A Unified Approach to Function Models, and the Transcription Problem

Abstract

The aim of this paper is to develop a general approach to function models of Hilbert space contractions. This approach has been drafted (mainly by the second author) in [1], [2] and in a latent form already in [3], [4], [5], The main idea of the method is to stop the standard construction of the function model half-way from a unitary dilation to final formulae of the model. In other words we do not fix a concrete spectral representation of the unitary dilation of a given contraction but work directly with an (abstract) dilation equipped with a special “function imbedding operator”. We hope you find such model more flexible to be adapting to various problems of spectral theory because as a “free parameter” for such an adaption it contains your choice of a spectral representation of the minimal unitary dilation. To demonstrate a few possibilities we consider as partial cases of our “coordinate-free” model the well-known models due to Sz.-Nagy — Poias (both in the original and Pavlov’s forms) and to de Branges — Rovnyak. Some other possibilities are considered.