Essential Spectra of Induced Operators on Subspaces and Quotients

TL;DR

This paper proves that the essential spectra of operators induced on invariant subspaces and quotients are contained within the polynomial hull of the original operator's essential spectrum, providing a simplified proof of this spectral property.

Contribution

It offers a straightforward proof demonstrating the containment of essential spectra of induced operators within the polynomial hull of the original spectrum.

Findings

Essential spectra of induced operators are contained in the polynomial hull of the original spectrum.

Provides a simplified proof of a known spectral inclusion property.

Clarifies the relationship between spectra of operators on subspaces, quotients, and the original operator.

Abstract

Let $X$ be a complex Banach space and let $T$ be a bounded linear operator on $X$. For any closed $T$-invariant subspace $F$ of $X$, $T$ induces operators $T_{|F}:F \longrightarrow F$ and $T/F:X/F\longrightarrow X/F$. In this note, we give a simple proof of the fact that the essential spectra of $T_{|F}$ and $T/F$ are always contained in the polynomial hull of the essential spectrum of $T$.