The Taylor joint spectrum and restriction to hyperinvariant subspaces

TL;DR

This paper constructs a specific example of a quadruple of commuting operators with a hyperinvariant subspace where the Taylor joint spectrum of the restriction is not contained in the original spectrum, challenging previous assumptions.

Contribution

It provides a counterexample showing that the Taylor joint spectrum of a restricted hyperinvariant subspace can differ from the original spectrum, revealing new complexities in spectral theory.

Findings

Counterexample of a quadruple of operators with unexpected spectral properties

Demonstration that spectral containment does not always hold for hyperinvariant subspaces

Highlights limitations of existing spectral inclusion results in multivariable operator theory

Abstract

It is well known that for a single bounded operator $A_0$ on a Hilbert $\mathfrak{H}$, if $\mathfrak{M}\subset \mathfrak{H}$ is hyperinvariant for $A_0$, then the spectrum of $A_0|_{\mathfrak{M}}$ is contained in the spectrum of $A_0$. In this note, we modify an example of Taylor to prove the following. There exist a quadruple $A=(A_1,A_2,A_3,A_4)$ of commuting bounded Hilbert space operators and a hyperinvariant subspace $\mathfrak{X}_1$ for $A$ such that the Taylor joint spectrum of $A$ restricted to $\mathfrak{X}_1$ is a not a subset of the Taylor joint spectrum of $A$.