A SIMPLE BOUND FOR THE REMAINDER OF THE NEUMANN SERIES IN THE CASE OFA SELF-ADJOINT COMPACT OPERATOR

We consider the functional equation of the second kind phi-lambda K ph i=f with K a compact self-adjoint linear operator on a Hilbert space: a Fredholm integral equation of the second kind, for example. We estab lish the simple bound \\phi-Sigma(n=0)(N) lambda(n)K(n)f\\/\\phi\\less than or equal to\lambda/lambda(1)\(N+1), where lambda is any regular value of K; phi is the solution of the equation corresponding to lambd a; lambda(1) is the characteristic value of K smallest in absolute val ue; and N = 0, 1, 2,.... For \lambda\ < \lambda 1\, this is an estimat e for the remainder of the partial sums of the Neumann series.