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

Let T = (T-1, ... T-n) be a commuting n-tuple of operators on a Hilbert space H, and let T-i equivalent to ViP (1 <= i <= n) be its canonical joint polar decomposition (i.e. P := root T-1*T-1+ ... + T-n*T-n, (V-1,..., V-n) a joint partial isometry, and boolean AND(n)(i-1) ker T-i = boolean AND(n)(i-1) ker V-i = ker P). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple (T) over cap := (root PV1 root P, ..., root PVn, root P). We prove that sigma(T)((T) over cap) = sigma(T)(T), where sigma(T) denotes the 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 T or (T) over cap 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. (C) 2019 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.