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

TL;DR

This paper investigates the decomposition of bounded operators into sums of operators similar to positive operators and explores the spectral properties of Lüders operators, revealing conditions under which their spectra are nonnegative.

Contribution

It introduces new results on representing operators as sums of positive-like operators and characterizes the spectra of Lüders operators based on their length and coefficient commutativity.

Findings

Operators can be expressed as sums of three similar-to-positive operators.

Spectra of Lüders operators with length ≥ 3 are not necessarily nonnegative.

Spectra of Lüders operators with length ≤ 2 are nonnegative if coefficients commute.

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.