Mathematical Programming and Linear Operators in Fréchet Spaces

1966 ◽  
Vol 3 (3) ◽  
pp. 367-371
Author(s):  
Wayne T. Ford
Keyword(s):  
Mathematical Programming ◽  
Linear Operators ◽  
Fréchet Spaces

Automatic Continuity for Linear Functions Intertwining Continuous Linear Operators on Frechet Spaces

1978 ◽  
Vol 30 (03) ◽  
pp. 518-530 ◽  
Author(s):  
Marc P. Thomas
Keyword(s):  
Banach Spaces ◽  
Functional Calculus ◽  
Linear Operators ◽  
Linear Functions ◽  
General Situation ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Continuity Theory ◽  
Continuous Linear Operators

Many results concerning the automatic continuity of linear functions intertwining continuous linear operators on Banach spaces have been obtained, chiefly by B. E. Johnson and A. M. Sinclair [1; 2; 3; 5]. The purpose of this paper is essentially to extend this automatic continuity theory to the situation of Fréchet spaces. Our motive is partly to be able to handle the more general situation, since for example, questions about Fréchet spaces and LF spaces arise in connection with the functional calculus.

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 Dynamics of multivalued linear operators

2017 ◽  
Vol 15 (1) ◽  
pp. 948-958 ◽  
Author(s):  
Chung-Chuan Chen ◽  
J. Alberto Conejero ◽  
Marko Kostić ◽  
Marina Murillo-Arcila
Keyword(s):  
Linear Operators ◽  
Fréchet Spaces ◽  
Topological Transitivity ◽  
Linear Dynamics ◽  
Devaney Chaos ◽  
Topologically Mixing ◽  
Mixing Property

Abstract We introduce several notions of linear dynamics for multivalued linear operators (MLO’s) between separable Fréchet spaces, such as hypercyclicity, topological transitivity, topologically mixing property, and Devaney chaos. We also consider the case of disjointness, in which any of these properties are simultaneously satisfied by several operators. We revisit some sufficient well-known computable criteria for determining those properties. The analysis of the dynamics of extensions of linear operators to MLO’s is also considered.

Linear Chaos on Fréchet Spaces

2003 ◽  
Vol 13 (07) ◽  
pp. 1649-1655 ◽  
Author(s):  
J. Bonet ◽  
F. Martínez-Giménez ◽  
A. Peris
Keyword(s):  
Linear Operators ◽  
Locally Convex Spaces ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Locally Convex ◽  
Continuous Linear Operators ◽  
Convex Spaces

This is a survey on recent results about hypercyclicity and chaos of continuous linear operators between complete metrizable locally convex spaces. The emphasis is put on certain contributions from the authors, and related theorems.

Investigating Distributional Chaos for Operators on Fréchet Spaces

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Zongbin Yin ◽  
Lianmei Li ◽  
Yongchang Wei
Keyword(s):  
Metric Spaces ◽  
Linear Operators ◽  
Fréchet Spaces ◽  
Continuous Linear ◽  
Complete Metric Spaces ◽  
Distributional Chaos ◽  
Shift Operators ◽  
Upper Density ◽  
Continuous Linear Operators

In this paper, various notions of chaos for continuous linear operators on Fréchet spaces are investigated. It is shown that an operator is Li–Yorke chaotic if and only if it is mean Li–Yorke chaotic in a sequence whose upper density equals one; that an operator is mean Li–Yorke chaotic if and only if it admits a mean Li–Yorke pair, if and only if it is distributionally chaotic of type 2, if and only if it has an absolutely mean irregular vector. As a consequence, mean Li–Yorke chaos is not conjugacy invariant for continuous self-maps acting on complete metric spaces. Moreover, the existence of invariant scrambled sets (with respect to certain Furstenberg families) of a class of weighted shift operators is proved.

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