Norm of the sum of two orthogonal projections

Abstract

In this note, we give a new proof of the following well-known norm formula whichholds for any two orthogonal projections PT , PS on a Hilbert H,|PT + PS| = 1 + |PT PS|,unless PT = PS = 0. This equality was proved by Duncan and Taylor (Proc R SocEdinb Sect A 75(2):119–129, 1975). We derive this formula from the relationshipbetween the spectra of the sum and product of any two idempotents, as well as variousnorm inequalities for positive operators defined on H. Applications of our results aregiven.