The Essential Norm of a Bloch-to-Qp Composition Operator

2004 ◽  
Vol 47 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Mikael Lindstróm ◽  
Shamil Makhmutov ◽  
Jari Taskinen
Keyword(s):  
Composition Operator ◽  
Bloch Space ◽  
Essential Norm

AbstractThe Qp spaces coincide with the Bloch space for p > 1 and are subspaces of BMOA for 0 < p ≤ 1. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into Qp, in particular from the Bloch space into BMOA.

scholarly journals Essential norm of weighted composition operator betweenα-Bloch space andβ-Bloch space in polydiscs

2004 ◽  
Vol 2004 (71) ◽  
pp. 3941-3950
Author(s):  
Li Songxiao ◽  
Zhu Xiangling
Keyword(s):  
Holomorphic Function ◽  
Composition Operator ◽  
Bloch Space ◽  
Weighted Composition Operator ◽  
Essential Norm ◽  
Weighted Composition

Letφ(z)=(φ1(z),…,φn(z))be a holomorphic self-map of&#x1D53B;nandψ(z)a holomorphic function on&#x1D53B;n, where&#x1D53B;nis the unit polydiscs ofℂn. Let0<α,β<1, we compute the essential norm of a weighted composition operatorψCφbetweenα-Bloch spaceℬα(&#x1D53B;n)andβ-Bloch spaceℬβ(&#x1D53B;n).

scholarly journals A New Essential Norm Estimate of Composition Operators from Weighted Bloch Space into -Bloch Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
René E. Castillo ◽  
Julio C. Ramos-Fernández ◽  
Edixon M. Rojas
Keyword(s):  
Weight Function ◽  
Unit Disk ◽  
Composition Operator ◽  
Special Function ◽  
Composition Operators ◽  
Bloch Space ◽  
Essential Norm ◽  
Bloch Spaces ◽  
Norm Estimate ◽  
Operator Mapping

Let be any weight function defined on the unit disk and let be an analytic self-map of . In the present paper, we show that the essential norm of composition operator mapping from the weighted Bloch space to -Bloch space is comparable to where for ,   is a certain special function in the weighted Bloch space. As a consequence of our estimate, we extend the results about the compactness of composition operators due to Tjani (2003).

A NEW ESSENTIAL NORM ESTIMATE OF COMPOSITION OPERATORS FROM α-BLOCH SPACES INTO μ-BLOCH SPACES

2013 ◽  
Vol 24 (14) ◽  
pp. 1350104 ◽  
Author(s):  
JULIO C. RAMOS-FERNÁNDEZ
Keyword(s):  
Weight Function ◽  
Unit Disk ◽  
Composition Operator ◽  
Besov Spaces ◽  
Special Function ◽  
Composition Operators ◽  
Bloch Space ◽  
Essential Norm ◽  
Bloch Spaces ◽  
Norm Estimate

Let μ be any weight function defined on the unit disk 𝔻 and let ϕ be an analytic self-map of 𝔻. In this paper, we show that the essential norm of composition operator Cϕ mapping from the α-Bloch space, with α > 0, to μ-Bloch space [Formula: see text] is comparable to [Formula: see text] where, for a ∈ 𝔻, σa is a certain special function in α-Bloch space. As a consequence of our estimate, we extend recent results, about the compactness of composition operators, due to Tjani in [Compact composition operators on Besov spaces, Trans. Amer. Math. Soc.355(11) (2003) 4683–4698] and Malavé Ramírez and Ramos-Fernández in [On a criterion for continuity and compactness of composition operators acting on α-Bloch spaces, C. R. Math. Acad. Sci. Paris351 (2013) 23–26, http://dx.doi.org/10.1016/j.crma.2012.11.013 ].

SPECTRA OF LINEAR FRACTIONAL COMPOSITION OPERATORS ON THE GROWTH SPACE AND BLOCH SPACE OF THE UPPER HALF-PLANE

2018 ◽  
Vol 107 (02) ◽  
pp. 199-214
Author(s):  
SHI-AN HAN ◽  
ZE-HUA ZHOU
Keyword(s):  
Spectral Radius ◽  
Half Plane ◽  
Composition Operator ◽  
Fractional Composition ◽  
Composition Operators ◽  
Bloch Space ◽  
Essential Norm ◽  
Essential Spectral Radius ◽  
Growth Space ◽  
Upper Half Plane

In this article, we provide a complete description of the spectra of linear fractional composition operators acting on the growth space and Bloch space over the upper half-plane. In addition, we also prove that the norm, essential norm, spectral radius and essential spectral radius of a composition operator acting on the growth space are all equal.

scholarly journals THE ESSENTIAL NORM OF A WEIGHTED COMPOSITION OPERATOR FROM THE BLOCH SPACE TO H∞

2009 ◽  
Vol 79 (3) ◽  
pp. 487-497 ◽  
Author(s):  
TAKUYA HOSOKAWA
Keyword(s):  
Weight Function ◽  
Composition Operator ◽  
Upper Bound ◽  
Bloch Space ◽  
Weighted Composition Operator ◽  
Essential Norm ◽  
Operator Norm ◽  
Hyperbolic Distance ◽  
Weighted Composition ◽  
Asymptotic Upper Bound

AbstractWe express the operator norm of a weighted composition operator which acts from the Bloch space ℬ to H∞ as the supremum of a quantity involving the weight function, the inducing self-map, and the hyperbolic distance. We also express the essential norm of a weighted composition operator from ℬ to H∞ as the asymptotic upper bound of the same quantity. Moreover we study the estimate of the essential norm of a weighted composition operator from H∞ to itself.

scholarly journals The essential norm of a composition operator mapping intoQktype spaces

2008 ◽  
Vol 6 (3) ◽  
pp. 241-258 ◽  
Author(s):  
Marko Kotilainen ◽  
Jouni Rättyä
Keyword(s):  
Asymptotic Formula ◽  
Composition Operator ◽  
Unit Disc ◽  
Bloch Space ◽  
Integral Condition ◽  
Essential Norm ◽  
Type Space ◽  
Operator Mapping ◽  
Dirichlet Type ◽  
Dirichlet Type Space

An asymptotic formula for the essential norm of the composition operatorCφ(f):=f∘φ, induced by an analytic self-mapφof the unit disc, mapping from theα-Bloch spaceℬαor the Dirichlet type spaceDαpintoQk(p,q)is established in terms of an integral condition.

scholarly journals Norm and Essential Norm of Composition Followed by Differentiation from Logarithmic Bloch Spaces toHμ∞

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Shanli Ye
Keyword(s):  
Lower Bound ◽  
Unit Disk ◽  
Weighted Space ◽  
Bloch Space ◽  
Essential Norm ◽  
Bloch Spaces

In this note we express the norm of composition followed by differentiationDCφfrom the logarithmic Bloch and the little logarithmic Bloch spaces to the weighted spaceHμ∞on the unit disk and give an upper and a lower bound for the essential norm of this operator from the logarithmic Bloch space toHμ∞.

On Analytic Functions of Bergman BMO in the Ball

1999 ◽  
Vol 42 (1) ◽  
pp. 97-103 ◽  
Author(s):  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Analytic Functions ◽  
Bloch Space ◽  
Bounded Mean Oscillation ◽  
Open Unit ◽  
Bergman Metric ◽  
Open Unit Ball ◽  
Mean Oscillation ◽  
Volume Measure

AbstractLet B = Bn be the open unit ball of Cn with volume measure v, U = B1 and B be the Bloch space on , 1 ≤ α < 1, is defined as the set of holomorphic f : B → C for whichif 0 < α < 1 and , the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic f : B → U for which the composition operator defined by , is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.

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