Operator Norm of Certain Sums of Orthogonal Projections in Hilbert Spaces

Abstract

Let N andMbe two closed linear subspaces in a complex Hilbert space H, and letPN and PM denote the orthogonal projections onto N andM, respectively. In thispaper, we study linear combinations of PN and PM. Specifically, we show that if PNand PM are non-zero and α, β ∈ R with αβ ≥ 0, then αPN + β PM =|α| + |β| + (α − β)2 + 4αβ PN PM 22.This provides an affirmative answer to a question recently posed by the secondauthor regarding the norm of linear combinations of orthogonal projections. By settingα = β = 1, we recover the well-known Duncan-Taylor formula for the sum of twoprojections. We also consider some cases where the scalars are complex.