The Schur test of compact operators

Abstract

Infinite matrix theory is an important branch of function analysis. Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space, but not every infinite matrix corresponds to an operator. The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators, which is considered a respectable mathematical accomplishment. In this paper, we prove the compact version of the Schur test. Moreover, we provide the Schur test for the Schatten class S₂. It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers. We finally provide the Schur test for compact operators from l_p into l_q.