Li–Yorke chaos in linear dynamics

2014 ◽  
Vol 35 (6) ◽  
pp. 1723-1745 ◽  
Author(s):  
N. C. BERNARDES ◽  
A. BONILLA ◽  
V. MÜLLER ◽  
A. PERIS
Keyword(s):  
Composition Operators ◽  
Linear Manifold ◽  
Holomorphic Functions ◽  
Linear Operators ◽  
Fréchet Spaces ◽  
Linear Dynamics ◽  
Spaces Of Holomorphic Functions ◽  
Dense Set

We obtain new characterizations of Li–Yorke chaos for linear operators on Banach and Fréchet spaces. We also offer conditions under which an operator admits a dense set or linear manifold of irregular vectors. Some of our general results are applied to composition operators and adjoint multipliers on spaces of holomorphic functions.

scholarly journals BOUNDARY-CHAOTIC BEHAVIOUR OF CONTINUOUS FUNCTIONS UNDER THE ACTION OF OPERATORS

2003 ◽  
Vol 46 (2) ◽  
pp. 451-464
Author(s):  
Luis Bernal-González ◽  
María del Carmen Calderón-Moreno
Keyword(s):  
Composition Operators ◽  
Primary 47B38 ◽  
Holomorphic Functions ◽  
Continuous Functions ◽  
Linear Operators ◽  
Multiplication Operators ◽  
Spaces Of Holomorphic Functions ◽  
Spaces Of Continuous Functions ◽  
Left And Right ◽  
Secondary 30D40

AbstractIn this paper we introduce two classes of operators on spaces of continuous functions with values in $F$-spaces under the action of which many functions behave chaotically near the boundary. Several examples—including onto linear operators, left and right composition operators, multiplication operators, and operators with pointwise dense range or with some stability property—are given. This new theory extends one recently developed on spaces of holomorphic functions.AMS 2000 Mathematics subject classification: Primary 47B38. Secondary 30D40; 46E10; 54D45

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 Mean ergodic composition operators on Banach spaces of holomorphic functions

2016 ◽  
Vol 270 (12) ◽  
pp. 4369-4385 ◽  
Author(s):  
María J. Beltrán-Meneu ◽  
M. Carmen Gómez-Collado ◽  
Enrique Jordá ◽  
David Jornet
Keyword(s):  
Banach Spaces ◽  
Composition Operators ◽  
Holomorphic Functions ◽  
Spaces Of Holomorphic Functions
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