On inequalities for eigenvalues of 2 × 2 matrices with Schatten–von Neumann entries

Abstract

Let SN_r (r ≥ 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix

the inequality

(p = 2; 3; … ) is proved, where λ_k(A) (k = 1; 2; … ) are the eigenvalues of A and N_r(.) is the norm in SN_r. Moreover, let P(z) = z²I + Bz + C (z ∈ ℂ) with B ∈ SN_2p, C ∈ SN_p. By z_k(P) (k = 1; 2; … ) the characteristic values of the pencil P are denoted. It is shown that

In the case p = 1, sharper results are established. In addition, it is derived that

Mathematics Subject Classification (2010):

• 47A75
• 47B10
• 47A56

Key words:

• Operator matrix
• operator pencil
• eigenvalues
• Schatten-von Neumann operators