Slant Hankel operators

Subhash Chander Arora; Ruchika Batra; M. P. Singh

Displaying similar documents to “Slant Hankel operators”

Bounded Toeplitz and Hankel products on weighted Bergman spaces of the unit ball

Małgorzata Michalska, Maria Nowak, Paweł Sobolewski (2010)

Annales Polonici Mathematici

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We prove a sufficient condition for products of Toeplitz operators $T_{f} T_{ḡ}$ , where f,g are square integrable holomorphic functions in the unit ball in ℂⁿ, to be bounded on the weighted Bergman space. This condition slightly improves the result obtained by K. Stroethoff and D. Zheng. The analogous condition for boundedness of products of Hankel operators $H_{f} H *_{g}$ is also given.

Carleson measures and Toeplitz operators on small Bergman spaces on the ball

Van An Le (2021)

Czechoslovak Mathematical Journal

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We study Carleson measures and Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip’s results from the unit disk of $ℂ$ to the unit ball of $ℂ^{n}$ . We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1 < p < \infty$ .

Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space

Xing-Tang Dong, Ze-Hua Zhou (2013)

Studia Mathematica

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We present here a quite unexpected result: If the product of two quasihomogeneous Toeplitz operators $T_{f} T_{g}$ on the harmonic Bergman space is equal to a Toeplitz operator $T_{h}$ , then the product $T_{g} T_{f}$ is also the Toeplitz operator $T_{h}$ , and hence $T_{f}$ commutes with $T_{g}$ . From this we give necessary and sufficient conditions for the product of two Toeplitz operators, one quasihomogeneous and the other monomial, to be a Toeplitz operator.

The quasi-canonical solution operator to $\bar{\partial}$ restricted to the Fock-space

Georg Schneider (2005)

Czechoslovak Mathematical Journal

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We consider the solution operator $S ℱ_{μ, (p, q)} \to L^{2} {(μ)}_{(p, q)}$ to the $\bar{\partial}$ -operator restricted to forms with coefficients in $ℱ_{μ} = \{f f is entire and \int_{ℂ^{n}} {| f (z) |}^{2} d μ (z) < \infty\}$ . Here $ℱ_{μ, (p, q)}$ denotes $(p, q)$ -forms with coefficients in $ℱ_{μ}$ , $L^{2} (μ)$ is the corresponding $L^{2}$ -space and $μ$ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula $S$ to $\bar{\partial}$ . This solution operator will have the property $S v ⊥ ℱ_{(p, q)} \forall v \in ℱ_{(p, q + 1)}$ . As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness...

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...

Reducing subspaces of Toeplitz operators on Dirichlet type spaces of the bidisk

Hongzhao Lin, Zhongming Teng (2021)

Czechoslovak Mathematical Journal

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The reducing subspaces of Toeplitz operators $T_{z_{1}^{N} {\bar{z}}_{2}^{M}}$ on Dirichlet type spaces of the $𝒟_{α} (𝔻^{2})$ are described, which extends the results for the corresponding operators on Bergman spaces of the bidisk.

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

Mats Andersson, Sebastian Sandberg (2003)

Studia Mathematica

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We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators $T_{g}$ on $H^{p} (D)$ , where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of $T_{g}$ provided that $g^{- 1} (0)$ is a compact subset of D.

Failure of Nehari's theorem for multiplicative Hankel forms in Schatten classes

Ole Fredrik Brevig, Karl-Mikael Perfekt (2015)

Studia Mathematica

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Ortega-Cerdà-Seip demonstrated that there are bounded multiplicative Hankel forms which do not arise from bounded symbols. On the other hand, when such a form is in the Hilbert-Schmidt class ₂, Helson showed that it has a bounded symbol. The present work investigates forms belonging to the Schatten classes between these two cases. It is shown that for every $p > {(1 - l o g π / l o g 4)}^{- 1}$ there exist multiplicative Hankel forms in the Schatten class $_{p}$ which lack bounded symbols. The lower bound on p is in a certain...

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...

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.

Compact operators on the weighted Bergman space A¹(ψ)

Tao Yu (2006)

Studia Mathematica

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We show that a bounded linear operator S on the weighted Bergman space A¹(ψ) is compact and the predual space A₀(φ) of A¹(ψ) is invariant under S* if and only if $S k_{z} \to 0$ as z → ∂D, where $k_{z}$ is the normalized reproducing kernel of A¹(ψ). As an application, we give conditions for an operator in the Toeplitz algebra to be compact.

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.

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...

Hankel determinant for a class of analytic functions of complex order defined by convolution

S. M. El-Deeb, M. K. Aouf (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain the Fekete-Szego inequalities for the functions of complex order defined by convolution. Also, we find upper bounds for the second Hankel determinant $| a_{2} a_{4} - a_{3}^{2} |$ for functions belonging to the class $S_{γ}^{b} (g (z); A, B)$ .