Finite Subspace Matrix Representation of the Multiplication Operator’s Resolvent in Terms of Fluctuation Matrices

Abstract

This work deals with the matrix representation of the resolvent for the operator which multiplies its operand by a single independent variable, and, aims to find its expression on a finite Hilbert subspace. The resulting expression deviates from the resolvent of the matrix representation of the considered operator on that finite Hilbert subspace by infinite number of fluctuation matrix containing terms. This can be used to develop fluctuation expansions for various purposes although the main goal of this paper is just to obtain the subspace matrix representation of the resolvent.