Dilations on locally Hilbert spaces

TL;DR

This paper extends the classical Sz.-Nagy dilation theorem to the setting of locally Hilbert spaces, introducing positive definiteness for kernels and deriving dilation results for various locally operator-valued functions.

Contribution

It introduces positive definiteness for locally Hilbert space kernels and proves a dilation theorem analogous to Sz.-Nagy's, applicable to locally contractions and semi-spectral measures.

Findings

Established a dilation theorem for locally Hilbert space operator-valued kernels.

Derived dilation results for locally contractions and locally ρ-contractions.

Extended classical dilation theory to the setting of locally Hilbert spaces.

Abstract

The principal theorem of Sz.-Nagy on dilation of a positive definite Hilbert space operator valued function has played a central role in the development of the non-self-adjoint operator theory. In this paper we introduce the positive definiteness for locally Hilbert space operator valued kernels, we prove an analogue of the Sz.-Nagy dilation theorem and, as application, we obtain dilation results for locally contractions and locally $\rho$ - contractions as well as for locally semi-spectral measures.