Models of {\bf Z}-Orbits of Unitary in Indefinite Inner Product Spaces Operators

TL;DR

This paper explores how to extend and modify indefinite inner product spaces to realize unitary operators as standard Hilbert space unitaries, providing a framework for analyzing Z-orbits of such operators.

Contribution

It introduces a method to construct Hilbert space topologies from indefinite inner products, enabling unitary operators to be extended and realized as classical unitaries.

Findings

Positive solutions for a wide class of operator-inner product pairs.

Construction of chains of transformations to achieve unitary extension.

Framework for representing indefinite inner product space operators as Hilbert space unitaries.

Abstract

Given a lineal H_0 and x_0\in H_0 and a linear injective operator U_0: H_0 \to H_0 such that all U_0^N, N \in {\bf Z} exist and all U_0^N x_0, N \in {\bf Z} are linearly independent, anyone can define on span{{U_0}^N x_0 | N \in {\bf Z}} a (pre)hilbert scalar product such that U_0 becomes a unitary operator. The problem under consideration is: Suppose there is specified an indefinite inner product {,}_0 on H_0 and U_0 is a {,}_0-unitary operator. Can one introduce a (pre)hilbert topology on L_{x_0} so that after completion and possible extension the resulting {,}_ext is continuous, the resulting U_ext is {,}_ext-unitary and there exists a pair L_{+}, L_{-} mutually {,}_ext-orthogonal, maximal strictly positive and respectively negative subspaces, so that they are U_ext-invariant? More generally, can one construct a sequence (chain) of transformations of the type "restrict, change topology, make completion, extend, restrict,..." with the same result? (And, as a result, after some transformations, which are natural in the field of indefinite inner product spaces, U_ext will become usual Hilbert space unitary operator). For a relatively wide class of pairs "operator, inner product" positive solutions proposed.