Notes on the History of the Uses of Analyticity in Operator Theory

Abstract

provided that d(X) #0. Heref and g are members of the function class C[[a, b]. The function d(X) is an entire analytic function of the complex parameter X and D(s, t; X), which is a continuous function of (s, t), is also an entire analytic function of X. The display of analyticity in the solution of the Fredholm equation of the second kind is an early signal of the important role which analyticity was destined to play in spectral theory. Anticipations of spectral theory itself can be traced at least as far back as 1836, which was the date of publication of the work of Sturm [38], [39] and Liouville [28], [29] on second-order ordinary differential equations and the associated eigenvalue and expansion theorems. In 1904 Hilbert [20] showed how to transform a differential boundary value problem into a problem of solving an integral equation. Moreover, the Green's function of the differential problem turned out, under certain conditions, to be a kernel of Hilbert-Schmidt type, from which followed a great many things of interest. The utility of analytic functions and the calculus of residues in connection with expansions in series of special functions has been known since the work of Cauchy. In 1894 Poincare [34] dealt with the problem of the vibrating membrane and related the eigenvalues and eigenfunction expansions to the meromorphic character of the Green's function's dependence on the parameter. In 1908 Birkhoff made explicit the use of the meromorphic behavior of the Green's