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 $\mathcal{H}(\Theta )=H_2^m\ominus \Theta H_2^m$ 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 $L_2^m$ 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.

16 pages, Revised, Section 5 is new, to appear in Integral Equations and Operator Theory