A note about the norm of the sum and the anticommutator of two orthogonal projections

Abstract

In this note, we prove that for any two orthogonal projections PT,PS on a Hilbert the well-known norm formulas ‖PT+PS‖=1+‖PTPS‖, unless PT=PS=0 and ‖PTPS+PSPT‖=‖PTPS‖2+‖PTPS‖, can be derived from each other. Such result is obtained from the relation between the spectra of the sum and product of any two idempotents in a Banach algebra. Applications of our results are given.