Bounded operators on topological vector spaces and their spectral radii

Abstract

In this paper, we consider three classes of bounded linear operators on a topological vector space with respect to three different topologies which are introduced by Troitsky. We obtain some properties for the spectral radii of a linear operator on a topological vector space. We find some sufficient conditions for the completeness of these classes of operators. Finally, as a special application, we deduce some sufficient conditions for invertibility of a bounded linear operator.

A few remarks on bounded homomorphisms acting on topological lattice groups and topological rings

“…Let us end this section with some motivation for the next section. It is observed in [6] that each class of bounded operators defined on a topological vector space X (with respect to the assigned topology) is complete if and only if so is X. The useful tool in this direction is the remarkable Hahn-Banach theorem that we lack in the category of all topological groups.…”

Topological algebras of statistical $τ$-bounded operators on ordered topological vector spaces

“…An operator T : X → Y is said to be bb-bounded if it maps every bounded set into a bounded set; see [10]. Motivated by this definition and by the ob-bounded operator in [5] and the nb-and bb-bounded operators in [12], we can give the following notions. Definition 2.3.…”

Topological groups of bounded homomorphisms on a topological group

“…Now, we are going to find conditions under which each class of considered bounded homomorphisms is topologically complete. In the case of bounded operators on topological vector spaces, absorbing neighborhoods and local convexity are two fruitful tools for discovering conditions (see [3]). In the topological group version, it turns out that boundedness of every singleton is a handy tool.…”

“…Bounded homomorphisms and their algebraic and topological structures on any topological algebraic structure are of interest for their own rights and also for their applications in other areas of mathematics. For examples, bounded operators on a topological vector space with suitable topologies form topological algebras; also, there is a spectral theory for these classes of bounded operators with some useful applications (see [3,6,7] for ample information). Therefore, it will be of interest to consider different types of bounded homomorphisms on a topological group and to investigate which topological and algebraic properties of the underlying topological group can be transferred to the mentioned classes of homomorphisms.…”