Inequalities for eigenvalues of compact operators in a Hilbert space

Abstract

Let [Formula: see text] be a compact operator in a separable Hilbert space and [Formula: see text] be the eigenvalues of [Formula: see text] with their multiplicities enumerated in the non-increasing order of their absolute values. We prove the inequality [Formula: see text] where [Formula: see text] and [Formula: see text] are the singular values of [Formula: see text] and of [Formula: see text], respectively, enumerated with their multiplicities in the non-increasing order. This result refines the classical inequality [Formula: see text]