Maximally dissipative and self-adjoint extensions of $K$-invariant operators
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arXiv
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| Format: | Preprint |
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2025
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| author | Fischbacher, Christoph Rosenzweig, Bart Stanfill, Jonathan |
| author_facet | Fischbacher, Christoph Rosenzweig, Bart Stanfill, Jonathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05178 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximally dissipative and self-adjoint extensions of $K$-invariant operators Fischbacher, Christoph Rosenzweig, Bart Stanfill, Jonathan Spectral Theory Functional Analysis Primary: 34B24, 47A05, 47B25, Secondary: 47A10, 47B44, 47B65 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. |
| title | Maximally dissipative and self-adjoint extensions of $K$-invariant operators |
| topic | Spectral Theory Functional Analysis Primary: 34B24, 47A05, 47B25, Secondary: 47A10, 47B44, 47B65 |
| url | https://arxiv.org/abs/2509.05178 |