On the Riesz Idempotent for a Class of Operators

Let A be a bounded linear operator acting on infinite dimensional separable Hilbert space H. Browder’s theorem and a-Browder’s theorem are related to an important property which has a leading role on local spectral theory: the single valued extension property see the recent monograph by Laursen and Neumann [19]. The study of operators satisfying Browder’s theorem is of significant interest, and is currently being done by a number of mathematicians around the world. In order to generalize some recent results in the literature, we prove that a-Browder’s theorem holds for a large class of operators containing the classes of normal, hyponormal, p-hyponormal and M hyponormal operators. In [22], Stampfli proved that if A is hyponormal and λ ∈ σ(A) is isolated, then the Riesz idempotent E with respect to λ is self-adjoint and satisfies EH = ker(A − λ) = ker(A − λ)∗. It is intersting to study whether Stampli’s result holds for other classes of operators containing the class of hyponormal operators. In this paper we prove that Stampfli’s result holds for algebraically class H(q) operators.