Pointwise Functional Calculi

SupposeAis a (possibly unbounded) closed linear operator on a Banach spaceX,x∈X, and F is a Banach algebra of functions. We introduce apointwiseFfunctional calculus for A at x. This is a bounded linear map from F intoX, with the properties that one would expect from a mapf↦f(A) x, ifAhad a F functional calculus; howeverAmay 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 whichAhas a (global) F functional calculus. We characterize being pointwise generalized scalar atxand 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.