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 $(Kf)(x)=A(x)f(\phi(x))$ under appropriate assumptions. Sufficient conditions on the coefficient functions for $K$-invariance to hold are shown to be related to Schr\"oder's equation and all $K$-invariant self-adjoint extensions are characterized. Explicit examples are discussed including a Bessel-type Schr\"odinger operator satisfying a nontrivial $K$-invariance on the half-line.