Spectral Radii of Bounded Operators on Topological Vector Spaces

TL;DR

This paper extends spectral theory to bounded operators on topological vector spaces, revealing multiple spectra and spectral radii, and adapting classical formulas like Gelfand's and Neumann series to this broader context.

Contribution

It introduces a novel spectral theory framework for operators on topological vector spaces, generalizing classical results from Banach spaces.

Findings

Operators have multiple spectra and spectral radii

Gelfand formula is adapted to topological vector spaces

Neumann series interpretation is extended

Abstract

In this paper we develop a version of spectral theory for bounded linear operators on topological vector spaces. We show that the Gelfand formula for spectral radius and Neumann series can still be naturally interpreted for operators on topological vector spaces. Of course, the resulting theory has many similarities to the conventional spectral theory of bounded operators on Banach spaces, though there are several important differences. The main difference is that an operator on a topological vector space has several spectra and several spectral radii, which fit a well-organized pattern.