For a symmetric operator invariant under a bounded invertible K, its Friedrichs and Krein-von Neumann extensions are automatically K-invariant, and K-invariant Sturm-Liouville extensions are characterized through Schroeder's equation.
Fischbacher,On the Theory of Dissipative Extensions, PhD Thesis, University of Kent, 2017
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Maximally dissipative and self-adjoint extensions of $K$-invariant operators
For a symmetric operator invariant under a bounded invertible K, its Friedrichs and Krein-von Neumann extensions are automatically K-invariant, and K-invariant Sturm-Liouville extensions are characterized through Schroeder's equation.