Dilations of Linear Maps on Vector Spaces

TL;DR

This paper extends the theory of dilations of linear maps on vector spaces, deriving several classical dilation results in a purely algebraic setting, highlighting differences from Hilbert space cases.

Contribution

It introduces vector space versions of key dilation theorems, expanding the algebraic framework beyond Hilbert space limitations.

Findings

Derived vector space versions of Wold decomposition and Halmos dilation.

Established N-dilation and inter-twining lifting theorems in vector spaces.

Showed that vector space Halmos dilation lacks a unique characterization.

Abstract

We continue the study dilation of linear maps on vector spaces introduced by Bhat, De, and Rakshit. This notion is a variant of vector space dilation introduced by Han, Larson, Liu, and Liu. We derive vector space versions of Wold decomposition, Halmos dilation, N-dilation, inter-twining lifting theorem and a variant of Ando dilation. It is noted further that unlike a kind of uniqueness of Halmos dilation of strict contractions on Hilbert spaces, vector space version of Halmos dilation can not be characterized.