Boundedness of the Cesàro averaging operators on Dirichlet spaces

2004 ◽  
Vol 134 (4) ◽  
pp. 609-616
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Dirichlet Space ◽  
Dirichlet Spaces ◽  
Averaging Operators ◽  
Cesaro Averaging ◽  
Associated Operators

It is shown that the Cesàro averaging operators Cα, Re α > −1, introduced by Stempak, are bounded on the Dirichlet space Da if and only if a > 0, while the associated operators Aα are bounded on Da if and only if −1 < a < 2. This extends results of Galanopoulos, who considered the particular case α = 0 for 0 ≤ a ≤ 1.

Adjoint of generalized Cesáro operators on analytic function spaces

2020 ◽  
Vol 26 (2) ◽  
pp. 185-192
Author(s):  
Sunanda Naik ◽  
Pankaj K. Nath
Keyword(s):  
Analytic Function ◽  
Convolution Operator ◽  
Composition Operators ◽  
Mixed Norm ◽  
Averaging Operators ◽  
Mixed Norm Spaces ◽  
Cesaro Averaging ◽  
Analytic Function Spaces ◽  
Appl Math ◽  
Two Parameter

AbstractIn this article, we define a convolution operator and study its boundedness on mixed-norm spaces. In particular, we obtain a well-known result on the boundedness of composition operators given by Avetisyan and Stević in [K. Avetisyan and S. Stević, The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc, Appl. Math. Comput. 213 2009, 2, 304–311]. Also we consider the adjoint {\mathcal{A}^{b,c}} for {b>0} of two parameter families of Cesáro averaging operators and prove the boundedness on Besov mixed-norm spaces {B_{\alpha+(c-1)}^{p,q}} for {c>1}.

Basic Properties and Examples of Dirichlet Forms

2011 ◽  
Author(s):  
Zhen-Qing Chen ◽  
Masatoshi Fukushima
Keyword(s):  
Markov Process ◽  
Potential Theory ◽  
Dirichlet Form ◽  
Dirichlet Space ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Basic Properties ◽  
Time Changes ◽  
Symmetric Operators ◽  
Analytic Potential

This chapter introduces the concepts of the transience, recurrence, and irreducibility of the semigroup for general Markovian symmetric operators and presents their characterizations by means of the associated Dirichlet form as well as the associated extended Dirichlet space. These notions are invariant under the time changes of the associated Markov process. The chapter then presents some basic examples of Dirichlet forms, with special attention paid to their basic properties as well as explicit expressions of the corresponding extended Dirichlet spaces. Hereafter the chapter discusses the analytic potential theory for regular Dirichlet forms, and presents some conditions for the demonstrated Dirichlet form (E,F) to be local.

scholarly journals Hyponormal Toeplitz Operators on the Dirichlet Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Puyu Cui ◽  
Yufeng Lu
Keyword(s):  
Toeplitz Operators ◽  
Dirichlet Space ◽  
Dirichlet Spaces ◽  
Harmonic Dirichlet Space

We completely characterize the hyponormality of bounded Toeplitz operators with Sobolev symbols on the Dirichlet space and the harmonic Dirichlet space.

scholarly journals Condensor Principle and the Unit Contraction

1967 ◽  
Vol 30 ◽  
pp. 9-28 ◽  
Author(s):  
Masayuki Itô
Keyword(s):  
Positive Measure ◽  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Dirichlet Space ◽  
Functional Space ◽  
Dirichlet Spaces ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Regular Functional ◽  
Locally Compact Hausdorff Space

Deny introduced in [4] the notion of functional spaces by generalizing Dirichlet spaces. In this paper, we shall give the following necessary and sufficient conditions for a functional space to be a real Dirichlet space.Let be a regular functional space with respect to a locally compact Hausdorff space X and a positive measure ξ in X. The following four conditions are equivalent.

Random Dirichlet Functions: Multipliers and Smoothness

1993 ◽  
Vol 45 (2) ◽  
pp. 255-268 ◽  
Author(s):  
W. George Cochran ◽  
Joel H. Shapiro ◽  
David C. Ullrich
Keyword(s):  
Power Series ◽  
Holomorphic Function ◽  
Dirichlet Series ◽  
Dirichlet Space ◽  
General Setting ◽  
Sufficient Condition ◽  
Dirichlet Spaces ◽  
Random Series ◽  
Weighted Dirichlet Spaces ◽  
Almost All

AbstractWe show that if is a holomorphic function in the Dirichlet space of the unit disk, then almost all of its randomizations are multipliers of that space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip1/2 of functions obeying a Lipschitz condition with exponent 1/2. However, unlike the lacunary situation, no corresponding “almost sure” Lipschitz result is possible for random series: we exhibit a Dirichlet function with norandomization in Lip1/2. We complement this result with a “best possible” sufficient condition for randomizations to belong almost surely to Lip1/2. Versions of our results hold for weighted Dirichlet spaces, and much of our work is carried out in this more general setting.

Cesàro averaging operators

1994 ◽  
Vol 124 (1) ◽  
pp. 121-126 ◽  
Author(s):  
Krzysztof Stempak
Keyword(s):  
Sequence Spaces ◽  
Averaging Operators ◽  
Cesaro Averaging

We define a family of Cesàro operators , Reα≧0, and consider the question of their boundedness on Hp spaces. We also consider discrete versions of these operators acting on sequence spaces.

Cesàro averaging operators on Hardy spaces

1996 ◽  
Vol 126 (3) ◽  
pp. 617-624 ◽  
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Hardy Spaces ◽  
Averaging Operators ◽  
Averaging Operator ◽  
Cesaro Averaging ◽  
Open Question

It is shown that the Cesàro averaging operatorℜα > – 1, satisfies an inequality which immediately implies that it is bounded on certain Hardy spaces including Hp, 0 < p < ∞. This answers an open question of Stempak, who introduced these operators and obtained their boundedness on Hp, 0 < p ≦ 2, for ℜα ≧ 0. The operator which is conjugate to on H2 is also shown to be bounded on Hp for 1 < p < ∞ and ℜα = – 1. This extends a result of Stempak who obtained this boundedness for 2 ≦ p≦ ∞ and ℜα ≧:0.

scholarly journals Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications

2021 ◽  
Author(s):  
Lorenzo Dello Schiavo
Keyword(s):  
Radon Measure ◽  
Measure Space ◽  
Probability Space ◽  
Dirichlet Space ◽  
Dirichlet Forms ◽  
Dirichlet Spaces ◽  
Ergodic Decomposition ◽  
Direct Integral ◽  
Unique Representation ◽  
Carré Du Champ

AbstractWe study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.

scholarly journals Cyclicity in Dirichlet Spaces

2019 ◽  
Vol 62 (02) ◽  
pp. 247-257 ◽  
Author(s):  
Y. Elmadani ◽  
I. Labghail
Keyword(s):  
Continuous Function ◽  
Unit Circle ◽  
Borel Measure ◽  
Dirichlet Space ◽  
Sufficient Condition ◽  
Dirichlet Spaces ◽  
Closed Unit Disk ◽  
Finite Borel Measure ◽  
Weighted Dirichlet Space ◽  
Uniformly Continuous

AbstractLet $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$ -capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ ), $f$ vanishes on $E$ , and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ .

Compact differences of composition operators on weighted Dirichlet spaces

2014 ◽  
Vol 12 (7) ◽  
Author(s):  
Robert Allen ◽  
Katherine Heller ◽  
Matthew Pons
Keyword(s):  
Analytic Functions ◽  
Composition Operators ◽  
Dirichlet Space ◽  
Complex Interpolation ◽  
Dirichlet Spaces ◽  
Space Of Analytic Functions ◽  
First Derivative ◽  
Weighted Dirichlet Spaces ◽  
The Difference

AbstractHere we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S 2, the space of analytic functions whose first derivative is in H 2, and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.

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