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 \(P_{\mathcal {T}}, P_{\mathcal {S}}\) on a Hilbert \({\mathcal {H}},\)

$$\begin{aligned} \Vert P_{\mathcal {T}}+P_{\mathcal {S}}\Vert = 1+\Vert P_{\mathcal {T}}P_{\mathcal {S}}\Vert , \end{aligned}$$

unless \(P_{\mathcal {T}}=P_{\mathcal {S}}=0.\) 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 \({\mathcal {H}}.\) Applications of our results are given.