An Example of J-unitary Operator. Solving a Problem Stated by M.G. Krein

TL;DR

This paper constructs a specific J-unitary operator with prescribed spectral properties and invariant subspaces, addressing a problem posed by M.G. Krein in operator theory.

Contribution

It provides an explicit example of a J-unitary operator satisfying particular spectral and invariant subspace conditions, solving a problem posed by Krein.

Findings

Existence of a J-unitary operator with prescribed spectral radius

Construction of invariant subspaces with specific orthogonality properties

Operator exhibits maximal positive and negative invariant subspaces

Abstract

Theorem 1. Given a number c >= 1, there exists a J-unitary operator \hat{V}, such that: (a) r(\hat{V})= r(\hat{V}^{-1})= c ; (b) S(c^{-1}\hat{V})=S(c^{-1}\hat{V}^{-1}) =S(c^{-1}\hat{V}^{*-1}) = S(c^{-1}\hat{V}^*)={0} (c) there exist maximal strictly positive and strictly negative \hat{V}^{\pm 1}-{invariant subspaces} L_{+}, L_{-}, such that they are mutually J-orthogonal and L_{+} + L_{-} is dense in the space. (d_1) if L_1 is non-zero \hat{V}-invariant subspace, then r(\hat{V}|L_1)=r(\hat{V}) (d_2) if L_2 is non-zero \hat{V}^{-1}-invariant subspace, then r(\hat{V}^{-1}|L_2)=r(\hat{V}^{-1}) .