Structure of the singularities of operator functions with a positive imaginary part

Abstract

In the theory of linear operators in a Hilbert space an important role is played by the so-called operator-value~ R-functions. By the latter we mean [i] functions T(1), analytic in the upper half-plane iC+, whose values are bounded operators, acting in a Hilbert space H and having positive imaginary part (Im T(%) ~ 0, Im% > 0). We elucidate briefly the manner in which this object arises in the theory of perturbations of self-adjoint (and also nonself-adjoint) operators. Namely, assume that in a Hilbert space H there are given two self-adjoint operators A, A + V, where V ~ 0 is a bounded operator. Then from Hilbert's identity we obtain the following representation for the "bordered" resolvent of the "perturbed" operator A + V: