A BOUND FOR THE REMAINDER OF THE HILBERT-SCHMIDT SERIES AND OTHER RESULTS ON REPRESENTATION OF SOLUTIONS TO THE FUNCTIONAL-EQUATION OF THE 2ND KIND WITH A SELF-ADJOINT COMPACT OPERATOR AS AN INFINITE

For the functional equation of the second kind (see (1)) phi - lambda K phi = f, with K a compact self-adjoint linear operator on a Hilbert space (a Fredholm integral equation of the second kind, for example), a bound for the remainder of the Hilbert-Schmidt series is found. It i s shown that the series solution to (1) introduced in the author's pre vious paper [1] is (much) more rapidly convergent than the Hilbert-Sch midt series and generally speaking, is a preferable way of expressing the solution to (1) for regular lambda as an infinite series. Other se ries solutions to (1) are given. The corresponding expressions for the inverse (I-lambda K)(-1) and the resolvent B-lambda, and also for the resolvent of the Fredholm integral equation of the second kind with s ymmetric kernel, are given too.