1.Faculté des Sciences Ain-Chock, Département de Mathématiques et Informatique,Université Hassan II-Casablanca,Maarif-Casablanca,Morocco;2.Faculté des Sciences de l’ Université Mohammed V de Rabat,Rabat,Morocco
Abstract:
Let T be a bounded operator with (SVEP) on its localizable spectrum (sigma _mathrm{loc}(T)). We show that for every open subset U of (sigma _mathrm{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.
