Compressions of Self-Adjoint Extensions of a Symmetric Operator and M.G. Krein’s Resolvent Formula

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 $$\widetilde{A}$$ .

Locations

• Integral Equations and Operator Theory - View - PDF