On the Schwarz derivative, the Bloch space and the Dirichlet space

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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POWERS OF COMPOSITION OPERATORS: ASYMPTOTIC BEHAVIOUR ON BERGMAN, DIRICHLET AND BLOCH SPACES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788719000235 ◽

2019 ◽

Vol 108 (3) ◽

pp. 289-320 ◽

Cited By ~ 2

Author(s):

W. ARENDT ◽

I. CHALENDAR ◽

M. KUMAR ◽

S. SRIVASTAVA

Keyword(s):

Asymptotic Behaviour ◽

Banach Spaces ◽

Composition Operators ◽

Bloch Space ◽

Bergman Spaces ◽

Dirichlet Space ◽

Complete Characterization ◽

Weighted Bergman Spaces ◽

Bloch Type Space ◽

Spaces Of Holomorphic Functions

We study the asymptotic behaviour of the powers of a composition operator on various Banach spaces of holomorphic functions on the disc, namely, standard weighted Bergman spaces (finite and infinite order), Bloch space, little Bloch space, Bloch-type space and Dirichlet space. Moreover, we give a complete characterization of those composition operators that are similar to an isometry on these various Banach spaces. We conclude by studying the asymptotic behaviour of semigroups of composition operators on these various Banach spaces.

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Qp Spaces on Riemann Surfaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1998-024-4 ◽

1998 ◽

Vol 50 (3) ◽

pp. 449-464 ◽

Cited By ~ 9

Author(s):

Rauno Aulaskari ◽

Yuzan He ◽

Juha Ristioja ◽

Ruhan Zhao

Keyword(s):

Banach Space ◽

Riemann Surface ◽

Unit Disk ◽

Complex Plane ◽

Riemann Surfaces ◽

Bloch Space ◽

Dirichlet Space ◽

Function Spaces ◽

Hyperbolic Riemann Surfaces ◽

Qp Spaces

AbstractWe study the function spaces Qp(R) defined on a Riemann surface R, which were earlier introduced in the unit disk of the complex plane. The nesting property Qp(R) ⊆Qq(R) for 0 < p < q < ∞ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space AD(R) ⊆ Qp(R) for any p, 0 < p < ∞, thus sharpening T. Metzger's well-known result AD(R) ⊆ BMOA(R). Also the first author's result AD(R) ⊆ VMOA(R) for a regular Riemann surface R is sharpened by showing that, in fact, AD(R) ⊆ Qp,0(R) for all p, 0 < p < ∞. The relationships between Qp(R) and various generalizations of the Bloch space on R are considered. Finally we show that Qp(R) is a Banach space for 0 < p < ∞.

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Composition operators on Qp spaces

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700002585 ◽

2001 ◽

Vol 70 (2) ◽

pp. 161-188 ◽

Cited By ~ 5

Author(s):

Zengjian Lou

Keyword(s):

Analytic Function ◽

Unit Disk ◽

Function Space ◽

Composition Operators ◽

Bloch Space ◽

Dirichlet Space ◽

Holomorphic Functions ◽

Bloch Spaces ◽

Holomorphic Map ◽

Analytic Function Space

AbstractA holomorphic map ψ of the unit disk ito itself induces an operator Cψ on holomorphic functions by composition. We characterize bounded and compact composition operators Cψ on Qp spaces, which coincide with the BMOA for p = 1 and Bloch spaces for p > 1. We also give boundedness and compactness characterizations of Cψ from analytic function space X to Qp spaces, X = Dirichlet space D, Bloch space B or B0 = {f: f′ ∈ H∞}.

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