Norms of basic operators in vector valued model spaces and 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.