Norms of basic operators in vector valued model spaces and de Branges spaces

TL;DR

This paper computes the norm of a basic operator in vector valued model and de Branges spaces, showing the norm is attained and can be expressed via singular values, extending previous results.

Contribution

It provides explicit formulas for the operator norm in vector valued model spaces and extends these results to de Branges spaces, including the case when the operator is contractive.

Findings

Norm of the operator is explicitly computed and attained.

Norm can be expressed in terms of singular values of the inner function at a point.

Extension of results to vector valued de Branges spaces.

Abstract

Let $\Omega_+$ be either the open unit disc or the open upper half plane or the open right half plane. In this paper, we compute the norm of the basic operator $A_\alpha=\Pi_\Theta T_{b_\alpha}|_{\mathcal{H}(\Theta)}$ in the vector valued model space $\mathcal{H}(\Theta)=H^m_2 \ominus \Theta H^m_2$ associated with an $m\times m$ matrix valued inner function $\Theta$ in $\Omega_+$ and show that the norm is attained. Here $\Pi_\Theta$ denotes the orthogonal projection from the Lebesgue space $L^m_2$ onto $\mathcal{H}(\Theta)$ and $T_{b_\alpha}$ is the operator of multiplication by the elementary Blaschke factor $b_{\alpha}$ of degree one with a zero at a point $\alpha\in \Omega_+$. We show that if $A_\alpha$ is strictly contractive, then its norm may be expressed in terms of the singular values of $\Theta(\alpha)$. We then extend this evaluation to the more general setting of vector valued de Branges spaces.