Width of the Gakhov class over the Dirichlet space is equal to 2

AbstractFor a bounded region G ⊂ ℂ and a compact set K ⊂ G, with area measure zero, we will characterize the invariant subspaces ℳ (under ƒ → zƒ) of the Bergman space (G \ K), 1 ≤ p < ∞, which contain (G) and with dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K. When G \ K is connected, we will see that dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K and thus in this case we will have a complete description of the invariant subspaces lying between (G) and (G \ K). When p = ∞, we will remark on the structure of the weak-star closed z-invariant subspaces between H∞(G) and H∞(G \ K). When G \ K is not connected, we will show that in general the invariant subspaces between (G) and (G \ K) are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under ƒ → ζƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space , there are invariant subspaces ℱ such that the dimension of ζℱ in ℱ is infinite.

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Kernels and spectrum of Toeplitz operators on the Dirichlet space

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2018.11.056 ◽

2019 ◽

Vol 472 (1) ◽

pp. 894-919 ◽

Cited By ~ 1

Author(s):

Yongning Li ◽

Ziliang Zhang ◽

Dechao Zheng

Keyword(s):

Toeplitz Operators ◽

Dirichlet Space

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Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space

Acta Mathematica Sinica English Series ◽

10.1007/s10114-008-7266-1 ◽

2008 ◽

Vol 24 (11) ◽

pp. 1843-1852 ◽

Cited By ~ 4

Author(s):

Tao Yu ◽

Shi Yue Wu

Keyword(s):

Toeplitz Operators ◽

Dirichlet Space ◽

Orthogonal Complement ◽

Algebraic Properties

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The n-th derivative characterisation of Möbius invariant Dirichlet space

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700031993 ◽

1998 ◽

Vol 58 (1) ◽

pp. 43-56 ◽

Cited By ~ 14

Author(s):

Rauno Aulaskari ◽

Maria Nowak ◽

Ruhan Zhao

Keyword(s):

Carleson Measure ◽

Bloch Space ◽

Dirichlet Space ◽

Function Spaces ◽

Multiplier Space ◽

Little Bloch Space ◽

Special Parameter

In this paper we give the n-th derivative criterion for functions belonging to recently defined function spaces Qp and Qp, 0. For a special parameter value p = 1 this criterion is applied to BMOA and VMOA, and for p > 1 it is applied to the Bloch space and the little Bloch space . Further, a Carleson measure characterisation is given to Qp, and in the last section the multiplier space from Hq into Qp is considered.

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Avkhadiev–Lehto Type Constants in the Study of the Gakhov Class

Russian Mathematics ◽

10.3103/s1066369x2103004x ◽

2021 ◽

Vol 65 (3) ◽

pp. 43-50

Author(s):

A. V. Kazantsev ◽

M. I. Kinder

Keyword(s):

Gakhov Class

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Avkhadiev-Lehto type constants in the study of the Gakhov class

Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika ◽

10.26907/0021-3446-2021-3-47-55 ◽

2021 ◽

pp. 47-55

Author(s):

Andrei Vitalievich Kazantsev ◽

◽

Michail Ivanovich Kinder ◽

Keyword(s):

Gakhov Class

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Basic Properties and Examples of Dirichlet Forms

Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35) ◽

10.23943/princeton/9780691136059.003.0002 ◽

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.

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Girsanov Transformations for Non-Symmetric Diffusions

Canadian Journal of Mathematics ◽

10.4153/cjm-2009-028-7 ◽

2009 ◽

Vol 61 (3) ◽

pp. 534-547 ◽

Cited By ~ 2

Author(s):

Chuan-Zhong Chen ◽

Wei Sun

Keyword(s):

Diffusion Process ◽

Dirichlet Form ◽

Dirichlet Space ◽

Explicit Representation ◽

Dual Process ◽

Sufficient Condition ◽

Dual Processes ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient ◽

Girsanov Transformations

Abstract.Let X be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form (ℰ, 𝓓 (ℰ)) on L2(E ;m). For u ∈ 𝓓(ℰ)e, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process Y of X . First, let be the dual process of X and Ŷ the Girsanov transformed process of . We give a necessary and sufficient condition for (Y , Ŷ to be in duality with respect to the measure e2um. We also construct a counterexample, which shows that this condition may not be satisfied and hence (Y , Ŷ ) may not be dual processes. Then we present a sufficient condition under which Y is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

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