Compressions of Self-Adjoint Extensions of a Symmetric Operator and MG Krein's Resolvent Formula

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