Spectra of anticommutator for two orthogonal projections

Abstract

In this note, for any two orthogonal projection P, Q on a Hilbert space, the characterization of spectrum of anticommutator P Q + QP has been obtained. As a corollary, the norm formula P Q + QP = P Q + P Q 2 has been got an alternative proof (see Sam Waltrs, Anticommutator norm formula for projection operators, arXiv:1604.00699vl [math.FA] 3 Apr 2016). AM S classif ication : 47A05

Robert Sheckley’s Answerer for two orthogonal projections

“…We therefore conclude that the spectrum of the anticommutator P Q + QP is the set {λ ± √ λ : λ ∈ σ(P QP )} from which the origin should be removed if 0, 1 / ∈ σ(H). This covers the result of [9]. Moreover, since P Q + QP is a positive semi-definite operator, its norm coincides with the maximum of its spectrum.…”

“…The approach of [1,9,10] and [24] was geometrical, using either (2) or its equivalents. In [17] a point was made to derive the existence criterion for the intertwining unitary U via the supersymmetric approach.…”