Does every contractive analytic function in a polydisk have a dissipative n-dimensional scattering realization?

TL;DR

This paper addresses a question about whether all contractive analytic functions in a polydisk admit a specific type of scattering realization, providing a negative answer to a previously posed problem.

Contribution

The authors demonstrate that not every contractive analytic function in a polydisk has a dissipative n-dimensional scattering realization, resolving an open problem.

Findings

Counterexample to the existence of dissipative scattering realizations

Clarification of limitations in multivariable operator theory

Advancement in understanding analytic functions in polydisks

Abstract

No. The title question was posed by D. Kalyuzhnyi-Verbovetskyi [1, Problem 1.3]. Let \mathcal{L(H,K)} denote the set of all bounded linear operators between a pair of Hilbert spaces \mathcal{H,K}, and let \mathbb{D}^{n} and \mathbb{T}^{n} denote the open unit polydisk, and the unit n-torus, respectively.