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

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.