Compressions of Self-Adjoint Extensions ofa Symmetric Operator and M.G. Krein’sResolvent Formula

Abstract

Let S be a symmetric operator with finite and equal defect numbers in the Hilbert space H. We study the compressions PHA˜∣∣H of the self-adjoint extensions A˜ of S in some Hilbert space H˜⊃H. These compressions are symmetric extensions of S in H. We characterize properties of these compressions through the corresponding parameter of A˜ in M.G. Krein’s resolvent formula. If dim(H˜⊖H) is finite, according to Stenger’s lemma the compression of A˜ is self-adjoint. In this case we express the corresponding parameter for the compression of A˜ in Krein’s formula through the parameter of the self-adjoint extension A˜.