Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators

TL;DR

This paper demonstrates that the Taylor spectrum of a commuting n-tuple of Hilbert space operators remains unchanged under the spherical Aluthge transform, extending spectral invariance results to this transform.

Contribution

It proves spectral invariance of the Taylor spectrum under the spherical Aluthge transform for commuting operator n-tuples, using novel techniques near the origin.

Findings

Taylor spectrum is preserved under the spherical Aluthge transform

Invertibility of the original and transformed tuples are equivalent at the origin

Results extend to other spectral systems based on the Koszul complex

Abstract

Let $\mathbf{T} \equiv (T_1,\cdots,T_n)$ be a commuting $n$-tuple of operators on a Hilbert space $\mathcal{H}$, and let $T_i \equiv V_i P \; (1 \le i \le n)$ be its canonical joint polar decomposition (i.e., $P:=\sqrt{T_1^*T_1+\cdots+T_n^*T_n}$, $(V_1,\cdots,V_n)$ a joint partial isometry, and $\bigcap_{i=1}^n \ker T_i = \bigcap_{i=1}^n \ker V_i = \ker P)$. \ The spherical Aluthge transform of $\mathbf{T}$ is the (necessarily commuting) $n$-tuple $\hat{\mathbf{T}}:=(\sqrt{P}V_1\sqrt{P},\cdots,\sqrt{P}V_n\sqrt{P})$. \ We prove that $\sigma_T(\hat{\mathbf{T}})=\sigma_T(\mathbf{T})$, where $\sigma_T$ denotes Taylor spectrum. \ We do this in two stages: away from the origin we use tools and techniques from criss-cross commutativity; at the origin we show that the left invertibility of $\mathbf{T}$ or $\hat{\mathbf{T}}$ implies the invertibility of $P$. \ As a consequence, we can readily extend our main result to other spectral systems that rely on the Koszul complex for their definitions.