Pseudospectra of the direct sum of linear operators in ultrametric Banach spaces

Abstract

In this paper, a characterization of essential pseudospectra of bounded linear operators on ultrametric Banach spaces over a spherically complete field was given and the notions of pseudospectra and condition pseudospectra of the direct sum of linear operators on ultrametric Banach spaces were introduced. In particular, we proved that the pseudospectra of the direct sum of bounded linear operators associated with various ε are nested sets and that the intersection of all the pseudospectra of bounded linear operators is the spectrum of the direct sum of bounded linear operators in the direct sum of ultrametric Banach spaces. In addition, many results were proved about them and examples were given.