Automatic Boundedness of Adjointable Operators on Barreled 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 $C^{\ast }$-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.

21 pages. arXiv admin note: text overlap with arXiv:1507.06840, arXiv:1701.04740

References in corpus (3)

• Operator Models for Hilbert Locally $C^{\ast }$-Modules
• Representations of $\ast$-semigroups associated to invariant kernels with values adjointable operators. I
• Representations of *-semigroups associated to invariant kernels with values continuously adjointable operators