Contractive upper triangular matrices with prescribed diagonals and superdiagonals

TL;DR

This paper characterizes a unique class of contractive upper triangular matrices with specified diagonals and superdiagonals, focusing on finite-dimensional model spaces and extending to infinite matrices.

Contribution

It provides a unique characterization of such matrices with spectral norm constraints and prescribed diagonals and superdiagonals, including an extension to infinite matrices.

Findings

Unique matrix with spectral norm ≤ 1 and given diagonals/superdiagonals

Characterization of the matrix representation of the compressed shift

Extension to infinite matrices

Abstract

A characterization of the matrix representation of the compressed shift acting on a finite-dimensional model space endowed with the Takenaka-Malmquist-Walsh basis, among all upper triangular matrices, is proved. For a fixed dimension, this matrix is the unique matrix with spectral norm no greater than one and with a prescribed diagonal and superdiagonal. We also discuss an extension to the setting of infinite matrices.