Inequalities for eigenvalues of compact operators in a Hilbert space

Let A be a compact operator in a separable Hilbert space and λk(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 ∑k=1m|λ k(A)|22 ≤ 2∑ 1≤k<j≤msk2(A)s j2(A) +∑ k=1ms k2(A2)(m = 2, 3,⋯), where sk(A) and sk(A2) are the singular values of A and of A2, respectively, enumerated with their multiplicities in the non-increasing order. This result refines the classical inequality ∑k=1m|λ k(A)|2 ≤∑ k=1ms k2(A)(m = 1, 2, 3,⋯).