Journal of Operator Theory

Volume 54, Issue 1, Summer 2005 pp. 125-136.

Similarity of perturbations of Hessenberg matrices

Authors: Leonel Robert (1)
Author institution: Department of Mathematics, University of Toronto, 100 St. George Street, Room 4072, Toronto, ON, Canada M5S 3G3

Summary: To every infinite lower Hessenberg matrix $D$ is associated a linear operator on $l_2$. In this paper we prove the similarity of the operator $D-\Delta$, where $\Delta$ belongs to a certain class of compact operators, to the operator $D-\Delta^\prime$, where $\Delta^\prime$ is of rank one. We first consider the case when $\Delta$ is lower triangular and has finite rank; then we extend this to $\Delta$ of infinite rank assuming that $D$ is bounded. We examine the cases when $D=S^t$ and $D=(S+S^t)/2$, where $S$ denotes the unilateral shift.

Keywords: Hessenberg matrix, similarity of operators, associated polynomials of $k$-th kind, perturbation determinants.

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