Let be a commuting n-tuple of operators on a Hilbert space , and let be its canonical joint polar decomposition (i.e. ⁎ ⁎ , a joint partial isometry, and ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple . We prove that , where 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 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.
Benhida, Chafiq; Curto, Raul E.; Lee, Sang Hoon; and Yoon, Jasang, "Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators" (2019). Mathematical and Statistical Sciences Faculty Publications and Presentations. 81.
Comptes Rendus Mathematique