Representations of Lie Algebras by non-Skewselfadjoint Operators in Hilbert Space

TL;DR

This paper develops a framework for representing finite-dimensional Lie algebras with non-selfadjoint operators in Hilbert spaces, introducing $g$-operator vessels and their frequency domain theory, with applications to the affine group.

Contribution

It introduces $g$-operator vessels for non-selfadjoint Lie algebra representations and develops their frequency domain theory, including a joint characteristic function, extending classical operator theory.

Findings

Established a new structure called $g$-operator vessel.

Developed the frequency domain theory for these vessels.

Applied the theory to the affine group of the line.

Abstract

We study non-selfadjoint representations of a finite dimensional real Lie algebra $\fg$. To this end we embed a non-selfadjoint representation of $\fg$ into a more complicated structure, that we call a $\fg$-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group $\fG$. We develop the frequency domain theory of the system in terms of representations of $\fG$, and introduce the joint characteristic function of a $\fg$-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the $ax+b$ group, the group of affine transformations of the line.