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

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