An observation on unbounded linear operators with an arbitrary spectrum

TL;DR

This paper presents a straightforward method to construct unbounded closed linear operators in complex Banach spaces with any specified nonempty closed spectrum, including compact sets.

Contribution

It introduces a simple construction technique for unbounded operators with arbitrary spectra in Banach spaces, expanding the understanding of spectral properties.

Findings

Constructs unbounded operators with any nonempty closed spectrum

Provides a method applicable to compact and non-compact spectra

Enhances spectral theory in Banach space operator analysis

Abstract

We furnish a simple way of constructing an unbounded closed linear operator in a complex Banach space, whose spectrum is an arbitrary nonempty closed, in particular compact, subset of the complex plane.