Robert Martin, University of Cape Town
"A multi-variable de Branges-Rovnyak model for row contractions"
In the operator-model theory of de Branges and Rovnyak, any completely non-coisometric (CNC) contraction on Hilbert space is represented as the adjoint of the restriction of the backward shift to a de Branges-Rovnyak subspace of the classical (vector-valued) Hardy space of analytic functions in the open unit disk. We provide a natural extension of this model to the setting of CNC (row) contractions from several copies of a Hilbert space into itself. A canonical extension of Hardy space to several complex dimensions is the Drury-Arveson space, and the appropriate analogue of the adjoint of the restriction of the backward shift to a de Branges-Rovnyak space is a Gleason solution, a row contraction which acts as a several-variable difference quotient. Our several-variable model completely characterizes the class of all CNC row contractions which can be represented as (extremal contractive) Gleason solutions for a multi-variable de Branges-Rovnyak subspace of (vector-valued) Drury-Arveson space.
- R.T.W. Martin and A. Ramanantoanina. A Gleason solution model for row contractions. To appear in Oper. Theory Adv. Appl., volume dedicated to J.A. Ball, 2018.
- A. Ramanantoanina. Gleason solutions and canonical models for row contractions. Master's thesis, University of Cape Town, 2017.