On the powers of quasihomogeneous Toeplitz operators

Aissa Bouhali; Zohra Bendaoud; Issam Louhichi

Displaying similar documents to “On the powers of quasihomogeneous Toeplitz operators”

On products of some Toeplitz operators on polyanalytic Fock spaces

Irène Casseli (2020)

Czechoslovak Mathematical Journal

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The purpose of this paper is to study the Sarason’s problem on Fock spaces of polyanalytic functions. Namely, given two polyanalytic symbols $f$ and $g$ , we establish a necessary and sufficient condition for the boundedness of some Toeplitz products $T_{f} T_{\bar{g}}$ subjected to certain restriction on $f$ and $g$ . We also characterize this property in terms of the Berezin transform.

Schatten class generalized Toeplitz operators on the Bergman space

Chunxu Xu, Tao Yu (2021)

Czechoslovak Mathematical Journal

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Let $μ$ be a finite positive measure on the unit disk and let $j \geq 1$ be an integer. D. Suárez (2015) gave some conditions for a generalized Toeplitz operator $T_{μ}^{(j)}$ to be bounded or compact. We first give a necessary and sufficient condition for $T_{μ}^{(j)}$ to be in the Schatten $p$ -class for $1 \leq p < \infty$ on the Bergman space $A^{2}$ , and then give a sufficient condition for $T_{μ}^{(j)}$ to be in the Schatten $p$ -class $(0 < p < 1)$ on $A^{2}$ . We also discuss the generalized Toeplitz operators with general bounded symbols. If $ϕ \in L^{\infty} (D, d A)$ and $1 < p < \infty$ , we define the generalized...

Compression of slant Toeplitz operators on the Hardy space of $n$ -dimensional torus

Gopal Datt, Shesh Kumar Pandey (2020)

Czechoslovak Mathematical Journal

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This paper studies the compression of a $k$ th-order slant Toeplitz operator on the Hardy space $H^{2} (𝕋^{n})$ for integers $k \geq 2$ and $n \geq 1$ . It also provides a characterization of the compression of a $k$ th-order slant Toeplitz operator on $H^{2} (𝕋^{n})$ . Finally, the paper highlights certain properties, namely isometry, eigenvalues, eigenvectors, spectrum and spectral radius of the compression of $k$ th-order slant Toeplitz operator on the Hardy space $H^{2} (𝕋^{n})$ of $n$ -dimensional torus $𝕋^{n}$ .

Coefficient inequality for a function whose derivative has a positive real part of order $α$

Deekonda Vamshee Krishna, Thoutreddy Ramreddy (2015)

Mathematica Bohemica

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The objective of this paper is to obtain sharp upper bound for the function $f$ for the second Hankel determinant $| a_{2} a_{4} - a_{3}^{2} |$ , when it belongs to the class of functions whose derivative has a positive real part of order $α$ $(0 \leq α < 1)$ , denoted by $R T (α)$ . Further, an upper bound for the inverse function of $f$ for the nonlinear functional (also called the second Hankel functional), denoted by $| t_{2} t_{4} - t_{3}^{2} |$ , was determined when it belongs to the same class of functions, using Toeplitz determinants.

The generalized Toeplitz operators on the Fock space $F_{α}^{2}$

Chunxu Xu, Tao Yu (2024)

Czechoslovak Mathematical Journal

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Let $μ$ be a positive Borel measure on the complex plane $ℂ^{n}$ and let $j = (j_{1}, \dots, j_{n})$ with $j_{i} \in ℕ$ . We study the generalized Toeplitz operators $T_{μ}^{(j)}$ on the Fock space $F_{α}^{2}$ . We prove that $T_{μ}^{(j)}$ is bounded (or compact) on $F_{α}^{2}$ if and only if $μ$ is a Fock-Carleson measure (or vanishing Fock-Carleson measure). Furthermore, we give a necessary and sufficient condition for $T_{μ}^{(j)}$ to be in the Schatten $p$ -class for $1 \leq p < \infty$ .

Complex symmetry of Toeplitz operators on the weighted Bergman spaces

Xiao-He Hu (2022)

Czechoslovak Mathematical Journal

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We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^{p} {\bar{z}}^{q}}$ on the weighted Bergman space $A^{2} (Ω)$ , where $Ω$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^{p} {\bar{z}}^{q}}$ to be complex symmetric. When $p, q \in ℕ^{2}$ , we prove that $T_{z^{p} {\bar{z}}^{q}}$ is complex symmetric on $A^{2} (Ω)$ if and only if $p_{1} = q_{2}$ and $p_{2} = q_{1}$ . Moreover, we completely characterize when monomial Toeplitz operators $T_{z^{p} {\bar{z}}^{q}}$ on $A^{2} (𝔻_{n})$ are $J_{U}$ -symmetric with the $n \times n$ symmetric unitary matrix $U$ .

Area differences under analytic maps and operators

Mehmet Çelik, Luke Duane-Tessier, Ashley Marcial Rodriguez, Daniel Rodriguez, Aden Shaw (2024)

Czechoslovak Mathematical Journal

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Motivated by the relationship between the area of the image of the unit disk under a holomorphic mapping $h$ and that of $z h$ , we study various $L^{2}$ norms for $T_{ϕ} (h)$ , where $T_{ϕ}$ is the Toeplitz operator with symbol $ϕ$ . In Theorem , given polynomials $p$ and $q$ we find a symbol $ϕ$ such that $T_{ϕ} (p) = q$ . We extend some of our results to the polydisc.

Separately radial and radial Toeplitz operators on the projective space and representation theory

Raul Quiroga-Barranco, Armando Sanchez-Nungaray (2017)

Czechoslovak Mathematical Journal

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We consider separately radial (with corresponding group $𝕋^{n}$ ) and radial (with corresponding group $U (n))$ symbols on the projective space $ℙ^{n} (ℂ)$ , as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^{*}$ -algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it...

The relationship between $K_{u}^{2} \cap v H^{2}$ and inner functions

Xiaoyuan Yang (2024)

Czechoslovak Mathematical Journal

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Let $u$ be an inner function and $K_{u}^{2}$ be the corresponding model space. For an inner function $v$ , the subspace $v H^{2}$ is an invariant subspace of the unilateral shift operator on $H^{2}$ . In this article, using the structure of a Toeplitz kernel $\ker T_{\bar{u} v}$ , we study the intersection $K_{u}^{2} \cap v H^{2}$ by properties of inner functions $u$ and $v$ $(v \neq u)$ . If $K_{u}^{2} \cap v H^{2} \neq {0}$ , then there exists a triple $(B, b, g)$ such that $\bar{u} v = \frac{λ b \bar{B O_{g}}}{g},$ where the triple $(B, b, g)$ means that $B$ and $b$ are Blaschke products, $g$ is an invertible function in $H^{\infty}$ , $O_{g}$ denotes the outer factor...

The boundedness of two classes of integral operators

Xin Wang, Ming-Sheng Liu (2021)

Czechoslovak Mathematical Journal

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The aim of this paper is to characterize the $L^{p} - L^{q}$ boundedness of two classes of integral operators from $L^{p} (𝒰, d V_{α})$ to $L^{q} (𝒰, d V_{β})$ in terms of the parameters $a$ , $b$ , $c$ , $p$ , $q$ and $α$ , $β$ , where $𝒰$ is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

Toeplitz operators on Bergman spaces and Hardy multipliers

Wolfgang Lusky, Jari Taskinen (2011)

Studia Mathematica

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We study Toeplitz operators $T_{a}$ with radial symbols in weighted Bergman spaces $A_{μ}^{p}$ , 1 < p < ∞, on the disc. Using a decomposition of $A_{μ}^{p}$ into finite-dimensional subspaces the operator $T_{a}$ can be considered as a coefficient multiplier. This leads to new results on boundedness of $T_{a}$ and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of $T_{a}$ for a satisfying an assumption on the positivity of certain...

Theoretical analysis for $ℓ_{1}$ - $ℓ_{2}$ minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of $k$ -sparse signals using the $ℓ_{1}$ - $ℓ_{2}$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$ -sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...