Riesz Idempotent and Browder's Theorem for Absolute-(p, r)-Paranormal Operators

An operator T is said to be paranormal if ‖T2x‖ ≥ ‖Tx‖2 holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paranormal introduced in [10], [14], respectively. Yamazaki and Yanagida [38] introduced the class of absolute-(p, r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators. An operator T ∈ B(H) is called absolute-(p, r)-paranormal operator if ‖|T|p|T*|rx‖r ≥ |‖T*|rx‖p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0. The famous result of Browder, that self adjoint operators satisfy Browder's theorem, isextended to several classes of operators. In this paper we show that for any absolute-(p, r)-paranormal operator T, T satisfies Browder's theorem and a-Browder's theorem. It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p, r)-paranormal operator T, then E is self-adjoint if and only if the null space of T − μ, N(T − μ) ⊆ N(T* − ).