The essential spectrum of holomorphic Toeplitz operators on $H^{p}$ spaces

Mats Andersson; Sebastian Sandberg

Displaying similar documents to “The essential spectrum of holomorphic Toeplitz operators on $H^{p}$ spaces”

The essential spectrum of Toeplitz tuples with symbols in $H^{\infty} + C$

Jörg Eschmeier (2013)

Studia Mathematica

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Let H²(D) be the Hardy space on a bounded strictly pseudoconvex domain D ⊂ ℂⁿ with smooth boundary. Using Gelfand theory and a spectral mapping theorem of Andersson and Sandberg (2003) for Toeplitz tuples with $H^{\infty}$ -symbol, we show that a Toeplitz tuple $T_{f} = (T_{f ₁}, . . ., T_{f ₘ}) \in L {(H ² (σ))}^{m}$ with symbols $f_{i} \in H^{\infty} + C$ is Fredholm if and only if the Poisson-Szegö extension of f is bounded away from zero near the boundary of D. Corresponding results are obtained for the case of Bergman spaces. Thus we extend results of McDonald (1977) and...

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 $𝕋^{n}$ .

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...

On the powers of quasihomogeneous Toeplitz operators

Aissa Bouhali, Zohra Bendaoud, Issam Louhichi (2021)

Czechoslovak Mathematical Journal

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We present sufficient conditions for the existence of $p$ th powers of a quasihomogeneous Toeplitz operator $T_{e^{i s θ} ψ}$ , where $ψ$ is a radial polynomial function and $p$ , $s$ are natural numbers. A large class of examples is provided to illustrate our results. To our best knowledge those examples are not covered by the current literature. The main tools in the proof of our results are the Mellin transform and some classical theorems of complex analysis.

Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk (2018)

Mathematica Bohemica

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We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $𝒜$ to have a non-void left joint spectrum $σ_{l} (S) .$ In particular, for a Lie subalgebra $L \subset 𝒜$ we have $σ_{l} (L) \neq \emptyset$ if and only if $[L, L]$ generates in $𝒜$ a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

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...

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.

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 $U$ .

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 $1 \leq p < \infty$ .

On the convergence and character spectra of compact spaces

István Juhász, William A. R. Weiss (2010)

Fundamenta Mathematicae

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An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

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.

Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee, Quoc-Phong Vu (2014)

Studia Mathematica

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We consider simultaneous solutions of operator Sylvester equations $A_{i} X - X B_{i} = C_{i}$ (1 ≤ i ≤ k), where $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C ₁, . . ., C_{k})$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

Graphs with small diameter determined by their $D$ -spectra

Ruifang Liu, Jie Xue (2018)

Czechoslovak Mathematical Journal

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Let $G$ be a connected graph with vertex set $V (G) = {v_{1}, v_{2}, ..., v_{n}}$ . The distance matrix $D (G) = {(d_{i j})}_{n \times n}$ is the matrix indexed by the vertices of $G$ , where $d_{i j}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$ . Suppose that $λ_{1} (D) \geq λ_{2} (D) \geq \dots \geq λ_{n} (D)$ are the distance spectrum of $G$ . The graph $G$ is said to be determined by its $D$ -spectrum if with respect to the distance matrix $D (G)$ , any graph having the same spectrum as $G$ is isomorphic to $G$ . We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs...

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

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For a linear operator T in a Banach space let $σ_{p} (T)$ denote the point spectrum of T, let $σ_{p, n} (T)$ for finite n > 0 be the set of all $λ \in σ_{p} (T)$ such that dim ker(T - λ) = n and let $σ_{p, \infty} (T)$ be the set of all $λ \in σ_{p} (T)$ for which ker(T - λ) is infinite-dimensional. It is shown that $σ_{p} (T)$ is $ℱ_{σ}$ , $σ_{p, \infty} (T)$ is $ℱ_{σ δ}$ and for each finite n the set $σ_{p, n} (T)$ is the intersection of an $ℱ_{σ}$ set and a $_{δ}$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

Displaying similar documents to “The essential spectrum of holomorphic Toeplitz operators on H p spaces”

Displaying similar documents to “The essential spectrum of holomorphic Toeplitz operators on $H^{p}$ spaces”