A simple bound for the remainder of the neumann series in the case of a self-adjoint compact operator

Abstract

We consider the functional equation of the second kind \\gf − λK\\gf = f with K a compact self-adjoint linear operator on a Hilbert space: a Fredholm integral equation of the second kind, for example. We establish the simple bound ▪ where λ is any regular value of K; ø is the solution of the equation corresponding to λ; λ1 is the characteristic value of K smallest in absolute value; and N = 0, 1, 2, .... For \\s|λ\\s| < \\s|λ1\\s|, this is an estimate for the remainder of the partial sums of the Neumann series.