The generic local spectrum of any operator is the full spectrum

Let T = (T-1,..., T-n) be a commuting n-tuple of bounded linear operators a cting on some complex Banach space X. We show that if T has the single-valu ed extension property, then the local spectrum sigma(T)(x) coincides with t he spectrum sigma(T), for all vectors x is an element of X, except on a set of the first Baire category.