The spectral picture and joint spectral radius of the generalized spherical Aluthge transform

Abstract

For an arbitrary commuting d–tuple T of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform Δt(T) and we prove that the spectral radius of T can be calculated from the norms of the iterates of Δt(T). Let T≡(T1,…,Td) be a commuting d–tuple of bounded operators acting on an infinite dimensional separable Hilbert space, let P:=T1⁎T1+⋯+Td⁎Td, and let (T1⋮Td)=(V1⋮Vd)P be the canonical polar decomposition, with (V1,…,Vd) a (joint) partial isometry and ⋂i=1dkerTi=⋂i=1dkerVi=kerP. For 0≤t≤1, we define the generalized spherical Aluthge transform of T by Δt(T):=(PtV1P1−t,…,PtVdP1−t). We also let ‖T‖2:=‖P‖. We first determine the spectral picture of Δt(T) in terms of the spectral picture of T; in particular, we prove that, for any 0≤t≤1, Δt(T) and T have the same Taylor spectrum, the same Taylor essential spectrum, the same Fredholm index, and the same Harte spectrum. We then study the joint spectral radius rT(T), and prove that rT(T)=limn⁡‖Δt(n)(T)‖2(0