On inequalities for eigenvalues of 2 x 2 matrices with Schatten-von Neumann entries

Let SNr (r >= 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix [GRAPHICS] the inequality (Sigma(infinity)(k=1)vertical bar lambda k(A)vertical bar(2p))(1/(2p)) <= (N-2p(2p)(A(11)) + N-2p(2p)(A(22)))(1/(2p)) + N-2p(2p)(A(21)))(1/(2p)) (p = 2; 3;...) is proved, where (k)(A) (k = 1; 2;...) are the eigenvalues of A and N-r(.) is the norm in SNr. Moreover, let P(z) = z(2)I + Bz + C (z C) with B SN2p, C SNp. By z(k)(P) (k = 1; 2;...) the characteristic values of the pencil P are denoted. It is shown that Sigma(infinity)(k=1)vertical bar z(k)(P)vertical bar(2p) <= (N-2p(B) + 2(1/(2p)) Np-1/2 (C))(2p) In the case p = 1, sharper results are established. In addition, it is derived that Sigma(infinity)(k=1)vertical bar Im lambda(k)(A)vertical bar(2) <= (N-2(2)(A(I)) -1/2(N-2(A(12))) - N-2(A(21)))(2) (A is an element of SN2; A(I) = 1/2i(A -A*)).