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

Abstract Let T ≡ ( T 1 , ⋯ , T n ) be a commuting n-tuple of operators on a Hilbert space H , and let T i ≡ V i P ( 1 ≤ i ≤ n ) be its canonical joint polar decomposition (i.e. P : = T 1 ⁎ T 1 + ⋯ + T n ⁎ T n , ( V 1 , ⋯ , V n ) a joint partial isometry, and ⋂ i = 1 n ker ⁡ T i = ⋂ i = 1 n ker ⁡ V i = ker ⁡ P ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple T ˆ : = ( P V 1 P , ⋯ , P V n P ) . We prove that σ T ( T ˆ ) = σ T ( T ) , where σ 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 ˆ 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.