Canonical decomposition of dissipative linear relations

TL;DR

This paper introduces two canonical decompositions for dissipative linear relations, extending classical theories and enabling the separation of selfadjoint and nonselfadjoint components through invariant subspace transformations.

Contribution

It provides new decompositions for dissipative linear relations based on classical theories, using the Z transform to separate selfadjoint parts.

Findings

Decompositions extend classical Sz. Nagy-Foiaș-Langer and von Neumann-Wold theories.

Invariant subspace transformations via Z transform realize the decompositions.

Facilitates analysis of dissipative relations by separating selfadjoint components.

Abstract

On the basis of Sz. Nagy-Foia\c{s}-Langer and von Neumann-Wold decompositions, two decompositions for dissipative linear relations are given and they are realized by transforming invariant subspaces for contractions, by means of the Z transform. These decompositions permit the separation of the selfadjoint and completely nonselfadjoint parts of a dissipative relation.