$mathcal{H}$ _-n-perturbations of Self-adjoint Operators and Krein's Resolvent Formula

Abstract.

Supersingular $mathcal{H}$ _-n rank one perturbations of an arbitrary positive self-adjoint operator A acting in the Hilbert space $mathcal{H}$ are investigated. The operator corresponding to the formal expression ¶¶$A_{\alpha}=A+\alpha(\varphi,\cdot)\,\varphi,\alpha\in\mathbf{R},\varphi\in \mathcal{H}$ _-n(A),¶¶is determined as a regular operator with pure real spectrum acting in a certain extended Hilbert space $\mathbf{H}\supset\mathcal{H}$. The resolvent of the operator so defined is given by a certain generalization of Krein's resolvent formula. It is proven that the spectral properties of the operator are described by generalized Nevanlinna functions. The results of [24] are extended to the case of arbitrary integer n $\geq$ 4.