In this paper we study the property $(gaz)$ for a bounded linear operator $T\in L(X)$ on a Banach space $X$, introduced by Zariouh in [\emph{Property $(gz)$ for bounded linear operators}, Mat. Vesnik, {\bf 65(1)}(2013), 94-103], through the methods of local spectral theory. This property is a stronger variant of generalized $a$-Browder's theorem. In particular, we shall give several characterizations of property $(gaz)$, by using the localized SVEP.