Nevanlinna-Pick type interpolation in a dual algebra
This beautifully written paper contains a matrix-valued version of Nevanlina-Pick interpolation for a dual algebra of operators on Hilbert space. This result, together with the Wold decomposition and its generalizations by \textit{M. B. Abrahamse} and \textit{R. G. Douglas} [Adv. Math. 19, 106-148 (1976; Zbl 0321.47019)], allows the author to recover the classical Nevanlinna-Pick interpolation on the unit disc and on finitely connected domains, and also a version of the abstract interpolation theorem of \textit{J. Agler} [Some interpolation theorems of Nevanlinna-Pick type, J. Oper. Theory (to appear)]. The main result of the paper is the following: if \(A\) is a weak\(^*\) closed subalgebra of \(L(H)\) containing the identity and \(I\) is a weak\(^*\) closed ideal of \(A\) then for every \(\phi\in A\) there exists \(\psi\in A\) such that \(\phi- \psi\in I\) and \(|\psi |= \text{dist}(\phi, I)\). This distance can be calculated as the supremum of all norms \(|P_{A(Q)} \phi^* P_{N(Q)}|\), where \(Q\) runs over the positive trace class operators on \(H\). Here, every such \(Q\) produces a sesquilinear form by \(\text{tr}(XQY^*)\). The resulting Hilbert space is \(K(Q)\). Each \(T\in L(H)\) defines by composition at left an operator \(T_Q\) on \(K(Q)\). The closure of \(A\) in \(K(Q)\) is the Hilbert space \(A(Q)\), \(N(Q)\) is the orthogonal complement of \(I(Q)\) in \(A(Q)\), \(P_{N(Q)}\) is the corresponding orthogonal projection and \(P_{A(Q)}\) is the orthogonal projection of \(K(Q)\) onto \(A(Q)\).
matrix-valued version of Nevanlina-Pick interpolation
scientific article; zbMATH DE number 847476
scientific article