On H-Contractions and the Extension Problem for Hermitian Block Toeplitz Matrices

Roland Freund; Thomas Huckle

doi:10.1080/03081088908817926

On H-Contractions and the Extension Problem for Hermitian Block Toeplitz Matrices

Roland Freund
, Thomas Huckle

University of Würzburg

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

2 Scopus citations

Abstract

It is well known that any positive semi-definite Hermitian block Toeplitz matrix T_n(C₀, C₁, …, C_n) has positive semi-definite Toeplitz extensions T_n+1(C₀, …, C_n, C_n+1). In this paper, we consider the more general problem of extending arbitrary Hermitian block Toeplitz matrices T_n to matrices T_n+1 with the same number of negative eigenvalues as T_n. A necessary and sufficient condition for the existence of such T_n+1 is derived, and a parametrization of all corresponding C_n+1 is presented. Our approach to the Hermitian block Toeplitz extension problem is based on a formulation as a completion problem for contractions in indefinite inner product spaces. A complete solution of this more general problem is given.

Original language	English
Pages (from-to)	27-37
Number of pages	11
Journal	Linear and Multilinear Algebra
Volume	25
Issue number	1
DOIs	https://doi.org/10.1080/03081088908817926
State	Published - 1 Jul 1989
Externally published	Yes

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abstract = "It is well known that any positive semi-definite Hermitian block Toeplitz matrix Tn(C0, C1, …, Cn) has positive semi-definite Toeplitz extensions Tn+1(C0, …, Cn, Cn+1). In this paper, we consider the more general problem of extending arbitrary Hermitian block Toeplitz matrices Tn to matrices Tn+1 with the same number of negative eigenvalues as Tn. A necessary and sufficient condition for the existence of such Tn+1 is derived, and a parametrization of all corresponding Cn+1 is presented. Our approach to the Hermitian block Toeplitz extension problem is based on a formulation as a completion problem for contractions in indefinite inner product spaces. A complete solution of this more general problem is given.",

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N2 - It is well known that any positive semi-definite Hermitian block Toeplitz matrix Tn(C0, C1, …, Cn) has positive semi-definite Toeplitz extensions Tn+1(C0, …, Cn, Cn+1). In this paper, we consider the more general problem of extending arbitrary Hermitian block Toeplitz matrices Tn to matrices Tn+1 with the same number of negative eigenvalues as Tn. A necessary and sufficient condition for the existence of such Tn+1 is derived, and a parametrization of all corresponding Cn+1 is presented. Our approach to the Hermitian block Toeplitz extension problem is based on a formulation as a completion problem for contractions in indefinite inner product spaces. A complete solution of this more general problem is given.

AB - It is well known that any positive semi-definite Hermitian block Toeplitz matrix Tn(C0, C1, …, Cn) has positive semi-definite Toeplitz extensions Tn+1(C0, …, Cn, Cn+1). In this paper, we consider the more general problem of extending arbitrary Hermitian block Toeplitz matrices Tn to matrices Tn+1 with the same number of negative eigenvalues as Tn. A necessary and sufficient condition for the existence of such Tn+1 is derived, and a parametrization of all corresponding Cn+1 is presented. Our approach to the Hermitian block Toeplitz extension problem is based on a formulation as a completion problem for contractions in indefinite inner product spaces. A complete solution of this more general problem is given.

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JO - Linear and Multilinear Algebra

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IS - 1

ER -

On H-Contractions and the Extension Problem for Hermitian Block Toeplitz Matrices

Abstract

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