 1 2 And x0 k

k=1 n k=1

≤ M1

2 1 n−k 2 Ad Akd x0 k n 1 Akd x0 2 ≤ M1 M2 x0 2 . k

k=1

So we have that 2

And ≤

M1 M2 . ln(n + 1)

Since for large n this will be less than one, we have that (And )n≥0 is power stable.

∞ 3. Since k=1 k1 Akd x0 − A˜kd x0 2 < ∞, for every ε > 0, there exists a nε

∞ such that k=nε k1 Akd x0 − A˜kd x0 2 < ε. Using (3.2) we ﬁnd n n 2 1 Akd x0 2 ≤ lim 1 2 Akd x0 − A˜kd x0 n→∞ n n→∞ n

lim

k=1

k=1 n

1 n→∞ n

+ lim

k=1

2 2 A˜kd x0

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nε −1 2 1 2 Akd x0 − A˜kd x0 n→∞ n

= lim

k=1

n 2 1 + lim 2 Akd x0 − A˜kd x0 + 0 n→∞ n

≤ 0 + lim

n→∞

k=nε n

k=nε

2 2 k Ad x0 − A˜kd x0 ≤ 2ε. k

Since this holds for all ε > 0, we have shown that 1 Akd x0 2 = 0, n→∞ n n

lim

k=1

and so

(And )n≥0

is strongly stable.

4. Equivalence of the Bergman Distances In the previous sections we have derived properties of operators with ﬁnite Bergman distance. In this section, we show that the Cayley transform preserves Bergman distances. First, we deﬁne the inner product space H. Definition 4.1. Let H denote the space of Lebesgue measurable functions f from [0, ∞) to the Hilbert space X such that: ∞

f (t)2X t dt < ∞.

0

On H we deﬁne the following inner product: ∞ f, g H = f (t), g(t) X t dt.

(4.1)

0

The following result is easy to see. Lemma 4.2. The inner product space H defined in Definition 4.1 is a Hilbert space. To create an orthonormal basis for this Hilbert space, we use the gen(1) eralised Laguerre polynomials Ln (t) [9, p. 99]. These are deﬁned by n−1 n (−2t)k (1) Ln−1 (2t) = , for n ≥ 1 and t ∈ [0, ∞). (4.2) n−k−1 k! k=0

Lemma 4.3. Let H be the Hilbert space defined by Definition 4.1 and let {em }m∈N be an orthonormal basis of X. The vectors ϕn,m defined by: qn (t) ϕn,m (t) = √ em , n

n, m ≥ 1,

(4.3)

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with (1)

qn (t) = −2e−t Ln−1 (2t),

(4.4)

form an orthonormal basis in H. ∞

Proof. We begin by showing that the sequence {ϕn,m }n,m=1 is orthonormal in H. Using Eq. (4.1), we ﬁnd: ∞ (1) (1) −2e−t Lν−1 (2t) −2e−t Ln−1 (2t) √ √ em , eμ ϕn,m , ϕν,μ H = t dt n ν X

0

4 = √ √ n ν 1 = √ √ n ν

∞

(1)

(1)

e−2t tLn−1 (2t)Lν−1 (2t) dt em , eμ X

0

∞

(1)

(1)

e−τ τ Ln−1 (τ )Lν−1 (τ ) dτ em , eμ X

0

n 1 δ(n−1)(ν−1) δmμ = δnν δmμ , = √ √ Γ(2) n−1 n ν where we use the orthogonality of the Laguerre polynomials, see [9, p. 99]. ∞ Next we show that the sequence {ϕn,m }n,m=1 is maximal in H. If h is orthogonal to every ϕn,m , then for all n and m ≥ 1: ∞ ϕn,m , h H = 0

−2e−t (1) Ln−1 (2t) √ em , h(t) X t dt = 0. n

(1)

From the maximality of {Ln−1 (2t)e−t t}n≥1 in L2 (0, ∞), see [9, p. 107], we conclude that for all m ≥ 1, em , h(t) X = 0

almost everywhere.

This, combined with the maximality of {em }m∈N in X, leads to the conclusion that the Lebesgue measurable function h(t) = 0 almost everywhere. So ∞ h = 0 in H and {ϕn,m }n,m=1 is maximal. Lemma 4.3 gives us the following Parseval equality: f 2H =

∞ ∞

|f, ϕn,m H |2 .

(4.5)

n=1 m=1

We use the Laguerre polynomials to write the Cayley transform as an integral. Lemma 4.4. Let qn be defined by Eq. (4.4), let A generate a C0 -semigroup and let Ad be the Cayley transform of A. Then, ∞ qn (t)eAt x0 dt = (−1)n And x0 − x0 , x0 ∈ X. (4.6) 0

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Proof. We rewrite qn (t) as follows: (1)

qn (t) = −2e−t Ln−1 (2t) n−1 n (−2t)k = −2e−t n−k−1 k! k=0

=2

n−1 k=0

n! (−1)k+1 (2t)k e−t (n − k − 1)!(k + 1)!k!

n

n! (−1) (2t)−1 e−t (n − )! !( − 1)! =1 n n t−1 −t e , = (−2) ( − 1)! =2

=1

where we introduce = k + 1 in the fourth equality sign. We insert this into the left-hand side of Eq. (4.6) and using, −

(A − I)

∞ t−1 −t At e e x0 dt, x0 = (−1) R(1, A) x0 = (−1) ( − 1)!

0

see [4, p. 57], gives: ∞

n n

∞

At

qn (t)e x0 dt =

=1

0

=

n n =1

=

n n =0

0

(−2)

t−1 −t At e e x0 dt ( − 1)!

2 (A − I)− x0 2 (A − I)− x0 − x0

n

= I + 2(A − I)−1 x0 − x0 = (−1)n And x0 − x0 .

Thus Eq. (4.6) holds.

The following theorem shows that the Cayley transform preserves the Bergman distances. Theorem 4.5. Let A and A˜ generate a C0 -semigroup and let Ad and A˜d be ˜ their Cayley transforms, then (eAt )t≥0 and (eAt )t≥0 have finite Bergman disn n ˜ tance if and only if (Ad )n≥0 and (Ad )n≥0 have finite Bergman distance. Furthermore, for all x0 ∈ X ∞ 0

˜

(eAt − eAt )x0 2X

∞ 2 1 1 n dt = (Ad − A˜nd )x0 . t n X n=1

(4.7)

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Proof. First, we write the left-hand side of (4.7) as a norm in H, see Definition 4.1. Next, we apply the Parseval identity of H, see Eq. (4.5): 2 ∞ ∞ ˜ (eAt − eAt )x0 ˜ At 2 1 At (e − e )x0 X dt = t dt t t 0

X

0

2 ˜ (eAt − eAt )x0 = t H ∞ ∞ ˜ At (e − eAt )x0 , ϕn,m = t n=1 m=1

H

2 .

Zooming in on the inner product, and applying Eq. (4.3) and Lemma 4.4, we ﬁnd ∞ At ˜ ˜ (e − eAt )x0 qn (t) (eAt − eAt )x0 , ϕn,m , √ em = t dt t t n H

0

1 = √ n 1 = √ n

X

∞

˜

qn (t)(eAt − eAt )x0 , em

0

∞

X

dt

At

qn (t)(e

˜ At

− e )x0 dt, em

0

X

1 (−1)n And − (−1)n A˜nd x0 , em = √ . n X (−1)n n = √ . Ad − A˜nd x0 , em n X We zoom out again and use the Parseval equation of X for the orthonormal basis {em }m∈N . ∞ ∞ ∞ 2 (−1)n n ˜ At At 2 1 n ˜ √ Ad − Ad x0 , em (e − e )x0 X dt = t n X n=1 m=1 0

∞ ∞ 2 1 n = Ad − A˜nd x0 , em n X n=1 m=1 ∞ 2 1 n = Ad − A˜nd x0 . n X n=1

Thus Eq. (4.7) holds.

5. Proof of the Main Result In this section, we return to Theorem 1.1. With the results from Sects. 2, 3, and 4, we are able to prove it. First, we reformulate Theorem 1.1 as follows:

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˜ Theorem 5.1. Let (eAt )t≥0 and (eAt )t≥0 be C0 -semigroups and let Ad and A˜d ˜ Then denote the Cayley transforms of A and A.

1. 2.

˜

if (eAt )t≥0 and (eAt )t≥0 have a finite Bergman distance: (And )n≥0 is strongly stable if and only if (A˜nd )n≥0 is strongly stable. n if (Ad )n≥0 and (A˜nd )n≥0 have a finite Bergman distance: ˜ (eAt )t≥0 is strongly stable if and only if (eAt )t≥0 is strongly stable.

Proof. We begin by recalling that from Theorem 4.5, we know that (eAt )t≥0 ˜ and (eAt )t≥0 have a ﬁnite Bergman distance if and only if (And )n≥0 and (A˜nd )n≥0 have a ﬁnite Bergman distance. So to prove item 1. the argument goes as follows. The ﬁnite Bergman ˜ distance of (eAt )t≥0 and (eAt )t≥0 implies the ﬁnite Bergman distance between n n (Ad )n≥0 and (A˜d )n≥0 . Using the third item of Theorem 3.3, we conclude that (And )n≥0 is strongly stable if and only if (A˜nd )n≥0 is strongly stable. The second item is proved similarly. Now, we return to the central question in this paper: If we know that the semigroup (eAt )t≥0 is strongly stable, what can be said about (And )n≥0 ? Or what can be said about sequences (A˜nd )n≥0 at a ﬁnite Bergman distance of (And )n≥0 . Before answering this question, we ﬁrst recall the following result by Guo and Zwart [5, Theorem 4.3]. Lemma 5.2. Let (eAt )t≥0 be a C0 -semigroup and let Ad denote the Cayley transform of A. If (eAt )t≥0 and (And )n≥0 are bounded, and (eAt )t≥0 is strongly stable, then (And )n≥0 is strongly stable. Hence, if we combine this lemma with Theorem 5.1, we ﬁnd that if (eAt )t≥0 and (And )n≥0 are bounded, then the strong stability of (eAt )t≥0 ˜ implies the strong stability of (eAt )t≥0 , (And )n≥0 and (A˜nd )n≥0 , provided the two semigroups or the two discrete operators have ﬁnite Bergman distance.

6. Applications In this section we present some examples of semigroups with a bounded Bergman distance. Lemma 6.1. Let A and A˜ generate exponentially stable semigroups and let ˜ A− A˜ be bounded, then (eAt )t≥0 and (eAt )t≥0 have a finite Bergman distance. Proof. Let M1 , M2 , ω1 and ω2 be positive constants s.t. eAt ≤ M1 e−ω1 t ˜ and eAt ≤ M2 e−ω2 t , respectively. We show that these semigroups satisfy Eq. (1.3) by cutting the time interval [0, ∞) into two parts, and showing, for each part, that the integral is ﬁnite.

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The ﬁrst time interval 1. We use the variation of constant t is from 0 to A(t−s) ˜ ˜ ˜ x0 ds. formula eAt x0 = eAt x0 + 0 eAs (A − A)e 2 1 1 t 2 1 1 ˜ ˜ At At A(t−s) As ˜ x0 dt = e (A − A)e x0 ds e −e t dt t 0 0 0 ⎛ ⎞2 1 t 1 ˜ ˜ A(t−s) dt ≤ ⎝ eAs (A − A)e x0 ds⎠ t 0

1 ≤

⎛ ⎝

0

≤

≤

t 0

1 0

0

⎞2 ˜ 2 x0 ds⎠ 1 dt M1 A − AM t

2 1 ˜ dt tM1 M2 A − Ax 0 t

M12 M22 A

2

2

1

˜ x0 − A

t dt < ∞. 0

This holds for all x0 ∈ X. The second time interval is from 1 to ∞. ∞ ∞ 1 ˜ ˜ eAt − eAt x0 2 dt ≤ eAt − eAt x0 2 dt t 1

1

∞ ≤

˜

2eAt x0 2 + 2eAt x0 2 dt

1

M2 M12 −2ω1 e x0 2 + 2 e−2ω2 x0 2 < ∞. ω1 ω2 This holds for all x0 ∈ X. Hence Eq. (1.3) holds. The proof for the adjoint operators goes the same, and hence, we conclude the proof. ≤

Next, we apply the previous lemma to the linear quadratic optimal control problem. Lemma 6.2. Let A generate an exponentially stable contraction semigroup, and let B be bounded. By Π we denote the stabilizing solution of the algebraic Riccati equation, corresponding to the optimal control problem ∞ min x(t)2 + u(t)2 dt, u

0

see [2, Chapter 6]. Then the Cayley transform of A−BB ∗ Π is strongly stable. ∗

Proof. By Lemma 6.1, the semigroups (eAt )t≥0 and (e(A−BB Π)t )t≥0 have a ﬁnite Bergman distance. Since (eAt )t≥0 is a contraction semigroup, each

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operator And has norm less than or equal to one. It is strongly stable a well, since (eAt )t≥0 is exponentially stable. Theorem 1.1 proves the assertion. In the next example, we show that a subset of the class of Desch– Schappacher perturbations leads to pairs of semigroups with ﬁnite Bergman distance. First, we introduce the class of Desch–Schappacher perturbations, see Engel and Nagel [4, Section III.3.a]. We start by deﬁning Xt0 as the space of all strongly continuous, L(X)-valued functions, Xt0 = C ([0, t0 ], Ls (X)),

with the norm F ∞ = sup F (r)L(X) . r∈[0,t0 ]

Note that Xt0 is a Banach space. For the C0 -semigroup (eAt )t≥0 and the operator B ∈ L(X, X−1 ) from X to the extrapolation space X−1 = D(A∗ ) we deﬁne the abstract Volterra operator VB on the space Xt0 by t (VB F )(t) =

eA−1 (t−r) BF (r) dr,

for all t ∈ [0, t0 ] and F ∈ Xt0 .

0

Note that we use the extended semigroup on X−1 in this definition. The class of Desch–Schappacher perturbations is deﬁned by = {B ∈ L(X, X−1 ) | VB ∈ L(Xt0 ), VB < 1}. StDS 0

(6.1)

If we restrict the class of Desch–Schappacher perturbations by two extra conditions, then a perturbation B in this restricted class leads to a ﬁnite Bergman distance. The perturbation is denoted by (A−1 + B)X which is deﬁned as follows: D((A−1 + B)X ) = {x ∈ X|A−1 x + Bx ∈ X} and for x ∈ D((A−1 + B)X ) (A−1 + B)X x = A−1 x + Bx. Lemma 6.3. Let A be the infinitesimal generator of an exponentially stable . If, for some M > 1 and α > 0 semigroup and let B ∈ StDS 0 (VB )L(Xt ) ≤ M tα ,

for t ∈ (0, t0 ),

and, for some q ∈ (0, 1) R(λ, A−1 )B ≤ q,

for all λ ∈ C+ ,

(6.2)

then the semigroups generated by A and (A−1 + B)X have a finite Bergman distance. Proof. First, we deﬁne A˜ = (A−1 + B)X . It follows from Eq. (6.2), that the semigroup generated by A˜ is exponentially stable, see [6, Proposition 5.8]. Now, the proof is similar to the proof of Lemma 6.2. Let M1 , M2 , ω1 and ˜ ω2 be positive constants such that eAt ≤ M1 e−ω1 t and eAt ≤ M2 e−ω2 t , respectively. We show that these semigroups satisfy Eq. (1.3) by cutting the time interval [0, ∞) into two parts, and showing, for each part, that the integral is ﬁnite.

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The ﬁrst time interval is from 0 to t0 . We use the variation of constant formula. 2 t0 t0 t 2 1 1 ˜ ˜ At At As A−1 (t−s) ˜ x0 dt = e (A − A)e dsx0 e −e t dt t 0

0

0

t0 = 0

t0 ≤

1 ˜ (VB eA· )(t)x0 2 dt t M 2 t2α−1 M22 x0 2 dt

0

=

M 2 M22 2α t x0 2 < ∞. 2α 0

This holds for all x0 ∈ X. For the adjoint operators we ∗ ∗ A˜ t − eA t x0 = sup e

y0 =1

(6.3)

make the following observation: ∗ ˜∗ y0 , (eA t − eA t )x0

˜ = sup (eAt − eAt )y0 , x0 y0 =1

t ˜ ˜ As = sup eA(t−s) (A − A)e ds y0 , x0 y0 =1 0 ˜ = sup VB (eA· )y0 , x0 y0 =1

≤ sup M tα M2 y0 x0 . y0 =1 α

= M t M2 x0 . Using this inequality, we ﬁnd similar to (6.3), that t0 2 1 M 2 M22 2α ˜∗ A∗ t dt ≤ t x0 2 . − eA t x0 e t 2α 0 0

The second time interval is from t0 to ∞. The proof for this interval is similar to the second part of the proof of Lemma 6.1, and is therefor ommitted. Concluding, we see that the semigroups generated by A and (A−1 +B)X have a ﬁnite Bergman distance.

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.

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References [1] van Casteren, J.A.: Boundedness properties of resolvents and semigroups of operators. In: Linear Operators, vol. 38, pp. 59–74. (Warsaw, 1994), Banach Center Publications (1997) [2] Curtain, R.F., Zwart, H.J.: An Introduction to Infinite-dimensional Linear Systems Theory. Springer, Berlin (1995) [3] Duren, P., Gallardo-Guti´errez, E.A., Montes-Rodr´ıguez, A.: A Paley–Wiener theorem for Bergman spaces with application to invariant subspaces. Bull. Lond. Math. Soc. 39, 459–466 (2007) [4] Engel, K.-J., Nagel, R.: One-parameter Semigroups for Linear Evolution Equations. Springer, Berlin (2000) [5] Guo, B.Z., Zwart, H.: On the relation between stability of continuous- and discrete-time evolution equations via the Cayley transform. Integral Equ. Oper. Theory 54, 349–383 (2006) [6] Paunonen, L.: Robustness of Stability of C0 -Semigroups. Tampere University of Technology (2006) [7] Richtmyer, R.D., Morton, K.W.: Difference methods for initial-value problems. Wiley, New York (1967) [8] Staffans, O.: Well-posed Linear Systems. Cambridge University Press, Cambridge (2005) [9] Szeg¨ o, G.: Orthogonal Polynomials. American Mathematical Society, Providence (1957) Niels Besseling (B) and Hans Zwart Department of Applied Mathematics University of Twente P. O. Box 217, 7500 AE Enschede The Netherlands e-mail: [email protected]; [email protected] Received: February 4, 2010. Revised: March 22, 2010.

Integr. Equ. Oper. Theory 68 (2010), 503–517 DOI 10.1007/s00020-010-1806-7 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Closed-Range Composition Operators on A2 and the Bloch Space John R. Akeroyd, Pratibha G. Ghatage and Maria Tjani Abstract. For any analytic self-map ϕ of {z : |z| < 1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cϕ to be closed-range on the Bloch space B. Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cϕ is closed-range on the Bergman space A2 , then it is closed-range on B, but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carath´eodory Theorem. Mathematics Subject Classification (2010). Primary 47B33, 47B38; Secondary 30D55. Keywords. Composition operator, analytic self-map, Blaschke product, univalent map, angular derivative, nontangential limit, Bergman space, Bloch space.

1. Preliminaries Let D denote the unit disk {z : |z| < 1} and let T denote the unit circle {z : |z| = 1}. We let A denote two-dimensional Lebesgue measure on D. The Bergman space A2 is the collection of functions f that are analytic in D such that ||f ||2A2 := |f |2 dA < ∞. D 2

As a closed subspace of L (A), A2 forms a Hilbert space with respect to the inner product := D f gdA. The Bloch space B is the collection of functions f that are analytic in D such that ||f ||B := |f (0)| + sup(1 − |z|2 )|f (z)| < ∞. z∈D

Now ||·||B deﬁnes a norm on B, and under this norm B forms a Banach space. Moreover, ||f ||A2 ≤ 3||f ||B for any function f that is analytic in D, and hence B ⊆ A2 . A function ϕ that is analytic in D and that satisﬁes ϕ(D) ⊆ D is

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called an analytic self-map of D. Analytic automorphisms of D are M¨ obius α−z , where c is some unimodular constant transformations of the form z → c 1− αz ¯ α−z and α is some point in D; we let ϕα (z) = 1− αz ¯ . The so-called pseudohyperbolic metric on D is given by ρ(z, w) = |ϕw (z)|; and is indeed a metric. For any z in D and any r, 0 < r < 1, we let D(z, r) denote the pseudohyperbolic disk of radius r about z, namely, {w ∈ D : ρ(z, w) < r}. Now if ϕ is an analytic self-map of D, then the composition operator Cϕ , given by Cϕ (f ) := f ◦ ϕ, is a bounded operator on both A2 and B. This result for the Bloch space is a simple consequence of the Schwarz-Pick Lemma (cf., [7, page 2]), and for the Bergman space case one may consult [13, page 17]. Moreover, if ϕ is not constant, then Cϕ is one-to-one on these spaces and hence, by the Open Mapping Theorem, is closed-range if and only if it is bounded below. For any analytic self-map ϕ of D, deﬁne τϕ on D by τϕ (z) :=

(1 − |z|2 )ϕ (z) . 1 − |ϕ(z)|2

For ε > 0, let Λε = {z ∈ D : |τϕ (z)| > ε} and let Fε = ϕ(Λε ). We say that Fε satisﬁes the reverse Carleson condition if there exist s and c, 0 < s, c < 1, such that A(Fε ∩ D(z, s)) ≥ cA(D(z, s)), for all z in D; cf., [10] for seminal work regarding this condition. It has been shown that Cϕ is closed-range on B if and only if there exists ε > 0 such that Fε satisﬁes the reverse Carleson condition; cf. [9] and [3]. In fact, in [3] it is shown that, what appears to be a weaker condition than the one stated above, is indeed equivalent. To be speciﬁc, if there exist ε > 0 and s, 0 < s < 1, such that Fε ∩ D(z, s) = ∅ for all z in D, then Cϕ is closed-range on B. One of the ﬁrst results of this paper adds one more equivalent condition to this list, and we give a brief and rather novel proof that each of the three conditions are equivalent to Cϕ being closed-range on B; see Theorem 2.2. We then turn to connections between the Bloch and Bergman space settings. In the Bergman space setting there is an analogue of Λε that takes center stage. Indeed, if ϕ is 1−|z|2 an analytic self-map of D and ε > 0, then we let Ωε = {z ∈ D : 1−|ϕ(z)| 2 > ε} and let Gε = ϕ(Ωε ). In [1] it is shown that Cϕ is closed-range on A2 if and only if there exists ε > 0 such that Gε satisﬁes the reverse Carleson condition; that is, there exist s and c, 0 < s, c < 1, such that A(Gε ∩ D(z, s)) ≥ cA(D(z, s)), for all z in D. Here, we establish an extension of the Julia-Carath´eodory Theorem (see Theorem 3.4) and use it to show that if Cϕ is closed-range on A2 , then there exist ε and s, 0 < ε, s < 1, such that {z : s ≤ |z| < 1} ⊆ Fε ; see Theorem 3.5. From this we easily have the implication that if Cϕ is closedrange on A2 , then it is also closed-range on B; see Corollary 3.6. We also (by examples) show that the converse of Corollary 3.6 fails, without remedy. Indeed, we construct a thin Blaschke product that ﬁxes zero and that has no angular derivative anywhere on T, whence CB is norm preserving on B and

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yet is compact on A2 ; see Example 3.8. And we also construct a univalent analytic self-map h of D that has no unimodular nontangential boundary values on T, and thus has no angular derivative anywhere on T (whence, Ch is compact on A2 ), such that Ch is closed-range on B; see Example 3.10. We close the paper with a result that follows easily from work done in [1] and a remark concerning Fredholm operators; see Sect. 4.

2. Regarding the Bloch Space Recall that, for any analytic self-map ϕ of D and any ε > 0, τϕ (z) :=

(1 − |z|2 )ϕ (z) , 1 − |ϕ(z)|2

and

Λε := {z ∈ D : |τϕ (z)| > ε}.

Lemma 2.1. For any ε > 0 there exist r and s, 0 < r, s < 1, such that if z ∈ Λε , then i) D(z, r) ⊆ Λ 2ε , ii) ϕ is univalent in D(z, r) and iii) D(ϕ(z), s) ⊆ ϕ(D(z, r)). Proof. (i) By [8], τϕ is Lipschitz with respect to the pseudohyperbolic metric. Indeed, there is a positive constant c, independent of ϕ and of z and w in D, such that |τϕ (z) − τϕ (w)| ≤ cρ(z, w). Let r =

ε 2c

and suppose that |τϕ (w)| ≤ 2ε . Then, for z in Λε ,

ε < ||τϕ (z)| − |τϕ (w)|| ≤ |τϕ (z) − τϕ (w)| ≤ cρ(z, w). 2 ε , then w ∈ Λ 2ε . Therefore, if z ∈ Λε and ρ(z, w) < 2c (ii) Suppose that a ∈ Λε and α := ϕ(a). Notice that ϕα ◦ ϕ ◦ ϕa is an analytic self-map of the unit disk that maps 0 to 0 and that |(ϕα ◦ ϕ ◦ ϕa ) (0)| = |τϕα ◦ϕ◦ϕa (0)| = |τϕ◦ϕa (0)| = |τϕ (a)| > ε. We argue that ϕα ◦ ϕ ◦ ϕa is univalent in {z : |z| < r}; where, as in (i), ε . Multiplying ϕα ◦ ϕ ◦ ϕa by an appropriate unimodular constant we r := 2c may assume that (ϕα ◦ ϕ ◦ ϕa ) (0) is a positive real number (greater than ε). And using the facts that τϕα ◦ϕ◦ϕa is Lipschitz with respect to the pseudohyperbolic metric, with the same Lipschitz constant c, and that ϕα ◦ ϕ ◦ ϕa maps 0 to 0, we ﬁnd that ε (2.1.1) Re((ϕα ◦ ϕ ◦ ϕa ) (z)) > , 2 whenever |z| < r. Now let z and w be distinct points both of which have modulus less than r, and deﬁne γ on [0, 1] by γ(t) = (1 − t)z + tw. Then, by (2.1.1), 1 0 = (w − z) · 0

(ϕα ◦ ϕ ◦ ϕa ) (γ(t))dt = (ϕα ◦ ϕ ◦ ϕa )(w) − (ϕα ◦ ϕ ◦ ϕa )(z),

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and hence ϕα ◦ ϕ ◦ ϕa is univalent in {z : |z| < r}. It now follows that ϕ is univalent in D(a, r). 1 (iii) Given the terminology of part (ii), h(z) := rε (ϕα ◦ ϕ ◦ ϕa )(rz) is analytic and univalent in D, h(0) = 0 and |h (0)| > 1. Therefore, by the Koebe One-Quarter Theorem (cf., [13, page 154]), 1 z : |z| < ⊆ h(D). 4 From this it follows that rε ⊆ (ϕα ◦ ϕ ◦ ϕa )({z : |z| < r}). z : |z| < 4 With s := rε 4 we then ﬁnd that D(ϕ(a), s) ⊆ ϕ(D(a, r)).

2

)ϕ (z) As before, let ϕ be an analytic self-map of D, let τϕ (z) = (1−|z| 1−|ϕ(z)|2 and for ε > 0, let Λε = {z ∈ D : |τϕ (z)| > ε} and let Fε = ϕ(Λε ). We now give two conditions, each of which is equivalent to Cϕ being closed-range on B; cf., [9] and [3], or Theorem 2.2 below. (∗) There exist ε > 0 and constants c and s, 0 < c, s < 1, such that A(Fε ∩ D(z, s)) ≥ cA(D(z, s)) for all z in D. (#) There exist ε > 0 and s, 0 < s < 1, such that Fε ∩ D(z, s) = ∅ for all z in D.

Theorem 2.2. Let ϕ be an analytic self-map of D. Then the following are equivalent. i) Cϕ is closed-range on B. ii) Condition (∗) holds. iii) Condition (#) holds. iv) There are constants r, s and c, 0 < r, s, c < 1, such that, for any w in D, there exists zw in D with the property: ϕ is univalent on D(zw , s), ϕ(D(zw , s)) ⊆ D(w, r) and A(ϕ(D(zw , s))) ≥ c(1 − |w|2 )2 . Proof. (i) =⇒ (iii). Since any Frostman shift of ϕ (i.e., ϕα ◦ ϕ, where α ∈ D) gives rise to a closed-range composition operator on B if and only if ϕ does, we may assume that ϕ(0) = 0. Now suppose that (iii) does not hold. Then we can ﬁnd sequences {rn }∞ n=1 , where 0 < rn < 1 and limr→∞ rn = 1, and in D, where lim |wn | = 1, such that {wn }∞ n→∞ n=1 sup{|τϕ (z)| : z ∈ ϕ−1 (D(wn , rn ))} −→ 0, as n → ∞. Let Δn = ϕ−1 (D(wn , rn )) and let Dn = D\Δn ; for n = 1, 2, 3, ... . Now ||ϕwn ◦ ϕ||B/C := sup{(1 − |z|2 )|(ϕwn ◦ ϕ) (z)| : z ∈ D} = sup{[1 − ρ2 (wn , ϕ(z))]|τϕ (z)| : z ∈ D} ≤ sup{[1 − ρ2 (wn , ϕ(z))]|τϕ (z)| : z ∈ Δn } + sup{[1 − ρ2 (wn , ϕ(z))]|τϕ (z)| : z ∈ Dn } −→ 0, as n → ∞. Yet ||ϕwn ||B/C = 1, for all n. By Theorem 0 of [9] it now follows that Cϕ is not closed-range on B.

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(iii) =⇒ (ii). We assume (iii), that (#) holds. Then, by Lemma 2.1, (∗) holds for 2ε . (ii) =⇒ (i). This follows immediately from Proposition 1 and Theorem 1 of [9]. At this point we have established the equivalence of (i), (ii) and (iii). (iii) =⇒ (iv). This follows immediately from Lemma 2.1. (iv) =⇒ (iii). Assuming (iv), |ϕ (z)|2 dA(z) ≥ c(1 − |w|2 )2 , D(zw ,s)

and hence

|ϕ (z)|2 dA(z) ≥ c. (1 − |w|2 )2

D(zw ,s)

Thus we can ﬁnd a positive constant ε, dependent only on r and s, such that |τϕ (z)|2 dA(z) ≥ ε2 A(D(zw , s)). D(zw ,s)

Therefore, |τϕ (z)| ≥ ε for some z in D(zw , s), and hence Fε ∩ D(w, r) = ∅ for each w in D; which gives us (iii). The proof is now complete. A special case of our next result is given by Theorem 2 of [9]; namely, the case that ϕ is a univalent, analytic self-map of D. As is indicated in the proof of Theorem 2.2, if f ∈ B, then ||f ||B/C := supz∈D (1 − |z|2 )|f (z)|. Corollary 2.3. Let ϕ be an analytic self-map of D. Then Cϕ is closed-range on B if and only if there exists δ > 0 such that, for all α in D, ||ϕα ◦ ϕ||B/C ≥ δ. Proof. We may assume that ϕ(0) = 0 here since any Frostman shift of ϕ gives rise to a closed-range composition operator on B if and only if ϕ does, and since the collection of analytic automorphisms of D forms a group under the operation of composition. Moreover, notice that ||ϕα ||B/C = 1 for all α in D. So, if Cϕ is closed-range on B, then, by Theorem 0 of [9], there exists δ > 0 such that ||ϕα ◦ ϕ||B/C ≥ δ for all α in D. Conversely, suppose that there exists δ > 0 such that ||ϕα ◦ ϕ||B/C ≥ δ for all α in D. Then, by Proposition 2 of [9], (iii) of Theorem 2.2 holds and hence Cϕ is closed-range on B.

3. The Context of A2 Versus that of B Let ϕ be an analytic self-map of D and, for ε > 0, let Ωε := {z ∈ D : 1−|z|2 eod1−|ϕ(z)|2 > ε}, let Gε = ϕ(Ωε ) and let K = T ∩ Ωε . By the Julia-Carath´ ory Theorem (cf., [13, page 57]), ϕ has an angular derivative at each point ξ in K, which we denote by ϕ (ξ). Indeed, ϕ (ξ) = ζξd, where ζ := ϕ(ξ) := ∠ limz→ξ ϕ(z) and d is given by 1 − |ϕ(z)| 1 − |ϕ(z)|2 d := lim inf . = lim inf z→ξ z→ξ 1 − |z| 1 − |z|2

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The Julia-Carath´eodory Theorem tells us that d > 0. And since ξ ∈ K, d ≤ 1ε . Proposition 3.1. Given the terminology of the above discussion, ϕ is continuous on Ωε and ϕ is continuous on K. Proof. The continuity of ϕ on Ωε was established in [1]; see Remark 2.6 in this reference. Now let {ξn }∞ n=1 be a sequence in K that converges to ξ0 in K, and let dn = |ϕ (ξn )|, for n = 0, 1, 2, . . .. Since ϕ is continuous on K, the continuity of ϕ on K will follow if we show that dn −→ d0 , as n → ∞. Now by the discussion just prior to this proposition, {dn }∞ n=1 is bounded. And so, passing to a subsequence if necessary, we may assume that dn −→ d, as n → ∞. Thus our goal here is to show that d = d0 . To this end, by the Julia-Carath´eodory Theorem we can ﬁnd a sequence {rn }∞ n=1 in (0, 1), such 1 n ξn )| | < , for n = 1, 2, 3, ... . Hence, that limn→∞ rn = 1 and |dn − 1−|ϕ(r 1−rn n 1−|ϕ(rn ξn )| ∞ {rn ξn }∞ }n=1 conn=1 is a sequence in D that converges to ξ0 and { 1−rn verges to d. Julia’s Theorem (cf., [13, page 63]) now tells us that d = d0 .

We now set the stage for two subsequent results. Discussion 3.2. For any point ξ in T and any θ, 0 < θ < π, we let S(ξ, θ) denote the interior of closed convex hull of {ξ} ∪ {z : |z| ≤ sin( θ2 )}. We call S(ξ, θ) the Stolz region based at ξ with vertex angle θ. For our purposes here it is sufﬁcient that we keep the vertex angles of our Stolz regions ﬁxed at π2 , though our arguments carry through for any ﬁxed θ in the aforementioned range. Let ϕ be an analytic self-map of D and, for ε > 0, let 1−|z|2 Ωε = {z ∈ D : 1−|ϕ(z)| 2 > ε} and let K = T ∩ Ωε . Deﬁne Wε by

Wε = S(ξ, π2 ). ξ∈K

{zn }∞ n=1

Suppose that is a sequence in Wε that converges to a point ξ0 in π K. So, we can ﬁnd a sequence {ξn }∞ n=1 in K such that zn ∈ S(ξn , 2 ) (for n = 1, 2, 3, . . .) and limn→∞ ξn = ξ0 . Now • ζn := ϕ(ξn ) := ∠ limz→ξn ϕ(z), and • ∠ limz→ξn ϕ (z) =: ϕ (ξn ) = ζn ξ n dn – the angular derivative of ϕ at ξn , where dn := |ϕ (ξn )|. By Proposition 3.1, ϕ (ξn ) −→ ϕ (ξ0 ) = ζ0 ξ 0 d0 , as n → ∞, where ζ0 := ϕ(ξ0 ) := ∠ limz→ξ0 ϕ(z) and d0 := |ϕ (ξ0 )|. Since 0 < d0 < ∞, we can ﬁnd 1 M > 1 such that M ≤ dn ≤ M for all n. Lemma 3.3. Assuming the terminology of Discussion 3.2, for any ε > 0, there exist s, 0 < s < 1, and N (in N) such that < ε, dn − 1−|ϕ(z)| 1−|z| whenever z ∈ S(ξn , π2 ), |z| > s and n ≥ N . ∞ Proof. If not, then we can ﬁnd d = d0 , a subsequence {ξnk }∞ k=1 of {ξn }n=1 ∞ and a sequence {zk }k=1 such that

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zk ∈ S(ξnk , π2 ) for all k, |zk − ξnk | −→ 0 and hence |zk − ξ0 | −→ 0 (as k → ∞), and (1 − |ϕ(zk )|)/(1 − |zk |) −→ d, as k → ∞. Julia’s Theorem this would then tell us that d = |ϕ (ξ0 )| = d0 ;

a contradiction.

Theorem 3.4. Assuming the terminology of Discussion 3.2, ϕ is continuous on W ε . Proof. Our proof here is based on Lemma 3.3 and some observations concerning the proof of the Julia-Carath´eodory Theorem in [13]. By Proposition 3.1, all we need to show is that, given the hypothesis of Discussion 3.2, ϕ (zn ) −→ ϕ (ξ0 ), as n → ∞. Claim A. For any ε > 0 there exist s, 0 < s < 1, and N (in N) such that ζn ξ¯n dn − ζn − ϕ(z) < ε, ξn − z whenever z ∈ S(ξn , π2 ), |z| > s and n ≥ N . To justify this claim we ﬁrst observe that, by Lemma 3.3, for any η > 0, there exist σ, 0 < σ < 1, and ν (in N) such that 2 dn − 1 − |ϕ(rξn )| < η and dn − 1 − |ϕ(rξn )| < η (3.4.1) 1−r 1 − r2 provided σ ≤ r < 1 and n ≥ ν. Mimicking the proof of JC (1) =⇒ JC (2) (in Sect. 4.5 of [13]), for n ≥ ν we carry the discussion to the right half-plane {w : Re(w) > 0}. Let ϕn and ψn be the M¨ obius transformations given by +z +z and ψn (z) := ζζnn −z . Deﬁne Φn and γn on {w : Re(w) > 0} ϕn (z) := ξξnn −z 1 by Φn (w) := (ψn ◦ ϕ ◦ ϕ−1 n )(w) and γn (w) := Φn (w) − cn w, where cn := dn . Now by (3.4.1), if n ≥ ν and σ ≤ r < 1, then 1 − |ϕ(rξn )| 1 − |ϕ(rξn )|2 , < dn + η 1−r 1 − r2 1+r ), and hence, by Julia’s Theorem and with wn,r := ϕn (rξn ) (= 1−r 1 − |ϕ(rξn )|2 (1 − r)2 1 Re(Φn (wn,r )) dn + η = ≤ < . 2 2 dn Re(wn,r ) 1−r |ζn − ϕ(rξn )| (dn − η)2 dn − η <

Therefore, if n is sufﬁciently large (allowing η to be sufﬁciently small), one can force Re(γn (wn,σ )) Re(wn,σ ) 1+σ , which to be less than any prescribed positive real number; and wn,σ = 1−σ clearly does not vary with n. We let wn,σ play the role of w0 in the proof of JC (1) =⇒ JC (2) (in Sect. 4.5 of [13]). And since the image under ϕ of any compact subset of D is a compact subset of D, {|γn (wn,σ )|}∞ n=1 is bounded. Thus, following through with the argument in [13], we ﬁnd that, for any

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τ > 0, there is a positive real number R such that if w ∈ ϕn (S(ξn , π2 )) and |w| > R, then γn (w) (3.4.2) w < τ, provided n is sufﬁciently large. Now, via the correspondence w = ϕn (z), routine calculations give that ζn − ϕ(z) w+1 ¯ = ξn ζn , Φn (w) + 1 ξn − z and hence,

γn (w) + 1 ξn − z ξn ζ¯n cn w = . − w + 1 ζn − ϕ(z) w+1

We now ﬁnd that Claim (A) follows from (3.4.2). Claim B. For any ε > 0 there exist s, 0 < s < 1, and N (in N) such that |ζn ξ¯n dn − ϕ (z)| < ε, whenever z ∈ S(ξn , π2 ), |z| > s and n ≥ N . Now Claim (B) follows directly from Claim (A) and the proof of JC (2) =⇒ JC (3) (in Sect. 4.6 of [13]). And by Claim (B) and the fact that ϕ (ξn ) −→ ϕ (ξ0 ), as n → ∞, we ﬁnd that ϕ (zn ) −→ ϕ (ξ0 ), as n → ∞; which completes our proof.

Theorem 3.5. Let ϕ be an analytic self-map of D. If Cϕ is closed-range on A2 , then there exist ε and s, 0 < ε, s < 1, such that {z : s ≤ |z| < 1} ⊆ Fε . Proof. Suppose that Cϕ is closed-range on A2 . Then there exists ε > 0 such that Gε := ϕ(Ωε ) satisﬁes the reverse Carleson condition; cf., [1]. In particular, T ⊆ Gε . So, for each point υ0 in T, we can ﬁnd a sequence {wn }∞ n=1 in Ωε such that {ϕ(wn )}∞ n=1 converges to υ0 . Passing to a subsequence if necessary, we may assume that {wn }∞ n=1 converges to some point ω0 in K := T ∩ Ωε . Therefore, by Julia’s Theorem, υ0 = ϕ(ω0 ) := ∠ limw→ω0 ϕ(w). Thus, ϕ(K) = T. We proceed indirectly and suppose that the conclusion of this the∞ orem fails. Then we can ﬁnd a sequence {zn }∞ n=1 in D\{0}, such that {|zn |}n=1 converges to 1 and sup{|τϕ (w)| : ϕ(w) = zn } −→ 0, {ξn }∞ n=1

(3.5.1)

as n → ∞. Since ϕ(K) = T, there exists in K such that ϕ(ξn ) = ζn := |zznn | , for n = 1, 2, 3, . . .. Passing to a subsequence if need be, we may assume that {ξn }∞ n=1 converges to some point ξ0 in K. Since, by Proposition 3.1, ϕ is continuous on K, indeed, continuous on Ωε , we ﬁnd that {ζn }∞ n=1 converges to ζ0 := ϕ(ξ0 ). Now, by Theorem 3.4 and its proof, there exist δ and s, 0 < δ, s < 1, and N in N such that |τϕ (z)| ≥ δ ,

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whenever z ∈ S(ξn , π2 ), |z| > s and n ≥ N . Moreover, by Claim (A) in the proof of Theorem 3.4 (that speaks to the conformality of ϕ at ξn ), we can ﬁnd σ, 0 < σ < 1, and ν in N such that π {rζn : σ ≤ r < 1} ⊆ ϕ({z ∈ S(ξn , ) : |z| > s}), 2 whenever n ≥ ν. Since zn ∈ {rζn : σ ≤ r < 1}, if n is sufﬁciently large, we ﬁnd that (3.5.1) above cannot occur; and our proof is complete. Our next result is an immediate consequence of Theorem 3.5 and Theorem 2.2; and so we state it without proof. Corollary 3.6. Let ϕ be an analytic self-map of D. If Cϕ is closed-range on A2 , then it is also closed-range on B. A slight modiﬁcation of the proof of Theorem 3.5 gives us the following rather surprising result. It also can be viewed as a byproduct of the nice behavior of ϕ on W ε , as indicated by Theorem 3.4. Theorem 3.7. Let ϕ be an analytic self-map of D. Then the following are equivalent. i) Cϕ is closed-range on A2 . ii) There exist ε, s and c, 0 < ε, s, c < 1, such that A(Gε ∩ D(z, s)) ≥ cA(D(z, s)), for all z in D. iii) There exist ε and s, 0 < ε, s < 1, such that {z : s ≤ |z| < 1} ⊆ Gε . Proof. The equivalence between (i) and (ii) was established in [1]. And clearly (iii) implies (ii). So we need only establish that (i) implies (iii). To this end, assume that Cϕ is closed-range on A2 and mimic the proof of Theorem 3.5, 1−|z|2 replacing |τϕ (z)| by 1−|ϕ(z)| 2 , throughout. The argument carries over with this modiﬁcation to gives us (iii). By Theorem 2.5 of [1], the only univalent analytic self-maps of D that give rise to closed-range composition operators on A2 are the analytic automorphisms of D. This is in contrast with the Bloch space setting. Indeed, if ψ is any conformal mapping from D one-to-one and onto D \ [0, 1), then Cψ is closed-range on B; cf., Example 2 of [9]. So, the converse of Corollary 3.6 fails. Our next two examples show that the converse fails with a vengeance. Our ﬁrst is an example of a thin Blaschke product B that ﬁxes zero and has no angular derivative at any point of the unit circle T; and by thin we mean that (1 − |an |2 )|B (an )| −→ 1, as n → ∞, where {an }∞ n=1 are the zeros of B. Therefore, CB is norm preserving on B (cf., [5], or [11]) and yet is compact on A2 (cf., [13, pages 52 and 195]). And since CB is compact and not of ﬁnite rank on A2 , it is not closed-range on A2 . This ﬁrst example is a factor of the one produced by J. Shapiro on page 185 of [13]. Example 3.8. Let B ∗ be the Blaschke product constructed by J. Shapiro on ∗ page 185 of [13] and let {an }∞ n=1 be the zeros of B . Associated with each an 1 iθ is an arc In of length n of the form In = {e : θn ≤ θ ≤ θn+1 }. The zeros an

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are given by: an := rn eiωn , where rn := 1 − n12 and ωn := 12 (θn + θn+1 ). A theorem of O. Frostman (cf., [13, page 183]) is then used to show that B ∗ has no angular derivative anywhere on T. For each positive integer ν, we deﬁne the ν th “layer” of zeros of B ∗ as [aν ] := {aν , aν+1 , . . . , aNν }, where Nν is the unique positive integer that satisﬁes: T⊆

Nν

In , yet T ⊆

n=ν

N

ν −1

In .

n=ν

Nν −1 1 Since n=ν n < 2π, it follows that Nν < 540ν. For any positive integer ν, let Bν be the Blaschke product with (simple) zeros [aν ]. For any ak in [aν ], let Bˆνk denote Bν with the Blaschke factor involving ak deleted. And choose ak∗ in [aν ] \ {ak } such that ρ(ak , ak∗ ) ≤ ρ(ak , al ), whenever al ∈ [aν ] \ {ak }. Then, for such l, ak − al 2 (1 − rk2 )(1 − rk2∗ ) −1≥− . 1 − a¯l ak 1 − 2rk rk∗ cos(θk − θk∗ ) + rk2 rk2∗ Now, |θk − θk∗ | ≥

1 4k

and so, for ν sufﬁciently large,

1 − 2rk rk∗ cos(θk − θk∗ ) + rk2 rk2∗ ≥ Hence,

1 . 20k 2

ak − al 2 80 80 1 − a¯l ak − 1 ≥ − (k ∗ )2 ≥ − ν 2 ,

independent of k and l in our range here. Therefore, Nν

ak − al 2 0> 1 − a¯l ak − 1 k=l=ν

80 43, 200 −→ 0, )=− 2 ν ν as ν → ∞; uniformly in k, ν ≤ k ≤ Nν . From this it follows that ≥ (540ν)(−

|Bˆνk (ak )| −→ 1,

(3.7.1) ∗

as ν → ∞; uniformly in k, ν ≤ k ≤ Nν . Now since B is a Blaschke product, |Bν | −→ 1

(3.7.2)

uniformly on compact subsets of D, as ν → ∞. And since, for any ﬁxed ν, Bν is a ﬁnite Blaschke product, |Bν (z)| −→ 1

(3.7.3)

uniformly in z, as |z| → 1− . Using (3.7.1)–(3.7.3), one can ﬁnd a (rapidly) increasing sequence {νj }∞ j=1 of positive integers such that [aνk ] ∩ [aνl ] = ∅ if k = l, and such that B :=

∞ j=1

Bνj ,

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whose (simple) zeros we enumerate as {αn }∞ n=1 , satisﬁes |Bˆn (αn )| −→ 1, as n → ∞; where Bˆn denotes B with the Blaschke factor involving αn deleted. And we may assume that ν1 = 1. Hence, B is a thin Blaschke product that ﬁxes zero. Since the zeros of B consist of inﬁnitely many disjoint layers of the zeros of B ∗ , one can argue as in [13, page 185], and ﬁnd that ∞

1 − |αn | = ∞, |ζ − αn |2 n=1

for each ζ in T. Thus, by a theorem of O. Frostman (cf., [13, page 183]), we conclude that B has no angular derivative at any point in T. Remark 3.9. The converse of Theorem 3.5 does not hold. Indeed, by Theorem 2.7 of [4], if B is the Blaschke product that we produced in Example 3.8, then D ⊆ F 12 ; and yet CB is far from closed-range on A2 . We now produce a univalent analytic self-map h of D that has no angular derivative at any point of T (whence, Ch is compact on A2 ) such that Ch is closed-range on B. This dramatically improves upon our understanding of what is possible in the univalent case; cf., Example 2 of [9]. And since h(D) contains no annulus with outer boundary equal to T (and similarly for Example 2 of [9]), there is no analogue of Theorem 3.7 in the context of the Bloch space. Example 3.10. Here we construct a conformal mapping h from D one-to-one and onto an infinite ribbon G that spirals out to T such that Ch is closedrange on B. So h will have no unimodular nontangential boundary values on T, and thus no angular derivative anywhere on T. We write h as the composition of three conformal mappings: 1+z + e , which maps D univalently onto G1 := {ζ : Im(ζ) > e}, • ζ = i 1−z •

ξ = log(ζ), which maps G1 univalently onto a smoothly bounded subregion G2 of the swath {ξ : Re(ξ) > 1 and 0 < Im(ξ) < π} that asymptotically approximates this swath, and • w = ξ i , which maps G2 univalently onto an inﬁnite ribbon G that spirals out to T. i 1+z +e . Clearly h has no unimodular nontangenThus, h(z) = log i 1−z tial boundary values on T and thus has no angular derivative anywhere on T. As we noted just prior to Example 3.8, this tells us that Ch is compact and hence not closed-range on A2 . One may also refer to Theorem 2.5 of [1] to obtain that Ch is not closed-range on A2 . Now let Γ = h([0, 1)), which is an arc of inﬁnite length that spirals out to T. Our strategy in showing that Ch is closed-range on B is to ﬁrst establish that there exists ε > 0 such that Γ ⊆ Fε and then establish that there exists s, 0 < s < 1, such that

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Γ ∩ D(z, s) = ∅ for all z in D. Theorem 2.2 then gives us the conclusion. In what follows we use the symbol ∼ between real-valued functions f and g deﬁned on [0, 1) (viz., f ∼ g) to indicate that there is a constant M > 1 such 1 f (x) ≤ g(x) ≤ M f (x) for all x in [0, 1). Now, for x in [0, 1), that M

Denoting log

1+x 1−x

i 1+x h(x) = log i +e 1−x i 1+x iπ +e + = log . 1−x 2 + e + iπ 2 by ξx , we have: h(x) = ei log(ξx ) = e− arg(ξx ) · ei log |ξx | .

Hence, 1 − |h(x)| ∼ arg(ξx ) ∼

log

1 1+x 1−x

. +e

Thus, for x in [0, 1), 1−x ∼ (1 − x) log 1 − |h(x)| And, for such x, h (x) =

ei log(ξx ) 1+x log( 1−x +e)+ iπ 2

|h (x)| ∼

·

1+x +e . 1−x

2 (1−x2 )+e(1−x)2 ;

whence

1 . 1+x (1 − x) log 1−x +e

Evidently, |τh (x)| ∼ 1, and so there exists ε > 0 such that Γ ⊆ Fε . Now, as x increases to 1 in [0, 1), h(x) traverses Γ through inﬁnitely many counterclockwise rotations about 0 as it works its way toward T. To complete our argument here it is important that we obtain a good estimate on the ratio between 1 − |h(x )| and 1 − |h(x)|, if [x, x ] is a subinterval of [0, 1) over which h makes precisely one rotation about 0. Recalling that h(x) = e− arg(ξx ) · ei log |ξx | , we ﬁnd that this reduces to an examination of 1

h∗ (y) := e− y · ei log(y) , as y in [1, ∞) increases to ∞. Notice that h∗ winds through 2π radians on any ∗ 2π (e y)| subinterval of [1, ∞) of the form [y, e2π y]. And, independent of y, 1−|h 1−|h∗ (y)| 1 is boundedly equivalent to e2π . This then tells us that D \ Γ does not contain pseudohyperbolic disks of radius arbitrarily near 1. Hence, there exists s, 0 < s < 1, such that Γ ∩ D(z, s) = ∅, for all z in D. Since, as we have shown, Γ ⊂ Fε , for some ε > 0, we can now refer to Theorem 2.2 and conclude that Ch is closed-range on B.

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4. Closing Remarks In this ﬁnal section we give a result in the context of A2 for singular inner functions and we point out some implications of our work here to the theory of Fredholm operators. In our discussion we let m denote normalized Lebesgue measure on T. Recall that a compact subset E of T is said to be porous if there exists ε, 0 < ε < 1, such that whenever I is a arc of T with I ∩ E = ∅, then there is a subarc J of I where m(J) > εm(I) and J ∩ E = ∅. In [12] it is shown that E is a porous subset of T if and only if E has the property: For any singular measure μ supported on E, every nontrivial Frostman shift of the singular inner function Sμ is a Carleson–Newman Blaschke product; that is, a ﬁnite product of interpolating Blaschke products. The proof of Corollary 3.11 in [1] also establishes our next result. Proposition 4.1. Let E be a porous subset of T. If μ is any singular measure with support in E, then CSμ is closed-range on A2 . Remark 4.2. We close the paper with some thoughts concerning Fredholm operators. We ﬁrst recall that the little Bloch space B0 is the collection of functions f in B for which lim sup (1 − |z|2 )|f (z)| = 0.

r→1 r<|z|<1

And the Dirichlet space D is the collection of functions f (z) = analytic in D, such that ||f ||2D :=

∞

∞ n=0

an z n ,

(n + 1)|an |2 < ∞.

n=0

An operator between two Banach spaces is called a Fredholm operator if its range is closed and both the operator and its adjoint have ﬁnite dimensional kernel. If ϕ is an analytic self-map of D and Cϕ is a Fredholm operator on a Hilbert space of analytic functions that contains D, then ϕ is a disk automorphism; cf., [6, page 153]. Now D ⊆ B0 , but we will show that the situation is different for B0 . Indeed, there exists Fredholm composition operators on B0 whose symbols are not disk automorphisms. The minimal Besov space B1 is the collection of all functions f that are analytic in D of the form f (z) = a0 +

∞

an ϕwn (z),

(4.2.1)

n=1 ∞ 1 where {wn }∞ n=1 ⊆ D, and {an }n=1 ∈ l . The norm on B1 is given by ∞

||f ||B1 := inf |an | : (4.2.1) holds . n=0

Now B1 is a Banach space with respect to this norm and is invariant under disk automorphisms. Under the pairing (f, g) = D f (z)g (z)dA(z), the dual

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of B0 is B1 and the dual of B1 is B; cf., [2]. Notice that, for g in B0 and w in D, 1 − |w|2 dA(z) = −(1 − |w|2 )g (w), (g, ϕw ) = − g (z) (1 − wz)2 D

and therefore, (g, Cϕ∗ (ϕw )) == −(1 − |w|2 ) (g ◦ ϕ) (w) = −τϕ (w)(g, ϕϕ(w) ). If w ∈ D, then Cϕ∗ (ϕw ) = −τϕ (w)ϕϕ(w) ,

(4.2.2)

and if |w| = 1, then ϕw = w and Cϕ∗ ϕw = 0.

(4.2.3)

By (4.2.2) and (4.2.3) it is easy to see that the kernel of Cϕ∗ : B1 → B1 consists of the constant functions. Also, a non-constant composition operator is always one-to-one, and therefore Cϕ : B0 −→ B0 will be a Fredholm operator if it is closed-range. It is shown in [9] that if ψ is a conformal mapping from D onto D \ [0, 1), then Cψ is bounded below on B. Any univalent self-map of D is in B0 , and thus ψ ∈ B0 and Cψ is a Fredholm operator on B0 . Acknowledgements The second author wishes to express gratitude to Wayne Smith and the University of Hawaii for the kind hospitality that was shown during a recent visit. And we thank the referee for comments that have led to an improvement of the exposition of this paper.

References [1] Akeroyd, J.R., Ghatage, P.G.: Closed-range composition operators on A2 . Illinois J. Math. 52, 533–549 (2008) [2] Arazy, J., Fisher, S., Peetre, J.: M¨ obius invariant function spaces. J. Reine Angew Math. 363, 110–145 (1985) [3] Chen, H.: Boundedness from below of composition operators on the Bloch spaces (English summary). Sci. China Ser. A 46, 838–846 (2003) [4] Cohen, J.M., Colonna, F.: Preimages of one-point sets of Bloch and normal functions. Mediterr. J. Math. 3, 513–532 (2006) [5] Colonna, F.: Characterisation of the isometric composition operators on the Bloch space. Bull. Austral. Math. Soc. 72, 283–290 (2005) [6] Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1995) [7] Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1982) [8] Ghatage, P.G., Zheng, D.: Hyperbolic derivatives and generalized Schwarz-Pick estimates. Proc. Amer. Math. Soc. 132, 3309–3318 (2004)

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[9] Ghatage, P.G., Zheng, D., Zorboska, N.: Sampling sets and closed-range composition operators on the Bloch space. Proc, Amer. Math. Soc. 133, 1371– 1377 (2005) [10] Luecking, D.H.: Inequalities on Bergman Spaces. Illinois J. Math. 25, 1– 11 (1981) [11] Mart´ın, M.J., Vukoti´c, D.: Isometries of the Bloch space among the composition operators. Bull. Lond. Math. Soc. 39, 151–155 (2007) [12] Mortini, R., Nicolau, A.: Frostman shifts of inner functions. J. Anal. Math. 92, 285–326 (2004) [13] Shapiro, J.H.: Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics. Springer-Verlag, New York (1993) John R. Akeroyd (B) Department of Mathematics University of Arkansas Fayetteville, AR 72701, USA e-mail: [email protected] Pratibha G. Ghatage Department of Mathematics Cleveland State University Cleveland, OH 44115, USA e-mail: [email protected] Maria Tjani Department of Mathematics University of Arkansas Fayetteville, AR 72701, USA e-mail: [email protected] Received: February 5, 2010. Revised: April 19, 2010.

Integr. Equ. Oper. Theory 68 (2010), 519–527 DOI 10.1007/s00020-010-1807-6 Published online June 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

On the Convexity of the Heinz Means Fuad Kittaneh Abstract. Let A, B, and X be operators on a complex separable Hilbert space such that A and B are positive, and let 0 ≤ v ≤ 1. The Heinz inequalities assert that for every unitarily invariant norm |||·||| , 2 A1/2 XB 1/2 ≤ Av XB 1−v + A1−v XB v ≤ |||AX + XB||| . Using the convexity of the function f (v) = Av XB 1−v + A1−v XB v on [0, 1], we obtain several refinements of these norm inequalities and we investigate their equality conditions. Mathematics Subject Classification (2010). 47A30, 47B15, 26D15. Keywords. Heinz mean, Heinz inequality, convex function, Hermite-Hadamard inequality, unitarily invariant norm.

1. Introduction For two nonnegative real numbers a and b, the Heinz mean in the parameter v, 0 ≤ v ≤ 1, is defined as av b1−v + a1−v bv . 2 Note that H0 (a, b) = H1 (a, b) = a+b 2 (the arithmetic mean of a and b) and √ H1/2 (a, b) = ab (the geometric mean of a and b). It is easy to see that as a function of v, Hv (a, b) is convex, attains its minimum at v = 12 , and attains its maximum at v = 0 and v = 1. Moreover, Hv (a, b) = H1−v (a, b) for 0 ≤ v ≤ 1. Thus, the Heinz mean interpolates between the geometric mean and the arithmetic mean: √ a+b for 0 ≤ v ≤ 1. ab ≤ Hv (a, b) ≤ 2 The operator version of this asserts that if A, B, and X are operators on a complex separable Hilbert space such that A and B are positive, for every unitarily invariant norm |||·||| , the function f (v) = v then A XB 1−v + A1−v XB v is convex on [0, 1], attains its minimum at v = 1 , 2 and attains its maximum at v = 0 and v = 1. Moreover, f (v) = f (1 − v) Hv (a, b) =

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for 0 ≤ v ≤ 1. Thus, for every unitarily invariant norm, we have the Heinz inequalities 2 A1/2 XB 1/2 ≤ Av XB 1−v + A1−v XB v ≤ |||AX + XB||| . (1.1) For comprehensive accounts on these norm inequalities including history, different proofs, refinements, and diverse applications, we refer to [1,2, 4,5], and references therein, which contain the proof ofthe mentioned properties of the function f (v) = Av XB 1−v + A1−v XB v on [0, 1]. In this note we utilize the convexity of the function f (v) = Av XB 1−v + A1−v XB v on [0, 1] to obtain new refinements of the inequalities (1.1). Our analysis enables us to discuss the equality conditions in (1.1 ) for certain unitarily invariant norms. When we consider |||T |||, we are implicitly assuming that the operator T belongs to the norm ideal associated with |||·|||.

2. Main Results The following lemma, which includes a basic property of convex functions, plays a central role in our investigation. It is known as the Hermite–Hadamard integral inequality for convex functions (see, e.g., [3, p. 122]). Lemma 1. Let g be a real-valued function which is convex on the interval [a, b]. Then g

a+b 2

1 ≤ b−a

b g(t)dt ≤ a

g(a) + g(b) . 2

Applying Lemma 1 to the function f (v) = Av XB 1−v + A1−v XB v on the interval [µ, 1 − µ] when 0 ≤ µ < 12 , and on the interval [1 − µ, µ] when 1 2 < µ ≤ 1, we obtain our first refinement of the first inequality in (1.1). Theorem 1. Let A, B, and X be operators such that A and B are positive. Then for 0 ≤ µ ≤ 1 and for every unitarily invariant norm, 1−µ 1 Av XB 1−v + A1−v XB v dv 2 A1/2 XB 1/2 ≤ |1 − 2µ| µ µ ≤ A XB 1−µ + A1−µ XB µ . (2.1) Proof. First assume that 0 ≤ µ < 12 . Then it follows by Lemma 1 that f

µ+1−µ 2

1 ≤ 1 − 2µ

1−µ

f (v)dv ≤ µ

f (µ) + f (1 − µ) , 2

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and so 1−µ 1 1 f f (v)dv ≤ f (µ). ≤ 2 1 − 2µ µ

Thus, 1−µ v 1 A XB 1−v + A1−v XB v dv 1 − 2µ µ µ (2.2) ≤ A XB 1−µ + A1−µ XB µ .

2 A1/2 XB 1/2 ≤

Now, assume that 12 < µ ≤ 1. Then it follows by symmetry (i.e., by applying (2.2) to 1 − µ) that µ v 1 A XB 1−v + A1−v XB v dv 2µ − 1 1−µ µ ≤ A XB 1−µ + A1−µ XB µ . (2.3)

2 A1/2 XB 1/2 ≤

Since

1−µ v 1 A XB 1−v + A1−v XB v dv = 2 A1/2 XB 1/2 , lim1 µ→ 2 |1 − 2µ| µ

the inequalities in (2.1) follow by combining the inequalities ( 2.2) and (2.3). We remark here that for 0 < µ < 12 , the second inequality in (2.1) has been obtained in [4, Proposition 3.2] using a completely different method. 1−v v Applying Lemma 1 to the function f (v) = Av XB 1−v + A 1 XB 1 1 on the interval [µ, 2 ] when 0 ≤ µ < 2 , and on the interval 2 , µ when 1 2 < µ ≤ 1, we obtain the following theorem. Theorem 2. Let A, B, and X be operators such that A and B are positive. Then for 0 ≤ µ ≤ 1 and for every unitarily invariant norm, 2µ+1 3−2µ 3−2µ 2µ+1 A 4 XB 4 + A 4 XB 4 1/2 2 v ≤ A XB 1−v + A1−v XB v dv |1 − 2µ| µ 1 ≤ Aµ XB 1−µ + A1−µ XB µ + A1/2 XB 1/2 . (2.4) 2 The inequality (2.4) and the first inequality in (1.1) yield the following refinement of the first inequality in (1.1).

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Corollary 1. Let A, B, and X be operators such that A and B are positive. Then for 0 ≤ µ ≤ 1 and for every unitarily invariant norm, 2µ+1 3−2µ 3−2µ 2µ+1 2 A1/2 XB 1/2 ≤ A 4 XB 4 + A 4 XB 4 1/2 2 v ≤ A XB 1−v + A1−v XB v dv |1 − 2µ| µ 1 ≤ Aµ XB 1−µ + A1−µ XB µ + A1/2 XB 1/2 2 ≤ Aµ XB 1−µ + A1−µ XB µ . (2.5) It should be noticed here that in the inequalities (2.4) and (2.5), 1/2 1/2 v 1 1−v 1−v v limµ→ 12 |1−2µ| µ A XB dv = A XB 1/2 . + A XB Applying Lemma 1 to the function f (v) = Av XB 1−v + A1−v XB v on the interval [0, µ] when 0 < µ ≤ 12 , and on the interval [µ, 1], when 1 2 ≤ µ < 1, we obtain the following related theorem. Theorem 3. Let A, B, and X be operators such that A and B are positive. Then (a)

for 0 ≤ µ ≤ 12 and for every unitarily invariant norm, µ/2 A XB 1−µ/2 + A1−µ/2 XB µ/2 1 ≤ µ

µ

v A XB 1−v + A1−v XB v dv

0

1 1 ≤ |||AX + XB||| + Aµ XB 1−µ + A1−µ XB µ . 2 2 (b)

for

1 2

(2.6)

≤ µ ≤ 1 and for every unitarily invariant norm, 1+µ 1−µ 1−µ 1+µ 2 XB 2 + A 2 XB 2 A ≤

1 1−µ

1

v A XB 1−v + A1−v XB v dv

µ

1 1 (2.7) ≤ |||AX + XB||| + Aµ XB 1−µ + A1−µ XB µ . 2 2 In view of the fact that the function f (v) = Av XB 1−v + A1−v XB v is decreasing on the interval [0, 12 ] and increasing on the interval [ 12 , 1], and based on the inequalities (2.6) and (2.7), we obtain our first refinement of the second inequality in (1.1). Corollary 2. Let A, B, and X be operators such that A and B are positive. Then

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On the Convexity of the Heinz Means

for 0 ≤ µ ≤ 12 and for every unitarily invariant norm, µ A XB 1−µ + A1−µ XB µ ≤ Aµ/2 XB 1−µ/2 + A1−µ/2 XB µ/2 1 ≤ µ

µ

v A XB 1−v + A1−v XB v dv

0

1 1 ≤ |||AX + XB||| + Aµ XB 1−µ + A1−µ XB µ 2 2 ≤ |||AX + XB||| . (b)

523

for

1 2

(2.8)

≤ µ ≤ 1 and for every unitarily invariant norm, µ A XB 1−µ + A1−µ XB µ 1+µ 1−µ 1−µ 1+µ ≤ A 2 XB 2 + A 2 XB 2 1 ≤ 1−µ

1

v A XB 1−v + A1−v XB v dv

µ

1 1 ≤ |||AX + XB||| + Aµ XB 1−µ + A1−µ XB µ 2 2 ≤ |||AX + XB||| .

(2.9)

It should noticed here that in the inequalities (2.6)–(2.9), 1 lim µ→0 µ

µ

v A XB 1−v + A1−v XB v dv

0

1 = lim µ→1 1 − µ

1

v A XB 1−v + A1−v XB v dv

µ

= |||AX + XB||| . We conclude this section with a norm inequality that leads to another refinement of the second inequality in (1.1). A special case of this inequality for the Hilbert–Schmidt norm has been obtained in [7, Theorem 3.4] using a different technique. Theorem 4. Let A, B, and X be operators such that A and B are positive. Then for 0 ≤ v ≤ 1 and for every unitarily invariant norm, v A XB 1−v + A1−v XB v ≤ 4r0 A1/2 XB 1/2 + (1 − 2r0 ) |||AX + XB||| ,

(2.10)

where r0 = min{v, 1 − v}. Proof. For v = 0, 12 , 1, theinequality (2.10) is obvious. First assume that 1 v 1−v 1−v v is convex on [0, 1], it 0 < v < 2 . Since f (v) = A XB + A XB

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follows by a slope argument that f ( 12 ) − f (0) f (v) − f (0) ≤ , 1 v−0 2 −0 and so

1 2v f (0) − f ≤ f (0) − f (v). 2

(2.11)

Now, assume that 12 < v < 1. Then it follows by symmetry (i.e., by applying (2.11) to 1 − v) that 1 2(1 − v) f (0) − f ≤ f (0) − f (v). (2.12) 2 Using (2.11) and (2.12), we obtain 1 ≤ f (0) − f (v) for 0 ≤ v ≤ 1, 2r0 f (0) − f 2 and so

2r0 (|||AX + XB||| − 2 A1/2 XB 1/2 ) ≤ |||AX + XB||| − Av XB 1−v + A1−v XB v ,

which is equivalent to the inequality (2.10).

(2.13)

In view of the inequalities (1.1) and (2.10), we have the following refinement of the second inequality in (1.1). Corollary 3. Let A, B, and X be operators such that A and B are positive. Then for 0 ≤ v ≤ 1 and for every unitarily invariant norm, v A XB 1−v + A1−v XB v ≤ 4r0 A1/2 XB 1/2 + (1 − 2r0 ) |||AX + XB||| ≤ |||AX + XB||| , where r0 = min{v, 1 − v}.

3. Equality Conditions Our simple proof of the operator arithmetic-geometric mean inequality [the case v = 12 of (1.1)] given in [6] allowed us to investigate the case when this inequality, for the Schatten p-norms with 1 < p < ∞, degenerates to equality. Recall that for 1 ≤ p ≤ ∞, the Schatten p-norm of an operator A is defined p as Ap = (tr|A| )1/p , where tr is the trace functional and |A| = (A∗ A)1/2 is the absolute value of A, which is positive. The Schatten p-norms are important examples of unitarily invariant norms, where p = 1, p = 2, and p = ∞ correspond to the trace norm, the Hilbert–Schmidt norm, and the spectral (usual operator) norm, respectively. If A, B, and X are operators such that A and B are positive and AX = XB, then it follows by the spectral therorem that Av X = XB v for 0 ≤ v ≤ 1,

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and so 2A1/2 XB 1/2 = Av XB 1−v + A1−v XB v = AX + XB. Thus, if AX = XB, then equality holds through the inequalities (1.1 ). It turns out that for the Schatten p-norms with 1 < p < ∞, this condition is necessary and sufficient for equality to hold in the second inequality of (1.1). If, in addition, A and B are invertible, then this is the case for the first inequality of (1.1). To achieve this, we need the following two lemmas from [6]. Lemma 2. Let A and B be operators such that AB is self-adjoint, and let 1 < p < ∞. Then ABp = Re(BA)p if and only if BA is self-adjoint. Lemma 3. Let A, B, and X be operators such that A and B are positive, and let 1 < p < ∞. Then 2 A1/2 XB 1/2 = AX + XBp p

if and only if AX = XB. Lemmas 2 and 3 are not true for the extreme cases p = 1 and p = ∞, as it has been demonostrated in [6]. Based on the inequality (2.13) and Lemma 3, we have the following theorem involving an equality condition of the second inequality of (1.1). Theorem 5. Let A, B, and X be operators such that A and B are positive, and let 1 < p < ∞. Then v A XB 1−v + A1−v XB v = AX + XB for some v with 0 < v < 1 p

p

if and only if AX = XB. For the special case p = 2, which corresponds to the Hilbert–Schmidt norm, this result has been inferred in [7] using the improved Heinz inequality v A XB 1−v + A1−v XB v 2 + 2r0 AX − XB2 ≤ AX + XB2 2 2 2 for 0 ≤ v ≤ 1, where r0 = min{v, 1 − v}. Finally, we employ Lemma 2 to investigate the equality condition in the first inequality of (1.1) when A and B are positive and invertible. Theorem 6. Let A, B, and X be operators such that A and B are positive and invertible, and let 1 < p < ∞. Then 2 A1/2 XB 1/2 = Av XB 1−v + A1−v XB v p for some v with 0 < v < 1, p

v =

1 2

if and only if AX = XB.

A 0 0 B

Proof. It is enough to prove the “only if” part. Let T = and

0 X Y = . Then T is positive and invertible, Y is self-adjoint, X∗ 0

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A1/2 XB 1/2 T YT = , and T v Y T 1−v + T 1−v Y T v = 0

0 Av XB 1−v + A1−v XB v . B v X ∗ A1−v + B 1−v X ∗ Av 0 1/2 Hence, T Y T 1/2 p = 21/p A1/2 XB 1/2 p and T v Y T 1−v + T 1−v Y T v p = 21/p Av XB 1−v + A1−v XB v p . Thus, if 2 A1/2 XB 1/2 = Av XB 1−v + A1−v XB v for some v with 1/2

1/2

0 1/2 ∗ 1/2 B X A

p

p

0 < v < 1, v = 12 , then 2 T 1/2 Y T 1/2 = T v Y T 1−v + T 1−v Y T v p , p

and so

1/2 T Y T 1/2 = Re(T v Y T 1−v ) p . p

1/2−v

v

1/2

1/2

1/2 1/2−v Since T (T Y T ) = T Y T (T v Y T 1/2 = is self-adjoint, )T v 1−v 1/2−v v 1/2 v 1/2 1/2−v T YT , and T (T Y T ) p = Re((T Y T )T ) p , it follows by Lemma 2 that T v Y T 1−v is self-adjoint, and so T v Y T 1−v = T 1−v Y T v . Since T is invertible, it follows that T 1−2v Y = Y T 1−2v , and hence it follows, by the spectral theorem, that T Y = Y T . This implies that AX = XB, as required.

It should be mentioned here that the invertibility assumption in Theorem 6is essential. To see

this, consider the two-dimensional example A = 0 1 1 0 and X = . Then 2A1/2 XA1/2 = Av XA1−v + A1−v XAv for 0 0 1 0 all v > 0 , but AX = XA. Acknowledgements Part of this work was done while the author was visiting the Institute of Mathematics of the Polish Academy of Sciences. This visit was sponsored by the European Community under the Marie Curie Host Fellowship of the Transfer of Knowledge: TODEQ (MTKD-CT-2005-030042).

References [1] Bhatia, R., Davis, C.: More matrix forms of the arithmetic-geometric mean inequality. SIAM J. Matrix Anal. Appl. 14, 132–136 (1993) [2] Bhatia, R.: Positive Definite Matrices. Princeton University Press, New Jersey (2007) [3] Bullen, P.S.: A Dictionary of Inequalities, Pitman Monogarphs and Surveys in Pure and Applied Mathematics, vol. 97. Longman, Harlow (1998) [4] Hiai, F., Kosaki, H.: Means for matrices and comparison of their norms. Indiana Univ. Math. J. 48, 899–936 (1999) [5] Hiai, F., Kosaki, H.: Means of Hilbert space operators. In: Lecture Notes in Mathematics, vol. 1820. Springer, New York (2003)

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[6] Kittaneh, F.: A note on the arithmetic-gemetric mean inequality for matrices. Linear Algebra Appl. 171, 1–8 (1992) [7] Kittaneh, F., Manasrah, Y.: Improved Young and Heinz inequalities for matrices. J. Math. Anal. Appl. 361, 262–269 (2010) Fuad Kittaneh (B) Department of Mathematics University of Jordan Amman, Jordan e-mail: [email protected] Received: February 9, 2010. Revised: April 15, 2010.

Integr. Equ. Oper. Theory 68 (2010), 529–550 DOI 10.1007/s00020-010-1833-4 Published online October 22, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Non-Commutative Carath´eodory Interpolation Sriram Balasubramanian Abstract. We prove a Carath´eodory–Fej´er type interpolation theorem for certain matrix convex sets in Cd using the Blecher–Ruan–Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S. Kalyuzhny˘i-Verbovetzki˘i for the d-dimensional non-commutative polydisc. Mathematics Subject Classification (2010). Primary 47A57, 47L30; Secondary 47A13. Keywords. Interpolation, Carath´eodory, Carath´eodory–Fej´er, abstract operator algebra, BRS, matrix convex set, formal power series.

1. Introduction A classical interpolation problem in function theory is the Carath´eodory– Fej´er interpolation problem (CFP): Given n+1 complex numbers ∞ c0 , c1 , . . . , cn does there exist a complex valued analytic function f (z) = j=0 fj z j deﬁned on the open unit disc D ⊂ C such that fj = cj for all 0 ≤ j ≤ n and |f (z)| ≤ 1 for all z ∈ D? The problem and some of its variants were studied by Carath´eodory and Fej´er [6], Schur [23] and Toeplitz [24]. A necessary and sufﬁcient condition, commonly referred to as the Schur criterion, for the existence of a solution to the problem is that the matrix ⎞ ⎛ c0 0 ··· 0 ⎜ .. ⎟ .. .. ⎜ c1 . . .⎟ ⎟ ⎜ (1) ⎟ ⎜. .. .. ⎠ ⎝ .. . . 0 c1 c0 cn · · · is a contraction. A detailed exposition of the CFP and its numerous function-theoretic and engineering applications can be found in the comprehensive book of Foias and Frazho [12]. The operator theoretic formulation of Sarason [22] has had a major impact on the CFP and the related Pick interpolation

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problem and the development of operator theory and the study of non-self adjoint operator algebras generally. From the operator theory/algebra point of view, the CFP is essentially unchanged if the coefﬁcients c0 , c1 , . . . , cn are taken to be elements of B(U) for some separable Hilbert space U. Indeed, The Schur n criterion in this case can be formulated in the following way. Let p(z) = j=0 cj z j : D → B(U) be (j) given. There exists an analytic function f : D → B(U) nsuch that jf (0) = cj for 0 ≤ j ≤ n if and only if the norm of p(T ) = j=0 cj ⊗ T is at most one for every contractive operator T on a Hilbert space which is nilpotent of order n + 1, i.e., T n+1 = 0. Several commutative multi-variable generalizations of the CFP have been obtained for different domains for example, the polydisc Dd ⊂ Cd , and for different interpolating classes of functions, for example the SchurAgler class of analytic functions that take contractive operator values on any d-tuple of commuting strict contractions in a manner discussed in [1]. For more details see [4,11]. Some results on the problem for bounded circular domains in Cd can also be found in [8]. The broad purpose of this article is to extend some of the results in the commutative case to the non-commutative setting of the free algebra on d generators. Some non-commutative generalizations of the CFP have already been studied in [3,7,14,18,19]. Here we pose the problem for domains that are matrix convex sets (see [10]) in Cd . The domains considered here include as speciﬁc examples, the d-dimensional non-commutative matrix polydisc, which is the domain used in [14], the d-dimensional non-commutative matrix polyball and the dd -dimensional non-commutative matrix mixed ball. Using noncommutative matrix (operator) valued analytic functions on matrix-convex sets—formal power series with matrix (operator) coefﬁcients that converge on some non-commutative neighborhood of 0 (see [13,15,17–19,25–27]), the creation operators acting on the non-commutative Fock space and the Blecher– Ruan–Sinclair characterization of abstract operator algebras, we prove an interpolation theorem from which a necessary and sufﬁcient condition for the existence of a minimum-norm solution to the CFP follows. It is to be noted that Popescu [20,21] in his works on non-commutative Hardy algebras Fn∞ associated with the non-commutative unit ball of B(H)n , has used the left creation operators (S1 , . . . , Sn ) on the Full Fock Space as a universal model for row contractions and has a Carath´eodory–Fej´er type theorem in that setting. Also quotients of Hardy algebras have been considered and used before to obtain interpolation theorems. For the non-commutative Hardy algebras Fn∞ , the ﬁrst results along this line were obtained independently in [2,9]. The article is structured as follows: In Sect. 2, matrix convex sets in Cd and the interpolating class A(K)∞ are introduced and a basic version of the main result is stated. In Sect. 3, it is established that the interpolating class A(K)∞ is an abstract operator algebra. In Sect.4, a weak-compactness property of the algebra A(K)∞ is proved. Section 5 begins with the definition of the ideal I(K) ⊂ A(K)∞ which plays a role analogous to that of the ideal of analytic functions in H ∞ of the unit disc which vanish to order n at 0 in the classical CFP. This section also contains the discussion of why A(K)∞ /I(K)

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is an abstract operator algebra. It is also shown in this section that the norms of classes in the quotient algebra are attained. In Sect.6, completely contractive representations of the algebra A(K)∞ are studied and it is also shown that ﬁnite-dimensional compressions of operators that give rise to completely contacting representations of A(K)∞ lie on the boundary of the underlying matrix convex set. Section 7 contains the matrix and operator versions of the CFP for (certain) matrix-convex sets in Cd and the main interpolation theorem. The article ends with some examples for which a variant of the main interpolation theorem from Sect. 7 holds even in the case of inﬁnite initial segments Λ. See Sect. 8.

2. Matrix Convex Sets in Cd A basic object of study in this article is a quantized, or non-commutative, version of a convex set. While the definitions easily extend to convex subsets of arbitrary vector spaces, here the focus is on subsets of Cd . In this section we review the definition of a matrix convex subset of Cd and introduce our standard assumptions regarding these sets. 2.1. Non-Commutative Sets Let Mm,n = Mm,n (C) denote the m × n matrices over C. In the case that m = n, we write Mn instead of Mn,n . Let Mn (Cd ) denote d-tuples with entries from Mn . Thus, an X ∈ Mn (Cd ) has the form X = (X1 , . . . , Xd ) where each Xj ∈ Mn . A non-commutative set L is a sequence (L(n)) where, for each positive integer n, L(n) ⊂ Mn (Cd ) which is closed with respect to direct sums, i.e., if X ∈ L(n) and Y ∈ L(m), then X ⊕ Y = (X1 ⊕ Y1 , . . . , Xd ⊕ Yd ) ∈ L(n + m) where

Xj ⊕ Yj =

Xj 0

0 Yj

(2)

.

A non-commutative set L = (L(n)) is open if each L(n) is open. 2.2. Convexity A matrix convex set K = (K(n)) is a non-commutative set which is closed with respect to conjugation by an isometry, i.e., if α ∈ Mm,n and α∗ α = In , and if X = (X1 , . . . , Xd ) ∈ K(m), then α∗ Xα = (α∗ X1 α, . . . , α∗ Xd α) ∈ K(n).

(3)

Note that, by choosing n = m and α a unitary matrix, the condition (3) implies that each L(n) is closed with respect to unitary conjugation. It is a simple matter to combine conditions (2) and (3) to conclude, if L is matrix convex, then each L(n) is itself convex.

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2.3. Circled Domains A subset U of Mn (Cd ) is circled if eiθ U ⊆ U for all θ ∈ R. A matrix convex set K is circled if each K(n) is circled. As a canonical example of a circled matrix convex set, suppose γ > 0 and consider the non-commutative γ-neighborhood Cγ = (Cγ (n)) of 0 ∈ Cd deﬁned by Cγ (n) = {X ∈ Mn (Cd ) :

d

Xj Xj∗ < γ 2 }.

j=1

For a matrix convex set K, unless otherwise noted, it is assumed there exist γ, Γ > 0 such that Cγ ⊆ K ⊆ CΓ ,

(4)

where the inclusions are interpreted termwise. Equivalently, K is bounded (contained in some non-commutative neighborhood of 0) and contains a noncommutative neighborhood of 0. A reason why we need the ﬁrst inclusion in condition (4) is to get some multiples of compressions of the creation operators (see Sect. 3) in the matrix convex set K and thereby to deduce some facts about formal power series. The second inclusion in condition (4) is used (implicitly) to prove that the formal power series deﬁned as the product of two formal power series of ﬁnite norm, is of ﬁnite norm as well. See Lemma 2. Condition (4) is important also because it is satisﬁed by several non-commutative domains of interest. Assumption 1. In this article, it is typically assumed that K (a) is open; (b) is bounded; (c) is circled; (d) is matrix convex; and (e) contains a non-commutative neighborhood of 0. 2.4. Examples of Matrix Convex Sets At this point we pause to consider some further examples of open, bounded matrix convex sets which contain a non-commutative neighborhood of 0. Example 1. √ Let K(n) = {(X1 , X2 , . . . , Xd ) : Xj ∈ Mn & Xj < 1} with γ ≤ 1 and Γ ≥ d. K = (K(n)) is the d-dimensional non-commutative matrix polydisc. Example 2. Let K(n) = {X = (X1,1 , X1,2 , . . . , Xd,d ) : Xi,j ∈ Mn & X op < 1}, where X op is the norm of the operator (Xij )d,d : (Cn )d → (Cn )d with i,j=1 √ 1 γ ≤ √dd and Γ ≥ dd is the d × d non-commutative matrix mixed ball. 2.5. The Interpolating Class A(K)∞ Let K denote a matrix convex set satisfying the conditions of Assumption 1. A central object of this article is an algebra of formal power series in noncommuting variables which converge uniformly on the matrix convex set K. These power series are deﬁned in terms of the free semi-group on d letters.

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2.5.1. The Free Semi-Group on d Letters. Let Fd denote the free semigroup of words generated by d symbols g1 , . . . , gd . The product on Fd is deﬁned by concatenation. Thus, if w = gi1 . . . gim and w = gj1 . . . gjn , then the product ww is given by gi1 . . . gim gj1 . . . gjn . The empty word ∅ acts as the identity so that w∅ = w = ∅w. The length of the word w = gi1 . . . gim is m and is denoted |w|. The length of ∅ is zero. 2.5.2. The Set A(K)∞ of Formal Power Series. A formal power series with entries from C is an expression of the form fw w, (5) w∈Fd

where fw ∈ C (more general coefﬁcients fw will be considered later). It is convenient to sum f according to its homogeneous of degree j terms, i.e., f=

∞

fw w =

j=0 |w|=j

∞

fj .

(6)

j=0

Given a d-tuple T = (T1 , . . . , Td ) of operators on a common Hilbert space H and a word w = gi1 gi2 . . . gik ∈ Fd , i1 , . . . , ik ∈ {1, 2, . . . , d}, deﬁne the evaluation of w at T by T w = Ti1 Ti2 . . . Tik . Given a formal power series f as above, deﬁne f (T ) =

∞

fw T w

(7)

j=0 |w|=j

provided the sum converges in the operator norm in the indicated order. Convergence in norm is not terribly important here and it is possible to use instead convergence in the strong or weak operator topologies for instance, [3]. Fix now a matrix convex set K = (K(n)) which satisﬁes the conditions of Assumption 1. We will write X ∈ K to denote X ∈ n∈N K(n). Let A(K) denote the collection

f= fw w : fw ∈ C and ∀ X ∈ K, the series deﬁning f (X) converges . w∈Fd

For f ∈ A(K), deﬁne f = sup{ f (X) : X ∈ K}. Of course, as it stands, this supremum can be inﬁnite. Let

∞ A(K) = f = fw w : f ∈ A(K), f < ∞ . w∈Fd

Thus, elements of A(K)∞ are in some sense analogous to H ∞ functions on the unit disk D. It is not hard to see that · is in fact a norm on A(K)∞ and not just a semi-norm.

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It will be shown that A(K)∞ is an algebra and it will also be necessary—and desirable—to consider formal power series with matrix and operator-valued coefﬁcients. Discussion of these topics is postponed until after stating a base version of the main result of this paper. 2.6. The Main Result A set Λ ⊂ Fd is an initial segment if its complement is an ideal in the semi-group Fd , i.e., if both gj w, wgj ∈ Fd \Λ (1 ≤ j ≤ d), whenever w ∈ Fd \Λ. In the case that d = 1 an initial segment is thus a set of the form {∅, g1 , g12 , . . . , g1m } for some m. A tuple X ∈ K is Λ-nilpotent provided X v = 0 whenever v ∈ Fd \Λ. If Λ is a ﬁnite initial segment, X ∈ K is Λ-nilpotent, and f is as in Eq. (6), then fw X w . f (X) = w∈Λ

Theorem 1. Fix a matrix convex set K satisfying the conditions of Assumption 1. Let Λ, a finite initial segment, and pw w ∈ A(K)∞ p= w∈Λ

be given. There exists f ∈ A(K)∞ such that fw = pw for w ∈ Λ and f ≤ 1 if and only if sup{ p(X) : X ∈ K, X is Λ-nilpotent} ≤ 1. The body of the paper contains a more general version of Theorem 1 allowing for operator-valued coefﬁcients pw and fw . A version of the Theorem for the case of inﬁnite initial segments Λ is also presented in Sect. 8 for some speciﬁc non-commutative domains.

3. The Operator Algebra A(K)∞ Broadly speaking, the strategy for proving Theorem 1 is to realize A(K)∞ as an operator algebra, note that Λ determines a closed ideal in A(K)∞ and then apply the important corollary of the BRS theorem (see [16]) which says that the quotient of an operator algebra by a closed (two-sided) ideal has a completely isometric representation as a subalgebra of the bounded operators on some Hilbert space. The norm on A(K)∞ deﬁned in Sect. 2.5.2 naturally generalizes to m×n matrices with entries from A(K)∞ and the resulting sequence of norms makes A(K)∞ an abstract operator algebra. This section contains the details of the construction beginning with proving that A(K)∞ is an algebra. 3.1. The Non-Commutative Fock Space Let Cg = Cg1 , . . . , gd denote the algebra of non-commuting polynomials in the variables {g1 , . . . , gd }. Thus elements of Cg are linear combinations

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of elements of Fd , i.e., an element of Cg of degree (at most) k has the form k

pw w,

j=0 |w|=j

where the pw are complex numbers. To construct the Fock space, F2 , deﬁne an inner product on Cg by deﬁning

0 if w = v w, v = 1 if w = v for w, v ∈ Fd and extending by linearity to all of Cg. The completion of Cg in this inner product is then the Hilbert space F2 . 3.2. The Creation Operators There are natural isometric operators on F2 called the creation operators which have been studied intensely in part because of their connection to the Cuntz algebra [5]. Given 1 ≤ j ≤ d, deﬁne Sj : F2 → F2 by Sj v = gj v for a word v ∈ Fd and extend Sj by linearity to all of Cg. It is readily veriﬁed that Sj is an isometric mapping of Cg into itself and it thus follows that Sj extends to an isometry on all of F2 . In particular Sj∗ Sj = I, the identity on F2 . Also of note is the identity, d

Sj Sj∗ = P,

(8)

j=1

where P is the projection onto the orthogonal complement of the one-dimensional subspace of F2 spanned by ∅, which follows by observing, for a word w ∈ Fd and 1 ≤ j ≤ d, that

w if w = gj w ∗ Sj (w) = 0 otherwise. Of course, as it stands the tuple S = (S1 , . . . , Sd ) acts on the inﬁnite dimensional Hilbert space F2 . There are however, ﬁnite dimensional subspaces which are essentially determined by ideals in Fd and which are invariant for each Sj∗ . The subset Λ() = {w : |w| ≤ } of Fd is a canonical example of a ﬁnite initial segment. Moreover, since each Sj∗ leaves Λ() invariant, the subspace F()2 of F2 spanned by Λ() is invariant for every Sj∗ . Let V () denote the inclusion of F()2 into F2 and let S() denote the operator V ()∗ SV (). Thus, S() = ((S())1 , . . . , (S())d ) where (S())j = V ()∗ Sj V (). Observe, with P and P () denoting the projection of F2 and F()2 onto the orthogonal

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complement of the span of ∅ in F2 and F()2 , respectively, Eq. (8) yields P () =V ()∗ P V () ⎛ ⎞ d =V ()∗ ⎝ Sj Sj∗ ⎠ V () j=1

=

d

(S())j (S())∗j .

j=1

It follows, for 0 ≤ t < γ, that tS() ∈ Cγ (n), where n = dimension of F()2 .

j=0

dj is the

3.3. The Algebra A(K)∞ In addition to the obvious pointwise addition and multiplication by scalars, there is a natural multiplication on A(K)∞ extending multiplication of polynomials which then turns A(K)∞ into an algebra over C. Since it will be necessary to consider, in the sequel, matrices with entries from A(K)∞ and their products, we deﬁne them here. Let Mp,q (A(K)) denote the collection

fw w : fw ∈ Mp,q and ∀ X ∈ K, the series deﬁning f (X) converges , f= w∈Fd

where f (X) =

∞

fw ⊗ X w .

j=0 |w|=j

For f ∈ Mp,q (A(K)), let f = f p,q = sup{ f (X) : X ∈ K}. Deﬁne

∞

Mp,q (A(K) ) =

f=

(9)

fw w : f ∈ Mp,q (A(K)), f < ∞ .

w∈Fd

The following Lemma plays an important role in the analysis to follow generally, and in proving that A(K)∞ is an algebra, in particular. Lemma 1. Suppose that f = w∈Fd fw w ∈ Mp,q (A(K)∞ ) and X ∈ K(n). Let fw ⊗ X w . Aj = |w|=j

If 0 < r < sup{0 < s : sX ∈ K(n)}, then rj Aj ≤ f . In particular, there is a ρ < 1 such that Aj ≤ ρj f .

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Proof. Because K(n) is open, convex, and circled, the function F (z) = f (zX) is deﬁned on a neighborhood of the closed unit disk {|z| ≤ 1}. Thus the series, F (z) =

∞

Aj z j

j=0

has radius of convergence exceeding one. Thus, for each j ∈ N, 1 Aj = 2π

2π

F (eit )e−ijt dt.

0

It follows that 1 Aj ≤ 2π

2π F (eit ) dt. 0

Since F (eit ) = f (eit X) and eit X ∈ K(n), it follows that F (eit ) ≤ f and the lemma follows. Given formal power series f = w∈Fd fw w ∈ Mp,q (A(K)∞ ) and g = ∞ w∈Fd gw w ∈ Mq,r (A(K) ), deﬁne the product f g of f and g as the convolution product, i.e.,

fu gv w. fg = uv=w

w∈Fd

This convolution product corresponds to pointwise product, extends the natural product of non-commutative polynomials (formal power series with only ﬁnitely many non-zero coefﬁcients), and makes A(K)∞ an algebra. Lemma 2. If f ∈ Mp,q (A(K)∞ ) and g ∈ Mq,r (A(K)∞ ) and X ∈ K, then (i) f g(X) converges; (ii) f g(X) = f (X)g(X); (iii) f g is in Mp,r (A(K)∞ ); and (iv) f g ≤ f g . Corollary 1. A(K)∞ is an algebra. Proof of Lemma 2. Fix X ∈ K. As in the proof of Lemma 1, let fw ⊗ X w , Aj = |w|=j

Bj =

|w|=j

Cj =

gw ⊗ X w ,

|w|=j

Observe that Cj =

j k=0

Ak Bj−k .

uv=w

fu gv

⊗ Xw.

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Let F (z) = f (zX) and G(z) = g(zX), both of which are deﬁned in a neighborhood of {|z| ≤ 1}. From Lemma 1, there is an ρ < 1 such that Am ≤ ρm f and Bk ≤ ρk g . Hence Cj ≤ (j + 1) f g ρj . It follows that, for |z| < ρ1 , the series j ∞ j=0

Ak Bj−k

zj

k=0

converges absolutely. In particular f g(X) = For |z| < ρ1 , one veriﬁes that F (z)G(z) =

j ∞

∞ j=0

Cj converges in norm.

Ak Bj−k z j

j=0 k=0

=f g(zX). Choosing z = 1 gives f (X)g(X) = f g(X). Since, for each X ∈ K, f g(X) = f (X)g(X) it follows that f g(X) ≤ f g . Thus f g ≤ f g and f g ∈ Mp,r (A(K)∞ ). 3.4. The Abstract Unital Operator Algebra A(K)∞ In this section, for the convenience of the reader, the definition of an abstract operator algebra is reviewed. Following that, it is shown that A(K)∞ with the norms · p,q on Mp,q (A(K)∞ ) given in Eq. (9) is an abstract operator algebra. 3.4.1. Abstract Operator Algebra. Let V be a complex vector space and Mp,q (V ) denote the set of all p × q matrices with entries from V . V is said to be a matrix normed space provided that there exist norms · p,q on Mp,q (V ) that satisfy A · X · B ,r ≤ A X p,q B , for all A ∈ M,p , X ∈ Mp,q (V ), B ∈ Mq,r . A matrix normed space V is said to be an abstract operator space if X ⊕ Y p+,q+r = max{ X p,q , Y ,r },

X 0 where X ∈ Mp,q (V ) and Y ∈ M,r (V ) and X ⊕ Y = . 0 Y V is an abstract unital operator algebra if V is a unital algebra, an abstract operator space and if the product on V is completely contractive, i.e., XY p ≤ 1 whenever X p ≤ 1 and Y p ≤ 1 for all X, Y ∈ Mp (V ) and for all p.

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3.4.2. The Abstract Unital Operator Algebra A(K)∞ . Consider A(K)∞ with · p,q being the norm on Mp,q (A(K)∞ ) deﬁned in Sect. 3.3 (see Eq. (9)). Theorem 2. A(K)∞ with the family of norms · p,q , is an abstract unital operator algebra. Proof. Let A ∈ M,p , F ∈ Mp,q (A(K)∞ ), B ∈ Mq,r . Interpret A and B as A∅ ∈ M,p (A(K)∞ ) and B∅ ∈ Mq,r (A(K)∞ ), respectively. For notation ease we will drop the subscripts that go with the norms. It follows from Lemma 2(ii) that for all X ∈ K(n), AF B(X) = A(X)F (X)B(X) ≤ A ⊗ In F (X) B ⊗ In ≤ A F B . Thus, AF B ≤ A F B . ∞

(10)

∞

Let F ∈ M,r (A(K) ), G ∈ Mp,q (A(K) ), X ∈ K(n). Observe that

F (X) 0 F ⊕ G (X) = 0 G(X) ≤ max{ F (X) , G(X) } ≤ max{ F , G }. Thus, F ⊕ G ≤ max{ F , G }.

(11)

Let > 0 be given. Without loss of generality assume that F ≥ G . Choose m ∈ N and R ∈ K(m) such that F (R) > F − . Therefore

F (R) 0 (12) F ⊕ G ≥ ≥ F (R) > F − . 0 G(R) Letting → 0 in the inequality (12) and from the inequality (11) it follows that, F ⊕ G = max{ F , G }.

(13) ∞

Lastly, complete contractivity of multiplication in Mp (A(K) ) follows directly from Lemma 2 (iv). Thus A(K)∞ is an abstract operator algebra. ∅ ∈ A(K)∞ is the multiplicative unit.

4. Weak Compactness and A(K)∞ In this section it is shown that every bounded sequence in A(K)∞ has a pointwise convergent subsequence. Indeed, A(K)∞ has weak compactness properties with respect to bounded pointwise convergence mirroring those for H ∞ , the usual space of bounded analytic functions on the unit disk D. The results easily extend to formal power series with matrix coefﬁcients and it is at this level of generality that they are needed in the sequel.

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Lemma 3. Suppose that fm = w∈Fd (fm )w w is a Mp,q (A(K)∞ ) sequence. If, for each X ∈ K the sequence fm (X) converges or if for each w ∈ Fd the sequence (fm )w converges, and if (fm ) is a bounded sequence (so there is a constant c such that fm ≤ c for all m), then there is an f ∈ Mp,q (A(K)∞ ) such that fm (X) converges to f (X) for each X ∈ K and moreover f ≤ c. Proof. If fm converges pointwise, then, by considering fm (tS()) where S() is deﬁned in Sect. 3.2 and 0 ≤ t < γ, it follows that (fm )w converges to some fw for each word w. Hence, to prove the Proposition it sufﬁces to prove, if (fm )w converges to fw for each w and fm ≤ c for each m, then for each X ∈ K, the series f (X) =

∞

fw ⊗ X w

j=0 |w|=j

converges and (fm (X)) converges to f (X). For any X ∈ K, (fm )w ⊗ X w → fw ⊗ X w . |w|=j

|w|=j

From Lemma 1, there is a ρ < 1 such that for all j ∈ N, (fm )w ⊗ X w ≤ ρj c, |w|=j

an estimate from which the conclusions of the proposition are easily seen to follow. Proposition 1. If fm = w∈Fd (fm )w w ∈ Mp,q (A(K)∞ ) satisfies fm ≤ c for all m ∈ N then, c (i) (fm )w ≤ γ |w| for all w ∈ Fd and for all m ∈ N; (ii) There exists a subsequence {fmk } of {fm } and fw ∈ Mp,q such that (fmk )w → fw for all w; (iii) Let f = w∈Fd fw w. For each X ∈ K the sequence (fmk (X)) converges to f (X) and moreover f (X) ≤ c. Proof. To prove item (i), recall γ from the definition of K. Let 0 ≤ t < γ and x ∈ Cq be a unit vector. For j = 0, 1, 2, . . . , , the hypothesis fm ≤ c together with the conclusion of Lemma 1 for X = tS() imply that (fm )w ⊗ S()w ≤ c. tj |w|=j

Hence c2 ≥ tj

(fm )w x ⊗ S()w ∅ 2

|w|=j

= t2j

(fm )w x 2

|w|=j 2j

≥ t (fm )w x 2 .

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c for Since x and are arbitrary, letting t ↑ γ it follows that (fm )w ≤ γ |w| all m ∈ N and all w ∈ Fd . The proof of item (ii) uses a standard diagonal argument. Let {w1 , w2 , . . .} be an enumeration of words in Fd which respects length (i.e., if v ≤ w, then |v| ≤ |w|). Since (fm )w1 ≤ γ |wc 1 | , there exists a subsequence say, {f1,m } of {fm } such that (f1,m )w1 → fw1 . Since (f1,m )w2 ≤ γ |wc 2 | , there exists a subsequence say, {f2,m } of {f1,m } and thereby of {fm }, such that (f2,m )w2 → fw2 . Continue this procedure to obtain a subsequence {fk,m } of {fk−1,m } and thereby of {fm } such that for all k ∈ N,

(fk,m )wk → fwk . Now consider the diagonal sequence {fm,m }. It follows that {fm,m } is a subsequence of {fm } and satisﬁes (fm,m )w → fw for all w ∈ Fd . In view of what has already been proved, an application of Lemma 3 proves item (iii).

5. The Operator Algebra A(K)∞ /I(K) In this section we consider the ideal I(K) of the algebra A(K)∞ determined by a ﬁnite initial segment Λ. It is shown that the quotient algebra A(K)∞ /I(K) is in fact an abstract unital operator algebra. It is also established that norms of classes in the quotient algebra are attained. 5.1. The Abstract Unital Operator Algebra A(K)∞ /I(K) For the initial segment Λ ⊂ Fd , let ⎫ ⎧ ⎬ ⎨ fw w : f < ∞ ⊂ A(K)∞ . I(K) = f = ⎭ ⎩ w∈Λ

Observe that I(K) is a closed two-sided ideal in the operator algebra A(K)∞ . The usual identiﬁcation of Mp,q (A(K)∞ /I(K)) with Mp,q (A(K)∞ )/ Mp,q (I(K)) yields the well-known fact that the quotient of an abstract operator algebra by a closed two-sided ideal is again an abstract operator algebra (see Exercises 13.3 and 16.3 in [16]). We formally record this fact. Theorem 3. A(K)∞ /I(K) is an abstract unital operator algebra. 5.2. Attainment of Norms of Classes in Mq (A(K)∞ )/Mq (I(K)) Let p ∈ Mq (A(K)∞ ). In this section it is shown that there exists f ∈ Mq (I(K)) such that p + f = p + Mq (I(K)) = inf{ p + g : g ∈ Mq (I(K))}. Let {fm } be a sequence in Mq (I(K)) such that p + Mq (I(K)) ≤ p + fm ≤ p + Mq (I(K)) +

1 . m

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It follows that the sequence {fm } is bounded and that p + fm → p + Mq (I(K)) . An application of Proposition 1 yields a subsequence {fmk } of {fm } and f ∈ Mq (I(K)) such that (p + fmk )(X) → (p + f )(X), for all X ∈ K. Proposition 2. If p, {fm k }, f are as above, then p + f = p + Mq (I(K)) . Proof. Let > 0 be given. Choose R ∈ K such that (14) p + f < (p + f )(R) + . 4 Since (p + fmk )(R) → (p + f )(R) , there exists K1 ∈ N such that, (15) (p + f )(R) < (p + fmk )(R) + , 4 for all k ≥ K1 . Combining the inequalities from Eqs. (14) and (15), implies that, for all k ≥ K1 , (16) p + f < p + fmk + . 2 Since p + fmk → p + Mq (I(K)) , there exists a natural number K2 such that for all k ≥ K2 , (17) p + fmk < p + Mq (I(K)) + . 2 Setting k = max{K1 , K2 } in Eqs. (16) and (17), and letting → 0 yields p + f ≤ p + Mq (I(K)) . On the other hand, since f ∈ Mq (I(K)), p + f ≥ p + Mq (I(K)) .

6. Representations of the Operator Algebra A(K)∞ In this section it is shown that completely contractive representations of the algebra A(K)∞ , when compressed to ﬁnite-dimensional subspaces end up in the boundary of the matrix-convex set K. 6.1. Completely Contractive/Isometric Representation Let V and W be abstract operator spaces and φ : V → W be a linear map. Deﬁne φq : Mq ⊗ V → Mq ⊗ W by φq = IMq ⊗ φ, where IMq : Mq → Mq is the identity map. The map φ is said to be completely contractive (isometric) if φq is a contraction (isometry) for each q ∈ N. A completely contractive (isometric) representation of an algebra A is a completely contractive (isometric) algebra homomorphism π : A → B(M) for some Hilbert space M.

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The following theorem due to Blecher, Ruan and Sinclair guarantees the existence of a completely isometric Hilbert space representation for an abstract unital operator algebra. Theorem 4 (BRS). Let A be an abstract unital operator algebra. There exists a Hilbert space M and a unital completely isometric algebra homomorphism π : A → B(M), i.e., a unital operator algebra isomorphism onto π(A). 6.2. Completely Contractive Representations of A(K)∞ Given a completely contractive unital representation π : A(K)∞ → B(M), let Tj = π(gj ) and let T = (T1 , . . . , Td ). For notation convenience, we will write πT for π. Further, we will also use πT to denote the map IMq ⊗ π : Mq (A(K)∞ ) → Mq ⊗ B(M). The main result of this section says, for a completely contractive representation πT of A(K)∞ , for any n ∈ N and ﬁnite dimensional subspace H of M of dimension n and 0 ≤ t < 1 the tuple tZ = tV ∗ T V = (tV ∗ T1 V, . . . , tV ∗ Td V ) is in K(n), where V : H → M is the inclusion map. The proof begins with a couple of lemmas. Given f ∈ Mq (A(K)∞ ) and 0 ≤ r < 1, let fr be deﬁned as follows. If ∞ ∞ fw w = fj , (18) f= j=0 |w|=j

j=0

then fr =

∞ j=0

rj

fw w =

|w|=j

∞

rj fj .

j=0

Lemma 4. If πT is a completely contractive representation of A(K)∞ and f ∈ Mq (A(K)∞ ), then fr (T ) converges in operator norm. Moreover πT (fr ) = fr (T ) and fr (T ) ≤ fr ≤ f . If in addition πT is completely isometric, then limr→1− fr (T ) = f . Proof. Write f as in Eq. (18). Lemma 1 implies that fj ≤ f . Because πT is completely contractive fj (T ) ≤ f . It follows that fr (T ) converges in norm. Since also the partial sums of fr converge (to fr ) in the norm of Mq (A(K)∞ ), it follows that πT (fr ) = fr (T ) and so fr (T ) ≤ fr . The inequality fr ≤ f is straightforward because rK ⊂ K. Now suppose that πT is completely isometric. In this case fr (T ) = fr . On the other hand limr→1− fr = f . Lemma 5. Given k × k matrices A1 , . . . , Ad , let L=

d

Aj gj .

j=1

Suppose 2 − L(X) − L(X)∗ 0,

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for all X ∈ K() and for all ∈ N. Let ΦL denote the formal power series, ΦL = L(2 − L)−1 =

∞ Lj+1 j=0

(a)

2j+1

.

If X ∈ K(), then ΦL (X) converges in norm, i.e., the series ∞ L(X)j+1 j=0

2j+1

converges. (b) ΦL (X) < 1 and hence ΦL is in Mk (A(K)∞ ) and has norm at most one. (c) If πT is a completely contractive representation of A(K)∞ , then 2 − (L(T ) + L(T )∗ ) 0. Proof. To prove part (a) of the lemma, let X ∈ K() be given. Because K() is circled, it follows that eiθ X ∈ K() for each θ. Hence, 2 − eiθ L(X) − e−iθ L(X)∗ 0,

(19)

for each θ. For notation ease, let Y = L(X). Thus Y is a k × k matrix and inequality (19) implies that the spectrum of Y lies strictly within the disc, i.e., each eigenvalue of Y has absolute value less than one. Thus, ∞ j 1 Y = (2 − Y )−1 2 j=0 2 converges in norm. It follows that −1

ΦL (X) = Y (2 − Y )

=

∞ Y j+1 j=0

2j+1

converges. To prove (b) observe that Y (2 − Y )−1 < 1 if and only if (2 − Y )∗ (2 − Y ) Y ∗ Y which is equivalent to 2 − (Y + Y ∗ ) 0. Thus ΦL (X) < 1 which implies that ΦL ∈ Mk (A(K)∞ ) with ΦL ≤ 1. This completes the proof of (b). To prove part (c), observe, Since πT is completely contractive and ΦL ∈ Mk (A(K)∞ ) with norm at most one, an application of Lemma 4 yields, ΦL (rT ) ≤ 1. Arguing as in the proof of part (b), it follows that 2 − (L(rT ) + L(rT )∗ ) 0. This inequality holds for all 0 ≤ r < 1 and thus the conclusion of part (c) follows. The following separation theorem plays a key role in the analysis to follow. A more general version of the theorem can be found in [10]. Theorem 5 (Eﬀros-Winkler). Let C = (C(n)) denote a closed matrix convex set in Cd which contains a non-commutative neighborhood of 0 ∈ Cd . If X ∈ C(n), then there exist n × n matrices A1 , A2 , . . . , Ad such that the nond commutative polynomial L = j=1 Aj gj satisfies the following conditions:

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2 − L(Y ) − L(Y )∗ 0 for all m ∈ N and all Y ∈ C(m), 2 − L(X) − L(X)∗ 0.

Proposition 3. If T = (T1 , . . . , Td ), and Tj ∈ B(M) for some Hilbert space M, and π(gj ) = Tj determines a completely contractive representation of A(K)∞ , then, for each positive integer n and finite dimensional subspace H of M of dimension n and each 0 ≤ t < 1 the tuple tZ = tV ∗ T V = (tV ∗ T1 V, . . . , tV ∗ Td V ) is in K(n), where V : H → M is the inclusion map. Proof. Let n and H be given and deﬁne Z as in the statement of the proposition. Suppose that L is as in the statement of Lemma 5. From part (c) of the previous lemma, it follows that 2 − (L(T ) + L(T )∗ ) 0. Applying Ik ⊗ V ∗ on the left and Ik ⊗ V on the right of this inequality gives, 2 − (L(Z) + L(Z)∗ ) = (Ik ⊗ V ∗ )(2 − (L(T ) + L(T )∗ )(Ik ⊗ V ) 0. An application of Theorem 5 implies that Z ∈ K(n). Hence tZ ∈ K(n) for all 0 ≤ t < 1. Lemma 6. Let Λ ⊂ Fd be a finite initial segment, f ∈ Mq (A(K)∞ ) be as in Eq. (18) and suppose that πT is a completely contractive representation of A(K)∞ into B(M) and T is Λ-nilpotent. Then fr (T ) ≤ sup{ f (X) : X ∈ K, X is Λ-nilpotent} for all 0 ≤ r < 1. Moreover if fw = 0 for all w ∈ Λ, then f (T ) ≤ sup{ f (X) : X ∈ K, X is Λ-nilpotent}. Proof. Since Λ is ﬁnite, fr (T ) = w∈Λ fw ⊗ (rT )w . Let {ej }qj=1 be the stanq dard basis of Cq and y = j=1 ej ⊗ hj ∈ Cq ⊗ M be a unit vector such that fr (T ) < fr (T )y + . Let H denote the ﬁnite-dimensional subspace of M spanned by the vectors {T w (hj ) : w ∈ Λ, 1 ≤ j ≤ q} and V : H → M be the inclusion map. Then Z = V ∗ T V is Λ-nilpotent and

V ∗ T w V if w ∈ Λ w Z = 0 otherwise. Proposition 3 implies that rZ ∈ K. Thus,

|w| w fr (T ) < fw ⊗ r T y + w∈Λ

= fr (Z)y + ≤ fr (Z) + ≤ sup{ f (X) : X ∈ K, X is Λ-nilpotent} + . Letting → 0 yields the desired inequality. If fw = 0 for all w ∈ Λ, then f = w∈Λ fw w is a non-commutative polynomial in which case we have limr→1− fr (T ) = f (T ) and this completes the proof.

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7. The Carath´eodory–Fej´er Interpolation Problem (CFP) The generalization of Theorem 1 allowing for operator coefﬁcients is proved in this section. The strategy is to ﬁrst prove the result for matrix coefﬁcients. This is done in Sect. 7.1 below. Passing from matrix to operator coefﬁcients is then accomplished using well-known facts about the weak operator topology (WOT) on the space of bounded operators on a separable Hilbert space. The details are in Sect. 7.2. 7.1. Matrix Version of the CFP For clarity, we begin by stating the matrix version of Theorem 1. Theorem 6. Fix a matrix convex set K satisfying the conditions of Assumption 1. Let Λ ⊂ Fd be a finite initial segment, and pw w ∈ Mq (A(K)∞ ) p= w∈Λ

be given. There exists f ∈ Mq (A(K)∞ ) such that fw = pw for w ∈ Λ and f ≤ 1 if and only if sup{ p(X) : X ∈ K, X is Λ-nilpotent} ≤ 1. Theorem 6 is easily seen to follow from the following proposition. Proposition 4. There exists f ∈ Mq (I(K)) such that p+f = p+Mq (I(K)) = sup{ p(X) : X ∈ K, X is Λ-nilpotent}. Proof. From Theorems 3 and 4 it follows that there exists a Hilbert space M and a completely isometric homomorphism θ : A(K)∞ /I(K) → B(M). As before, identify Mq (A(K)∞ /I(K)) with Mq (A(K)∞ )/Mq (I(K)). Let θq denote the map IMq ⊗ θ : Mq (A(K)∞ )/Mq (I(K)) → Mq ⊗ B(M). Let R be the d-tuple (R1 , R2 , . . . , Rd ), where Rj = π(gj + I(K)) ∈ B(M), for 1 ≤ j ≤ d. Observe that R is Λ-nilpotent. Let η : A(K)∞ → A(K)∞ /I(K) be the quotient map. The composition map π = θ ◦ η : A(K)∞ → B(M) is a completely contractive representation of A(K)∞ . Also since π(gj ) = Rj , consistent with the notation introduced earlier, we will use πR to denote the map π = θ ◦ η. It follows from Theorem 2 that there exists f ∈ Mq (I(K)) such that p + f = p + Mq (I(K)) .

(20)

The fact that θ is completely isometric implies that p + Mq (I(K)) = θq (p + Mq (I(K)) = p(R) .

(21)

Since πR is a completely contractive representation of A(K)∞ , Lemma 6 implies that p(R) ≤ sup{ p(X) : X ∈ K, X is Λ-nilpotent}. Combining the Eqs. (20), (21) and (22), it follows that p + f ≤ sup{ p(X) : X ∈ K, X is Λ-nilpotent}.

(22)

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But the definition of p + f implies that p + f ≥ sup{ p(X) : X ∈ K, X is Λ-nilpotent}

and this completes the proof.

7.2. Operator Version of the CFP As before, let Λ ⊂ Fd be a ﬁnite initial segment. Departing from the previoussubsection, let U be a separable Hilbert space and let the polynomial p = w∈Λ pw w, where {pw }w∈Λ ⊂ B(U) be given. ˜w w such that Theorem 7. There exists a formal power series x ˜ = w∈Fd x x = sup{ p(X) : X ∈ K, X is Λ-nilpotent}. x ˜w = pw for all w ∈ Λ and ˜ Proof. Let {u1 , u2 , . . .} denote an orthonormal basis for the separable Hilbert space U and Um be the subspace of U spanned by the vectors {uj }m j=1 . Note that one can identify Um with Mm . For notation ease, let C = sup{ p(X) : X ∈ K, X is Λ-nilpotent}. For w ∈ Λ, deﬁne Mm (pm )w = Vm∗ pw Vm where Vm : Um → U is the inclusion map. Let pm denote the non-commutative polynomial (pm )w w. pm = w∈Λ

For each X ∈ K, observe that pm (X) ≤ p(X) . Thus pm ≤ p and pm ∈ Mm (A(K)∞ ) for all m ∈ N. By Proposition 4, there exists fm ∈ Mm (I(K)) such that xm = pm + fm ∈ Mm (A(K)∞ ) and xm = sup{ pm (X) : X ∈ K, X is Λ-nilpotent}. For w ∈ Fd , deﬁne xm )w = Vm (xm )w Vm∗ . Let x ˜m denote the B(U) (˜ xm )w w. For X ∈ K and j = 0, 1, 2, . . ., it follows formal power series w∈Fd (˜ from Lemma 1 that there exists 0 ≤ ρ < 1 such that (˜ xm )w ⊗ X w ≤ (xm )w ⊗ X w |w|=j

|w|=j

≤ ρ xm j

≤ Cρj .

(23)

This implies that series for x ˜m (X) converges for each X ∈ K and moreover we have ˜ xm ≤ xm ≤ C.

(24)

Recall γ and S() from Sect. 3.2. Let u ∈ U be an arbitrary unit vector. For each 0 ≤ t < γ, 0 ≤ j ≤ and X ∈ K, it follows that (˜ xm )w ⊗ S()w (u ⊗ ∅) 2 C 2 ≥ tj |w|=j

≥ t

j

(˜ xm )w u ⊗ w 2

|w|=j

≥ t2j

|w|=j

(˜ xm )w u 2 .

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Since is arbitrary, letting t ↑ γ implies that (˜ xm )w ≤ γC |w| for all w ∈ Fd and m ∈ N. Since U is a separable Hilbert space and the sequence {(˜ xm )w }∞ m=1 is C xm )w }∞ bounded (by γ |w| ), for each w ∈ Fd , there exists a subsequence of {(˜ m=1 that converges with respect to the WOT on B(U). By a diagonal argument similar to the one in Proposition 1, it follows that there exists a subsequence xm } and {˜ xw }w∈Fd ⊂ B(U) such that for each w ∈ Fd {˜ xmk } of {˜ (˜ xmk )w → x ˜w with respect to the WOT on B(U). ˜w w. The proof of the Let x ˜ denote the formal power series w∈Fd x theorem is completed by showing that ˜ x ≤ C and that noting that x ˜w = lim(˜ xmk )w = lim Vmk (pmk )w Vm∗ k = pw (WOT limits) for w ∈ Λ. The details are omitted. Remark 1. Lemma 6, Proposition 4 and Theorem 6 can easily be extended to the case where the given coefficients pw are p × q matrices, by applying the respective results after embedding the coefficient matrices pw in large enough square matrices of fixed size having zeros as their additional entries. Theorem 7 can be extended to the case where the given coefficients pw are in B(U, V), by applying it first to the case with initial coefficients pw where 0 0 pw = ∈ B(U ⊕ V), pw 0 and by observing that, if x ˜ is a solution to the CFP for this specific case, then ˜|U is a solution to the CFP with initial data pw . PV x

8. Examples and the Case of Infinite Initial Segments Λ Of course the results of this paper apply to the examples in Sect. 2.4. In the case of the non-commutative matrix polydisc, the operators obtained by applying the representation of the quotient algebra to the generators [gj ] = gj + I(K); 1 ≤ j ≤ d, are automatically contractions and thus certain technical details of the proof of Theorem 1 are absent. Consequently, the argument easily extends to handle inﬁnite initial segments, provided the underlying domain is expanded to include operators on a separable Hilbert space. Fix a separable inﬁnite dimensional Hilbert space H. Let C d denote the operator non-commutative polydisc C d = {(T1 , T2 , . . . , Td ) : Tj ∈ B(H) & Tj < 1}. The following variant of Theorem 7 holds. Theorem 8. Let Ube a separable Hilbert space, Λ ⊂ Fd be an infinite initial segment and p = w∈Λ pw w be a formal power series with coefficients pw ∈ d } < ∞. There exists operators B(U) such that p = sup{ p(X) : X ∈ C ˜ = w∈Fd x ˜w w such that x ˜ w = pw x ˜w ∈ B(U) and a formal power series x d for all w ∈ Λ and ˜ x = sup{ p(T ) : T ∈ C , T is Λ-nilpotent}.

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Similarly, consider the dd -dimensional operator non-commutative mixed ball,

Ddd = {T = (T11 , T12 , . . . , Tdd ) : Tij ∈ B(H) & T op < 1},

d d where T op is the norm of the operator (Tij )d,d i,j=1 : B(H ) → B(H ). A variant of Theorem 7 holds in this case as well, the statement of which can be obtained by replacing C d in the statement of Theorem 8 by Ddd .

Acknowledgement I would like to thank my advisor Scott McCullough for his guidance in the preparation of this article.

References [1] Agler, J.: On the representation of certain holomorphic functions deﬁned on a polydisc. In: Topics in Operator Theory: Ernst D. Hellinger Memorial Volume. Operator Theory: Advances and Applications, vol. 48, pp. 47–66. Birkh¨ auser, Basel (1990) [2] Arias, A., Popescu, G.: Noncommutative interpolation and Poisson transforms. Israel J. Math. 115, 205–234 (2000) [3] Ball, J.A., Groenewald, G., Malakorn, T.: Conservative structured noncommutative multidimensional linear systems. In: The State Space Method Generalizations and Applications. Operator Theory: Advances and Applications, vol. 161, pp. 179–223. Birkh¨ auser, Basel (2006) [4] Ball, J.A., Li, W.S., Timotin, D., Trent, T.T.: A commutant liftint theorem on the polydisc. Indiana Univ. Math. J. 48(2), 653–675 (1999) [5] Cuntz, J.: Simple C*-algebras generated by isometries. Comm. Math. Phys. 57, 173–185 (1977) [6] Carath´eodory, C., Fej´er, L.: Uber den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koefﬁzienten und uber den PicardLandauschen Satz. Rend. Circ. Mat. Palermo 32, 218–239 (1911) [7] Constantinescu, T., Johnson, J.L.: A note on noncommutative interpolation. Can. Math. Bull. 46(1), 59–70 (2003) [8] Dautov, Sh.A., Khuda˘iberganov, G.: The Carath´eodory-Fej´er problem in higher-dimensional complex analysis. Sibirsk. Mat. Zh. 23(2), 58–64, 215 (1982) [9] Davidson, K.R., Pitts, D.R.: Nevanlinna-Pick interpolation for noncommutative analytic Toeplitz algebras. Integr. Equ. Oper. Theory 31(3), 321– 337 (1998) [10] Eﬀros, E.G.: Winkler, Soren Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems. J. Funct. Anal. 144(1), 117–152 (1997) [11] Eschmeier, J., Patton, L., Putinar, M.: Carath´eodory-Fej´er interpolation on polydisks. Math. Res. Lett. 7(1), 25–34 (2000) [12] Foias, C., Frazho, A.E.: The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, vol. 44. Birkh¨ auser, Basel (1990)

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[13] Helton, J.W., Klep, I., McCullough, S., Slinglend, N.: Noncommutative ball maps. J. Funct. Anal. 257(1), 47–87 (2009) [14] Kalyuzhny˘i-Verbovetzki˘i, D.: Carath´eodory interpolation on the noncommutative polydisk. J. Funct. Anal. 229, 241–276 (2005) [15] Kalyuzhnyi-Verbovetski, D., Vinnikov, V.: Foundations of noncommutative function theory (in preparation) [16] Paulsen, V.: Completely Bounded Maps and Operator Algebras, 1st edn. Cambridge University Press, Cambridge (2003) [17] Popescu, G.: Free holomorphic functions on the unit ball of B(H)n . J. Funct. Anal. 241, 268–333 (2006) [18] Popescu, G.: Free holomorphic functions and interpolation. Math. Ann. 342, 1–30 (2008) [19] Popescu, G.: Interpolation problems in several variables. J. Math. Anal. Appl. 227(1), 227–250 (1998) [20] Popescu, G.: Von Neumann inequality for (B(H)n )1 . Math. Scand. 68(2), 292–304 (1991) [21] Popescu, G.: Multi-analytic operators on Fock spaces. Math. Ann. 303(1), 31–46 (1995) [22] Sarason, D.: Generalized interpolation in H ∞ . Trans. Am. Math. Soc. 127, 179–203 (1967) [23] Schur, I.: Uber Potenzreihen die im Innern des E inheitskreises beschrankt sind. J. Reine Angew. Math. 147, 205–232 (1917) [24] Toeplitz, O.: Uber die Fouriersche Entwickelung positiver Funktionen. Rend. Circ. Mat. Palermo 32, 191–192 (1911) [25] Voiculescu, D.V.: Free Probability Theory. American Mathematical Society, Providence (1997) [26] Voiculescu, D.V.: Free analysis questions. I: Duality transform for the coalgebra of X:B. Int. Math. Res. Notes 16, 793–822 (2004) [27] Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables: a Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras, and Harmonic Analysis on Free Groups. American Mathematical Society, Providence (1992) Sriram Balasubramanian (B) Department of Mathematics University of Florida Florida, USA e-mail: [email protected] Received: February 19, 2010. Revised: September 15, 2010.

Integr. Equ. Oper. Theory 68 (2010), 551–572 DOI 10.1007/s00020-010-1828-1 Published online September 8, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Lidskii-Type Formulae for Dixmier Traces A. A. Sedaev, F. A. Sukochev and D. V. Zanin Abstract. We establish several analogues of the classical Lidskii Theorem for some special classes of singular traces (Dixmier traces and Connes– Dixmier traces) used in noncommutative geometry. Mathematics Subject Classification (2010). 46L52, 47B10, 46E30. Keywords. Dixmier traces, Lidskii formula.

1. Introduction and Preliminaries 1.1. Dixmier–Macaev Ideal and Dixmier Traces An important role in noncommutative geometry [7] is played by the set of compact operators whose partial sums of singular values are logarithmically divergent. This set can be adequately described using the terminology of Marcinkiewicz spaces. Consider the Marcinkiewicz sequence space m1,∞ := {x = {xn }∞ n=1 : xm1,∞ < ∞}, where we set xm1,∞ = sup N

N 1 x∗ . log(N + 1) n=1 n

∞ Here, {x∗n }∞ n=1 is the sequence {|xn |}n=1 rearranged in nonincreasing order. Fix an inﬁnite-dimensional separable complex Hilbert space H and consider the set M1,∞ of all compact operators x on H such that the sequence of its singular values {sn (T )}∞ n=1 falls into the space m1,∞ (recall that the singular values of a compact operator T are the eigenvalues of the operator |T | = (T ∗ T )1/2 ). We set

T M1,∞ := {sn (T )}m1,∞ . It is well known that the ideal of compact operators M1,∞ equipped with the norm · M1,∞ is a Banach space. We refer to the recent paper [16] by Pietsch for additional references and information on these spaces. A. A. Sedaev was partially supported by RFBR 08-01-00226. F. A. Sukochev was partially supported by the ARC.

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We describe brieﬂy a construction of singular traces on the ideal M1,∞ due to Dixmier [8] and its various modiﬁcations which are of importance in noncommutative geometry [7]. For a more detailed treatment we refer to [6]. Let σn , n ≥ 1 be the operator on l∞ deﬁned by σn (x1 , . . . , xk , . . .) = (x1 , . . . , x1 , x2 , . . . , x2 , . . . , xk , . . . , xk , . . .). n-times

n-times

n-times

Let ω be a σn -invariant generalised limit on l∞ , that is, ω is a positive normalised functional on l∞ such that ω(σn (x)) = ω(x) for all x ∈ l∞ and such that ω|c0 = 0, where c0 is the subspace of all vanishing sequences. For an element 0 ≤ T ∈ M1,∞ we set τω (T ) := ω

∞

N 1 sn (T ) . log(N + 1) n=1 N =1

It is well known (see e.g. § 5 in [6] and additional references therein) that τω is an additive functional on the positive part of M1,∞ . Thus, τω admits a linear extension to a unitarily invariant functional (trace) on M1,∞ . This trace vanishes on all ﬁnite-dimensional operators from B(H). Such singular traces are called Dixmier traces (see [8]). A smaller subclass of Dixmier traces was introduced by Connes in [7] by observing that a functional ω = γ ◦ M is σn -invariant state on l∞ for all n ≥ 1. Here, γ is an arbitrary generalised limit on the space L∞ (0, ∞) of all bounded measurable functions and the operator M is a Cesaro operator deﬁned by the formula 1 log(t)

(M x)(t) =

t 1

x(s)ds . s

Referring to ω above as a functional on l∞ , we tacitly apply an isometric embedding i : l∞ → L∞ (0, ∞) given by i

{xj }∞ j=1 →

∞

xj χ[j−1,j) ,

j=1

where χ[j−1,j) is the characteristic function of the interval [j − 1, j). Dixmier traces τω deﬁned such ω’s are termed Connes–Dixmier traces. We refer to [6,7,15] for discussion of their properties. Finally, various formulae of noncommutative geometry (in particular, those involving heat kernel estimates and generalised ζ-function) were established in [3,5,7] for yet a smaller subset of Connes–Dixmier traces, when the functional ω was assumed to be M -invariant. This class (and its further modiﬁcations) was ﬁrst introduced in [3] (see also [10]) and further studied and used in [1,2,5]. For brevity we refer to the latter class (a proper subclass of Connes–Dixmier traces) as a class of M -invariant Dixmier traces.

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1.2. Lidskii Formula for M -Invariant Dixmier Traces in [1, 2, 3] In the case, when we deal with the standard trace Tr and the standard trace class S1 of compact operators from B(H), the classical Lidskii Theorem asserts that the trace λn (T ) Tr(T ) = n≥1

for any T ∈ S1 . Here, {λn (T )}n≥1 is the sequence of eigenvalues of T, taken in any order. This arbitrariness of the order is due to the absolute convergence of the series n≥1 |λn (T )|. In particular, we can choose the decreasing order of absolute values of λn (T ) and counting multiplicities. The core difference of this situation with the setting of Dixmier traces living on the ideal M1,∞ consists in the fact that the series n≥1 |λn (T )| generally speaking diverges for every T ∈ M1,∞ . For simplicity, we explain the emerging obstacle in the case of a self-adjoint operator T = T ∗ = T+ − T− ∈ M1,∞ . For such T, by the definition, τω (T ) = τω (T+ ) − τω (T− ), where

N 1 τω (T± ) = ω λn (T± ) . log(N ) n=1 Even in this case, it is not clear why the equality

N 1 λn (T ) τω (T ) = ω log(N ) n=1 should hold for the special enumeration of the set {λn (T )}n≥1 given by the decreasing order of absolute values of |λn (T )|; or for that matter for any enumeration of this set. The following result from [1] establishes the equality above under significant additional constraints on τω and T ∈ M1,∞ . Theorem 1. Let ω be M -invariant and let T ∈ M1,∞ satisfy the assumption sn (T ) ≤ C/n for some C > 0 and all n ≥ 1. We have ⎛ ⎞ 1 λ⎠ , τω (T ) = ω ⎝ log(n) |λ|>1/n,λ∈σ(T )

where σ(T ) is the spectrum of T. In the case when T is a positive arbitrary element from M1,∞ and ω is taken from a rather special subset of all M -invariant generalised limits (termed in [4] “maximally invariant Dixmier functionals”) this result can be already found in [3, Proposition 2.4]. In [2, Theorem 1], the assertion from [3, Proposition 2.4] was extended to an arbitrary M -invariant ω. Another modiﬁcation of the class of ω’s for which the result of [3, Proposition 2.4] and [2, Theorem 1] holds is given in [5, Proposition 4.3].

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1.3. Statement of Main Results In this paper we prove significant extensions and generalisations of Theorem 1 from [1], [3, Proposition 2.4], [2, Theorem 1] and [5, Proposition 4.3]. Many of our results are established for a general class of Marcinkiewicz ideals. Here, for convenience of the reader, we restate these results for traces on M1,∞ . Our ﬁrst main result shows that the assertion of Theorem 1 holds for an arbitrary Connes–Dixmier trace τω . Theorem 2. Let τω be a Connes–Dixmier trace on M1,∞ . We have ⎛ ⎞ 1 λ⎠ , T ∈ M1,∞ . τω (T ) = ω ⎝ log(n)

(1)

|λ|>1/n,λ∈σ(T )

Theorem 2 follows immediately from Theorem 14 below. Our second main result is the answer to a natural question whether formula (1) holds for every Dixmier trace. This question is answered in negative in Theorem 5. Our third (and the last) main result answers in the aﬃrmative the question whether there exists a modiﬁcation of the summation method used in formula (1) ensuring that it holds for all Dixmier traces. Theorem 3. Let τω be a Dixmier trace on M1,∞ . We have ⎛ ⎞ 1 λ⎠ , T ∈ M1,∞ . τω (T ) = ω ⎝ log(n) λ∈σ(T ),|λ|>log(n)/n

Theorem 3 follows immediately from Theorem 31 below. At the end of the paper we also provide an application of our results. The result proved in the last section concerns heat kernel type formulae from noncommutative geometry (see [2,3,5,7]) and has been already established in [18] with a rather arcane argument. We present here a very simple approach to these formulae. 1.4. Marcinkiewicz Spaces and Singular Traces It is convenient to consider the general class of Marcinkiewicz spaces since many of our results hold for this class with no extra eﬀort. We frequently use commutative results as a stepping stone to obtain their noncommutative analogues. Recall that the distribution function nx of a bounded measurable function x is deﬁned by the formula nx (t) = m({s, |x(s)| > t,

t > 0}).

We write x∗ for the decreasing rearrangement of the function x: x∗ is the right continuous non-increasing function whose distribution function coincides with that of |x| (see [14]).

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The following formula is frequently used in the proofs below sometimes without explicit referencing. nx (t)

∞

∗

x (s)ds = −

λdnx (λ).

(2)

t

0

Here, z is any positive number. Marcinkiewicz spaces are a special case of fully symmetric function and sequence spaces, see [14]. Denote by Ψ the class of all concave increasing functions such that ψ(∞) = ∞, ψ(t) = O(t) as t → 0 and ψ(t) = o(t) as t → ∞. For every ψ ∈ Ψ, Marcinkiewicz space Mψ is a set of all bounded measurable functions x on [0, ∞) such that xMψ

1 := sup t>0 ψ(t)

t

x∗ (s)ds < ∞.

(3)

0

Marcinkiewicz sequence space mψ is a set of sequences (see e.g. [6,16]) satisfying the condition n

xmψ = sup n

1 ∗ xn < ∞. ψ(n) k=1

In this paper, we mainly work with functions ψ ∈ Ψ satisfying the following condition. lim sup t→∞

ψ(2t) < 2. ψ(t)

(4)

Let K(H) be the ideal of all compact operators. If mψ is a Marcinkiewicz sequence space, then the corresponding Marcinkiewicz operator space Mψ is the set of all T ∈ K(H) such that {sn (T )} ∈ mψ equipped with the norm T Mψ := {sn (T )}mψ . Let ψ ∈ Ψ and let ω be a dilation invariant generalised limit. The mapping τω deﬁned by the formula ⎛ ⎞ t 1 τω (x) := ω ⎝ x∗ (s)ds⎠ ψ(t) 0

is a subadditive homogeneous functional on Mψ+ . If τω is additive on Mψ+ , then τω is called Dixmier trace generated by ω. We refer the reader to [9–11] for conditions which guarantee the additivity of τω . It is well known that τω is additive for any ω as above when lim

t→∞

ψ(2t) = 1. ψ(t)

(5)

Similarly, the definitions of Connes–Dixmier traces and M -invariant traces naturally extend to denote corresponding singular traces on Marcinkiewicz ideals Mψ (see [15]).

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Our main result for general Dixmier traces on ideals Mψ is given in Theorem 31 which asserts that for any Dixmier trace τω on Mψ with ψ ∈ Ψ satisfying condition (4) we have ⎛ ⎞ 1 λ⎠ , T ∈ Mψ . (6) τω (T ) = ω ⎝ ψ(n) λ∈σ(T ),|λ|>ψ(n)/n

The result of Theorem 3 follows immediately from the formula above, if we set ψ(t) = log(t) for all t ≥ 2. 1.5. Failure of (1) for Dixmier Traces Here, we show that there are Dixmier traces τω on M1,∞ for which formula (1) fails. To this end we use ω provided by the lemma below. Deﬁne a subadditive functional π : L∞ (0, ∞) → R by the formula 1 π(x) = lim sup log(log(N )) N →∞

N log(N )

N

x(s)ds . s

Clearly, π is positive and homogeneous. The following lemma is routine. We include the proof for convenience of the reader. Lemma 4. Let x ∈ L∞ (0, ∞) be an arbitrary positive element. 1. If ω ∈ L∞ (0, ∞)∗ such that ω ≤ π, then ω is dilation invariant generalised limit. 2. If π(x) > 0, then there exists a dilation invariant generalised limit ω such that ω(x) > 0. Proof. We prove the ﬁrst assertion and then derive the second one from it. 1. At ﬁrst we note that by assumption − π(−y) ≤ ω(y) ≤ π(y)

(7)

for every y ∈ L∞ (0, ∞). Note that π(−y) ≤ 0 for every 0 ≤ y ∈ L∞ (0, ∞). It follows that ω is positive. Further, for every y ∈ L∞ (0, ∞) we have N log(N ) N ) (y − σ y)(s)ds N log(N y(s)ds y(s)ds n − = . s s s N N log(N )/n N/n Therefore, |π(y − σn y)| ≤ lim sup N →∞

1 · 2y∞ · | log(n)| = 0. N log(N )

Hence, ω(y − σn y) ≤ π(y − σn y) = 0,

ω(σn y − y) ≤ π(σn y − y) = 0.

Thus, ω is dilation invariant. If y ∈ L∞ (0, ∞) is such that y(t) → 0 as t → ∞, then π(y) = π(−y) = 0. It follows from (7) that ω(y) = 0.

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Noting that ω(1) = 1, we conclude that ω is dilation invariant generalised limit. 2. Consider linear space xR spanned by element x. Set ω(λx) = λπ(x) for every λ ∈ R. It follows that ω ≤ π on xR. By the Hahn–Banach theorem, there exists a functional ω ∈ L∞ (0, ∞)∗ such that ω(x) = π(x) and ω ≤ π. It follows from above that ω is a dilation invariant generalised limit. Theorem 5. There exist a positive function x ∈ M1,∞ and a Dixmier trace τω such that ⎛ ⎞ ∞ ⎜ −1 ⎟ λdnx (λ)⎠ . τω (x) = ω ⎝ (8) log(t) 1/t

Proof. Deﬁne a function x by the formula k

x = sup e−e χ[1,ek+ek ] . k

k−1

If t ∈ [ek−1+e 1 log(t)

t

k

, ek+e ], then

x∗ (s)ds ≤ e1−k

1

k+e e

k

x∗ (s)ds ≤ e1−k

Thus, x ∈ M1,∞ . We claim that

N →∞

⎛

1 log(log(N ))

n

n

e−e · en+e ≤

n=1

1

lim sup

k

N log(N )

⎜ ⎝

N

1 log(t)

⎞

nx(1/t)

t

k

e2 . e−1

⎟ dt > 0. x∗ (s)ds⎠ t

k

Set N = ee . It is clear that nx (1/t) = ek+e for every t ∈ [N, N log(N )]. k Since x∗ (s) = e−e for every s ∈ [t, nx (1/t)] and every t ∈ [N, N log(N )], we can rewrite the expression under the limit in the left-hand side as 1 k

k+e e

ee k

k k

ek+e − t ek dt = k k ee t log(t) k

= This proves the claim. Thus, ⎛ ⎛ ⎜ π⎝

1 ⎜ ⎝ log(t)

k+e e

ee k

k

1 dt − t log(t) keek

e k log 1 + k k e

nx(1/t)

x∗ (s)ds −

1

t

k+e e

k

dt = log(t)

ee k

− o(1) = 1 + o(1).

⎞⎞ ⎟⎟ x∗ (s)ds⎠⎠ > 0.

1

The assertion of the theorem now follows from Lemma 4 and (2).

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2. Lidskii Formula for Connes–Dixmier Traces In this section, we extend results of [1] (and, partially, those of [2]) to a wider class of Marcinkiewicz spaces and Connes–Dixmier traces. To this end, we need some extra assumptions on ψ ∈ Ψ. The need of such additional conditions is seen from the example below, which shows that analogue of formula (1) for an arbitrary ψ ∈ Ψ fails. Example 6. Let ψ(t) = exp( log(t)) and let x = ψ . If τω is a Dixmier trace on Mψ , then ⎛ ⎞ ∞ ⎜ −1 ⎟ e1/2 τω (x) ≤ ω ⎝ λdnx (λ)⎠ . ψ(t) 1/t

Proof. It is clear that x(t) = exp( log(t))/2t log(t). We have t exp( log(t)) ≤ nx (1/t). 2 log(t) for all sufﬁciently large t. Hence, e1/2 + o(1) ≤

ψ(nx (1/t)) . ψ(t)

The assertion follows immediately.

Thus, some additional restrictions on the function ψ are needed. We require the following condition ψ(tψ(t)) lim = 1. (9) t→∞ ψ(t) It is clear that (5) holds and, therefore, Marcinkiewicz space Mψ admits nonzero Dixmier traces (see [9–11]). Now we show that formula (1) holds for all Connes–Dixmier traces on Mψ . Lemma 7. Let ψ ∈ Ψ satisfy condition (9). If c > xMψ , we have dx (1/t) ≤ ctψ(t) for every x ∈ Mψ and every sufficiently large t. Proof. Assume the contrary. Hence, there exists a sequence tk → ∞ such that x∗ (s) ≥ 1/tk for every s ∈ [0, ctk ψ(tk )]. By the definition of Marcinkiewicz norm, xMψ

1 ≥ ψ(ctk ψ(tk ))

ctkψ(tk )

x∗ (s)ds ≥

0

cψ(tk ) . ψ(ctk ψ(tk ))

It follows from (9) that cψ(tk ) → c. ψ(ctk ψ(tk )) The contradiction proves the Lemma.

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Remark 8. Let 0 ≤ x, y ∈ L∞ (0, ∞) and let y(t) = x(t) · (1 + o(1)) as t → ∞. If x ∈ / L1 (0, ∞), we have T

T y(s)ds = (1 + o(1))

x(s)ds.

1

1

Lemma 9. Let ψ ∈ Ψ satisfy condition (9). We have 1 log(T )

T 1

ctψ(t)

dt tψ(t)

1 x(s)ds = log(T )

0

T 1

dt tψ(t)

t x(s)ds + o(1) 0

∗

as T → ∞ for every positive x = x ∈ Mψ and every c > 0. Proof. The assertion is linear with respect to x. Since the assertion holds for x(t) = ψ (t), it is sufﬁcient to verify it for x + ψ instead of x. Hence, we may assume that x(t) ≥ ψ (t). Thus, integral in the right-hand side is unbounded as T → ∞. Make a substitution z = ctψ(t) in the left-hand side integral. It follows from the condition (9) that dz dt = (1 + o(1)). tψ(t) zψ(z) Indeed, by Lagrange theorem, we have dz dt = z t

ψ(z) = ψ(t)(1 + o(1)),

tψ (t) 1+ ψ(t)

=

dt (1 + o(1)). t

It follows from Remark 8 that T 1

dt tψ(t)

ctψ(t)

cTψ(T )

∗

x (s)ds = (1 + o(1)) 0

cψ(1)

Evidently, cTψ(T )

T

dz zψ(z)

z

⎛ ⎜ x∗ (s)ds = O ⎝

0

cTψ(T )

T

dz zψ(z)

z

x∗ (s)ds.

(10)

0

⎞ dz ⎟ ⎠ = o(log(T )). z

(11)

Noting that cTψ(T )

= 1

cTψ(T )

T + 1

T

the combination of (10) and (11) yields the assertion.

Lemma 10. Let ψ ∈ Ψ satisfy the condition (9) and let τω be a Connes– Dixmier trace on Mψ . We have

560

A. A. Sedaev, F. A. Sukochev and D. V. Zanin ⎛ ⎜ −1 ω⎝ ψ(t)

∞

IEOT

⎞ ⎟ λdnx (λ)⎠ ≤ τω (x)

1/t

for every positive x ∈ M1,∞ . Proof. Due to (2) and Lemma 7 we have ⎛ ⎞ ⎞ ⎛ nx(1/t) ctψ(t) ⎜ 1 ⎟ ⎟ ⎜ 1 ω⎝ x∗ (s)ds⎠ ≤ ω ⎝ x∗ (s)ds⎠ ψ(t) ψ(t) 0

0

⎞ ⎞ t 1 x∗ (s)ds⎠ + o(1)⎠ = γ ⎝M ⎝ ψ(t) 0 ⎛ ⎛ ⎞⎞ t 1 = γ ⎝M ⎝ x∗ (s)ds⎠⎠ = τω (x). log(t) ⎛

⎛

0

Lemma 11. Let ψ ∈ Ψ and let τω be a Dixmier trace on Mψ . We have ⎛ ⎞ ∞ ⎜ −1 ⎟ τω (x) ≤ ω ⎝ λdnx (λ)⎠ ψ(t) 1/t

for every positive x ∈ Mψ . Proof. We claim that t

nx(1/t)

∗

x∗ (s)ds + 1.

x (s)ds ≤ 0

0

The inequality is evident if t ≤ nx (1/t). If t > nx (1/t), then x∗ (s) ≤ 1/t for every s ∈ [nx (1/t), t]. It follows that t

nx(1/t)

∗

x (s)ds = 0

t

∗

x (s)ds + 0

x∗ (s)ds

nx (1/t)

nx(1/t)

x∗ (s)ds + (t − nx (1/t)) · t−1 .

≤ 0

Thus, claim holds in either case. It follows that ⎛ ⎜ 1 τω (x) ≤ ω ⎝ ψ(t)

nx(1/t)

⎞

⎟ x (s)ds⎠ + ω

1

The assertion follows immediately.

∗

1 ψ(t)

.

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The next theorem follows immediately from Lemmas 10 and 11. Theorem 12. Let ψ ∈ Ψ satisfy the condition (9) and let τω be a Connes– Dixmier trace on Mψ . We have ⎛ ⎞ ∞ ⎜ −1 ⎟ λdnx (λ)⎠ τω (x) = ω ⎝ ψ(t) 1/t

for every positive x ∈ Mψ . Remark 13. Consider weak space Mψw (the smallest symmetric ideal containing ψ ). Suppose that ψ satisﬁes the condition (9). If τω is an arbitrary Dixmier trace on Mψ , then we have ⎛ ⎞ ∞ ⎜ −1 ⎟ λdnx (λ)⎠ τω (x) = ω ⎝ ψ(t) 1/t

for every positive x ∈ Mψw . Using Lemma 7, the equality above follows immediately. Arguing as in the Sect. 4 below, we obtain a noncommutative version of Theorem 12, which strengthens [2, Theorem 1] and [1, Corollary 2.12] (see Theorem 1). Theorem 14. Let ψ ∈ Ψ satisfy the condition (9) and let τω be a Connes– Dixmier trace on Mψ . We have ⎛ ⎞ 1 τω (T ) = ω ⎝ λ⎠ ψ(n) |λ|>1/n,λ∈σ(S)

for every operator T ∈ Mψ .

3. Adjusted Lidskii Formula for Dixmier Traces: Commutative Setting As we have seen in Theorem 5, formula (1) does not hold for Dixmier traces τω . In this section, we consider a modiﬁcation of formula (1) which holds for all Dixmier traces τω on a commutative Marcinkiewicz space Mψ , ψ ∈ Ψ. Lemma 15. Let ψ ∈ Ψ satisfy condition (4). If 0 ≤ x ∈ Mψ , then there exists a constant c(x) ∈ N such that ψ(t) nx ≤ c(x)t t for every sufficiently large t. Proof. Set ϕ(t) = t/ψ(t). It follows from (4) that there exists a constant α > 0 and t0 > 0 such that ϕ(2t) ≥ 2α ϕ(t)

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for every t ≥ t0 . Thus, ϕ(2n t) ≥ 2nα ϕ(t) for t ≥ t0 . Consider sets A and B deﬁned by the formula ψ(t) ψ(s) ψ(t) A := s : x∗ (s) > > ⊂ s : xMψ =: B. t s t Fix c = 2n such that 2nα ≥ max{1, xMψ }. It follows that ϕ(ct) > xMψ ϕ(t) for all t ≥ t0 . Therefore, ct ∈ / B if t ≥ t0 . Since ϕ is an increasing function (see [14]), we have sup B ≤ ct for t ≥ t0 . Since B is an interval, we have m(B) ≤ ct provided that t ≥ t0 . Thus, for t ≥ t0 , we have nx (ψ(t)/t) = m(A) ≤ m(B) ≤ ct. Remark 16. Let ψ ∈ Ψ and let τω be a Dixmier trace on Mψ . We have ψ(nt) ω =1 ψ(t) for every n ≥ 1. Indeed, if τω is linear then (see [11]) ψ(nt) ω = τω (nσ1/n ψ ) = τω (ψ ) = 1. ψ(t) This remark is frequently used below together with the following lemma from [9]. Lemma 17. Let ω ∈ L∞ (0, ∞)∗ be an arbitrary generalised limit. If x, y ∈ L∞ (0, ∞) are such that ω(|x − 1|) = 0, then ω(xy) = ω(y). Lemma 18. Let ψ ∈ Ψ satisfy the condition (4) and let τω be a Dixmier trace on Mψ . We have ⎛ ⎞ ∞ ⎜ −1 ⎟ ω⎝ λdnx (λ)⎠ ≤ τω (x) ψ(t) ψ(t)/t

for every positive x ∈ Mψ . Proof. Let c(x) be the constant deﬁned in Lemma 15. Clearly, ⎞ ⎛ nx (ψ(t)/t) nx (ψ(t)/t) ψ(c(x)t) 1 1 ⎟ ⎜ x∗ (s)ds = x∗ (s)ds⎠ . ·⎝ ψ(t) ψ(t) ψ(c(x)t) 0

0

It follows from Remark 16 and Lemma 17 that ⎞ ⎛ ⎛ nx (ψ(t)/t) 1 ⎟ ⎜ 1 ⎜ x∗ (s)ds⎠ = ω ⎝ ω⎝ ψ(t) ψ(c(x)t) 0

nx (ψ(t)/t)

⎞

⎟ x∗ (s)ds⎠ .

0

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It follows from Lemma 15 that ⎛ ⎞ ⎞ ⎛ nx (ψ(t)/t) c(x)t 1 1 ⎜ ⎟ ⎟ ⎜ ω⎝ x∗ (s)ds⎠ ≤ ω ⎝ x∗ (s)ds⎠ . ψ(c(x)t) ψ(c(x)t) 0

0

However, since ω is dilation invariant, we have ⎞ ⎛ ⎞ ⎛ c(x)t t 1 1 ⎟ ⎜ x∗ (s)ds⎠ = ω ⎝ x∗ (s)ds⎠ . ω⎝ ψ(c(x)t) ψ(t) 0

0

Lemma 19. Let ψ ∈ Ψ and let τω be a Dixmier trace on Mψ . We have ⎛ ⎞ ∞ ⎜ −1 ⎟ λdnx (λ)⎠ τω (x) ≤ ω ⎝ ψ(t) ψ(t)/t

for every positive x ∈ Mψ . Proof. Fix n ∈ N. Clearly, 1 ψ(t)

t

x∗ (s)ds =

0

ψ(nt) ψ(t)

⎛ 1 ·⎝ ψ(nt)

t

⎞ x∗ (s)ds⎠ .

0

It follows from Remark 16 and Lemma 17 that ⎛ ⎞ t 1 τω (x) = ω ⎝ x∗ (s)ds⎠ . ψ(nt)

(12)

0

We claim that t

nx (ψ(nt)/nt)

∗

x∗ (s)ds +

x (s)ds ≤ 0

0

1 ψ(nt). n

The inequality is evident if t ≤ nx (ψ(nt)/nt). If t > nx (ψ(nt)/nt), then x∗ (s) ≤ ψ(nt)/nt for every s ∈ [nx (ψ(nt)/nt), t]. Thus, t

nx (ψ(nt)/nt)

∗

x (s)ds =

t

∗

x∗ (s)ds

x (s)ds +

0

0

nx (ψ(nt)/nt)

nx (ψ(nt)/nt)

≤

x (s)ds + t − nx 0

and the claim follows.

∗

ψ(nt) nt

·

ψ(nt) nt

564

A. A. Sedaev, F. A. Sukochev and D. V. Zanin Hence, ⎛ ω⎝

1 ψ(nt)

t

⎞

⎛

⎜ x∗ (s)ds⎠ ≤ ω ⎝

0

1 ψ(nt)

nx (ψ(nt)/nt)

0

IEOT

⎞

⎟ 1 x∗ (s)ds⎠ + . n

It follows from (12) and the dilation-invariance of ω that ⎛ ⎞ nx (ψ(t)/t) ⎜ 1 ⎟ 1 x∗ (s)ds⎠ + . τω (x) ≤ ω ⎝ ψ(t) n 0

Since n is arbitrary large, we are done.

The following theorem is the principal result of this section. It follows immediately from Lemmas 18 and 19. Theorem 20. Let ψ ∈ Ψ satisfy the condition (4) and let τω be a Dixmier trace on Mψ . We have ⎛ ⎞ ∞ ⎜ −1 ⎟ τω (x) = ω ⎝ λdnx (λ)⎠ ψ(t) ψ(t)/t

for every positive x ∈ Mψ . Arguing similarly, one can obtain similar assertion for Marcinkiewicz sequence spaces. Theorem 21. Let ψ ∈ Ψ satisfy the condition (4) and let τω be a Dixmier trace on mψ . We have ⎞ ⎛ 1 xk ⎠ τω (x) = ω ⎝ ψ(n) xk ≥ψ(n)/n

for every positive x ∈ mψ .

4. Adjusted Lidskii Formula for Dixmier Traces: Noncommutative Setting In this section, we extend preceding results to Dixmier traces on Marcinkiewicz operator ideals. 4.1. Adjusted Lidskii Formula for Dixmier Traces: Normal Operators The following assertion follows directly from the Theorem 21. Lemma 22. Let ψ ∈ Ψ satisfy the condition (4) and let τω be a Dixmier trace on Mψ . We have ⎛ ⎞ 1 λ⎠ τω (S) = ω ⎝ ψ(n) λ∈σ(S),|λ|>ψ(n)/n

for every self-adjoint operator S ∈ Mψ .

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The following three lemmas are used to extend the formula above to the case of normal operators. Lemma 23. Let ψ ∈ Ψ satisfy the condition (4) and let τω be a Dixmier trace on Mψ . We have 1 ψ(t) ω nx =0 t t for every positive x ∈ Mψ . A similar assertion holds for Marcinkiewicz sequence space mψ . Proof. Fix n ∈ N. It follows from the dilation-invariance of ω that 1 ψ(t) ψ(nt) 1 nx nx ω =ω . t t nt nt It is clear that nx (ψ(nt)/nt) ψ(nt) 1 ψ(nt) 1 1 nx ds. = + nt nt n ψ(nt) nt t

If t > nx (ψ(nt)/nt), we have nx (ψ(nt)/nt)

t

ψ(nt) ds ≤ 0. nt

If t ≤ nx (ψ(nt)/nt), then nx (ψ(nt)/nt)

t

ψ(nt) ds ≤ nt

nx (ψ(nt)/nt)

c(x)nt

x∗ (s)ds ≤

x∗ (s)ds.

t

t

The last inequality holds for all sufﬁciently large t by Lemma 15. In either case, 1 0 ≤ nx nt

ψ(nt) nt

1 1 ≤ + n ψ(nt)

It follows now from the (13) that ω

1 nx t

It is clear that ⎛ ⎜ ω⎝

1 ψ(nt) ⎛

ψ(t) t

≤

⎛

1 ⎜ 1 +ω⎝ n ψ(nt)

c(x)nt

x∗ (s)ds.

t

⎞

c(x)nt

⎟ x∗ (s)ds⎠ .

t

⎞

c(x)nt

⎟ x∗ (s)ds⎠

t

⎜ 1 = ω⎝ ψ(nt)

c(x)nt

0

⎞

⎛

1 ⎟ x∗ (s)ds⎠ − ω ⎝ ψ(nt)

t 0

⎞ x∗ (s)ds⎠ .

(13)

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It follows from the dilation-invariance of ω that ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ c(x)nt c(x)t t 1 ⎜ 1 ⎟ ⎟ ⎜ 1 ω⎝ x∗ (s)ds⎠ = ω ⎝ x∗ (s)ds⎠ −ω ⎝ x∗ (s)ds⎠ . ψ(nt) ψ(t) ψ(nt) t

0

0

It follows from Remark 16 and Lemma 17 that both terms in the right-hand side of the equality above are equal to τω (x). Therefore, 1 ψ(t) 1 nx ω ≤ . t t n Since n is arbitrary large, we are done.

Lemma 24. Let ψ ∈ Ψ satisfy the condition (4) and let τω be a Dixmier trace on Mψ . We have ⎞ ⎛ 1 λ⎠ = 0, ω⎝ ψ(n) |λ|>ψ(n)/n,| λ|≤ψ(n)/n,λ∈σ(S) ⎛ ⎞ 1 ω⎝

λ⎠ = 0 ψ(n) |λ|≤ψ(n)/n,| λ|>ψ(n)/n,λ∈σ(S)

for any normal operator S ∈ Mψ . Proof. We prove the ﬁrst assertion only. Proof of the second one is identical. Note that λ ∈ σ(S) if and only if |λ| ∈ σ(|S|). It follows immediately that ψ(n) λ ≤ |λ|>ψ(n)/n,| λ|≤ψ(n)/n,λ∈σ(S) |λ|>ψ(n)/n,| λ|≤ψ(n)/n,λ∈σ(S) n ψ(n) ψ(n) ψ(n) ψ(n) ≤ n|S| 1= 1= . n n n n |λ|>ψ(n)/n,λ∈σ(S)

λ>ψ(n)/n,λ∈σ(|S|)

The assertion follows now from Lemma 23.

Lemma 25. Let ψ ∈ Ψ satisfy the condition (4) and let τω be a Dixmier trace on Mψ . We have ⎞ ⎛ 1 λ⎠ = 0 ω⎝ ψ(n) |λ|,| λ|≤ψ(n)/n,|λ|>ψ(n)/n,λ∈σ(S)

for any normal operator S ∈ Mψ .

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Proof. It is clear that λ ≤ |λ| |λ|,| λ|≤ψ(n)/n,|λ|>ψ(n)/n,λ∈σ(S) |λ|,| λ|≤ψ(n)/n,|λ|>ψ(n)/n,λ∈σ(S) 2ψ(n) ≤ |λ| ≤ 1 n ψ(n)/n<|λ|≤2ψ(n)/n,λ∈σ(S) |λ|>ψ(n)/n,λ∈σ(S) ψ(n) 2ψ(n) 2ψ(n) n|S| 1= = . n n n λ>ψ(n)/n,λ∈σ(|S|)

The assertion follows now from Lemma 23.

The following theorem extends result of Lemma 22 to normal operators from Mψ . Theorem 26. Let ψ ∈ Ψ satisfy the condition (4) and let τω be a Dixmier trace on Mψ . We have ⎛ ⎞ 1 λ⎠ τω (S) = ω ⎝ ψ(n) |λ|>ψ(n)/n,λ∈σ(S)

for any normal operator S ∈ Mψ . Proof. It follows from Lemma 22 that ⎛ ⎞ ⎛ 1 1 τω ( S) = ω ⎝ λ⎠ = ω ⎝ ψ(n) ψ(n) |λ|>ψ(n)/n),λ∈σ(S)

⎞

λ⎠ .

|λ|>ψ(n)/n,λ∈σ(S)

By Lemma 24, ⎛ τω ( S) = ω ⎝

1 ψ(n)

⎞

λ⎠ .

max{|λ|,| λ|}>ψ(n)/n,λ∈σ(S)

The same is valid for S. By the linearity, ⎛ 1 τω (S) = ω ⎝ ψ(n)

⎞ λ⎠ .

max{|λ|,| λ|}>ψ(n)/n,λ∈σ(S)

It follows from Lemma 25 that ⎛ τω (S) = ω ⎝

1 ψ(n)

⎞ λ⎠ .

|λ|>ψ(n)/n,λ∈σ(S)

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4.2. Adjusted Lidskii Formula for Dixmier Traces: General Case Recall the following result of Ringrose (see Theorems 1, 6 and 7 from [17]). Theorem 27. Let T ∈ B(H) be a compact operator. There exists a projection-valued measure Eλ such that 1. T E λ = Eλ T E λ . 2.

Either Eλ = Eλ−0 or rank(Eλ − Eλ−0 ) = 1.

3.

If, in addition, T Eλ = Eλ−0 T Eλ , then T is quasi-nilpotent.

Corollary 28. Let T ∈ B(H) be a compact operator. There exist compact normal operator S and compact quasi-nilpotent operator Q such that T = S + Q and σ(S) = σ(T ). Proof. Deﬁne an operator S by the following formula S= (Eλ − Eλ−0 )T (Eλ − Eλ−0 ). Eλ =Eλ−0

A straightforward computation shows that the operator Q = T − S satisﬁes the condition 3 of the Theorem above. Hence, Q is quasi-nilpotent. Evidently, S is a diagonal operator with eigenvalues of T on the diagonal. Hence, σ(S) = σ(T ). By the Weil theorem, sequence of eigenvalues of T is majorized by the sequence of its singular values (see Theorem 3.1 from [12]). Hence, for T ∈ Mψ , we obtain S, Q ∈ Mψ . The following assertion directly follows from the Theorem 3.3 from [13]). Theorem 29. If Q ∈ Mψ is a quasi-nilpotent operator, then Q belongs to the commutator [Mψ , B(H)]. Corollary 30. If Q ∈ Mψ is a quasi-nilpotent operator and τω is an arbitrary Dixmier trace on Mψ , then τω (Q) = 0. Indeed, due to [7], we have τω ([A, B]) = 0 for every A ∈ Mψ and every B ∈ B(H). The following theorem is the main result of this section. Theorem 31. Let ψ ∈ Ψ satisfy condition (4) and let τω be a Dixmier trace on Mψ . We have ⎛ ⎞ 1 τω (T ) = ω ⎝ λ⎠ ψ(n) λ∈σ(T ),|λ|>ψ(n)/n

for any operator T ∈ Mψ .

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Proof. Let S be a normal operator constructed in Corollary 28. The assertion holds for S by Theorem 26. Note that τω (T ) = τω (S) by Corollaries 28 and 30. Since σ(S) = σ(T ), we are done.

5. Applications to Heat Kernel Formula In this section, we provide a simple proof of one of the heat semigroup formulae from [18] (see also earlier results in [3,5]). Our hypothesis on ω is very mild. Lemma 32. For any positive x ∈ M1,∞ we have ⎛ ⎞ nx(1/t) ⎜ 1 ⎟ M⎝ (x∗ (s) − 1/t)ds⎠ = o(1). log(t) t

Proof. If t > nx (1/t), we have nx(1/t) ∗ (x (s) − 1/t)ds ≤ 1. t If t ≤ nx (1/t), then x∗ (s) ≥ 1/t for every s ∈ [t, nx (1/t)]. Therefore, nx(1/t)

nx(1/t)

∗

x∗ (s)ds.

(x (s) − 1/t)ds ≤

0≤ t

t

The assertion follows now from the Lemma 9.

Theorem 33. Let τω be a Dixmier trace on M1,∞ such that ω = ω ◦ M. We have ⎛ ⎞ 1 α ω⎝ τω (T ) = exp(−(tλ)−α )⎠ Γ(1/α) t λ∈σ(T )

for every positive operator T ∈ M1,∞ . Proof. Let x = x∗ ∈ M1,∞ be the rearrangement of T, that is x = i({sn (T )}). Without loss of generality, x ≤ 1. Since distributions of T and x coincide, we have ⎛ ⎞ ⎛ ∞ ⎞ 1 1 ω⎝ exp(−(tλ)−α )⎠ = ω ⎝ exp(−(tx(s))−α )ds⎠ . t t λ∈σ(T )

0

Setting 1/x(s) = u, we obtain ⎛ ∞ ⎞ ⎛ ∞ ⎞ 1 1 ω⎝ exp(−(tx(s))−α )ds⎠ = ω ⎝ exp(−(u/t)α )dnx (1/u)⎠ . t t 0

0

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It follows from the weak version of Karamata Theorem (see [3,18] for details) that ⎛ ⎞ 1 1 α ω⎝ nx (1/t) . exp(−(tλ)−α )⎠ = ω Γ(1/α) t t λ∈σ(T )

It is clear that

⎛ ⎞ t 1 1 M2 nx (1/t) − M ⎝ x(s)ds⎠ t log(t) 1 ⎛ t ⎛ ⎞⎞ t 1 1 ⎝ =M⎝ nx (1/s)ds − x(s)ds⎠⎠ . log(t) s2

1

1

Integrating by parts, we obtain t 1

1 1 nx (1/s)ds = − nx (1/t) + s2 t

t 1

1 dnx (1/s) = s

nx(1/t)

1

1 x(s)ds − nx (1/t). t

Hence, t 1

1 nx (1/s)ds − s2

nx(1/t)

t

(x(s) − 1/t)ds − 1.

x(s)ds = 1

t

It follows from the Lemma 32 that ⎛ ⎞ t 1 1 2 nx (1/t) − M ⎝ M x(s)ds⎠ = o(1). t log(t) 1

The assertion follows now from the M -invariance of ω.

References [1] Azamov, N., Sukochev, F.: A Lidskii type formula for Dixmier traces. C. R. Math. Acad. Sci. Paris 340(2), 107–112 (2005) [2] Benameur, M., Fack, T.: Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras. Adv. Math. 199(1), 29–87 (2006) [3] Carey, A., Phillips, J., Sukochev, F.: Spectral flow and Dixmier traces. Adv. Math. 173(1), 68–113 (2003) [4] Carey, A., Phillips, J., Rennie, A., Sukochev, F.: The Hochschild class of the Chern character for semifinite spectral triples. J. Funct. Anal. 213(1), 111–153 (2004) [5] Carey, A., Rennie, A., Sedaev, A., Sukochev, F.: The Dixmier trace and asymptotics of zeta functions. J. Funct. Anal. 249(2), 253–283 (2007) [6] Carey, A., Sukochev, F.: Dixmier traces and some applications in noncommutative geometry. Russ. Math. Surv. 61(6), 1039–1099 [7] Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994)

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[8] Dixmier, J.: Existence de traces non normales. C. R. Acad. Sci. Paris 262, A1107–A1108 (1966) [9] Dodds, P., de Pagter, B., Sedaev, A., Semenov, E., Sukochev, F.: Singular symmetric functionals, (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 290, (2002), Issled. po Linein. Oper. i Teor. Funkts. 30, 42–71, 178; translation in J. Math. Sci. (N.Y.) 124(2), 4867–4885 (2004) [10] Dodds, P., de Pagter, B., Sedaev, A., Semenov, E., Sukochev, F.: Singular symmetric functionals and Banach limits with additional invariance properties (Russian). Izv. Ross. Akad. Nauk Ser. Mat. 67(6), 111–136 (2003) [11] Dodds, P., de Pagter, B., Semenov, E., Sukochev, F.: Symmetric functionals and singular traces. Positivity 2(1), 47–75 (1998) [12] Gohberg, I., Krein, M.: Introduction to the theory of linear nonselfadjoint operators. In: Translations of Mathematical Monographs, vol. 18. American Mathematical Society, Providence (1969) [13] Kalton, N.: Spectral characterization of sums of commutators, I. J. Reine Angew. Math. 504, 115–125 (1998) [14] Krein, S., Petunin, J., Semenov, E.: Interpolation of linear operators. Nauka, Moscow (1978, in Russian); English translation in Translations of Math. Monographs, vol. 54. Amer. Math. Soc., Providence (1982) [15] Lord, S., Sedaev, A., Sukochev, F.: Dixmier traces as singular symmetric functionals and applications to measurable operators. J. Funct. Anal. 224(1), 72–106 (2005) [16] Pietsch, A.: About the Banach envelope of l1,∞ . Rev. Mat. Complut. 22(1), 209–226 (2009) [17] Ringrose, J.: Super-diagonal forms for compact linear operators. Proc. Lond. Math. Soc. (3) 12, 367–384 (1962) [18] Sedaev, A.: Generalized limits and related asymptotic formulas. Math. Notes 86(4), 612–627 (2009) A. A. Sedaev Department of Mathematics Voronezh State University of Architecture and Civil Engineering 20-letiya Oktyabrya 84 Voronezh 394006 Russia e-mail: [email protected] F. A. Sukochev (B) School of Mathematics and Statistics University of New South Wales Kensington NSW 2052 Australia e-mail: [email protected]

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D. V. Zanin School of Computer Science, Engineering and Mathematics Flinders University Bedford Park SA 5042 Australia e-mail: [email protected] Received: February 21, 2010. Revised: August 10, 2010.

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Integr. Equ. Oper. Theory 68 (2010), 573–599 DOI 10.1007/s00020-010-1829-0 Published online September 23, 2010 c Springer Basel AG 2010

Integral Equations and Operator Theory

Optimal Extension of Fourier Multiplier Operators in Lp(G) G. Mockenhaupt, S. Okada and W. J. Ricker Abstract. Given 1 ≤ p < ∞, a compact abelian group G and a p-multiplier ψ : Γ → C (with Γ the dual group), we study the optimal domain (p) of the multiplier operator Tψ : Lp (G) → Lp (G). This is the larg(p)

est Banach function space, denoted by L1 (mψ ), with order continuous p

(p)

norm into which L (G) is embedded and to which Tψ has a continuous Lp (G)-valued extension. Compactness conditions for the optimal exten(p) sion are given, as well as criteria for those ψ for which L1 (mψ ) = Lp (G) (p)

is as small as possible and also for those ψ for which L1 (mψ ) = L1 (G) is as large as possible. Several results and examples are presented for (p) cases when Lp (G) L1 (mψ ) L1 (G). Mathematics Subject Classification (2010). 28B05, 43A15, 43A22, 46G10, 47B07. Keywords. p-Multiplier, optimal extension, Banach function space, vector measure.

1. Introduction Let X be a Banach function space (brieﬂy, B.f.s.) over a positive, ﬁnite measure space (Ω, Σ, μ), E be a Banach space and T : X → E be a linear operator. Assume the additive, E-valued set function mT : A → T (χA ) is μ-determined, i.e., mT and μ have the same null sets. For each Σ-simple function s we have Ω s dmT = T (s). If T has the property, for each + increases f ∈ X + , that T (fn ) → T (f ) weakly in E whenever {fn }∞ n=1 ⊆ X μ-a.e. to f , then mT is σ-additive (i.e., a vector measure). The advantage of σ-additivity is that the space L1 (mT ) of all mT -integrable functions is a B.f.s. with σ-order continuous norm, X is continuously embedded in L1 (mT ), and 1 the integration map ImT : L (mT ) → E, i.e., f → Ω f dmT , is a continuous extension of T . Also, L1 (mT ) is optimal : if T has a continuous extension S. Okada was supported by the Maximilian Bickhoﬀ Universit¨ atsstiftung.

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T : Y → E, with Y any σ-order continuous B.f.s. (over (Ω, Σ, μ)) containing X, then Y is continuously embedded in L1 (mT ) and ImT coincides with T on Y . This “optimal extension process” is known for kernel operators ([2, 7,46]), Sobolev imbeddings ([8,9,14,25]), the Hardy operator, [10], and the Hausdorﬀ-Young inequality, [35]. For convolutions with measures (in Lp -spaces), which form a proper subclass of all p-multiplier operators, see [39,40], [41, Ch.7]. Given an inﬁnite compact abelian group G, with normalized Haar measure μ and dual group Γ, and a p-multiplier ψ : Γ → C (1 ≤ p < ∞), there is (p) (p) the associated multiplier operator Tψ : Lp (G) → Lp (G). If ψ = 0, then Tψ (p)

(p)

is μ-determined. For T := Tψ denote the above vector measure mT by mψ , (p)

i.e., X = E = Lp (G). Then Lp (G) ⊆ L1 (mψ ) ⊆ L1 (G), with both inclusions the natural ones and continuous. Moreover, the Lp (G)-valued extension Im(p) ψ (p) (p) (p) (p) of Tψ satisﬁes ( G f dmψ ) = ψ · f, for f ∈ L1 (mψ ). Whenever Tψ is (p)

(p)

(p)

Fredholm, L1 (mψ ) = Lp (G) whereas if Tψ

is compact, Lp (G) L1 (mψ ). (p) We will see L1 (mψ ) = L1 (G) is as large as possible iﬀ ψ = f for some (p)

(p)

f ∈ Lp (G) iﬀ mψ has ﬁnite variation. The B.f.s. L1 (mψ ) is weakly sequen(p)

tially complete, translation invariant and homogeneous. Also, if f ∈ L1 (mψ ) (p)

and g satisﬁes |g| ≤ |f |, then g ∈ L1 (mψ ) with g L1 (m(p) ) ≤ f L1 (m(p) ) . ψ

(p)

ψ

Moreover, f ∈ L1 (G) belongs to L1 (mψ ) iﬀ ψ · (χA f ) ∈ (Lp (G)) , for all Borel sets A ⊆ G. For 1 < p ≤ 2 ﬁxed, we show there exist p-multipliers (p) (p) ψ satisfying Lp (G) L1 (mψ ) L1 (G) such that L1 (mψ ) contains Lr (G) (p) for some 1 < r < p, others for which L1 (mψ ) contains 1
others for which L1 (mψ ) fails to contain Lr (G) for all 1 < r < p. (p)

(p)

2. The Vector Measure mψ and L1 (mψ ) of a Multiplier ψ For 1 ≤ p ≤ ∞, let Lp (G) be as usual. Since μ is ﬁnite, L∞ (G) Lq (G) Lp (G) L1 (G), for 1 < p < q. The Banach spaces c0 (Γ) and p (Γ), 1 ≤ p ≤ ∞, also arise. Given γ ∈ Γ, its value at x ∈ G is written (x, γ). The unit of G is denoted by 0 and that of Γ by e. Let T (G) := span{(·, γ) : γ ∈ Γ} be the trigonometric polynomials on G. The Banach space of all C-valued, regular measures on the Borel σ-algebra B(G) is denoted by M (G) and has its total variation norm · M (G) . Given λ ∈ M (G), its Fourier–Stieltjes transform The Fourier transform F1 : L1 (G) → ∞ (Γ) is continuous is denoted by λ. and c0 (Γ)-valued. Considering F1 as c0 (Γ)-valued, we denote it by F1,0 . The restriction of F1,0 to Lp (G) is denoted by Fp,0 . Actually, f ∈ p (Γ), with 1 1 p p p + p = 1, whenever 1 ≤ p ≤ 2, f ∈ L (G), with f p (Γ) ≤ f L (G) (Hausdorﬀ–Young inequality, [23, p.227], [28, Theorem F.8.4]). The operator f → f, from Lp (G) into p (Γ), is denoted by Fp .

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Given 1 ≤ p < ∞, λ ∈ M (G) and f ∈ Lp (G), the convolution operator (p) Cλ : f → (f ∗ λ)(x) := f (x − y) dλ(y), x ∈ G, (2.1) G p

is linear, L (G)-valued and continuous, because of the inequality f ∗ λ Lp (G) ≤ λ M (G) · f Lp (G) ,

f ∈ Lp (G),

(2.2) (p)

[28, Appendix F]. Let δa be the Dirac measure at a ∈ G. Then Cδa is the translation operator τa on Lp (G), i.e., x → f (x − a), for f ∈ Lp (G), which (p) (p) is an isometry satisfying τa ◦ Cλ = Cλ ◦ τa for a ∈ G, λ ∈ M (G). For any continuous translation invariant operator T : Lp (G) → Lp (G) there is ψ ∈ ∞ (Γ) satisfying (T f )= ψ · f,

f ∈ Lp (G),

(2.3) (p)

[4, Theorem 4.4], [28, Corollary 4.1.2]. We denote T by Tψ . Then Mp (G) := {ψ ∈ ∞ (Γ) : ψ satisﬁes (2.3)} is the space of all p-multipliers for G. Given λ ∈ M (G), Cλ ((·, γ)) = (·, γ) ∗ λ = λ(γ) · (·, γ), (p)

γ ∈ Γ.

(2.4)

(p)

Equip Op (G) := {Tψ : ψ ∈ Mp (G)} with the operator norm as a subalgebra of the space L(Lp (G)) of all bounded linear operators on Lp (G). Then (p)

|||ψ|||p := Tψ ,

ψ ∈ Mp (G),

(2.5)

is a norm in Mp (G) and ψ ∞ (Γ) ≤ |||ψ|||p , for ψ ∈ Mp (G), [28, Corollary ∈ c0 (Γ)}, 4.1.2]. A relevant subalgebra of M (G) is M0 (G) := {λ ∈ M (G) : λ [21,29,31]. Then L1 (G) M0 (G) M (G); see [29, p.422]. For Banach spaces X, E, the dual operator of T ∈ L(X, E), with L(X, E) the space of bounded operators from X to E, is the operator T ∗ ∈ L(E ∗ , X ∗ ) given by x, T ∗ (u∗ ) := T (x), u∗ ,

x ∈ X, u∗ ∈ E ∗ ,

(2.6)

: γ → ψ(−γ), for γ ∈ Γ. with X the dual space of X. For ψ ∈ (Γ), deﬁne ψ ∈ Mp (G) iﬀ ψ ∈ Mp (G). Recall Then (f)∼ = (f), for f ∈ L1 (G), and ψ Mp (G) is isometrically isomorphic to Mp (G) via T → T ∗ , [28, Theorem 4.1.2]. If ψ ∈ Mp (G), 1 ≤ p ≤ 2, then ψ ∈ Mr (G) for p ≤ r ≤ 2 and |||ψ|||r ≤ |||ψ|||p , [28, Corollary 4.1.3]. To summarize: ∗ (p) (p ) Lemma 2.1. Let 1 < p < ∞ and ψ ∈ Mp (G). Then Tψ = Tψ , i.e., ∗

∞

(p)

(p )

Tψ (f ), h = f, Tψ (h) , (p) · h, In particular, [(Tψ )∗ (h)]= ψ

f ∈ Lp (G), h ∈ Lp (G).

(2.7)

for h ∈ Lp (G).

Let E be a Banach space and m : B(G) → E a vector measure. Its variation measure |m| is deﬁned in [12, Ch.I, Deﬁnition 1.4]. For x∗ ∈ E ∗ , deﬁne m, x∗ : A → m(A), x∗ . A B(G)-measurable function f is m-integrable, if

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(I-1): f is m, x∗ -integrable (i.e., G |f | d| m, x∗ | < ∞) for each x∗ ∈ E ∗ , and each A ∈ B(G), there exists an element A f dm ∈ E satisfying (I-2): for A f dm, x∗ = A f d m, x∗ , for each x∗ ∈ E ∗ , [26, Ch.II], [41, Ch.3]. The vector space L1 (m) of all m-integrable functions has the seminorm |f | d| m, x∗ |, f ∈ L1 (m); (2.8) f L1 (m) := sup x∗ ≤1

G

it is complete and the subspace simB(G) of all C-valued, B(G)-simple functions is dense. A function f ∈ L1 (m) is called m-null if f L1 (m) = 0. We identify L1 (m) with its quotient Banach space modulo m-null functions. The semivariation m is deﬁned by m (A) := χA L1 (m) , for A ∈ B(G), and satisﬁes m(A) ≤ m (A) ≤ |m|(A). A set A ∈ B(G) is m-null if χA is m-null. If |f | ≤ |g| in L1 (m) (i.e., m-a.e. on G), then f L1 (m) ≤ g L1 (m) ; cf. (2.8). Also, if f ∈ L1 (m) and g measurable satisfy |g| ≤ |f |, then g ∈ L1 (m). Hence, L1 (m) is a Banach function space (relative to (G, B(G), λ) for any control measure λ of m), [41, Ch.3]. The norm of L1 (m) is σ-order continuous, i.e., limn→∞ fn L1 (m) = 0 whenever L1 (m) fn ↓ 0 in the order of L1 (m). The space of B(G)-measurable functions f which satisfy (I-1) is denoted by L1w (m). It is a Banach lattice for the norm (2.8) and L1 (m) is a closed subspace. The integration map Im : L1 (m) → E is deﬁned by (2.9) Im (f ) := f dm, f ∈ L1 (m). G

It is linear and continuous as Im (f ) E ≤ f L1 (m) for f ∈ L1 (m). The above facts concerning m, L1 (m), L1w (m), and Im occur in [26], [41, Ch.3]. For 1 ≤ p < ∞ and ψ ∈ Mp (G), deﬁne the Lp (G)-valued vector measure (p)

(p)

mψ : A → Tψ (χA ),

A ∈ B(G).

Applying (2.7) with f := χA , for A ∈ B(G), we obtain (p) (p ) (p ) mψ , h (A) = χA , Tψ (h) = Tψ (h) dμ,

(2.10)

(2.11)

A p

p

∗

for each h ∈ L (G) = L (G) ; for p = 1 see [40, Lemma 2.2(i)]. Hence, (p) (p ) (2.12) | mψ , h |(A) = |Tψ (h)| dμ, A ∈ B(G), h ∈ Lp (G). A

p

older’s inequality and Lemma 2.1 yield Given f ∈ L (G) and h ∈ Lp (G), H¨ (p ) |f | · |Tψ (h)| dμ ≤ |||ψ|||p f Lp (G) h Lp (G) . (2.13) G

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Proposition 2.2. Let 1 ≤ p < ∞ and ψ ∈ Mp (G). (i)

(p)

For each A ∈ B(G), the semivariation mψ (A) is given by ⎧ ⎫ ⎨ ⎬ (p) (p ) |Tψ (h)| dμ : h Lp (G) ≤ 1 mψ (A) = sup ⎩ ⎭

(2.14)

A

and satisﬁes the inequalities (p)

ψ ∞ (Γ) μ(A) ≤ mψ (A) ≤ |||ψ|||p · [μ(A)]1/p . (ii)

(p)

(2.15) (p)

Always mψ μ. Conversely, if ψ = 0, then also μ mψ . In partic(p)

ular, Tψ

∈ L(Lp (G)) is μ-determined and μ is a control measure for

(p) mψ .

Proof. For p = 1, see [40, Proposition 2.4]. So, assume that 1 < p < ∞. (i) Setting f = χA in (2.8), together with (2.12), yields (2.14). For the ﬁrst inequality in (2.15), let γ ∈ Γ. As (·, −γ) Lp (G) = 1 and (p )

Tψ ((·, −γ)) = ψ(γ)(·, −γ),

(2.16)

(p ) (p) we see from (2.14) that |ψ(γ)|·μ(A) = A |Tψ ((·, −γ))| dμ ≤ mψ (A), for A ∈ B(G). But, γ ∈ Γ is arbitrary, and so the ﬁrst inequality in (2.15) follows. For the second inequality in (2.15), let h ∈ Lp (G) satisfy (p ) h Lp (G) ≤ 1. Fix A ∈ B(G). Then (2.13) gives A |Tψ (h)| dμ = (p ) χ · |Tψ (h)| dμ ≤ χA Lp (G) |||ψ|||p = [μ(A)]1/p |||ψ|||p . The second G A inequality in (2.15) then follows from (2.14). (ii) Follows directly from (2.15). For 1 ≤ r ≤ s ≤ ∞, let js,r : Ls (G) → Lr (G) be the natural inclusion. In order to use interpolation for compactness, note that μ is non-atomic, [41, Lemma 7.97]. Proposition 2.3. Let 1 < p ≤ 2 and ψ ∈ Mp (G). The following are equivalent. (i) ψ ∈ c0 (Γ). (p) (p) (ii) R(mψ ) := {mψ (A) : A ∈ B(G)} is a relatively compact set in Lp (G). (2)

(iii) Tψ : L2 (G) → L2 (G) is a compact operator. (r)

(iv) Tψ : Lr (G) → Lr (G) is compact for some/all p < r ≤ 2. (2)

(v) j2,q ◦ Tψ : L2 (G) → Lq (G) is compact for some/all p ≤ q ≤ 2. (2)

(vi) R(mψ ) is a relatively compact subset of L2 (G). (q)

(vii) R(mψ ) is relatively compact in Lq (G) for some/all p ≤ q ≤ 2. Proof. (i) ⇔ (iii). The map F2 : L2 (G) → 2 (Γ) is an isometric isomor(2) phism and S := F2 ◦ Tψ ◦ F2−1 ∈ L(2 (Γ)) is multiplication by ψ ∈ ∞ (Γ).

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So, (i) ⇔ (iii) is equivalent to: S is a compact operator iﬀ ψ ∈ c0 (Γ). This is routine to verify. (p) (ii) ⇒ (i). Observe Tψ ((·, γ)) = ψ(γ)(·, γ) and (·, γ) = χ{γ} , for γ ∈ Γ. (p)

As F1,0 ◦ jp,1 ◦ Tψ (p)

(p)

= Fp,0 ◦ Tψ

belongs to L(Lp (G), c0 (Γ)), we see η :=

F1,0 ◦ jp,1 ◦ mψ is a vector measure whose range is relatively compact in c0 (Γ). Via the previous identities the proof can be completed as that of (iii) ⇒ (i) in Proposition 2.3 of [40]. (iii) ⇒ (v). This is clear via the continuity of j2,q . (q) (2) (v) ⇒ (vii). Use mψ = j2,q ◦ mψ together with the containment (2)

(2)

R(mψ ) ⊆ {Tψ (f ) : f L2 (G) ≤ 1}.

(p)

(q)

(vii) ⇒ (ii). Note that jq,p ∈ L(Lq (G), Lp (G)) and use mψ = jq,p ◦mψ . (iii) ⇒ (iv). Since μ is non-atomic, we can apply, [27, Theorem 3.10]. (r) (iv) ⇒ (iii). Clearly (iv) implies that R(mψ ) is a relatively compact (p)

(r)

set in Lr (G). By continuity of jr,p : Lr (G) → Lp (G) and mψ = jr,p ◦ mψ , we see that (ii) follows. Hence, so does (iii), as (ii) ⇒ (i) ⇒ (iii) has already been veriﬁed. (iii) ⇒ (vi). This is obvious. (vi) ⇒ (vii). Use continuity of j2,q : L2 (G) → Lq (G) and the identity (q) (2) mψ = j2,q ◦ mψ . (p)

(i) If 1 < p ≤ 2 and ψ ∈ Mp (G) \ c0 (Γ), i.e., R(mψ ) is not

Remark 2.4.

(p)

relatively compact, then Lemma 3.53(iv) of [41] implies that |mψ | is not σ-ﬁnite. (p) (p) (p) (ii) Let σ(Tψ ) be the spectrum of Tψ . Then ψ(Γ) ⊆ σ(Tψ ), [47]. We (p)

say Tψ

(p)

satisﬁes the spectral mapping property if ψ(Γ) = σ(Tψ ). The (p)

(p)

total disconnectedness of σ(Tψ ) (e.g., if Tψ

(p) Tψ

is compact) sufﬁces for

to be decomposable (cf. proof of [33, Lemma 2.2]), in which case (p)

ψ(Γ) = σ(Tψ ), [1, Lemma 3.2]. For 1 < p < 2, n ∈ N, there is ψ ∈ (p)

c0 (Zn ) ∩ Mp (Tn ) with Tψ Such a

(p) Tψ

failing the spectral mapping property, [48]. (r)

cannot be compact, whereas Proposition 2.3 implies Tψ is (q)

compact for p < r ≤ 2 and R(mψ ) is compact in Lq (G) for p ≤ q ≤ 2 (p)

(p)

(here q := p is included). So, R(mψ ) can be compact without Tψ being compact. For ψ ∈ (M0 (G))⊆ c0 (Γ), the case r := p can be included in (iv) of Proposition 2.3, [40, Proposition 2.3]. Crucial is (M (G))⊆ Mp (G) for every 1 ≤ p < ∞. Corollary 2.5. For ψ ∈ 1<s≤2 Ms (G), the following assertions are equivalent. (i) ψ ∈ c0 (Γ). (p) (ii) R(mψ ) is relatively compact in Lp (G) for some/all p ∈ (1, 2].

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(2)

(iii) Tψ : L2 (G) → L2 (G) is a compact operator. (p)

(iv) Tψ : Lp (G) → Lp (G) is compact for some/all 1 < p ≤ 2. (2)

(v) j2,p ◦ Tψ : L2 (G) → Lp (G) is compact for some/all 1 < p ≤ 2. (2)

(vi) R(mψ ) is relatively compact in L2 (G). Proof. (i) ⇔ (ii) and (i) ⇔ (iii) follow from Proposition 2.3. For (iii) ⇔ (iv) ⇔ (v) ⇔ (vi) observe, for ﬁxed 1 < p < 2, that we can choose s with (s) 1 < s < p. Since ψ ∈ c0 (Γ) iﬀ R(mψ ) is relatively compact (by (i) ⇔ (ii)), apply Proposition 2.3. Remark 2.6. (i) Concerning Corollary 2.5, (M (G)) ⊆ 1<s≤2 Ms (G) is always strict. Indeed, Γ contains an inﬁnite Sidon set, [16, p.157]. Such sets are Λ(p)-sets for every 1 < p < ∞, [16, p.156]. So, the standing assumption in Section (37.22) of [23] is fulﬁlled. Then part (c) of [23, Section (37.22)] is the stated claim. (ii) For 1 < p ≤ 2, let mp (G) be the closure of (L1 (G)) in the space (Mp (G), ||| · |||p ). The Banach space mp (G) consists of those ψ ∈ Mp (G) (p) such that Tψ ∈ L(Lp (G)) is compact, [19, Theorem 4.2.2], [13, Theorems 4 & 5]. Then (i) ⇔ (iv) of Corollary 2.5 asserts that Ms (G) ∩ c0 (Γ) = 1<s≤2 ms (G). The inclusion (M0 (G)) ⊆ 1<s≤2 1<s≤2 ms (G) may be strict. Indeed, for G := T and ψ : Z → C deﬁned by ψ(n) := −isgn(n)χZ\{−1,0,1} (n)/2 log(|n|),

n ∈ Z,

(2.17)

this is so; see [34, Example 5(b) & Proposition 6] for the details.

For a function H : G → E, with E a Banach space, to be strongly μ-measurable see [12, p.41]. Then H is Bochner μ-integrable iﬀ we have H(x) dμ(x) < ∞; its integrals, for A ∈ B(G), are denoted by G (B)- A H dμ ∈ E, [12, Ch.II]. (p)

Let 1 ≤ p < ∞ and ψ ∈ Mp (G)\{0}. Proposition 2.2(ii) implies Tψ (p)

is μ-determined. Theorem 4.14 of [41] implies L1 (mψ ) is the largest B.f.s. with σ-order continuous norm (over (G, B(G), μ)) into which Lp (G) is con(p) tinuously embedded and to which Tψ admits a continuous extension Im(p) : ψ

(p)

L1 (mψ ) → Lp (G) with (p) (p) Im(p) (f ) := f dmψ = Tψ (f ), ψ

(p)

f ∈ Lp (G) ⊆ L1 (mψ ).

(2.18)

G

So, with

(p) Jψ

p

(p)

: L (G) → L1 (mψ ) denoting the natural inclusion, we have (p)

(p)

Tψ = Im(p) ◦ Jψ , ψ

(p)

(p)

with Jψ = Tψ , [41, Proposition 4.4].

(2.19)

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Lemma 2.7. Let 1 ≤ p < ∞ and ψ ∈ Mp (G)\{0}. With continuous inclusions, (p)

Lp (G) ⊆ L1 (mψ ) ⊆ L1 (G).

(2.20) (p)

Proof. The ﬁrst inclusion follows from the optimality of L1 (mψ ); see (2.18). For p = 1 see [40, Theorem 1.1(v)]. Let 1 0. (p)

(p)

Then L1 (mψ ) ⊆ L1 ( mψ , h ) = L1 (aμ) = L1 (G); see (I-1) and (2.8) for the inclusion.

Proposition 2.8. For 1 < p < ∞, ψ ∈ Mp (G)\{0} the following are equivalent. (p)

(i) The vector measure mψ : B(G) → Lp (G) has ﬁnite variation. (ii) There exists f ∈ Lp (G) such that ψ = f, i.e., ψ ∈ (Lp (G)). (iii) There is a Bochner μ-integrable function H : G → Lp (G) with (p) mψ (A) = (B) − H dμ, A ∈ B(G).

(2.21)

A (p)

(p)

(iv) The extension Im(p) : L1 (mψ ) → Lp (G) of Tψ ψ

(v) (vi) (vii)

is compact.

(p) L (|mψ |) = L1 (G). (p) L1 (mψ ) = L1 (G). (p) (p) L1 (|mψ |) = L1 (mψ ). 1

(p)

Proof. For (iii) ⇒ (i) see [12, p.46, Theorem 4]. As mψ μ and Lp (G) has the RN-property, [12, p.218], (i) ⇒ (iii) is immediate, [12, p. 61, Deﬁnition 3]. (ii) ⇔ (iii). Given (ii), ψ ∈ (M (G))and (iii) follows, [40, Lemma 4.1]. Conversely, suppose that (iii) holds with H : G → Lp (G) Bochner μ-integrable. Deﬁne L(x, γ) := (−x, γ)ψ(γ) = (x, γ)ψ(γ), for x ∈ G, γ ∈ Γ. We show that [H(x)](γ) dμ(x) = L(x, γ) dμ(x), A ∈ B(G). (2.22) A

A

Indeed, for γ ∈ Γ ﬁxed, πγ ∈ (c0 (Γ))∗ by ξ → ξ(γ), for ξ ∈ c0 (Γ), so πγ ◦ Fp,0 ∈ Lp (G). Hence, A H, πγ ◦ Fp,0 dμ = (B)- A H dμ, πγ ◦ Fp,0 , for (p)

(p)

A ∈ B(G). This identity and (2.21) yield [Tψ (χA )](γ) = [mψ (A)](γ) = (p) mψ (A), πγ ◦Fp,0 = A H, πγ ◦Fp,0 dμ = A [H(x)](γ) dμ(x). Then (2.22) follows via (p) L(x, γ) dμ(x) = (x, γ)ψ(γ) dμ(x) = ψ(γ) · χ A (γ) = Tψ (χA ) (γ). A

A

The function H has μ-essentially separable range, [12, p.42, Theorem 2]. So, we may assume that its range H(G) ⊆ Lp (G) is separable. Then (Fp,0 ◦ H)(G) is separable in c0 (Γ). As elements of c0 (Γ) have countable

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support, there is Γ0 ⊆ Γ countable such that, whenever γ ∈ Γ\Γ0 , we have [H(x)](γ) = [(Fp,0 ◦ H)(x)](γ) = 0, for x ∈ G. Via (2.22) and the definition of L we have, for γ ∈ Γ\Γ0 , that A ψ(γ)(x, γ) dμ(x) = 0 for A ∈ B(G). So, given γ ∈ Γ\Γ0 we have ψ(γ)(x, γ) = 0 for all x ∈ G, i.e., ψ(γ) = 0. So, for each γ ∈ Γ\Γ0 , it follows L(x, γ) = 0 = [H(x)](γ),

μ-a.e. x ∈ G.

(2.23)

Given γ ∈ Γ0 , by (2.22) there is Aγ ∈ B(G) with μ(Aγ ) = 1 such that [H(x)] (γ) = L(x, γ), for x ∈ Aγ . As Γ0 is countable, AΓ0 := ∩γ∈Γ0 Aγ ∈ B(G). Moreover, μ(AΓ0 ) = 1 and L(x, γ) = [H(x)](γ), for x ∈ AΓ0 , γ ∈ Γ0 . This and (2.23) imply L(x, ·) = [H(x)](·), for x ∈ AΓ0 , with equality as functions on Γ. Choose any w ∈ AΓ0 . Then L(w, ·) = (−w, ·)ψ(·) = [H(w)](·), i.e., ψ = (w, ·)[H(w)](·) = δ−w (·)[H(w)](·) = (δ−w ∗ H(w))(·). But, H(w) ∈ Lp (G) and δ−w ∈ M (G) and so h := δ−w ∗ H(w) ∈ Lp (G); see (2.2). Since h = ψ, this yields (ii). (ii) ⇒ (iv). By (ii), ψ ∈ (M (G)). So, (iv) follows from [40, Theorem 1.2]. (iv) ⇒ (vii). See Theorems 1 and 4 of [38]. (ii) ⇒ (v) by [40, Theorem 1.2] (p) (p) (v) ⇒ (vi). Apply (2.20) and L1 (|mψ |) ⊆ L1 (mψ ), [30, Theorem 4.1]. (vi) ⇒ (ii). According to (2.18), T := jp,1 ◦ Im(p) ∈ L(L1 (G)). Fix ψ

(p)

(p)

a ∈ G. Then τa ◦ jp,1 = jp,1 ◦ τa . Since L1 (mψ ) = L1 (G), the map Jψ (p)

speciﬁed in (2.19) coincides with jp,1 . So, (2.19) becomes Tψ = Im(p) ◦ jp,1 , ψ

as elements of L(Lp (G)). Hence, as an equality in L(Lp (G), L1 (G)), we have (p) jp,1 ◦ Tψ = T ◦ jp,1 . It follows that (T ◦ τa )(f ) = (τa ◦ T )(f ), for f ∈ T (G), (1) λ

i.e., T = T

(1)

= Cλ

(p) λ

for some λ ∈ M (G); see (2.1). Moreover, T

(p)

(p)

(p)

= Tψ .

Since L1 (G) = L1 (mψ ) = L1 (m ), the equivalence (i) ⇔ (viii) in Theorem λ 1.2 of [40] yields (ii). (p) (p) (p) (vii)⇒(i). As χG ∈ L1 (mψ ), also χG ∈ L1 (|mψ |), i.e., |mψ |(G) < ∞. Remark 2.9.

(p)

(i) Let 1 < p < ∞ and ψ ∈ Mp (G)\{0}. If |mψ |(G) < ∞, (p)

(p)

then Lp (G) L1 (mψ ); cf. Proposition 2.8. Moreover, if Tψ p

1

pact, then this is so. Indeed, suppose L (G) = L (p)

(p) (mψ ).

is com-

Then Im(p) : ψ

(p)

L1 (mψ ) → Lp (G) is compact (as it equals Tψ ). So, Proposition 2.8 implies Lp (G) = L1 (G); contradiction! (p) (p) (ii) By (2.19), Tψ ∈ L(Lp (G)) is compact if Im(p) ∈ L(L1 (mψ ), Lp (G)) is ψ

compact. The “converse in false”. The function ψ in (2.17) is in Mp (T), (p) 1 < p ≤ 2, with Tψ ∈ L(Lp (T)) compact; cf. Remark 2.6(ii). But, ψ ∈ / p (Z). Hence, ψ fails (ii) of Proposition 2.8 and so, Im(p) is not 1

compact, i.e., L

implies Lp (T)

(p) (mψ ) (p) L1 (mψ ).

ψ

1

(p)

L (T). By (i) above, compactness of Tψ (p)

So, Lp (T) L1 (mψ ) L1 (T). Examples

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(p)

of ϕ ∈ (M (G)) are known for which Lp (G) L1 (mϕ ) L1 (G), [41, / (M (T)). Ch.7]. But, by Remark 2.6(ii), for ψ ∈ Mp (T) as in (2.17), ψ ∈ (iii) Let 1 < p ≤ 2. By (ii) of Proposition 2.8, ψ ∈ p (Γ) if ψ ∈ Mp (G) (p) satisﬁes |mψ |(G) < ∞. The “converse is false”, even if ψ belongs to (M (G)) ∩ p (Γ) ⊆ (M0 (G)) . Such ψ always belong to mp (G), [40, (p) Proposition 2.3], whereas Im(p) ∈ L(L1 (mψ ), Lp (G)) fails to be compact (recall

(p) |mψ |(G)

ψ

< ∞ and so apply Proposition 2.8). By (i) above (p)

and Proposition 2.8, such a ψ satisﬁes Lp (G) L1 (mψ ) L1 (G); see (2.20). For speciﬁc examples, let T := Fp : Lp (Tn ) → p (Zn ), for ﬁxed 1 < p < 2 and n ∈ N. Then mT : B(Tn ) → p (Zn ) satisﬁes Lp (Tn ) L1 (mT ) L1 (Tn ), [35]. Choose f ∈ L1 (mT )\Lp (Tn ), so that ImT (f ) = f ∈ p (Zn ), [35]. As f ∈ / Lp (Tn ), ψ := f fails (ii) of (p) n Proposition 2.8, i.e., |mψ |(T ) = ∞. (p)

3. The Optimal Domain L1 (mψ ) and Im(p) ψ

Basic properties of

(p) L1 (mψ )

and Im(p) : ψ

(p) L1 (mψ )

→ Lp (G) were given in (p)

Sect. 2. Here we treat these objects in more detail. We show L1 (mψ ) is weakly sequentially complete, translation invariant and is a homogeneous Banach space. Proposition 3.1. Let 1 < p < ∞ and ψ ∈ Mp (G)\{0}. (p)

(i) A B(G)-measurable function f : G → C is mψ -integrable iﬀ (p ) |f | · |Tψ (h)| dμ < ∞, h ∈ Lp (G) = Lp (G)∗.

(3.1)

G (p)

Moreover, the norm of f ∈ L1 (mψ ) is given by ⎧ ⎫ ⎨ ⎬ (p ) |f | · |Tψ (h)| dμ : h Lp (G) ≤ 1 . f L1 (m(p) ) = sup ψ ⎩ ⎭

(3.2)

G

(p)

(p)

(ii) We have L1 (mψ ) = L1w (mψ ) and both the natural inclusions in (2.20) (p)

(p)

hold and are continuous. More precisely, if Jψ : Lp (G) → L1 (mψ ) is (p)

(p)

as in (2.19) and Λψ is the injection of L1 (mψ ) into L1 (G), then (p)

ψ ∞ (Γ) ≤ Jψ = |||ψ|||p and

(3.3)

−1 −1 (p) (p) mψ (G) ≤ Λψ ≤ ψ ∞ (Γ) . (p)

(iii) Both subspaces T (G) and Lp (G) are dense in L1 (mψ ).

(3.4)

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(p)

(iv) The extension Im(p) ∈ L(L1 (mψ ), Lp (G)) has Im(p) = 1. If f ∈ ψ

ψ

L1 (mψ ), then ψ f ∈ c0 (Γ) and Im(p) (f ) ∈ Lp (G) satisﬁes ψ Im(p) (f ) = ψ · f. (p)

(3.5)

ψ

(p)

(v) L1 (mψ ) = {f ∈ L1 (G) : ψ · (χA f )∈ (Lp (G)) ∀A ∈ B(G)}.

(3.6)

(p)

(i) The codomain Lp (G) of mψ does not contain a copy of c0 and

Proof. so

(p) L1 (mψ )

(p) = L1w (mψ ), [26, (p) definition of L1w (mψ ) this

p.31], [30, Theorem 5.1]. By (2.12) and

the is equivalent to (3.1). Then (3.2) follows via (2.8) and (2.12). (p) (p) (ii) L1 (mψ ) = L1w (mψ ) by the proof of (i). Continuity is Lemma 2.7. (p)

(p)

(p)

Since Jψ = Tψ , [41, Proposition 4.4(ii)], and Tψ = |||ψ|||p , the inequality (3.3) follows from ψ ∞ (Γ) ≤ |||ψ|||p ; see (2.15). (p) Concerning (3.4), let f ∈ L1 (mψ ) ⊆ L1 (G). For γ ∈ Γ, (2.16) and (3.2) yield (p ) (3.7) |ψ(γ)| · |f | dμ = |f | · Tψ ((·, −γ)) dμ ≤ f L1 (m(p) ) . ψ

G

G

Hence, ψ ∞ (Γ) f L1 (G) ≤ f L1 (m(p) ) . As ψ ∞ (Γ) > 0, the definiψ

(iii)

(p) (p) tion of Λψ yields Λψ ≤ [ ψ ∞ (Γ) ]−1 . Observe that the function (p) (p) (p) f := [ mψ (G)]−1 . χG ∈ L1 (mψ ) and f L1 (m(p) ) = 1. So, Λψ ≥ ψ (p) (p) Λψ (f ) L1 (G) = [ mψ (G)]−1 . (p) As simB(G) ⊆ Lp (G) with simB(G) dense in L1 (mψ ), also Lp (G) is (p) (p) dense in L1 (mψ ). So, T (G) is dense in L1 (mψ ) as it is dense in Lp (G),

[44, p.24]. (p) (iv) Let f ∈ L1 (mψ ). Set A(n) := |f |−1 ([0, n)) ∈ B(G), for n ∈ N. The (p)

Dominated Convergence Theorem for mψ , [41, Theorem 3.7], yields (p) (p) (p) Tψ (f χA(n) ) = G f χA(n) dmψ → G f dmψ in Lp (G). Hence, ⎞ ⎛ (p) (p) (3.8) lim Tψ (f χA(n) ) = lim ψ · (f χA(n) )= ⎝ f dmψ ⎠ , n→∞

n→∞

G

in c0 (Γ). Also f χA(n) → f in L (G) and so, (f χA(n) ) → f in c0 (Γ). Since multiplication by ψ in c0 (Γ) is continuous, we have limn→∞ ψ · (p) (f χA(n) )= ψ · f, in c0 (Γ). This, and (3.8), imply ψ · f = ( G f dmψ )as (p) elements of c0 (Γ). Moreover, Im(p) (f ) := G f dmψ implies [Im(p) (f )]= 1

ψ

ψ

ψ · f. For Im(p) = 1, see [41, §3.3]. ψ

1

(p)

(p)

(v) Let f ∈ L (mψ ). As |χA f | ≤ |f |, also χA f ∈ L1 (mψ ), for A ∈ B(G). Via part (iv) we see that ψ · (χA f )= [Im(p) (χA f )] with Im(p) (χA f ) ∈ ψ

ψ

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Lp (G), i.e., ψ · (χA f ) ∈ (Lp (G)) for all A ∈ B(G). This yields one inclusion in (3.6). For the other inclusion, note Λ := {(·, γ) : γ ∈ Γ} ⊆ Lp (G) = Lp (G)∗ is a total set for Lp (G). Let f be in the right-side of (3.6). Given A ∈ B(G), let νf (A) ∈ Lp (G) be determined by [νf (A)]= ψ · (χA f ). n Then [νf (A)] = [ j=1 νf (Aj )] , whenever {Aj }nj=1 is a partition of n A ∈ B(G), i.e., νf (A) = j=1 νf (Aj ). So, νf : A → νf (A) is ﬁnitely additive. Fix γ ∈ Γ. For A ∈ B(G), we have νf (A), (·, γ) = (x, γ)ν (A)(x) dμ(x) = [ν (A)] (−γ) = ψ(−γ)[χA f ] (−γ) with f f G χA f ∈ L1 (G). If A(n) ↓ ∅, then f χA(n) → 0 in L1 (G) and so (f χA(n) ) → 0 in c0 (Γ). Then ψ(−γ)[f χA(n) ] (−γ) → 0 in C. The previous equalities show νf (A(n)), (·, γ) → 0 in C. So, νf , h is σadditive for all h ∈ Λ. As Λ is total and Lp (G) contains no copy of ∞ , we imply νf : B(G) → Lp (G) is σ-additive, [11, Theorem 1.1]. Deﬁne B(n) := |f |−1 ([0, n)), for n ∈ N. Then (A ∩ B(n)) ↑ A (μ(p) a.e.) for A ∈ B(G) ﬁxed. Since χA∩B(n) f ∈ L∞ (G) ⊆ L1 (mψ ), (iv) implies [Im(p) (χA∩B(n) f )] = ψ · (χA∩B(n) f ) = [νf (A ∩ B(n))] , i.e., ψ

Im(p) (χA∩B(n) f ) = νf (A∩B(n)) for n ∈ N. By σ-additivity of νf we have ψ (p) νf (A ∩ B(n)) = A f χB(n) dmψ → νf (A) in Lp (G). Then, f χB(n) → f (p)

pointwise implies f ∈ L1 (mψ ), [41, Theorem 3.5]. Remark 3.2. (p)

(p)

(p)

(i) Since L1 (mψ ) = L1w (mψ ), 1 ≤ p < ∞, the space

L1 (mψ ) is weakly sequentially complete, [41, Proposition 3.38]. If G is metrizable, arguing as in the proof of Proposition 7.21(iv) in [41] shows (p) that L1 (mψ ) is separable. (ii) It follows immediately from (3.1) and the identity (p ) |f (x − a)| · Tψ (h) (x) dμ(x) G

=

(p ) |f (x)| · Tψ (τ−a (h)) (x) dμ(x)

(3.9)

G (p)

(p)

for a ∈ G, h ∈ Lp (G) and f ∈ L1 (mψ ), that L1 (mψ ) is translation invariant. As τ−a is an isometric isomorphism in Lp (G), via (3.2) and (3.9) we have τa (f ) L1 (m(p) ) = f L1 (m(p) ) , ψ

ψ

(p)

f ∈ L1 (mψ ),

a ∈ G.

(3.10)

(p)

By (3.1), L1 (mψ ) is invariant for complex conjugation. Also, (Im(p) ◦ ψ

(p)

τa )(f ) = (τa ◦ Im(p) )(f ), for f ∈ L1 (mψ ), is valid; just check it for T (G) ⊆ (p)

ψ

L1 (mψ ). This identity implies the range of Im(p) and its closure are transψ

lation invariant.

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A homogeneous Banach space on G, [3], [24, p.14], is a linear subspace B of L1 (G) having a norm · B under which B is a Banach space satisfying: f ∈ B. (H-1) f L1 (G) ≤ f B , (H-2) If f ∈ B and a ∈ G, then τa (f ) ∈ B and τa (f ) B = f B . (H-3) For all f ∈ B and a0 ∈ G we have lima→a0 τa (f ) − τa0 (f ) B = 0. (p) To see L1 (mψ ) is homogeneous ﬁx ψ ∈ Mp (G)\{0}. For (H-2), see (p)

(3.10). For (H-3) we require, for f ∈ L1 (mψ ), that lima→0 τa (f ) = f in (p)

L1 (mψ ). This follows via the argument of the proof of Proposition 7.25(iii) in [41]; note Fp (G) there has the form L1 (mT ). Concerning (H-1), (3.7) implies, (p) for γ ∈ Γ, that |ψ(γ)| · f L1 (G) ≤ f L1 (m(p) ) , for f ∈ L1 (mψ ). So, (H-1) ψ

holds whenever |ψ(γ)| ≥ 1,

for some γ ∈ Γ.

(3.11)

If ψ fails (3.11), select γ0 ∈ Γ with ψ(γ0 ) = 0. Since (H-2), (H-3) always hold (i.e., whether or not ψ satisﬁes (3.11)), it can be veriﬁed that | · | := (p) (1/|ψ(γ0 )|) · L1 (m(p) ) is an equivalent lattice norm for which (L1 (mψ ), | · | ) ψ

is homogeneous.

Concerning the space P (G) of all pseudomeasures on G, see [28]. Fix 1 < p < ∞. We say σ ∈ P (G) is in Lp (G) if there is h ∈ Lp (G) with ψ = ψ. σ = h. Let ψ ∈ Mp (G). As ψ ∈ ∞ (Γ), there is σψ ∈ P (G) with σ Given h ∈ L1 (G), denote σψ ∗ σh ∈ P (G) by ˇ ψ ∗ h. Then (ψ ˇ ∗ h) = ψh. If σψ ∗ σh ∈ P (G) is in Lp (G), we write ˇ ψ ∗ h ∈ Lp (G). Which f ∈ L1 (G) satisfy ˇ ψ ∗ (χA f ) ∈ Lp (G), for A ∈ B(G)? Each f ∈ Lp (G) has this property (p)

as ˇ ψ ∗ (χA f ) ∈ P (G) corresponds to Tψ (χA f ) ∈ Lp (G) ⊆ P (G). The answer is the following version of Proposition 3.1(v). Corollary 3.3. Let 1 < p < ∞ and ψ ∈ Mp (G)\{0}. Then (p) L1 (mψ ) = f ∈ L1 (G) : ˇ ψ ∗ (χA f ) ∈ Lp (G), ∀A ∈ B(G) . (3.12) (p)

Given f ∈ L1 (mψ ), the pseudomeasure ˇ ψ ∗ (χA f ) in (3.12) corre (p) sponds to the element Im(p) (χA f ) = A f dmψ ∈ Lp (G) for A ∈ B(G); see ψ

Proposition 3.1(iv). The proof of the next result proceeds as in [23, Lemma (35.11)] and is omitted. Proposition 3.4. A pseudomeasure σ belongs to Lp (G) iﬀ Kσ := sup σ ∗ h Lp (G) : h ∈ T (G), h L1 (G) ≤ 1 < ∞. An immediate consequence of Propositions 3.4 and 2.8 is as follows. Corollary 3.5. Let 1 < p < ∞ and ψ ∈ Mp (G)\{0} ⊆ P (G). Each assertion in (i)–(vii) of Proposition 2.8 is equivalent to sup{ ψ ˇ ∗ h Lp (G) : h ∈ T (G), h L1 (G) ≤ 1} < ∞.

(viii)

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4. Further Results and Examples In this section we analyze particular p-multipliers ψ. Examples from (M (G))are treated in [41, Ch.7]. Results of Sect. 2 are useful in determin(p) (p) ing when L1 (mψ ) coincides with Lp (G) or L1 (G). For Lp (G) L1 (mψ ) (p)

L1 (G) we present additional criteria which identify L1 (mψ ) more precisely. Consider a Banach space E and an operator T ∈ L(E). Then T is upper semi-Fredholm if ker(T ) := {u ∈ E : T (u) = 0} is ﬁnite-dimensional and R(T ) := {T (u) : u ∈ E} is closed in E. If E/R(T ) has ﬁnite dimension, then T is called lower semi-Fredholm, in which case R(T ) is necessarily closed, [36, p.150, Lemma 2]. If T is both upper and lower semi-Fredholm, then T is called Fredholm. For the basic properties of such operators we refer to [36, Section 16], for example. For multiplier operators, it turns out that these three notions coincide. First we require a preliminary result. Lemma 4.1. Let 1 < p < ∞ and ψ ∈ Mp (G). (p) (i) ker(T ) = {f ∈ Lp (G) : supp(f) ∩ supp(ψ) = ∅}. ψ

(i)

The following assertions are equivalent. (p) (a) ker(Tψ ) is ﬁnite dimensional. (b) Γ\supp(ψ) is a ﬁnite set. In this case (p)

ker(Tψ ) = span ({(·, γ) : γ ∈ Γ\supp(ψ)} ∪ {0}) .

(4.1)

Proof. (i) Follows from injectivity of the Fourier transform map (p) Fp,0 : Lp (G) → c0 (Γ) and the fact that f ∈ Lp (G) satisﬁes Tψ (f ) = 0 iﬀ ψ f = 0 iﬀ supp(ψ) ∩ supp(f) = ∅. (ii) (a) ⇒ (b). Observe (by part (i)) that if Γ\supp(ψ) is an inﬁnite set, then the inﬁnite linearly independent set {(·, γ) : γ ∈ Γ\supp(ψ)} is (p) contained in ker(Tψ ). (b) ⇒ (a) is clear from part (i) which implies that (p) (p) supp(f) ⊆ Γ\supp(ψ) for every f ∈ ker(Tψ ). Hence, ker(Tψ ) ⊆ span{(·, γ) : γ ∈ Γ\supp(ψ)}. The formula (4.1) is clear from part (i) and the ﬁniteness of Γ\supp(ψ). Proposition 4.2. Let 1 < p < ∞ and ψ ∈ Mp (G). The following assertions are equivalent. (p)

(i) Tψ

is upper semi-Fredholm.

(ii)

is lower semi-Fredholm.

(iii)

(p) Tψ (p) Tψ

is Fredholm.

Proof. We make use of the following facts: (p)

(p)

(Lp (G)/R(Tψ ))∗ is isomorphic to the annihilator (R(Tψ ))⊥ ⊆ Lp (G), (4.2)

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(p)

with Lp (G)/R(Tψ ) denoting the quotient space, [36, p.343, Theorem 20], and (p )

(p)

(p)

ker(Tψ ) = ker(Tψ )∗ = (R(Tψ ))⊥ ;

(4.3)

see Lemma (2.1) (for the ﬁrst equality) and [36, p. 342, Theorem 14]. Moreover, the reﬂection ψ of ψ clearly satisﬁes is a ﬁnite set iﬀ Γ\supp(ψ) is a ﬁnite set . Γ\supp(ψ)

(4.4)

is a ﬁnite (i) ⇒ (ii). Lemma 4.1(ii) and (4.4) imply that Γ\supp(ψ) (p ) p set and hence, applying Lemma 4.1(ii) to Tψ ∈ L(L (G)), we conclude (p )

(p)

that ker(Tψ ) = ker(Tψ )∗ ⊆ Lp (G) is ﬁnite dimensional. Then (4.3) yields (p)

(p)

dim(R(Tψ ))⊥ < ∞ which, by (4.2), implies dim(Lp (G)/R(Tψ ))∗ < ∞. (p)

(p)

Then also dim(Lp (G)/R(Tψ ))∗∗ < ∞ and, since Lp (G)/R(Tψ ) is a sub(p)

space of (Lp (G)/R(Tψ ))∗∗ , we can conclude that (ii) holds.

(p )

(ii) ⇒ (iii). According to [36, p. 150, Theorem 4], Tψ

(p )

upper semi-Fredholm and hence, by (i) ⇒ (ii) applied to Tψ (p ) Tψ

is also lower semi-Fredholm, i.e., (p ) (Tψ )∗

(p) Tψ

(p ) Tψ

(p)

= (Tψ )∗ is we see that

is Fredholm. Then also its dual

= is Fredholm, [36, p. 150, Theorem 4]. operator (iii) ⇒ (i) is obvious.

An immediate application is the following result. (p)

Proposition 4.3. Let 1 < p < ∞ and ψ ∈ Mp (G). Then Tψ is Fredholm iﬀ it is a ﬁnite rank perturbation of an invertible p-multiplier operator. (p)

Proof. Let Tψ

be Fredholm. Then D := Γ\supp(ψ) is a ﬁnite set. Deﬁne (p)

(p)

(p)

(p)

ϕ := ψ +χD in which case ϕ ∈ Mp (G). Since Tψ = Tϕ +T−χ , with T−χ (p)

a ﬁnite rank operator, it sufﬁces to show that Tϕ According to (4.1), the index (p)

D

D

is invertible in L(Lp (G)).

(p)

α(Tψ ) := dim(ker(Tψ )) = dim(span({(·, γ) : γ ∈ D} ∪ {0})) and by [36, p. 150, Theorem 4], we have that the index (p)

(p )

(p )

β(Tψ ) = α(Tψ ) = dim(ker(Tψ )) ∪ {0})), = dim(span({(·, γ) : γ ∈ Γ\supp(ψ)} (p )

(p)

where the last equality follows from (4.1) applied to Tψ . Hence, α(Tψ ) = (p)

(p)

β(Tψ ) so that Tψ (p) Tϕ

(p)

has index 0. It follows from Tψ

(p)

(p)

= Tϕ + T−χ

D

that

is Fredholm and has index 0; see Proposition 4.2 and [36, p. 155, also (p) Theorem 16]. Since Γ\supp(ϕ) = ∅, it follows from (4.1) applied to Tϕ that (p) (p) (p) dim(ker(Tϕ )) = 0, i.e., Tϕ is injective. Then ind(Tϕ ) = 0 implies that (p) (p) R(Tϕ ) = Lp (G), i.e., Tϕ is also surjective.

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Since invertible operators are Fredholm, the converse is well known, [36, p. 155, Theorem 16]. Another application of Proposition 4.2, in combination with [41, Proposition 4.18], is the following fact. Proposition 4.4. Let 1 < p < ∞ and ψ ∈ Mp (G) have the property that (p) (p) Tψ ∈ L(Lp (G)) is Fredholm. Then L1 (mψ ) = Lp (G). Perhaps the best known multipliers in this regard are the following ones. Remark 4.5. (i) If ρ ∈ bv(Z), the functions of bounded variation, then ρ(Z) ⊆ C is countable with at most two limit points, viz L+ ρ := := lim ρ(n), [34, Proposition 8(i)]. By limn→∞ ρ(n) and L− n→−∞ ρ Steˇckin’s Theorem, [17, Theorem 6.3.5], bv(Z) ⊆ 1
Tψ 1

is Fredholm, [34, Proposition 8(ii)]. Proposition 4.4 implies that (p)

L (mψ ) = Lp (T). A classical example is the Hilbert transform ϕ given by n → isgn(n) := i(n/|n|)χZ\{0} (n), for n ∈ Z. Observe that for some λ ∈ M (T), then ν := δ1 − iλ ϕ ∈ / (M (T)) . For, if ϕ = λ satisﬁes ν = 2χN + χ{0} . As ν(k) = 0 for k < 0, the F. & M. Riesz Theorem, [24, p. 89], implies ν(A) = A f dμ, for A ∈ B(T) and some f ∈ L1 (T), i.e., ν ∈ c0 (Z) contradicting ν = 2χN + χ{0} . (ii) Fredholm multiplier operators need not be decomposable. For 1 < p < 2 (p) and n ∈ N, let ψ ∈ c0 (Zn ) ∩ Mp (Tn ) be such that Tψ ∈ L(Lp (Tn )) is (p)

(p)

not decomposable; see Remark 2.4(ii). Then Tψ−λ = Tψ − λI is invert(p)

(p)

ible, hence Fredholm, for each λ ∈ / σ(Tψ ). But, Tψ − λI cannot be decomposable, [5, p. 37, Corollary 1.11]. Let (Ω, Σ) be a measurable space, ν : Σ → E be a vector measure and 0 < r < ∞. We say ν has ﬁnite r-variation if |ν|r (Ω) := 1/r n r sup ν(A ) < ∞, with the supremum taken over all ﬁnite j E j=1

Σ-partitions {Aj }nj=1 of Ω and all n ∈ N.

(p)

Proposition 4.6. Let 1 0 but, mψ has ﬁnite p-variation. (p)

Proof. Since mψ is non-atomic (see Proposition 2.2(ii)), Corollary 3.23(ii) (p)

(p)

of [41] implies |mψ |(A) = ∞ for all A with μ(A) > 0. As L1 (mψ ) = Lp (G), (p)

(p)

the inclusion Jψ : Lp (G) → L1 (mψ ) is the identity operator on Lp (G) and (p)

so, is p-concave. By Proposition 3.70(ii) of [41] we conclude that mψ has ﬁnite p-variation.

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Optimal extension of multiplier operators (p)

Idempotent p-multiplier operators Tψ (p) L1 (mψ )

(i.e., ψ = χD ) provide further

= L (G). If D ⊆ Γ has ﬁnite complement Dc , then there is f ∈ T (G) with f = −χDc . So, χD = δ0 + f ∈ (M (G)) with (p) f ∈ (M0 (G)). It follows that L1 (mχ ) = Lp (G) for all 1 < p < ∞, [41,

examples satisfying

p

589

D

Lemma 7.71]. P ⊆ Z is periodic if P = r∈P ((kZ) + r) for some k ∈ N with P := P ∩ [0, k). If wk (j) := exp(−2πij/k), for 0 ≤ j < k, then χP = k−1 1 r j=0 (wk (j)) δwk (j) , i.e., χP = λ for a ﬁnitely supported λ ∈ M (T). r∈P k Hence, αλ = ν + δa for some α ∈ C\{0}, a ∈ T and measure ν with (p) supp(ν) = T and a ∈ / supp(ν). By [41, Remark 7.77(i)], L1 (mχP ) = Lp (T), (p)

(p)

(p)

as L1 (mχP ) = L1 (mαχP ) = L1 (αmχP ).

Proposition 4.7. Let 1 < p < ∞ and D ⊆ Z satisfy χD ∈ (M (T)). (p)

(i) If D is ﬁnite, then χD ∈ (Lp (T))and so L1 (mχD ) = L1 (T). (p)

(ii) If D is an inﬁnite set, then L1 (mχD ) = Lp (T).

Proof. (i) From above χD ∈ (Lp (T)); then apply Proposition 2.8. (ii) By Helson’s Theorem, [44, p. 61], D = P ∪ F is the disjoint union of a periodic set P and a ﬁnite set F . Then χD = χP + χF = ν1 + ν2 = (ν1 + ν2 ) with ν1 ∈ M (T) ﬁnitely supported and ν2 ∈ M (T) satisfying ν2 ∈ (Lp (T)) . Set λ := ν1 + ν2 and η := −ν2 , i.e., ν1 = λ + η. By [41, Lemma 7.69] (or Lemma 4.8 below), we (p) (p) (p) (p) have that Lp (T) ⊆ L1 (mχD ) ∩ L1 (m−ν2 ) ⊆ L1 (mν1 ). But, L1 (m−ν2 ) = (p)

(p)

(p)

L1 (−mν2 ) = L1 (mν2 ) = L1 (mχF ) = L1 (T) (by (i) applied to F ) and (p)

(p)

L1 (mν1 ) = L1 (mχP ) = Lp (T).

(p)

(p)

(p)

Lemma 4.8. Let 1 < p < ∞ and ϕ, ψ ∈ Mp (G). Then mψ+ϕ = mψ + mϕ . If none of ϕ, ψ, (ϕ + ψ) are zero, then also (p)

(p)

1 Lp (G) ⊆ L1 (mψ ) ∩ L1 (m(p) ϕ ) ⊆ L (mψ+ϕ ). (p)

(p)

(4.5)

(p)

Proof. The identity mψ+ϕ = mψ + mϕ is routine to establish. (p)

As Lp (G) ⊆ L1 (mκ ), κ ∈ Mp (G), the ﬁrst inclusion in (4.5) is clear. (p ) (p ) (p ) Let h ∈ Lp (G). Then |T(ψ+ϕ)∼ (h)| ≤ |Tψ (h)| + |Tϕ (h)|. So, for (p) (p) f : G → C measurable we have G |f | d| mψ+ϕ , h | ≤ G |f | d| mψ , h | + (p) (p) (p) |f | d| mϕ , h |; see (2.12), i.e., if f ∈ L1 (mψ ) ∩ L1 (mϕ ), then we have G (p) that G |f | d| mψ+ϕ , h | < ∞, for h ∈ Lp (G). So, Proposition 3.1(i) applied (p)

(p)

to mψ+ϕ yields f ∈ L1 (mψ+ϕ ).

An adaption of the argument used in the proof of Proposition 4.7(ii), where Lemma 7.69 of [41] is replaced with Lemma 4.8 above, yields

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Proposition 4.9. Let 1 < p < ∞ and ψ, ϕ ∈ Mp (G) satisfy L1 (mψ ) = Lp (G) and

(p) L1 (mϕ )

= L1 (G). If ϕ + ψ = 0, then

(p) L1 (mψ+ϕ )

= Lp (G).

Fix 1 < p < ∞, ψ ∈ Mp (G) and γ ∈ Γ. Let ψγ (β) := ψ(β −γ) for β ∈ Γ. (p) Let Mγ ∈ L(Lp (G)) denote the operator of multiplication by (·, γ) ∈ C(G). (p) (p) (p) Then Mγ is an isometric isomorphism on Lp (G) with (Mγ )−1 = M−γ and (p)

(p)

(p)

Tψγ = Mγ(p) ◦ Tψ ◦ M−γ ,

γ ∈ Γ.

(4.6)

Via (4.6) we have |||ψγ |||p = |||ψ|||p . It follows from (2.8) that (·, γ)f ∈ (p) (p) (p) L1w (mψ ) = L1 (mψ ), for f ∈ L1 (mψ ), and (·, γ)f L1 (m(p) ) = f L1 (m(p) ) . [p]

So, Mγ

(p)

ψ

(p)

ψ

: f → (·, γ)f , for f ∈ L1 (mψ ), is an isometric isomorphism of

L1 (mψ ) onto itself. Proposition 4.10. Let 1 < p < ∞ and ψ ∈ Mp (G). For each γ ∈ Γ, the (p) (p) spaces L1 (mψγ ) and L1 (mψ ) are isometrically isomorphic and [p]

Im(p) = Mγ(p) ◦ Im(p) ◦ M−γ , ψγ

that is,

⎛

(p)

f dmψγ = Mγ(p) ⎝

G

(4.7)

ψ

⎞ (p) (·, −γ)f dmψ ⎠ ,

(p)

f ∈ L1 (mψγ ).

(4.8)

G (p )

(p)

Proof. Fix γ ∈ Γ. By direct calculation, (Mγ )∗ = Mγ . It follows from (p) (p) (p) (p) (p ) (p) (p ) (4.6) that (Tψγ )∗ = (M−γ )∗ ◦ (Tψ )∗ ◦ (Mγ )∗ = M−γ ◦ (Tψ )∗ ◦ Mγ . For

h ∈ Lp (G), ∗ (p) ∗ (p) ∗ (p) Mγ(p ) (h) = Tψ Mγ(p ) (h) (h) = (·, −γ) Tψ T ψγ

(4.9) pointwise μ-a.e. on G. It follows from Lemma 2.1 and the identities (2.11), (p) (2.12) and (4.9) that, for A ∈ B(G), the value | mψγ , h |(A) coincides with (p) ∗ (p) ∗ (p) T dμ = Mγ(p ) (h) dμ = mψ , Mγ(p ) (h) (A). (h) ψγ Tψ A

A

(p )

Since Mγ ∈ L(Lp (G)) is an isometric isomorphism, a B(G)-measurable function f : G → C satisﬁes (3.1) for ψ iﬀ it satisﬁes (3.1) for ψγ . So, (p) (p) Proposition 3.1(i) yields L1 (mψγ ) = L1 (mψ ). Moreover, it follows from (4.9), (p )

the fact that Mγ

is an isometric isomorphism, and via (3.2) that · L1 (m(p) ) ψγ

and · L1 (m(p) ) coincide. ψ

[p]

(p)

It follows from (4.6) and M−γ (f ) = M−γ (f ), for every f ∈ Lp (G), (p)

that (4.7) holds on the dense subspace Lp (G) of L1 (mψγ ) and hence, also on

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(p)

L1 (mψγ ). Then (4.8) follows from (4.7) via the definition of the integration map; see (2.9). Given a function f : G → C let R(f ) : G → C be its reﬂection, i.e., x → f (−x), for x ∈ G. If 1 < p < ∞ and f ∈ Lp (G), denote R(f ) by R(p) (f ). Then (p)

(p)

R(p) ◦ Tψ = Tψ ◦ R(p) ,

ψ ∈ Mp (G).

(4.10)

Proposition 4.11. Let 1 < p < ∞ and ψ ∈ Mp (G). Then Rp : f → R(f ) is (p) (p) an isometric isomorphism of L1 (mψ ) onto L1 (mψ ) and Im(p) = R(p) ◦ Im(p) ◦ Rp , ψ

that is,

(p)

f dmψ G

(4.11)

ψ

⎛ ⎞ (p) = R(p) ⎝ Rp (f ) dmψ ⎠ ,

(p)

f ∈ L1 (mψ ).

(4.12)

G

Proof. Let f be B(G)-measurable and h ∈ Lp (G). Via (4.10) (with p in place (p ) (p ) of p) we have that G |f | · |Tψ (h)| dμ = G |f | · |R(p ) (R(p ) Tψ (h))| dμ = (p ) (p ) |R2 (f )|·|R(p ) (Tψ (R(p ) (h)))| dμ = G |R(f )|·|T(ψ )∼ (R(p ) (h))| dμ. So, if G (p) (p ) f ∈ L1 (mψ ), then Proposition 3.1(i) implies G |f |·|Tψ (h)| dμ < ∞ for h ∈

Lp (G). The above equalities, with R(p ) an isometric isomorphism, imply via (p) Proposition 3.1(i) that R(f ) ∈ L1 (mψ ) with f L1 (m(p) ) = R(f ) L1 (m(p) ) ; ψ

(p) ψ

ψ

(p)

see (3.2). Similarly, if f ∈ L1 (m ), then R(f ) ∈ L1 (mψ ) with f L1 (m(p) ) = ψ

f L1 (m(p) ) . ψ

Because of (4.10), the identity (4.11) holds on the dense subspace Lp (G) (p) (p) 1 of L (mψ ) and so, also holds on L1 (mψ ). Then (4.12) follows from (4.11) and (2.9). Let 1 < p ≤ 2 and D ⊆ Z satisfy χD ∈ Mp (T). We say D contains arbitrarily long intervals if there are sequences m(j) ↑ ∞ and n(j) ↑ ∞ in N with ∞ m(j) < n(j) and supj (n(j) − m(j)) = ∞ such that either j=1 ([m(j), n(j)] ∩ ∞ Z) ⊆ D or j=1 ([−n(j), −m(j)] ∩ Z) ⊆ D or both. Each Jn+ := [n, ∞) ∩ Z and Jn− := (−∞, n] ∩ Z, for n ∈ Z, satisﬁes χ + , χ − ∈ Mp (T) and contains Jn

Jn

arbitrarily long intervals. Via Littlewood–Paley theory there exist sets D, with χD ∈ Mp (T), which contain arbitrarily long intervals but, are not of the form Jn+ or Jn− for any n ∈ Z. Proposition 4.12. Let 1 < p < ∞ and D ⊆ Z satisfy χD ∈ Mp (T). (i) (ii)

(p)

If D contains arbitrarily long intervals, then L1 (mχD ) = Lp (T). / (M (T)). If, additionally, D ⊆ Jn+ or D ⊆ Jn− for some n ∈ Z, then χD ∈

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∞ Proof. (i) Suppose j=1 ([m(j), n(j)] ∩ Z) ⊆ D for sequences m(j) ↑ ∞ and n(j) ↑ ∞ with supj (n(j) − m(j)) = ∞. According to (2.12), (p) (p ) | mχ , h |(A) = A |TχD (h)| dμ, for A ∈ B(T), h ∈ Lp (T). Let D

h ∈ T (T) satisfy h Lp (T) ≤ 1. There exist j, k ∈ N (depending on h) such that (·, k)h ∈ T (T) satisﬁes supp(((·, k)h) ) ⊆ (p ) [m(j), n(j)] ∩ Z. Then [TχD ((·, k)h)] = χD · ((·, k)h) = ((·, k)h) , (p )

(p)

that is, TχD ((·, k)h) = (·, k)h. So, for ﬁxed f ∈ L1 (mχ ), we have D (p ) |f | · |h| dμ = |f | · |(·, k)h| dμ = |f | · |T ((·, k)h)| dμ = χ T T D T (p) |f | d| mχ , (·, k)h | ≤ f L1 (m(p) ) , where the inequality follows from T χ

D

D

(2.8) and (·, k)h Lp (T) ≤ 1. Let h ∈ Lp (T) be arbitrary. Choose {hn }∞ n=1 ⊆ T (T) with hn → h in Lp (T)and also μ-a.e. on T. Then |f |·|h | → |f |·|h| μ-a.e. By Fatou’s n inf |f | · |h | dμ. But, for each n ∈ N, Lemma, T |f | · |h| dμ ≤ lim n n T we have from above that T |f | · |hn | dμ ≤ hn Lp (T) f L1 (m(p) ) ≤ χ

D

Kh f L1 (m(p) ) , where Kh := supn hn Lp (T) < ∞. It follows that χ D |f | · |h| dμ ≤ Kh f L1 (m(p) ) < ∞, for all h ∈ Lp (T), i.e., f ∈ T χ

D

(p)

Lp (T). This shows that L1 (mχ ) ⊆ Lp (T). Since the reverse inclusion D

(p)

always holds, we have L1 (mχ ) = Lp (T). By Proposition 4.11 also D

(p)

L1 (mχD ) = Lp (T). ∞ The case j=1 ([−n(j), −m(j)] ∩ Z) ⊆ D reduces to the previous ∞ one by noting that j=1 ([m(j), n(j)] ∩ Z) ⊆ (−D), and then applying Proposition 4.11. (ii) If D ⊆ Jn+ for some n ∈ Z, then F := D ∩ (−N) is ﬁnite and so / c0 (Z). Arguing via the χF ∈ (M0 (T)). Since D ∩ N0 is inﬁnite, χD∩N ∈ 0 F. & M. Riesz Theorem (cf. Remark 4.5(i)) shows χD∩N ∈ / (M (T)) . 0 Then χD = χF + χD∩N implies χD ∈ / (M (T)). As χ−D = χ D , the same 0 conclusion holds if D ⊆ Jn− . Let 1 0 with f Lp (T) ≤ Cq f Lq (T) , for f ∈ T (T) with supp(f) ⊆ D, [15, Ch.15, §5], [43]. Examples are Sidon sets, [15, p. 259], [43]. Inﬁnite Λ(p)-sets / (M (T)), [43, Theorem 5.1]. If D is Sidon, then D ⊆ Z are relevant as χD ∈ χD ∈ Mp (T) for 1 < p < ∞, [28, F.11(d), p.255]. For 2 ≤ p < ∞, every Λ(p)-set D satisﬁes χD ∈ Mp (T); cf. the proof of (b) in [23, Section (37.22), p.434] with E := χD . Proposition 4.13. Let D ⊆ Z be any inﬁnite set which is a Λ(q)-set for every 1 < q < ∞. For 1 < p < ∞ we have the strict containments 1 Lp (T) Lr (T) L1 (m(p) (4.13) χ ) L (T). 1
D

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/ (M (T)), Proof. The ﬁrst inclusion in (4.13) is known to be proper. As χD ∈ also χD ∈ / (Lp (T)). So, Proposition 2.8 implies the last inclusion in (4.13) is strict. For the middle inclusion observe χD ∈ Mq (T), 1 < q < ∞. Fix 1 0 satisfy h Lp (T) ≤ Cr h Lr (T) , for h ∈ T (T) with supp( h) ⊆ D. If f ∈ T (T), then supp(χD · f) ⊆ D ∩ (p) (r) supp(f) with supp(f) ﬁnite. As (Tχ (f )) = (Tχ (f )) = χ · f, apply the (p)

D

(r)

D

D

(p)

previous inequality to h := TχD (f ) = TχD (f ) to deduce TχD (f ) Lp (T) ≤ (r)

Cr TχD (f ) Lr (T) . By density of T (T) in Lr (T) there is T ∈ L(Lr (T), Lp (T)) (p)

(p)

(p)

extending TχD . By optimality of L1 (mχD ) we have Lr (T) ⊆ L1 (mχD ). As 1 < r < p is arbitrary, the middle inclusion in (4.13) holds. To verify the middle inclusion in (4.13) is proper, let 1 < r(n) < p be strictly decreasing with r(n) ↓ 1. The Banach spaces Fn := Lr(n) (T), (p) for n ∈ N, satisfy Fn Fn+1 and Fn ⊆ F := L1 (mχA ), for n ∈ N, with continuous inclusions. So, Lemma 4.15 below yields ( 1
Proof. Let Jr,ψ : Lr (G) → L1 (mψ ) be the natural inclusion. A closed graph argument ensures Jr,ψ is continuous. So, S := Jp,r ◦ Im(p) ◦ Jr,ψ ∈ L(Lr (G)). ψ

) = I If f ∈ Lr (G), then (3.5) implies S(f (p) (f ) = ψ · f , i.e., ψ ∈ Mr (G). m ψ

To apply Proposition 4.16 we require the next result; also [6] is relevant. Lemma 4.17. Let 1 < p < 2 be arbitrary.

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The inclusion mp (G) (Mp (G) ∩ c0 (Γ)) is strict. M (G) Mp (G). 1≤s

Let r satisfy 1 < r < p. Then the following assertions hold. (v) (vi) (vii) (viii)

Mr (G) Mp (G). mr (G) mp (G). mp (G) Mr (G). Each element of (Mp (G) ∩ c0 (Γ))\mp (G) fails to belong to 1≤s

Proof. (v) By [20, Remark (a), p.384], there is ψ ∈ (Mp (G) ∩ c0 (Γ)) which is not approximable in (Mp (G), |||·|||p ) by elements from Mr (G)∩c0 (Γ). / Mr (G). This establishes (v). See Since ψ ∈ c0 (Γ), it follows that ψ ∈ also [21, p. 287]. (i) Choose any r ∈ (1, p) and let ψ be as in the proof of (v). Since (L1 (G)) ⊆ (Mr (G) ∩ c0 (Γ)) and (L1 (G)) is dense in (mp (G), ||| · |||p ), it follows that ψ ∈ / mp (G). Hence, the stated inclusion is strict. We also refer to [21, p. 289]. (vi) Proposition 2.3 ((i) ⇔ (iv)) yields Mr (G) ∩ c0 (Γ) ⊆ mp (G). This and part (i) give mr (G) Mr (G) ∩ c0 (Γ) ⊆ mp (G). (ii) Fix 1 < p < 2 and choose ψ ∈ (Mp (G) ∩ c0 (Γ))\mp (G); see part (i). Fix any 1 < r < p. If it were the case that ψ ∈ Mr (G), then actually ψ ∈ (Mr (G) ∩ c0 (Γ)). But, we saw in the proof of (vi) that always (Mr (G) ∩ c0 (Γ)) ⊆ mp (G) and so ψ ∈ mp (G) would follow. This contradicts the choice of ψ, i.e., ψ ∈ / Mr (G). (iii) Let 1 < r(n) < p satisfy r(n) ↑ p and deﬁne Fn := mr(n) (G), for n ∈ N. Let F := (mp (G), ||| · |||p ). Then Fn Fn+1 (strictness is via part (vi)) N, with all the indicated inclusions continuous. and Fn ⊆ F , for n ∈ ∞ Lemma 4.15 implies ( 1≤r

Remark 4.18. Let 1 < r < p ≤ 2. Proposition 4.16 implies Lr (G) L1 (mψ ) (p)

whenever ψ ∈ Mp (G)\Mr (G), e.g., if L1 (mψ ) = Lp (G). More interesting (p)

(p)

is ψ ∈ Mp (G)\Mr (G) with L1 (mψ ) = Lp (G) but, L1 (mψ ) contains no

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space Ls (G) for 1 ≤ s ≤ r. To exhibit such ψ ﬁx 1 < r < 2. For p ∈ (r, 2), Lemma 4.17(vii) ensures there is ψ ∈ mp (G)\Mr (G). The previous (p) (p) discussion and compactness of Tψ imply Lp (G) L1 (mψ ) and Ls (G) (p) L1 (mψ ) for 1 ≤ s ≤ r. Or, ﬁx p ∈ (1, 2) and ψ ∈ mp (G)\( 1≤s

see Lemma 4.17(iv). Remark 2.9(i) implies Lp (G) L1 (mψ ) and Proposi(p)

tion 4.16 ensures Lr (G) L1 (mψ ) for 1 < r < p.

(p)

(p)

There also exist ψ ∈ Mp (G) satisfying L1 (mψ ) = Lp (G) with L1 (mψ ) containing Lr (G) for some/all 1 < r < p. Indeed, let 1 ≤ q < ∞. An Lq -improving measure λ ∈ M (G), [22], provides such an example. Let 1 < −1 −1 = 1 + p−1 . Then, for f ∈ r < p < ∞ and u ∈ (1, p) satisfy u +r r p L (G)\L (G), the measure μf : A → A f dμ, for A ∈ B(G), is Lq -improving (p)

with Lp (G) Lu (G) ⊆ L1 (m ), [41, Proposition 7.96]. Or, ﬁx 1 < p < ∞ f and let f ∈ ( 1<s

1<s
[41, Proposition 7.98]. As noted, μf is Lq -improving. In this case, μ f = f ∈ mp (G). Non-compact multipliers can have similar properties. Consider the ∞ Riesz-product measure λ := j=1 (1 + aj cos(nj t)) ∈ M (T) as constructed in [45, Example 2, p.310]. For aj = 1, for all j, λ is positive, non-atomic and ∈ λ∈ / M0 (T), [45]. In particular, λ / 1<s<∞ ms (T). Via [41, Remark 7.104], (p) λ is Lq -improving and Lp (T) L1 (m ) L1 (T), for 1 < p < ∞. λ ∞ Cantor–Lebesgue measure λ := ∗ k=1 ( 12 δ3−k + 12 δ−3−k ), satisﬁes λ ∈ / M0 (T) with λ(Z) countable, [45, Example 3 & Proposition 3.4]. Fix

(p) (2) (2) 1 < p < ∞. As σ(T ) = σ(T ), [45, Theorem 3.7], and σ(T ) = λ(Z), we (p)

λ

λ

(p)

λ

see σ(T ) is totally disconnected. By Remark 2.4(ii), T is decomposable. λ λ Also, λ is Lq -improving, [37]. Argue as in [41, Remark 7.104(ii)] gives the (p) containments Lp (T) L1 (m ) L1 (T). λ

(p)

Lemma 4.19. Let 1 < p < ∞ and ψ ∈ Mp (G)\{0}. Then L1 (mψ ) ⊆ (p)

L1 (mϕψ ) for all ϕ ∈ Mp (G) such that ϕψ = 0. (p)

(p )

Proof. Let f ∈ L1 (mψ ). For h ∈ Lp (G), note that hϕ := Tϕ (h) ∈ Lp (G) (p ) (p ) and so G |f | · |T(ϕψ)∼ (h)| dμ = G |f | · |Tψ (hϕ )| dμ < ∞; see Proposition (p)

3.1(i). Again Proposition 3.1(i) gives f ∈ L1 (mϕψ ).

Remark 4.20. The condition ϕψ = 0 is essential as the zero operator in Lp (G) is not μ-determined. This condition should be inserted in Lemma 7.80 of [41] to correct it; it is a special case of Lemma 4.19 above when both ϕ, ψ ∈ (M (G)).

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with λ Cantor–Lebesgue measure or the Riesz-product meaLet ψ := λ ∈ sure of Remark 4.18. Fix 1 0 with F := {n ∈ Z : |ψ(n)| ≥ α} inﬁnite. Via [23, Theorem (37.18)] choose an inﬁnite Sidon set D ⊆ F . Then ϕ := χD ∈ Mp (T)\(M (T)) , [28, F.11(d), (p)

p.255] and ϕψ = 0. As Lp (T) L1 (mψ ) (see Remark 4.18), Lemma 4.19 (p)

implies Lp (T) L1 (mϕψ ). But ϕψ = χD ψ ∈ / c0 (Z) and so ϕψ ∈ / (Lp (T)). (p)

(p)

By Proposition 2.8, L1 (mϕψ ) L1 (T). So, Lp (T) L1 (mψχ ) L1 (T) for D every inﬁnite Sidon set D ⊆ F . (p) By the same argument, if ψ ∈ Mp (G)\c0 (Γ) satisﬁes Lp (G) L1 (mψ ), (p)

then every ϕ ∈ Mp (G) with ϕψ ∈ / c0 (Γ) satisﬁes Lp (G) L1 (mϕψ ) L1 (G). Example 4.21. For α > 0, deﬁne ψ[α] : n → n−α e−iπα/2 χN (n), for n ∈ Z. Then ψ[α] ∈ bv(Z) ∩ c0 (Z) ⊆ mp (T) for 1 < p ≤ 2 (cf. Remark 4.5(i)). (p)

The operator Tψ[α] corresponds to the Weyl fractional integral operator of order α, whose kernel has Fourier transform ψ[α] , [49, Ch.XII, Section 9]. (p)

Remark 2.9(i) implies Lp (T) L1 (mψ[α] ) for α > 0. If α > 12 , then ψ[α] ∈ (Lp (T)) for 1 < p ≤ 2 and so, by Proposition 2.8, the optimal domain space is L1 (T). For 0 < α ≤ 12 the Hardy-Littlewood–Sobolev inequality shows (p) Lr (T), with 1r = α + p1 , is contained in L1 (mψ[α] ); [49, Ch.XII, Theorem

9.22]. O’Neil’s extension to Orlicz spaces, [42, Section V] implies for α = 12 that the Orlicz space LA (T) corresponding to A(s) := s(log(1 + s))1/2 , which (2) properly contains L2 (T), is contained in L1 (mψ[1/2] ).

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[9] Curbera, G.P., Ricker, W.J.: Compactness properties of Sobolev imbeddings for rearrangement invariant norms. Trans. Am. Math. Soc. 359, 1471–1484 (2007) [10] Delgado, O., Soria, J.: Optimal domains for the Hardy operator. J. Funct. Anal. 244, 119–133 (2007) [11] Diestel, J., Faires, B.: On vector measures. Trans. Am. Math. Soc. 198, 253–271 (1974) [12] Diestel, J., Uhl, J.J. Jr.: Vector Measures. American Mathematical Society, Providence (1977) [13] Dunkl, C.F., Ramirez, D.E.: Multipliers on compact groups. Proc. Am. Math. Soc. 28, 456–460 (1971) [14] Edmunds, D., Kerman, R., Pick, L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal. 170, 307–355 (2000) [15] Edwards, R.E.: Fourier Series: A Modern Introduction, vol. 2. Holt, Rinehart and Winston, New York (1967) [16] Edwards, R.E.: Integration and Harmonic Analysis on Compact Groups. Cambridge University Press, Cambridge (1972) [17] Edwards, R.E., Gaudry, G.I.: Littlewood–Paley and Multiplier Theory. Springer, Berlin (1977) [18] Fig` a-Talamanca, A., Gaudry, G.I.: Multipliers of Lp which vanish at inﬁnity. J. Funct. Anal. 7, 475–486 (1971) [19] Gaudry, G.I.: Quasimeasures and Multiplier Problems. Ph.D. Thesis, The Australian National University (1965) [20] Gaudry, G.I., Inglis, I.R.: Approximation of multipliers. Proc. Am. Math. Soc. 44, 381–384 (1974) [21] Graham, C.C., McGehee, O.C.: Essays in Commutative Harmonic Analysis. Springer, New York (1979) [22] Graham, C.C., Hare, K.E., Ritter, D.L.: The size of Lp -improving measures. J. Funct. Anal. 84, 472–495 (1989) [23] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis II. Springer, Berlin (1970) [24] Katznelson, Y.: An Introduction to Harmonic Analysis. Dover, New York (1976) [25] Kerman, R., Pick, L.: Optimal Sobolev imbeddings. Forum Math. 18, 535–570 (2006) [26] Kluv´ anek, I., Knowles, G.: Vector Measures and Control Systems. North Holland, Amsterdam (1976) [27] Krasnoselskii, M.A., Pustylnik, E.I., Sbolevskii, P.E., Zabreiko, P.P.: Integral Operators in Spaces of Summable Functions (English translation). Noordhoﬀ, Leyden (1976) [28] Larsen, R.: An Introduction to the Theory of Multipliers. Springer, Berlin (1971) [29] Laursen, K.B., Neumann, M.M.: An Introduction to Local Spectral Theory. Clarendon Press, Oxford (2000) [30] Lewis, D.R.: On integrability and summability in vector spaces. Illinois J. Math. 16, 294–307 (1972) [31] Lyons, R.: Sur l’historie de M0 (T). In: Kahane, J.-P., Salem, R. (eds.) Ensembles Parfaits et Series Trigonometriques, pp. 231–237. Hermann, Paris (1994)

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S. Okada 112 Marconi Crescent Kambah, ACT 2902 Australia e-mail: [email protected] W. J. Ricker (B) Math.-Geogr. Fakult¨ at Katholische Universit¨ at Eichst¨ att-Ingolstadt 85072 Eichst¨ att Germany e-mail: [email protected] Received: May 2, 2009. Revised: June 15, 2010.

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