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.