scholarly journals Toeplitz operators with distributional symbols on Bergman spaces

2011 ◽  
Vol 54 (2) ◽  
pp. 505-514 ◽  
Author(s):  
Antti Perälä ◽  
Jari Taskinen ◽  
Jani Virtanen
Keyword(s):  
Sobolev Space ◽  
Hyperbolic Geometry ◽  
Toeplitz Operators ◽  
Weighted Sobolev Space ◽  
Bergman Spaces ◽  
Sufficient Condition ◽  
Negative Order ◽  
Natural Relation

AbstractWe study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces $A^p(\mathbb{D})$, 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space $\smash{W_\nu^{-m,\infty}(\mathbb{D})}$ of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

scholarly journals Hyponormality on general Bergman spaces

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5737-5741 ◽  
Author(s):  
Houcine Sadraoui
Keyword(s):  
Hilbert Space ◽  
Unit Disk ◽  
Toeplitz Operators ◽  
Bounded Operator ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Necessary Condition ◽  
Sufficient Condition

A bounded operator T on a Hilbert space is hyponormal if T*T-TT* is positive. We give a necessary condition for the hyponormality of Toeplitz operators on weighted Bergman spaces, for a certain class of radial weights, when the symbol is of the form f+g?, where both functions are analytic and bounded on the unit disk. We give a sufficient condition when f is a monomial.

BOUNDED TOEPLITZ AND HANKEL PRODUCTS ON THE WEIGHTED BERGMAN SPACES OF THE UNIT BALL

2015 ◽  
Vol 99 (2) ◽  
pp. 237-249
Author(s):  
MAŁGORZATA MICHALSKA ◽  
PAWEŁ SOBOLEWSKI
Keyword(s):  
Unit Ball ◽  
Toeplitz Operators ◽  
Hankel Operator ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Berezin Transform ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Necessary Condition ◽  
Sufficient Condition

Let $A_{{\it\alpha}}^{p}$ be the weighted Bergman space of the unit ball in ${\mathcal{C}}^{n}$, $n\geq 2$. Recently, Miao studied products of two Toeplitz operators defined on $A_{{\it\alpha}}^{p}$. He proved a necessary condition and a sufficient condition for boundedness of such products in terms of the Berezin transform. We modify the Berezin transform and improve his sufficient condition for products of Toeplitz operators. We also investigate products of two Hankel operators defined on $A_{{\it\alpha}}^{p}$, and products of the Hankel operator and the Toeplitz operator. In particular, in both cases, we prove sufficient conditions for boundedness of the products.

scholarly journals Eigenvalues of K-invariant Toeplitz Operators on Bounded Symmetric Domains

2021 ◽  
Vol 93 (3) ◽  
Author(s):  
Harald Upmeier
Keyword(s):  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Bounded Symmetric Domains

AbstractWe determine the eigenvalues of certain “fundamental” K-invariant Toeplitz type operators on weighted Bergman spaces over bounded symmetric domains $$D=G/K,$$ D = G / K , for the irreducible K-types indexed by all partitions of length $$r={\mathrm {rank}}(D)$$ r = rank ( D ) .

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.

scholarly journals Toeplitz Operators on the Bergman Space of Planar Domains with Essentially Radial Symbols

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Roberto C. Raimondo
Keyword(s):  
Bergman Space ◽  
Toeplitz Operators ◽  
Boundary Behavior ◽  
Berezin Transform ◽  
Planar Domain ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Planar Domains ◽  
Necessary And Sufficient ◽  
The Relationship

We study the problem of the boundedness and compactness of when and is a planar domain. We find a necessary and sufficient condition while imposing a condition that generalizes the notion of radial symbol on the disk. We also analyze the relationship between the boundary behavior of the Berezin transform and the compactness of

Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

2021 ◽  
pp. 1-13
Author(s):  
Kita Naoyasu ◽  
Sato Takuya
Keyword(s):  
Sobolev Space ◽  
Global Solution ◽  
Initial Value Problem ◽  
Trivial Solution ◽  
Decay Estimate ◽  
Weighted Sobolev Space ◽  
The Other ◽  
Initial Value ◽  
Decay Of Solutions ◽  
Dissipative Nonlinearity

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

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