Abstract. A localized version of the single-extension property is studied, for a bounded linear operator $T$ acting on a Banach space, at the points $\lambda\in {\msbm C}$ such that $\lambda I-T$ is quasi-Fredholm. This property is also studied at the points $\lambda\in {\msbm C}$ which are not limit points of the approximate point spectrum and the surjectivity spectrum. As a consequence, we improve a classical Putnam result about the non-isolated boundary points of the spectrum. From the characterizations of this property we shall also deduce several results on cluster points of some distinguished parts of the spectrum.

AMS Subject Classification (1991): 47A10, 47A11; 47A53, 47A55

Keyword(s): Localized SVEP, quasi-Fredholm operators, B, -Fredholm operators

Received July 7, 2006, and in final form January 15, 2007. (Registered under 5965/2009.)