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

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