THE ESSENTIAL NORMS OF COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

2018 ◽  
Vol 97 (2) ◽  
pp. 297-307
Author(s):  
YUFEI LI ◽  
YUFENG LU ◽  
TAO YU
Keyword(s):  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Dirichlet Space ◽  
Essential Norm ◽  
Angular Derivative ◽  
Dirichlet Spaces ◽  
Closed Unit Disc ◽  
Weighted Dirichlet Spaces

Let $\unicode[STIX]{x1D711}$ be an analytic self-map of the unit disc. If $\unicode[STIX]{x1D711}$ is analytic in a neighbourhood of the closed unit disc, we give a precise formula for the essential norm of the composition operator $C_{\unicode[STIX]{x1D711}}$ on the weighted Dirichlet spaces ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D6FC}>0$. We also show that, for a univalent analytic self-map $\unicode[STIX]{x1D711}$ of $\mathbb{D}$, if $\unicode[STIX]{x1D711}$ has an angular derivative at some point of $\unicode[STIX]{x2202}\mathbb{D}$, then the essential norm of $C_{\unicode[STIX]{x1D711}}$ on the Dirichlet space is equal to one.

scholarly journals Composition operators

1978 ◽  
Vol 18 (3) ◽  
pp. 439-446 ◽  
Author(s):  
R.K. Singh ◽  
B.S. Komal
Keyword(s):  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Unit Disc ◽  
Necessary And Sufficient ◽  
Made In

A study of centered composition operators on l2 is made in this paper. Also the spectrum of surjective composition operators is computed. A necessary and sufficient condition is obtained for the closed unit disc to be the spectrum of a surjective composition operator.

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.

scholarly journals Self-Commutators of Composition Operators with Monomial Symbols on the Dirichlet Space

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
A. Abdollahi ◽  
S. Mehrangiz
Keyword(s):  
Positive Integer ◽  
Essential Spectrum ◽  
Composition Operator ◽  
Point Spectrum ◽  
Composition Operators ◽  
Dirichlet Space ◽  
Essential Norm

Let , for some positive integer and the composition operator on the Dirichlet space induced by . In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators and self-commutators of , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.

Weighted Composition Operators on Bergman and Dirichlet Spaces

1997 ◽  
Vol 4 (4) ◽  
pp. 373-383
Author(s):  
G. Mirzakarimi ◽  
K. Seddighi
Keyword(s):  
Composition Operator ◽  
Hilbert Spaces ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Sufficient Conditions ◽  
Weighted Composition Operators ◽  
Dirichlet Spaces ◽  
Weighted Dirichlet Spaces ◽  
Weighted Composition ◽  
Functional Hilbert Spaces

Abstract Let 𝐻(Ω) denote a functional Hilbert space of analytic functions on a domain Ω. Let 𝑤 : Ω → 𝐂 and ϕ : Ω → Ω be such that 𝑤 𝑓 ○ ϕ is in 𝐻(Ω) for every 𝑓 in 𝐻(Ω). The operator 𝑤𝐶 ϕ Given by 𝑓 → 𝑤 𝑓 ○ ϕ is called a weighted composition operator on 𝐻(Ω). In this paper we characterize such operators and those for which (𝑤𝐶 ϕ )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.

scholarly journals Composition operators on weighted Bergman spaces of a half-plane

2011 ◽  
Vol 54 (2) ◽  
pp. 373-379 ◽  
Author(s):  
Sam J. Elliott ◽  
Andrew Wynn
Keyword(s):  
Spectral Radius ◽  
Half Plane ◽  
Composition Operator ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Essential Norm ◽  
Weighted Bergman Space ◽  
Weighted Bergman Spaces ◽  
Angular Derivative ◽  
The Right

AbstractWe use induction and interpolation techniques to prove that a composition operator induced by a map ϕ is bounded on the weighted Bergman space $\mathcal{A}^2_\alpha(\mathbb{H})$ of the right half-plane if and only if ϕ fixes the point at ∞ non-tangentially and if it has a finite angular derivative λ there. We further prove that in this case the norm, the essential norm and the spectral radius of the operator are all equal and are given by λ(2+α)/2.

scholarly journals COMPOSITION OPERATORS BELONGING TO SCHATTEN CLASS 𝒮p

2010 ◽  
Vol 81 (3) ◽  
pp. 465-472
Author(s):  
CHENG YUAN ◽  
ZE-HUA ZHOU
Keyword(s):  
Bergman Space ◽  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Schatten Class

AbstractWe investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

Composition operators on weighted spaces of holomorphic functions on the upper half plane

2018 ◽  
Vol 122 (1) ◽  
pp. 141
Author(s):  
Wolfgang Lusky
Keyword(s):  
Half Plane ◽  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Bounded Operator ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Weight Functions ◽  
Spaces Of Holomorphic Functions ◽  
Upper Half Plane

We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

scholarly journals Normal weighted composition operators on weighted Dirichlet spaces

2015 ◽  
Vol 423 (1) ◽  
pp. 758-769 ◽  
Author(s):  
Lu Li ◽  
Yukihide Nakada ◽  
Douglas Nestor ◽  
Wendy Shang ◽  
Rachel Weir
Keyword(s):  
Composition Operators ◽  
Weighted Composition Operators ◽  
Dirichlet Spaces ◽  
Weighted Dirichlet Spaces ◽  
Weighted Composition
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