Riesz idempotent of (n, k)-quasi-*-paranormal operators

A bounded linear operator T on a complex Hilbert space H is called (n, k)-quasi-*-paranormal if ‖T1+n(Tkx)‖1/(1+n)‖Tkx‖n/(1+n)≥‖T*(Tkx)‖ for all x∈H,‖T1+n(Tkx)‖1/(1+n)‖Tkx‖n/(1+n)≥‖T*(Tkx)‖ for all x∈H,where n, k are nonnegative integers. This class of operators has many interesting properties and contains the classes of n-*-paranormal operators and quasi-*-paranormal operators. The aim of this note is to show that every Riesz idempotent Eλ with respect to a non-zero isolated spectral point λ of an (n, k)-quasi-*-paranormal operator T is self-adjoint and satisfies ran Eλ = ker(T – λ) = ker(T – λ)*.