Interpolation and Commutant Lifting with Weights

TL;DR

This paper extends classical theorems in operator theory to weighted Hardy algebras derived from $W^*$-correspondences, broadening their applicability and connecting to recent developments in the field.

Contribution

It generalizes the commutant lifting and Nevanlinna-Pick interpolation theorems to weighted Hardy algebras, building on and extending Muhly and Solel's earlier work.

Findings

Generalized commutant lifting theorem with weights

Extended Nevanlinna-Pick interpolation to weighted contexts

Connected theorems to Parrott's Lemma and Arias' work

Abstract

Our two principle goals are generalizations of the commutant lifting theorem and the Nevanlinna-Pick interpolation theorem to the context of Hardy algebras built from $W^*$-correspondences endowed with a sequence of weights. These theorems generalize theorems of Muhly and Solel from 1998 and 2004, respectively, which were proved in settings without weights. Of special interest is the fact that commutant lifting in our setting is a consequence of Parrott's Lemma; it is inspired by work of Arias.