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

Michael Gil’

doi:10.2989/16073606.2015.1031849

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

Michael Gil’

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let SN_r (r ≥ 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix (Formula presented.) the inequality (Formula presented.) (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 (Formula presented.) In the case p = 1, sharper results are established. In addition, it is derived that (Formula presented.)

Original language	English
Pages (from-to)	145-152
Number of pages	8
Journal	Quaestiones Mathematicae
Volume	39
Issue number	2
DOIs	https://doi.org/10.2989/16073606.2015.1031849
State	Published - 31 Mar 2016

Keywords

Operator matrix
Schatten-von Neumann operators
eigenvalues
operator pencil

ASJC Scopus subject areas

Mathematics (miscellaneous)

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10.2989/16073606.2015.1031849

Cite this

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abstract = "Let SNr (r ≥ 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix (Formula presented.) the inequality (Formula presented.) (p = 2; 3; … ) is proved, where λk(A) (k = 1; 2; … ) are the eigenvalues of A and Nr(.) is the norm in SNr. Moreover, let P(z) = z2I + Bz + C (z ∈ ℂ) with B ∈ SN2p, C ∈ SNp. By zk(P) (k = 1; 2; … ) the characteristic values of the pencil P are denoted. It is shown that (Formula presented.) In the case p = 1, sharper results are established. In addition, it is derived that (Formula presented.)",

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AB - Let SNr (r ≥ 1) denote the Schatten-von Neumann ideal of compact operators in a separable Hilbert space. For the block matrix (Formula presented.) the inequality (Formula presented.) (p = 2; 3; … ) is proved, where λk(A) (k = 1; 2; … ) are the eigenvalues of A and Nr(.) is the norm in SNr. Moreover, let P(z) = z2I + Bz + C (z ∈ ℂ) with B ∈ SN2p, C ∈ SNp. By zk(P) (k = 1; 2; … ) the characteristic values of the pencil P are denoted. It is shown that (Formula presented.) In the case p = 1, sharper results are established. In addition, it is derived that (Formula presented.)

KW - Operator matrix

KW - Schatten-von Neumann operators

KW - eigenvalues

KW - operator pencil

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DO - 10.2989/16073606.2015.1031849

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ER -

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

Abstract

Keywords

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