Automatic Boundedness of Adjointable Operators on Barreled VH-Spaces

TL;DR

This paper proves that adjointable operators on barreled VH-spaces are inherently bounded, extending known results from locally Hilbert $C^*$-modules and impacting dilation theory and reproducing kernel VH-spaces.

Contribution

It establishes the automatic boundedness of adjointable operators on barreled VH-spaces, generalizing prior results and simplifying conditions for VH-space linearizations.

Findings

Adjointable operators on barreled VH-spaces are automatically bounded.

A boundedness condition for VH-space linearisations is inherently satisfied in barreled VH-spaces.

The result extends the theory from locally Hilbert $C^*$-modules to VH-spaces.

Abstract

We consider the space of adjointable operators on barreled VH (Vector Hilbert) spaces and show that such operators are automatically bounded.This generalizes the well known corresponding result for locally Hilbert $C^*$-modules.We pick a consequence of this result in the dilation theory of VH-spaces and show that, when barreled VH-spaces are considered, a certain boundedness condition for the existence of VH-space linearisations, equivalently, of reproducing kernel VH-spaces, is automatically satisfied.