Decomposing numerical ranges along with spectral sets

TL;DR

This paper explores the application of spectral set methods to numerical ranges, expanding their usefulness in operator theory and functional analysis.

Contribution

It extends existing spectral set techniques to new contexts involving numerical ranges, building on prior work by Mlak and the author.

Findings

New applicability of spectral set methods to numerical ranges

Enhanced understanding of $K$-spectral sets in operator theory

Potential for broader use in functional analysis

Abstract

This note is to indicate the new sphere of applicability of the method developed by Mlak as well as by the author. Restoring those ideas is summoned by current developments concerning $K$-spectral sets on numerical ranges.