On dissipative and non-unitary solutions to operator commutation relations

TL;DR

This paper investigates generalized semi-Weyl commutation relations involving operators and affine group actions, establishing fixed point theorems and exploring solutions that include dissipative and non-unitary cases, revealing diverse representations.

Contribution

It introduces fixed point theorems for flows induced by affine group actions on operators and characterizes solutions to generalized Weyl relations, including dissipative and non-unitary representations.

Findings

Fixed point theorems for flows on operator unit balls.

Existence of non-unitarily equivalent dissipative solutions.

Strengthened results for deficiency indices (1,1).

Abstract

We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A) \quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and $G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the affine group $\cG$, the group of affine transformations of the real axis preserving the orientation. If $A$ is a symmetric operator, the group $G$ induces an action/flow on the operator unit ball of contractive transformations from $\Ker (A^*-iI)$ to $\Ker (A^*+iI)$. We establish several fixed point theorems for this flow. In the case of one-parameter continuous subgroups of linear transformations, self-adjoint (maximal dissipative) operators associated with the fixed points of the flow give rise to solutions of the (restricted) generalized Weyl commutation relations. We show that in the dissipative setting, the restricted Weyl relations admit a variety of non-unitarily equivalent representations. In the case of deficiency indices $(1,1)$, our general results can be strengthened to the level of an alternative.