2 - Generalized rank one perturbations

Summary

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 P_ℋ the projector in the space H onto the space ℋ.