Complete continuity of the inverse of a positive symmetric operator.
James P. Fink · Proceedings of the American Mathematical Society · 1970
Let A A be a symmetric positive definite linear transformation defined on a dense subset of a Hilbert space H H , and let H A {H_A} . be the Hilbert space completion of the domain of A A with respect to the inner product ( u , v ) A = ( A u , v ) {(u,v)_A} = (Au,v) . It is shown that the inverse of A A is completely continuous on H A {H_A} if and only if it is completely continuous on H H .