Property (.) and its perturbations

A Hilbert space operator T is said to have property (. 1) if . a (T)\. aw (T) . .00(T), where . a (T) and . aw (T) denote the approximate point spectrum and the Weyl essential approximate point spectrum of T respectively, and . 00(T) = {. . iso .(T), 0 < dimN(T . .I) < .}. If . a (T)\. aw (T) = .00(T), we say T satisfies property (.). In this note, we investigate the stability of the property (. 1) and the property (.) under compact perturbations, and we characterize those operators for which the property (. 1) and the property (.) are stable under compact perturbations.