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

In this work, we introduce an algebraic operation between bounded Hessenberg matrices and we analyze some of its properties. We call this operation mm-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 mm-sum of a weighted shift. In particular, we construct an interesting example: a subnormal Hessenberg matrix obtained as the mm-sum of two not subnormal Hessenberg matrices.

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