Linear Operations Which Depend Analytically on a Parameter

Abstract

It has long been customary in the theory of linear transformations-integral equations, linear equations in an infinite number of unknowns, etc.-to make use of certain notions from the theory of functions of a complex variable. In particular, the well known Neumann expansion of the resolvent is a formal power series whose coefficients, instead of being numbers, are iterates of a certain linear transformation.' These resemblances to the classical CauchyWeierstrass theory can be described accurately in terms of a theory of analytic functions whose values lie in a complex Banach space. The possibility of such a theory, derived in the usual manner from Cauchy's theorem and integral formula, was recognized by Wiener.2 Still further generalizations, in which the independent variable may also range over a complex Banach space, have been considered by a number of people.3 It therefore seems desirable to outline a general abstract theory of linear operations depending analytically on a parameter, which we shall here suppose to be a complex variable. DEFINITION 1.1 A function f(X) defined in an open set A of the complex plane, with values in a complex Banach space E, is called analytic in A if the limit