Borel Spectrum of Operators on Banach Spaces

TL;DR

This paper studies how the spectrum of bounded operators on Banach spaces varies, showing it is a Borel measurable function with respect to the strong operator topology, and explores related topological properties.

Contribution

It establishes the Borel measurability of the spectrum function for operators on separable Banach spaces, extending understanding of spectral variation in infinite dimensions.

Findings

Spectrum function is Borel measurable in the strong operator topology.

Results extend to other topologies on operator spaces.

Provides insights into spectral variation in infinite-dimensional analysis.

Abstract

The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. In particular, it is shown that the spectrum function is Borel from the space of bounded operators on a separable Banach space; equipped with the strong operator topology, into the Polish space of compact subsets of the closed unit disc of the complex plane; equipped with the Hausdorff topology. Remarks and results are given when other topologies are used.