Norm of the sum of two orthogonal projections

Abstract

In this note, we give a new proof of the following well-known norm formula which holds for any two orthogonal projections on a Hilbert

unless This equality was proved by Duncan and Taylor (Proc R Soc Edinb Sect A 75(2):119–129, 1975). We derive this formula from the relationship between the spectra of the sum and product of any two idempotents, as well as various norm inequalities for positive operators defined on Applications of our results are given