Inequalities for eigenvalues of compact operators in a Hilbert space

Let A be a compact operator in a separable Hilbert space and lambda(k)(A) (k = 1, 2,...) be the eigenvalues of A with their multiplicities enumerated in the non-increasing order of their absolute values. We prove the inequality (Sigma(m)(k=1) vertical bar lambda(k)(A)vertical bar(2) )(2) <= 2 Sigma(1 <= k< j <= m) s(k)(2) (A)s(j)(2) (A) + Sigma(m)(k=1) s(k)(2)(A(2)) (m = 2, 3,...), where s(k)(A) and s(k)(A(2)) are the singular values of A and of A(2), respectively, enumerated with their multiplicities in the non-increasing order. This result refines the classical inequality Sigma(m)(k=1) vertical bar lambda(k)(A)vertical bar(2) <= Sigma(m)(k=1) s(k)(2) (A) (m = 1, 2, 3,...).