On the essential spectrum of the sum of self-adjoint operators and the closedness of the sum of operator ranges

TL;DR

This paper establishes criteria for when zero is in the essential spectrum of a sum of self-adjoint operators with compact pairwise products and characterizes the closedness of the sum of their ranges.

Contribution

It provides new necessary and sufficient conditions for the essential spectrum inclusion and the closedness of the sum of operator ranges in the context of self-adjoint operators.

Findings

Zero in the essential spectrum characterized by a new criterion.

Necessary and sufficient conditions for the sum of ranges to be closed.

Application to sums of self-adjoint operators with compact pairwise products.

Abstract

We get a criterion for 0 to be in the essential spectrum of a sum of self-adjoint operators whose pairwise products are compact. Using this result, we obtain necessary and sufficient conditions for the sum of ranges of such operators to be closed.