Spectra of anticommutator for two orthogonal projections

TL;DR

This paper characterizes the spectrum of the anticommutator of two orthogonal projections on a Hilbert space and provides an alternative proof for its norm formula, advancing understanding of operator spectra.

Contribution

It offers a new characterization of the spectrum of the anticommutator of two orthogonal projections and an alternative proof of its norm formula.

Findings

Spectrum of anticommutator characterized

Norm formula for anticommutator proved

Alternative proof provided for existing formula

Abstract

In this note, for any two orthogonal projection $P,Q$ on a Hilbert space, the characterization of spectrum of anticommutator $PQ+QP$ has been obtained. As a corollary, the norm formula $$\parallel PQ+QP\parallel=\parallel PQ\parallel+\parallel PQ\parallel^2$$ has been got an alternative proof (see Sam Waltrs, Anticommutator norm formula for projection operators, arXiv:1604.00699vl [math.FA] 3 Apr 2016)