A note on partial isometries on pseudo-Hilbert spaces

TL;DR

This paper investigates conditions under which two accessible subspaces in a Loynes space can be realized as initial and final spaces of a partial gramian isometry, based on the norm difference of associated projections.

Contribution

It establishes a criterion involving the norm of the difference of gramian projections for the existence of a partial gramian isometry linking two subspaces.

Findings

Subspaces are initial and final spaces of a partial gramian isometry if the projection difference norm is less than 1.

Provides a new condition for the existence of partial isometries in pseudo-Hilbert spaces.

Extends the theory of partial isometries to the setting of Loynes $ ext{ extbf{Z}}$-spaces.

Abstract

The aim of this paper is to show that two accessible subspaces in the Loynes $\mathcal{Z}$ - space $\Hc$ are the initial and final space of a partial gramian isometry, respectively if the norm of the difference of the associated gramian selfadjoint projections is strictly less than 1.