Maximally dissipative and self-adjoint extensions of K-invariant operators

We introduce the notion of K-invariant operators, S, (in a Hilbert space) with respect to a bounded and boundedly invertible operator K defined via K^*SK=S. Conditions such that self-adjoint and maximally dissipative extensions of K-invariant symmetric operators are also K-invariant are investigated. In particular, the Friedrichs and Krein--von Neumann extensions of a nonnegative K-invariant symmetric operator are shown to always be K-invariant, while the Friedrichs extension of a K-invariant sectorial operator is as well. We apply our results to the case of Sturm--Liouville operators where K is given by $(Kf)(x)=A(x)f(ϕ(x))$ under appropriate assumptions. Sufficient conditions on the coefficient functions for K-invariance to hold are shown to be related to Schröder's equation and all K-invariant self-adjoint extensions are characterized. Explicit examples are discussed including a Bessel-type Schrödinger operator satisfying a nontrivial K-invariance on the half-line.