Krein's formula for the generalized resolvents

Generalized self–adjoint extensions and generalized resolvents

Consider an arbitrary symmetric operator A0 acting in a certain Hilbert space ℋ. Let the deficiency indices of the operator A0 be equal. Then the self–adjoint extensions of A0 acting in the same Hilbert space can be described using von Neumann theory. The case of extensions of the operators having unit deficiency indices was studied in the previous chapter. Our goal in this chapter is to investigate self–adjoint extensions in extended Hilbert spaces. Such extensions are needed to obtain operators with a richer analytical structure of the spectrum. Let us introduce the following definitions.

Definition 2.1.1Let A0 be a symmetric operator acting in the Hilbert space ℋ An operatorAis calleda generalized self–adjoint extensionof the operator A0 if there exists a Hilbert spaceH ⊃ ℋ such that the operatorAis a self–adjoint operator in this Hilbert space and the operator A0 is its symmetric restriction. All extensions of the operator A0 inside the Hilbert space ℋ will be calledstandard extensions.

Obviously the set of generalized extensions includes the set of standard extensions. Only standard self–adjoint extensions have been considered in the previous chapter. Let us denote by Pℋ the projector in the space H onto the space ℋ.