arXiv subjects

Miguel Angel Rodriguez Rodriguez

Publications and source records attributed to Miguel Angel Rodriguez Rodriguez.

5 recordsLinked to original sources

arXiv2026

A localization framework for $\mathbb{T}^m$-Invariant Toeplitz Algebras: Applications to Gelfand Theory

We study $\mathbb T^m$-invariant Toeplitz operators on weighted Bergman spaces over the unit ball. A natural unitary transformation represents each such operator as a direct sum of restrictions of Toeplitz operators acting on a polyball. Under an $L^\infty$-valued angular continuity assumption, the corresponding symbols extend from $\mathbb N_0^m$ to a norm-continuous family indexed by the maximal ideal space of the $C^*$-algebra generated by $k$-quasi-radial Toeplitz operators. This extension yields a localization framework for Toeplitz operator algebras. As an application, we consider commutative Banach algebras obtained by adjoining Toeplitz operators whose symbols are invariant under mixed unitary and circle actions. We identify their local quotient algebras on finite-coordinate strata and at infinity and use these identifications to describe their maximal ideal spaces and Gelfand transforms explicitly.

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arXiv2026

Commutative topological algebras on translation-invariant reproducing kernel Hilbert spaces

We study commutative topological algebras naturally associated with translation-invariant reproducing kernel Hilbert spaces whose direct integral decomposition has one-dimensional fibers. Starting from the bounded algebra of translation-invariant operators, we pass to a common dense domain generated by reproducing kernels and identify the corresponding diagonalizable operators with multiplication by symbols in an intersection of weighted $L^2$-spaces. On the symbol side this gives a canonical space $\mathcal F_0$ and a maximal multiplicative subalgebra $\mathcal F_M$, which is a complete locally convex $*$-algebra. Transporting the structure back yields corresponding algebras of operators and integral kernels. We also discuss when the inclusions $L^\infty(Ω)=\mathcal F_\infty\subset \mathcal F_M\subset \mathcal F_0$ are strict, and illustrate the results with vertical and radial operators on classical Bergman and Fock spaces.

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arXiv2024

Operators in the Fock-Toeplitz algebra

We consider various classes of bounded operators on the Fock space $F^2$ of Gaussian square integrable entire functions over the complex plane. These include Toeplitz (type) operators, weighted composition operators, singular integral operators, Volterra-type operators and Hausdorff operators and range from classical objects in harmonic analysis to more recently introduced classes. As a leading problem and closely linked to well-known compactness characterizations we pursue the question of when these operators are contained in the Toeplitz algebra. This paper combines a (certainly in-complete) survey of the classical and more recent literature including new ideas for proofs from the perspective of quantum harmonic analysis (QHA). Moreover, we have added a number of new theorems and links between known results.

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arXiv2023

Commutative $G$-invariant Toeplitz C$^\ast$ algebras on the Fock space and their Gelfand theory through Quantum Harmonic Analysis

We discuss the notion of spectral synthesis for the setting of Quantum Harmonic Analysis. Using these concepts, we study subalgebras of the full Toeplitz algebra with certain invariant symbols and their commutators. In particular, we find a new class of commutative Toeplitz C$^\ast$ algebras on the Fock space. In the end, we investigate the Gelfand theory of those commutative C$^\ast$ algebras.

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arXiv2022

Commutative Banach Algebras Generated by Toeplitz Operators on the Bergman Space

We present and study commutative Banach algebras generated by Toeplitz operators with generalized quasi-radial pseudo-homogeneous symbols acting on the Bergman space over the unit ball. We develop the Gelfand theory of these algebras and give some structural information about them. In particular, we provide a description of the radical of these algebras. This paper generalizes and completes the results from previous works related to Toeplitz operators with quasi-radial quasi-homogeneous symbols.

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