A functional calculus for unbounded generalized scalar operators on banach spaces

Abstract

We define weak and ultraweak functional calculus and construct using the Fourier transform technique an ultraweak functional calculus for an unbounded k-tuple of commuting generalized scalar operators Tj acting on a Banach space X. This functional calculus comprises the functions from the subset of Sobolev space, where α > 0. It also contains some unbounded functions. We also give examples and related results. © 2011 by Pacific Journal of Mathematics.