Berger-Coburn Theorem, Localized Operators, and the Toeplitz Algebra

Bauer, Wolfram; Robert, Fulsche,

doi:10.1007/978-3-030-44651-2_8

Cited by 9 publications

(13 citation statements)

References 54 publications

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“…So if T was band-dominated, then T restricted to F 2 1 would be compact by Theorem 16, which it obviously is not. Further, maybe more interesting examples are provided in [5,Example 2], for instance.…”

Section: Compact Toeplitz and Hankel Operatorsmentioning

confidence: 99%

“…We have seen in Corollary 6 that every Toeplitz operator with bounded symbol is banddominated. Bauer and Fulsche [5] showed that the algebra of band-dominated operators on F 2 is generated by Toeplitz operators. A natural question is therefore: Question 33.…”

Section: Remarks and Open Problemsmentioning

confidence: 99%

“…Consequently, in analogy to the sequence space case, another C * -algebra, called the band-dominated operators, was introduced in [9], which formalized the limit operator idea. It was later shown in [5] that this would be again the same algebra, but the band-dominated approach of [9] also provided a characterization of Fredholm operators. Moreover, it allowed to study Hankel operators, providing quick proofs for well-known compactness results as well as some new insights [13].…”

Section: Introductionmentioning

confidence: 96%

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Toeplitz and related operators on polyanalytic Fock spaces

Hagger¹

2022

Preprint

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We give a characterization of compact and Fredholm operators on polyanalytic Fock spaces in terms of limit operators. As an application we obtain a generalization of the Bauer-Isralowitz theorem using a matrix valued Berezin type transform. We then apply this theorem to Toeplitz and Hankel operators to obtain necessary and sufficient conditions for compactness. As it turns out, whether or not a Toeplitz or Hankel operator is compact does not depend on the polyanalytic order. For Hankel operators this even holds on the true polyanalytic Fock spaces.

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Section: Compact Toeplitz and Hankel Operatorsmentioning

confidence: 99%

Section: Remarks and Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Toeplitz and related operators on polyanalytic Fock spaces

Hagger¹

2022

Preprint

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“…Corollary 7.12 provides us a tool to study the C*-algebra VT generated by Toeplitz operators with translation-invariant generating symbols. A natural problem is to find the C*-algebra generated by all Toeplitz operators with bounded symbols (not necesarily translation-invariant), acting in a RKHS H. Various characterizations of this Toeplitz algebra have been found for the Bergman and Segal-Bargmann-Fock spaces, see Xia [38], Bauer and Fulsche [4], and Hagger [17]. Much earlier, Engliš [8] proved that Toeplitz operators acting in the Bergman space L 2 hol (D) are weakly dense in B(L 2 hol (D)).…”

Section: Spectral Functions Of Toeplitz Operators With Translation-in...mentioning

confidence: 99%

Translation-invariant operators in reproducing kernel Hilbert spaces

Yañez¹,

Maximenko²,

Ramos-Vazquez³

2021

Preprint

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Let G be a locally compact abelian group with a Haar measure, and Y be a measure space. Suppose that H is a reproducing kernel Hilbert space of functions on G × Y , such that H is naturally embedded into L 2 (G × Y ) and is invariant under the translations associated with the elements of G. Under some additional technical assumptions, we study the W*-algebra V of translation-invariant bounded linear operators acting on H. First, we decompose V into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces H ξ , ξ ∈ G, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of V. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to V, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors.

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“…Hachadi and Youssfi [14] developed a general scheme for computing the reproducing kernels of the spaces of polyanalytic functions on radial plane domains (disks or the whole plane) with radial measures. There are general investigations about bounded linear operators in reproducing kernel Hilbert spaces (RKHS), especially about Toeplitz operators in Bergman or Fock spaces [5,40,41], but the complete description of the spectral properties is found only for some special classes of operators, in particular, for Toeplitz operators with generating symbols invariant under some group actions, see Vasilevski [39], Grudsky, Quiroga-Barranco, and Vasilevski [11], Dawson, Ólafsson, and Quiroga-Barranco [8]. The simplest class of this type consists of Toeplitz operators with bounded radial generating symbols.…”

Section: Introduction Backgroundmentioning

confidence: 99%

Radial operators on polyanalytic weighted Bergman spaces

Barrera-Castelán,

Maximenko,

Ramos-Vazquez

2020

Preprint

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Let µ α be the Lebesgue plane measure on the unit disk with the radial weight α+1 π (1 − |z| 2 ) α . Denote by A n the space of the n-analytic functions on the unit disk D, square integrable with respect to µ α , and by A (n) the true-n-analytic space defined as A n ∩ A ⊥ n−1 . Extending results of Ramazanov (1999Ramazanov ( , 2002, we find the orthonormal basis of L 2 (D, µ α ) that can be obtained by the orthogonalization of the monomials in z and z. In the polar coordinates, this basis is expressed in terms of Jacobi polynomials. Using this basis, we decompose the von Neumann algebra of radial operators, acting in A n , into the direct sum of some matrix algebras, i.e., radial operators are represented as matrix sequences. Furthermore, we prove that the radial operators, acting in A (n) , are diagonal with respect to the same basis restricted to A (n) . In particular, we provide explicit representations of the Toeplitz operators with bounded radial symbols, acting in A n or in A (n) . Moreover, using ideas by Engliš (1996), we show that the set of the Toeplitz operators with bounded generating symbols is not weakly dense in B(A n ).

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Berger-Coburn Theorem, Localized Operators, and the Toeplitz Algebra

Cited by 9 publications

References 54 publications

Toeplitz and related operators on polyanalytic Fock spaces

Toeplitz and related operators on polyanalytic Fock spaces

Translation-invariant operators in reproducing kernel Hilbert spaces

Radial operators on polyanalytic weighted Bergman spaces

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