A formula to calculate the spectral radius of a compact linear operator

Abstract

There is a formula (Gelfand's formula) to find the spectral radius of a linear operator defined on a Banach space. That formula does not apply even in normed spaces which are not complete. In this paper we show a formula to find the spectral radius of any linear and compact operator T defined on a complete topological vector space, locally convex. We also show an easy way to find a non-trivial T-invariant closed subspace in terms of Minkowski functional.

Keywords

• compact linear operator
• spectral radius
• locally convex topological linear space
• invariant subspace
• nets
• ultimately bounded nets.