Dissipative and nonunitary solutions of operator commutation relations

We study the (generalized) semi-Weyl commutation relations $U_gAU_g^*=g(A)$ on $\operatorname{Dom}(A)$, 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 $\mathcal G$, the group of affine orientation-preserving transformations of the real axis. If $A$ is a symmetric operator, then the group $G$ induces an action/flow on the operator unit ball of contracting transformations from $\operatorname{Ker}(A^*-iI)$ to $\operatorname{Ker}(A^*+iI)$. 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 yield solutions of the (restricted) generalized Weyl commutation relations. We show that in the dissipative setting, the restricted Weyl relations admit a variety of representations that are not unitarily equivalent. For deficiency indices $(1,1)$, the basic results can be strengthened and set in a separate case.