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, Raúl E. Curto, Sang Hoon Lee, and Jasang Yoon. 2019. “Joint Spectra of Spherical Aluthge Transforms of Commuting N-Tuples of Hilbert Space Operators.” Comptes Rendus Mathematique 357 (10): 799–802. https://doi.org/10.1016/j.crma.2019.10.003.
Comptes Rendus Mathematique