Pointwise functional calculi

Suppose A is a (possibly unbounded) closed linear operator on a Banach space X, x is an element of X, and F is a Banach algebra of functions. We introduce a pointwise F functional calculus for A at x. This is a bounded linear map from F into X, with the properties that one would expect from a map f --> f(A) s, if A had a F functional calculus; however A may not have such a functional calculus. We show that the existence of a pointwise F functional calculus is equivalent to the existence of a continuously embedded Banach subspace on which A has a (global) F functional calculus We characterize being pointwise generalized scalar at x and give simple sufficient conditions. We also discuss the relationship between pointwise functional calculi and the many physical problems that may be modelled as an abstract Cauchy problem. (C) 1996 Academic Press, Inc.