Essential Norm of Toeplitz Operators and Hankel Operators on the Weighted Bergman Space

2012 ◽  
Vol 75 (4) ◽  
pp. 517-525 ◽  
Author(s):  
Fengying Li
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Essential Norm ◽  
Hankel Operators ◽  
Weighted Bergman Space

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.

scholarly journals (m,λ)-Berezin Transform on the Weighted Bergman Spaces over the Polydisk

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Ran Li ◽  
Yufeng Lu
Keyword(s):  
Linear Operator ◽  
Bergman Space ◽  
Bounded Linear Operator ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Main Tool ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Bounded Linear

We prove that every bounded linear operator on weighted Bergman space over the polydisk can be approximated by Toeplitz operators under some conditions. The main tool here is the so-called(m,λ)-Berezin transform. In particular, our results generalized the results of K. Nam and D. C. Zheng to the case of operators acting onAλ2(Dn).

scholarly journals Finite-rank intermediate Hankel operators on the Bergman space

2001 ◽  
Vol 25 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Takahiko Nakazi ◽  
Tomoko Osawa
Keyword(s):  
Bergman Space ◽  
Orthogonal Projection ◽  
Finite Rank ◽  
Lebesgue Space ◽  
Unit Disc ◽  
Hankel Operator ◽  
Hankel Operators ◽  
Weighted Bergman Space ◽  
Open Unit ◽  
Open Unit Disc

LetL2=L2(D,r dr dθ/π)be the Lebesgue space on the open unit disc and letLa2=L2∩ℋol(D)be the Bergman space. LetPbe the orthogonal projection ofL2ontoLa2and letQbe the orthogonal projection ontoL¯a,02={g∈L2;g¯∈La2,   g(0)=0}. ThenI−P≥Q. The big Hankel operator and the small Hankel operator onLa2are defined as: forϕinL∞,Hϕbig(f)=(I−P)(ϕf)andHϕsmall(f)=Q(ϕf)(f∈La2). In this paper, the finite-rank intermediate Hankel operators betweenHϕbigandHϕsmallare studied. We are working on the more general space, that is, the weighted Bergman space.

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 The Essential Norm of the Generalized Hankel Operators on the Bergman Space of the Unit Ball inCn

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Luo Luo ◽  
Yang Xuemei
Keyword(s):  
Unit Ball ◽  
Bergman Space ◽  
Hankel Operator ◽  
Equivalent Form ◽  
Essential Norm ◽  
Hankel Operators

In 1993, Peloso introduced a kind of operators on the Bergman spaceA2(B)of the unit ball that generalizes the classical Hankel operator. In this paper, we estimate the essential norm of the generalized Hankel operators on the Bergman spaceAp(B)  (p>1)of the unit ball and give an equivalent form of the essential norm.

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