Sums of products of positive operators and spectra of L\" uders operators

Abstract

Each bounded operator T on an infinite dimensional Hilbert space H is a sum of three operators that are similar to positive operators; two such operators are sufficient if T is not a compact perturbation of a scalar. The spectra of L\"uders operators (elementary operators on B(H) with positive coefficients) of lengths at least three are not necessarily contained in the set of all nonnegative real numbers. On the other hand, the spectra of such operators of lengths at most two contain only nonnegative real numbers, if the coefficients on one side commute.