scholarly journals OPERATORS ON BERGMAN SPACES

2021 ◽  
Vol 56 (5) ◽  
pp. 399-403
Author(s):  
Kifah Y. Alhami
Keyword(s):  
Invariant Subspace ◽  
Bergman Space ◽  
Hardy Spaces ◽  
Complex Analysis ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Space Theory ◽  
Shift Operators ◽  
The Core ◽  
Vector Valued

Bergman space theory has been at the core of complex analysis research for many years. Indeed, Hardy spaces are related to Bergman spaces. The popularity of Bergman spaces increased when functional analysis emerged. Although many researchers investigated the Bergman space theory by mimicking the Hardy space theory, it appeared that, unlike their cousins, Bergman spaces were more complex in different aspects. The issue of invariant subspace constitutes one common problem in mathematics that is yet to be resolved. For Hardy spaces, each invariant subspace for shift operators features an elegant description. However, the method for formulating particular structures for the large invariant subspace of shift operators upon Bergman spaces is still unknown. This paper aims to characterize bounded Hankel operators involving a vector-valued Bergman space compared to other different vector value Bergman spaces.

The range of block Hankel operators

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3237-3243
Author(s):  
In Hwang ◽  
In Kim ◽  
Sumin Kim
Keyword(s):  
Hardy Space ◽  
Invariant Subspace ◽  
Hankel Operators ◽  
Coprime Factorization ◽  
Backward Shift ◽  
Vector Valued

In this note we give a connection between the closure of the range of block Hankel operators acting on the vector-valued Hardy space H2Cn and the left coprime factorization of its symbol. Given a subset F ? H2Cn, we also consider the smallest invariant subspace S*F of the backward shift S* that contains F.

scholarly journals Multipliers on Vector Valued Bergman Spaces

2002 ◽  
Vol 54 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
Oscar Blasco ◽  
José Luis Arregui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bergman Space ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Taylor Coefficients ◽  
Vector Valued

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

scholarly journals Toeplitz operators on generalized Bergman-Hardy spaces

2001 ◽  
Vol 88 (1) ◽  
pp. 96
Author(s):  
Wolfgang Lusky
Keyword(s):  
Hardy Spaces ◽  
Toeplitz Operators ◽  
Fock Space ◽  
Bergman Spaces ◽  
Trigonometric Polynomials ◽  
Essential Spectra ◽  
Radial Functions ◽  
Hardy And Bergman Spaces ◽  
Schmidt Operator ◽  
Vector Valued

We study the Toeplitz operators $T_f: H_2 \to H_2$, for $f \in L_\infty$, on a class of spaces $H_2$ which in- cludes, among many other examples, the Hardy and Bergman spaces as well as the Fock space. We investigate the space $X$ of those elements $f \in L_\infty$ with $\lim_j \|T_f-T_{f_j}\|=0$ where $(f_j)$ is a sequence of vector-valued trigonometric polynomials whose coefficients are radial functions. For these $T_f$ we obtain explicit descriptions of their essential spectra. Moreover, we show that $f \in X$, whenever $T_f$ is compact, and characterize these functions in a simple and straightforward way. Finally, we determine those $f \in L_\infty$ where $T_f$ is a Hilbert-Schmidt operator.

scholarly journals Hankel operators induced by radial Bekollé–Bonami weights on Bergman spaces

2019 ◽  
Vol 296 (1-2) ◽  
pp. 211-238 ◽  
Author(s):  
José Ángel Peláez ◽  
Antti Perälä ◽  
Jouni Rättyä
Keyword(s):  
Bergman Space ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Weighted Bergman Space ◽  
Doubling Condition ◽  
Weighted Mean ◽  
Mean Oscillation ◽  
Radial Weight ◽  
Classical Setting

Abstract We study big Hankel operators $$H_f^\nu :A^p_\omega \rightarrow L^q_\nu $$ H f ν : A ω p → L ν q generated by radial Bekollé–Bonami weights $$\nu $$ ν , when $$1<p\le q<\infty $$ 1 < p ≤ q < ∞ . Here the radial weight $$\omega $$ ω is assumed to satisfy a two-sided doubling condition, and $$A^p_\omega $$ A ω p denotes the corresponding weighted Bergman space. A characterization for simultaneous boundedness of $$H_f^\nu $$ H f ν and $$H_{{\overline{f}}}^\nu $$ H f ¯ ν is provided in terms of a general weighted mean oscillation. Compared to the case of standard weights that was recently obtained by Pau et al. (Indiana Univ Math J 65(5):1639–1673, 2016), the respective spaces depend on the weights $$\omega $$ ω and $$\nu $$ ν in an essentially stronger sense. This makes our analysis deviate from the blueprint of this more classical setting. As a consequence of our main result, we also study the case of anti-analytic symbols.

scholarly journals ON R-SECTORIAL DERIVATIVES ON BERGMAN SPACES

2008 ◽  
Vol 77 (2) ◽  
pp. 305-313 ◽  
Author(s):  
TOMÁŠ BÁRTA
Keyword(s):  
Bergman Space ◽  
Integrodifferential Equation ◽  
Bergman Spaces ◽  
Bergman Projections ◽  
Vector Valued

AbstractIn this paper we show boundedness of vector-valued Bergman projections on simple connected domains. With this result we show R-sectoriality of the derivative on the Bergman space on C+ and maximal Lp-regularity for an integrodifferential equation with a kernel in the Bergman space.

Litte Hankel Operators Between Vector-Valued Bergman Spaces on the Unit Ball

2021 ◽  
Vol 93 (3) ◽  
Author(s):  
David Békollé ◽  
Hugues Olivier Defo ◽  
Edgar L. Tchoundja ◽  
Brett D. Wick
Keyword(s):  
Unit Ball ◽  
Bergman Spaces ◽  
Hankel Operators ◽  
Vector Valued

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

scholarly journals Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sumin Kim ◽  
Jongrak Lee
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

AbstractIn this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator $T_{\varphi }$ T φ on the Bergman space $A^{2}(\mathbb{D})$ A 2 ( D ) with non-harmonic symbols under certain assumptions.

scholarly journals The Berezin Transform of Toeplitz Operators on the Weighted Bergman Space

2021 ◽  
Author(s):  
Cezhong Tong ◽  
Junfeng Li ◽  
Hicham Arroussi
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Reproducing Kernel ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Kernel Estimates ◽  
Berezin Transforms

AbstractIn this paper, we obtain some interesting reproducing kernel estimates and some Carleson properties that play an important role. We characterize the bounded and compact Toeplitz operators on the weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, we estimate the essential norm of them assuming that they are bounded.

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