A FUNCTIONAL CALCULUS FOR UNBOUNDED GENERALIZED SCALAR OPERATORS ON BANACH SPACES

摘要

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 T_j acting on a Banach space X. This functional calculus comprises the functions from the subset = {f : R~k→ C | f is a measure such that ∫_(R~k) (1 + |t|~a)d|f|< ∞ } ,of Sobolev space W_~α(R), where α > 0. It also contains some unbounded functions. We also give examples and related results.