scholarly journals Geometric properties of composition operators belonging to Schatten classes

2001 ◽  
Vol 26 (4) ◽  
pp. 239-248 ◽  
Author(s):  
Yongsheng Zhu
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Composition Operator ◽  
Composition Operators ◽  
Schatten Classes ◽  
Geometric Properties ◽  
Image Domain ◽  
Function Mapping

We investigate the connection between the geometry of the image domain of an analytic function mapping the unit disk into itself and the membership of the composition operator induced by this function in the Schatten classes. The purpose is to provide solutions to Lotto's conjectures and show a new compact composition operator which is not in any of the Schatten classes.

scholarly journals Isometric multiplication operators and weighted composition operators of BMOA

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ligang Geng
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Composition Operator ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Multiplication Operator ◽  
Multiplication Operators ◽  
Weighted Composition Operators ◽  
Weighted Composition

AbstractLet u be an analytic function in the unit disk $\mathbb{D}$ D and φ be an analytic self-map of $\mathbb{D}$ D . We give characterizations of the symbols u and φ for which the multiplication operator $M_{u}$ M u and the weighted composition operator $M_{u,\varphi }$ M u , φ are isometries of BMOA.

scholarly journals Characterisation of the isometric composition operators on the Bloch space

2005 ◽  
Vol 72 (2) ◽  
pp. 283-290 ◽  
Author(s):  
Flavia Colonna
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Composition Operator ◽  
Blaschke Product ◽  
Analytic Functions ◽  
Composition Operators ◽  
Bloch Space ◽  
Open Unit ◽  
Open Unit Disk ◽  
Identity Function

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator Cϕ: f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = gB where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{zn} containing 0 and a subsequence satisfying the conditions , and

Commutants of composition operators induced by a parabolic linear fractional automorphisms of the unit disk

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Yuriy Linchuk
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Analytic Functions ◽  
Composition Operators ◽  
Linear Fractional Transformation ◽  
Space Of Analytic Functions ◽  
Continuous Operators

AbstractThe commutant of composition operator induced by a parabolic linear fractional transformation of the unit disk onto itself in the class of linear continuous operators acting on the space of analytic functions is described.

scholarly journals On composition operators inQKtype spaces

2007 ◽  
Vol 5 (2) ◽  
pp. 103-122 ◽  
Author(s):  
Marko Kotilainen
Keyword(s):  
Banach Space ◽  
Analytic Function ◽  
Composition Operator ◽  
Composition Operators ◽  
Bloch Space ◽  
Function Spaces ◽  
Analytic Mapping ◽  
Analytic Function Spaces

Letp≥1,q>-2and letK:[0,∞)→[0,∞)be nondecreasing. With a different choice ofp,q,K, the Banach spaceQK(p,q)coincides with many well-known analytic function spaces. Boundedness and compactness of the composition operatorCφfromα-Bloch spaceBαintoQK(p,q)are characterized by a condition depending only on analytic mappingφ:𝔻→𝔻. The same properties are also studied in the caseCφ:QK(p,q)→Bα.

scholarly journals Composition operators on some Möbius invariant Banach spaces

2000 ◽  
Vol 62 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Shamil Makhmutov ◽  
Maria Tjani
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Disk ◽  
Composition Operator ◽  
Composition Operators ◽  
Bloch Space ◽  
Holomorphic Functions ◽  
Mean Oscillation

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

scholarly journals COMPACT DIFFERENCES OF COMPOSITION OPERATORS

2008 ◽  
Vol 77 (1) ◽  
pp. 161-165 ◽  
Author(s):  
ELKE WOLF
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Infinite Order ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Open Unit ◽  
Open Unit Disk ◽  
The Difference

AbstractLet ϕ and ψ be analytic self-maps of the open unit disk. Each of them induces a composition operator, Cϕ and Cψ respectively, acting between weighted Bergman spaces of infinite order. We show that the difference Cϕ−Cψ is compact if and only if both operators are compact or both operators are not compact and the pseudohyperbolic distance of the functions ϕ and ψ tends to zero if ∣ϕ(z)∣→1 or ∣ψ(z)∣→1.

scholarly journals Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces

2020 ◽  
Vol 40 (4) ◽  
pp. 495-507
Author(s):  
Ching-on Lo ◽  
Anthony Wai-keung Loh
Keyword(s):  
Hardy Space ◽  
Unit Disk ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Sufficient Conditions ◽  
Weighted Composition Operators ◽  
Weighted Composition

Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\).

scholarly journals Composition operators fromℬαtoQKtype spaces

2008 ◽  
Vol 6 (1) ◽  
pp. 88-104 ◽  
Author(s):  
Jizhen Zhou
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Analytic Functions ◽  
Composition Operators ◽  
Nondecreasing Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Analytic Functions ◽  
Type Spaces ◽  
Necessary And Sufficient

Suppose thatϕis an analytic self-map of the unit diskΔ. Necessary and sufficient condition are given for the composition operatorCϕf=fοϕto be bounded and compact fromα-Bloch spaces toQKtype spaces which are defined by a nonnegative, nondecreasing functionk(r)for0≤r<∞. Moreover, the compactness of composition operatorCϕfromℬ0toQKtype spaces are studied, whereℬ0is the space of analytic functions offwithf′∈H∞and‖f‖ℬ0=|f(0)|+‖f′‖∞.

scholarly journals Composition operators from Bloch type spaces to F(p, q, s) spaces

Filomat ◽  
2007 ◽  
Vol 21 (2) ◽  
pp. 11-20 ◽  
Author(s):  
Xiangling Zhu
Keyword(s):  
Unit Disk ◽  
Composition Operator ◽  
Composition Operators ◽  
Type Space ◽  
Bloch Type Space ◽  
Type Spaces

Suppose that if is an analytic self-map of the unit disk, the compactness of the composition operator C? from the Bloch type space into the space F(p, q, s) is investigated .

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).

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