Hessenberg matrix for sums of Hermitian positive definite matrices and weighted shifts

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Abstract

In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation m-sum and we obtain an expression for it that involves the Cholesky factorization of the corresponding Hermitian positive definite matrices associated with the Hessenberg components.

This work extends a method to obtain the Hessenberg matrix of the sum of measures from the Hessenberg matrices of the individual measures, introduced recently by the authors for subnormal matrices, to matrices which are not necessarily subnormal.

Moreover, we give some examples and we obtain the explicit formula for the m-sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the m-sum of two not subnormal Hessenberg matrices.

Highlights

► We introduce an algebraic operation, m-sum, between bounded Hessenberg matrices. ► The m-sum is obtained from sums of Hermitian positive definite matrices associated. ► The m-sum expression involves the Cholesky factors of the associated HPD matrices. ► We analyze some properties of the m-sum as subnormality or hyponormality. ► We obtain the explicit formula for the m-sum of a weighted shift.

Keywords

Hessenberg matrix
Orthogonal polynomials
Moment matrix
Moment problem
Weighted shift

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