Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols

Rodríguez, Miguel Ángel Rodríguez

doi:10.1007/s40590-020-00299-8

Cited by 5 publications

(1 citation statement)

References 9 publications

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“…It turns out that, in many cases, the right choice of set of symbols S yields commutative algebras, or at least some setup where studying the commutativity of Toeplitz operators is quite interesting. We refer to [1,2,3,6,8,9,23,29,36] for just a few examples of this fact.…”

Section: Introductionmentioning

confidence: 99%

Radial-like Toeplitz operators on Cartan domains of type I

Quiroga-Barranco¹

2022

Preprint

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Let D I n×n be the Cartan domain of type I which consists of the complex n×n matrices Z that satisfy Z * Z < In. For a symbol a ∈ L ∞ (D I n×n ) we consider three radial-like type conditions: 1) left (right) U(n)-invariant symbols, which can be defined by the condition a, respectively), and 2) U(n) × U(n)-invariant symbols, which are defined by the condition a(A −1 ZB) = a(Z) for every A, B ∈ U(n). We prove that, for n ≥ 2, these yield different sets of symbols. If a satisfies 1), either left or right, and b satisfies 2), then we prove that the corresponding Toeplitz operators Ta and T b commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for n ≥ 2, symbols a whose corresponding Toeplitz operators Ta are non-normal. We use these facts to prove the existence, for n ≥ 2, of commutative Banach non-C * algebras generated by Toeplitz operators.

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Section: Introductionmentioning

confidence: 99%

Radial-like Toeplitz operators on Cartan domains of type I

Quiroga-Barranco¹

2022

Preprint

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Toeplitz Operators, $$\mathbb {T}^m$$-Invariance and Quasi-homogeneous Symbols

Quiroga-Barranco

2021

Integr. Equ. Oper. Theory

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Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

Bauer

Rodríguez

2022

Complex Anal. Oper. Theory

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We present a survey and new results on the construction and Gelfand theory of commutative Toeplitz algebras over the standard weighted Bergman and Hardy spaces over the unit ball in $$\mathbb {C}^n$$ C n . As an application we discuss semi-simplicity and the spectral invariance of these algebras. The different function Hilbert spaces are dealt with in parallel in successive chapters so that a direct comparison of the results is possible. As a new aspect of the theory we define commutative Toeplitz algebras over spaces of functions in infinitely many variables and present some structural results. The paper concludes with a short list of open problems in this area of research.

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Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols

Cited by 5 publications

References 9 publications

Radial-like Toeplitz operators on Cartan domains of type I

Radial-like Toeplitz operators on Cartan domains of type I

Toeplitz Operators, $$\mathbb {T}^m$$-Invariance and Quasi-homogeneous Symbols

Commutative Toeplitz Algebras and Their Gelfand Theory: Old and New Results

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