Maximal Norms of Orthogonal Projections and Closed-Range Operators

Abstract

Using the Dixmier angle between two closed subspaces of a complex Hilbert space H,we establish the necessary and sufficient conditions for the operator norm of the sum oftwo orthogonal projections, PW1 and PW2 , onto closed subspaces W1 and W2, to attainits maximum, namely ∥PW1 + PW2∥ = 2. These conditions are expressed in terms of thegeometric relationship and symmetry between the ranges of the projections. We applythese results to orthogonal projections associated with a closed-range operator via itsMoore–Penrose inverse. Additionally, for any bounded operator T with closed range in H,we derive sufficient conditions ensuring ∥TT† + T†T∥ = 2, where T† denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges andtheir algebraic structure governs norm extremality and extends a recent finite-dimensionalresult to the general Hilbert space setting.