Abstract
Let S be a symmetric operator with finite and equal defect numbers in the Hilbert space \({{\mathfrak {H}}}\). We study the compressions \(P_{{\mathfrak {H}}}\widetilde{A}\big |_{{\mathfrak {H}}}\) of the self-adjoint extensions \(\widetilde{A}\) of S in some Hilbert space \(\widetilde{{\mathfrak {H}}}\supset {{\mathfrak {H}}}\). These compressions are symmetric extensions of S in \({{\mathfrak {H}}}\). We characterize properties of these compressions through the corresponding parameter of \(\widetilde{A}\) in M.G. Krein’s resolvent formula. If \(\dim \, (\widetilde{{\mathfrak {H}}}\ominus {{\mathfrak {H}}})\) is finite, according to Stenger’s lemma the compression of \(\widetilde{A}\) is self-adjoint. In this case we express the corresponding parameter for the compression of \(\widetilde{A}\) in Krein’s formula through the parameter of the self-adjoint extension \(\widetilde{A}\).
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Dedicated to Professor Rien Kaashoek on the occasion of his 80th birthday.
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Dijksma, A., Langer, H. Compressions of Self-Adjoint Extensions of a Symmetric Operator and M.G. Krein’s Resolvent Formula. Integr. Equ. Oper. Theory 90, 41 (2018). https://doi.org/10.1007/s00020-018-2465-3
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DOI: https://doi.org/10.1007/s00020-018-2465-3
Keywords
- Hilbert space
- Symmetric and self-adjoint operators
- Self-adjoint extension
- Compression
- Generalized resolvent
- Krein’s resolvent formula
- Q-function