Vertical symbols, Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions

Yañez, Crispin Herrera; Hutník, Ondrej; Maximenko, Egor A.

doi:10.1016/j.crma.2013.12.004

Cited by 15 publications

(8 citation statements)

References 5 publications

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“…This formula coincides with Vasilevski [36, Theorem 3.1] and [37, Theorem 5.2.1], see also Grudsky, Karapetyants, and Vasilevski [12]. The C*-algebra G in this example consists of all bounded functions on R + , uniformly continuous with respect to the log-distance, see [18,19].…”

Section: Examplessupporting

confidence: 76%

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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Section: Examplessupporting

confidence: 76%

Translation-invariant operators in reproducing kernel Hilbert spaces

Yañez¹,

Maximenko²,

Ramos-Vazquez³

2021

Preprint

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“…(4.1) Therefore, if f ∈ VSO(R), then f | R+ ∈ VSO(R + ), where the class of functions VSO(R + ) was defined in [12] and mentioned in the Introduction. Furthermore, Herrera Yañez, Hutník, and Maximenko [11] have shown that for every σ ∈ VSO(R + ) and ε > 0 there exists b ∈ L ∞ (R + ) such that…”

Section: Density Of γ λ In Vso(r)mentioning

confidence: 99%

“…The papers [11,12] continue this program and give a description of the commutative algebra generated by Toeplitz operators with bounded vertical symbols. The result states that their spectral functions form a dense subset in the C * -algebra VSO(R + ) of very slowly oscillating functions on R + , i.e.…”

Section: Introductionmentioning

confidence: 99%

C*-Algebra Generated by Angular Toeplitz Operators on the Weighted Bergman Spaces Over the Upper Half-Plane

Esmeral

Maximenko

Vasilevski

2015

Integr. Equ. Oper. Theory

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We consider the set of all Toeplitz operators acting on the weighted Bergman space over the upper half-plane whose L∞-symbols depend only on the argument of the polar coordinates. The main result states that the uniform closure of this set coincides with the C * -algebra generated by the above Toeplitz operators and is isometrically isomorphic to the C * -algebra of bounded functions that are very slowly oscillating on the real line in the sense that they are uniformly continuous with respect to the arcsinh-metric on the real line.Mathematics Subject Classification. Primary: 47B35; Secondary: 32A36.

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“…This result extends to weighted Bergman spaces over the unit ball [3,20]. Recent studies [11,17,19,25] have given explicit descriptions of commutative C*-algebras generated by vertical and angular Toeplitz operators on Bergman spaces. This paper focuses on studying the C*-algebra generated by radial Toeplitz operators acting on Fock spaces.…”

Section: Introductionmentioning

confidence: 61%

Radial Toeplitz Operators on the Fock Space and Square-Root-Slowly Oscillating Sequences

Esmeral

Maximenko

2016

Complex Anal. Oper. Theory

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In this paper we show that the C*-algebra generated by radial Toeplitz operators with L ∞ -symbols acting on the Fock space is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the squareroot-metric ρ( j, k) = | j − k |. More precisely, we prove that the sequences of eigenvalues of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.

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Vertical symbols, Toeplitz operators on weighted Bergman spaces over the upper half-plane and very slowly oscillating functions

Cited by 15 publications

References 5 publications

Translation-invariant operators in reproducing kernel Hilbert spaces

Translation-invariant operators in reproducing kernel Hilbert spaces

C*-Algebra Generated by Angular Toeplitz Operators on the Weighted Bergman Spaces Over the Upper Half-Plane

Radial Toeplitz Operators on the Fock Space and Square-Root-Slowly Oscillating Sequences

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