Bounded Local Resolvents for Operators with Single-Valued Extension Property

Let T be a bounded operator with (SVEP) on its localizable spectrum sigma(loc) (T). We show that for every open subset U of a sigma(loc) (T), there exists a unit vector x whose local spectrum coincides with the closure of U, and such that its local resolvent function is bounded. This result answers positively to an open question stated by several authors, and extends the both cases of operators with trivial divisible subspace and operators whose point spectrum has empty interior.