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

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., $P:=\sqrt{T_1^{\ast }{T}_1+\cdots +T_n^{\ast }{T}_n}$, $(V_1,\cdots ,V_n)$ 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 $\widehat{\mathbf{T}}:=(\sqrt{P}{V}_1\sqrt{P},\cdots ,\sqrt{P}{V}_n\sqrt{P})$. \ We prove that ${\sigma }_T(\widehat{\mathbf{T}})={\sigma }_T(\mathbf{T})$, 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 $\widehat{\mathbf{T}}$ 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.