Around multivariate Schmidt-Spitzer theorem

TL;DR

This paper introduces a polynomial basis linked to complex matrices, explores spectral properties of associated submatrices, and proposes a multivariate generalization of the Schmidt-Spitzer theorem for banded Toeplitz matrices.

Contribution

It develops a new polynomial basis related to matrices, studies spectral asymptotics for banded Toeplitz matrices, and proposes a multivariate extension of the Schmidt-Spitzer theorem.

Findings

Defined a polynomial basis B_A associated with matrix A.

Analyzed the spectral locus of submatrices of A.

Proposed and partially proved a multivariate Schmidt-Spitzer conjecture.

Abstract

Given an arbitrary complex-valued infinite matrix A and a positive integer n we introduce a naturally associated polynomial basis B_A of C[x0...xn]. We discuss some properties of the locus of common zeros of all polynomials in B_A having a given degree m; the latter locus can be interpreted as the spectrum of the m*(m+n)-submatrix of A formed by its m first rows and m+n first columns. We initiate the study of the asymptotics of these spectra when m goes to infinity in the case when A is a banded Toeplitz matrix. In particular, we present and partially prove a conjectural multivariate analog of the well-known Schmidt-Spitzer theorem which describes the spectral asymptotics for the sequence of principal minors of an arbitrary banded Toeplitz matrix. Finally, we discuss relations between polynomial bases B_A and multivariate orthogonal polynomials.