Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains

Mehmet Çelik; Yunus E. Zeytuncu

Displaying similar documents to “Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on complete pseudoconvex Reinhardt domains”

Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on a class of unbounded complete Reinhardt domains

Le He, Yanyan Tang (2024)

Czechoslovak Mathematical Journal

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We consider a class of unbounded nonhyperbolic complete Reinhardt domains $D_{n, m, k}^{μ, p, s} : = \{(z, w_{1}, \dots, w_{m}) \in ℂ^{n} \times ℂ^{k_{1}} \times \dots \times ℂ^{k_{m}} : \frac{∥ w_{1} ∥^{2 p_{1}}}{e^{- μ_{1} {∥ z ∥}^{s}}} + \dots + \frac{∥ w_{m} ∥^{2 p_{m}}}{e^{- μ_{m} {∥ z ∥}^{s}}} < 1\},$ where $s$ , $p_{1}, \dots, p_{m}$ , $μ_{1}, \dots, μ_{m}$ are positive real numbers and $n$ , $k_{1}, \dots, k_{m}$ are positive integers. We show that if a Hankel operator with anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $A^{2} (D_{n, m, k}^{μ, p, s})$ , then it must be zero. This gives an example of high dimensional unbounded complete Reinhardt domain that does not admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.

$L ²_{h}$ -domains of holomorphy and the Bergman kernel

Peter Pflug, Włodzimierz Zwonek (2002)

Studia Mathematica

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We give a characterization of $L ²_{h}$ -domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.

On boundary behaviour of the Bergman projection on pseudoconvex domains

M. Jasiczak (2005)

Studia Mathematica

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It is shown that on strongly pseudoconvex domains the Bergman projection maps a space $L v_{k}$ of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space $E \supset L^{\infty} (Ω)$ defined by weighted-sup seminorms and equipped...

Certain partial differential subordinations on some Reinhardt domains in $ℂ^{n}$

Gabriela Kohr, Mirela Kohr (1997)

Annales Polonici Mathematici

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We obtain an extension of Jack-Miller-Mocanu’s Lemma for holomorphic mappings defined in some Reinhardt domains in $ℂ^{n}$ . Using this result we consider first and second order partial differential subordinations for holomorphic mappings defined on the Reinhardt domain $B_{2 p}$ with p ≥ 1.

Rigidity of the holomorphic automorphism of the generalized Fock-Bargmann-Hartogs domains

Ting Guo, Zhiming Feng, Enchao Bi (2021)

Czechoslovak Mathematical Journal

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We study a class of typical Hartogs domains which is called a generalized Fock-Bargmann-Hartogs domain $D_{n, m}^{p} (μ)$ . The generalized Fock-Bargmann-Hartogs domain is defined by inequality $e^{{μ ∥ z ∥}^{2}} \sum_{j = 1}^{m} {| ω_{j} |}^{2 p} < 1$ , where $(z, ω) \in ℂ^{n} \times ℂ^{m}$ . In this paper, we will establish a rigidity of its holomorphic automorphism group. Our results imply that a holomorphic self-mapping of the generalized Fock-Bargmann-Hartogs domain $D_{n, m}^{p} (μ)$ becomes a holomorphic automorphism if and only if it keeps the function $\sum_{j = 1}^{m} {| ω_{j} |}^{2 p} e^{{μ ∥ z ∥}^{2}}$ invariant.

Weighted sub-Bergman Hilbert spaces

Maria Nowak, Renata Rososzczuk (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces $A_{α}^{2}$ , $- 1 < α < \infty$ . These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.

The space $N_{*}$ of holomorphic functions on bounded symmetric domains

Manfred Stoll (1976)

Annales Polonici Mathematici

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The equation $\bar{\partial}u=f$ the intersection of pseudoconvex domains

Alessandro Perotti (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Viene studiata l'equazione $\bar{\partial}u=f$ per le forme regolari sulla chiusura dell'intersezione di $k$ domini pseudoconvessi. Si costruisce un operatore soluzione in forma integrale e sotto ipotesi opportune si ottengono stime della soluzione nelle norme $\mathbf{C}^{k}$ .

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.

The $\bar{\partial}$ -Neumann operator and commutators of the Bergman projection and multiplication operators

Friedrich Haslinger (2008)

Czechoslovak Mathematical Journal

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We prove that compactness of the canonical solution operator to $\bar{\partial}$ restricted to $(0, 1)$ -forms with holomorphic coefficients is equivalent to compactness of the commutator $[𝒫, \bar{M}]$ defined on the whole $L_{(0, 1)}^{2} (Ω),$ where $\bar{M}$ is the multiplication by $\bar{z}$ and $𝒫$ is the orthogonal projection of $L_{(0, 1)}^{2} (Ω)$ to the subspace of $(0, 1)$ forms with holomorphic coefficients. Further we derive a formula for the $\bar{\partial}$ -Neumann operator restricted to $(0, 1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the...

On n-circled $ℋ^{\infty}$ -domains of holomorphy

Marek Jarnicki, Peter Pflug (1997)

Annales Polonici Mathematici

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We present various characterizations of n-circled domains of holomorphy $G \subset ℂ^{n}$ with respect to some subspaces of $ℋ^{\infty} (G)$ .

Balanced domains of holomorphy of type $H^{α}$

J. Siciak (1985)

Matematički Vesnik

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Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids

Romaric Tytgat (2015)

Czechoslovak Mathematical Journal

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Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes $\bar{f}$ tels que l’opérateur de Hankel $H_{\bar{f}}$ sur l’espace de Bergman à poids soit dans l’idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten $𝒮^{p}$ quand $p$ tend vers $1$ et nous nous appuyons sur le résultat de Engliš et Rochberg sur l’espace de Bergman. Nous parlons aussi des puissances de tels opérateurs....

Operator positivity and analytic models of commuting tuples of operators

Monojit Bhattacharjee, Jaydeb Sarkar (2016)

Studia Mathematica

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We study analytic models of operators of class $C_{\cdot 0}$ with natural positivity assumptions. In particular, we prove that for an m-hypercontraction $T \in C_{\cdot 0}$ on a Hilbert space , there exist Hilbert spaces and ⁎ and a partially isometric multiplier θ ∈ ℳ (H²(),A²ₘ(⁎)) such that $≅_{θ} = A ² ₘ (⁎) ⊖ θ H ² ()$ and $T ≅ P_{_{θ}} M_{z} |_{_{θ}}$ , where A²ₘ(⁎) is the ⁎-valued weighted Bergman space and H²() is the -valued Hardy space over the unit disc . We then proceed to study analytic models for doubly commuting n-tuples of operators and investigate their...

Strict plurisubharmonicity of Bergman kernels on generalized annuli

Yanyan Wang (2014)

Annales Polonici Mathematici

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Let $A_{ζ} = Ω - \bar{ρ (ζ) \cdot Ω}$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{ζ} (z)$ of $A_{ζ}$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{ζ}$ is non-pseudoconvex when the dimension of $A_{ζ}$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ² l o g K_{ζ} / \partial ζ \partial ζ ̅$ , as well as its boundary behavior.

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...

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik, Karabi Rajbangshi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator $_{a, b}$ by $_{a, b} (f) (z) = Γ (a + 1) / Γ (b + 1) \int_{0}^{1} (f (t) {(1 - t)}^{b}) / ({(1 - t z)}^{a + 1}) d t$ , where a and b are non-negative real numbers. In particular, for a = b = β, $_{a, b}$ becomes the generalized Hilbert operator $_{β}$ , and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that $_{a, b}$ is bounded on Dirichlet-type spaces $S^{p}$ , 0 < p < 2, and on Bergman spaces $A^{p}$ , 2 < p < ∞. Also we find an upper bound for the norm of the operator $_{a, b}$ ....