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On a Solution of the Multidimensional Truncated Matrix-Valued Moment Problem

Lookup NU author(s): Dr David KimseyORCiD, Matina Trachana

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Abstract

© 2022, The Author(s). We will consider the multidimensional truncated p× p Hermitian matrix-valued moment problem. We will prove a characterisation of truncated p× p Hermitian matrix-valued multisequence with a minimal positive semidefinite matrix-valued representing measure via the existence of a flat extension, i.e., a rank preserving extension of a multivariate Hankel matrix (built from the given truncated matrix-valued multisequence). Moreover, the support of the representing measure can be computed via the intersecting zeros of the determinants of matrix-valued polynomials which describe the flat extension. We will also use a matricial generalisation of Tchakaloff’s theorem due to the first author together with the above result to prove a characterisation of truncated matrix-valued multisequences which have a representing measure. When p= 1 , our result recovers the celebrated flat extension theorem of Curto and Fialkow. The bivariate quadratic matrix-valued problem and the bivariate cubic matrix-valued problem are explored in detail.

Publication metadata

Author(s): Kimsey DP, Trachana M

Publication type: Article

Publication status: Published

Journal: Milan Journal of Mathematics

Year: 2022

Volume: 90

Pages: 17-101

Print publication date: 01/06/2022

Online publication date: 19/03/2022

Acceptance date: 23/11/2021

Date deposited: 19/04/2022

ISSN (print): 1424-9286

ISSN (electronic): 1424-9294

Publisher: Birkhauser

URL: https://doi.org/10.1007/s00032-021-00346-7

DOI: 10.1007/s00032-021-00346-7

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