Maximal Norms of Orthogonal Projections and Closed-Range Operators

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 of two orthogonal projections, PW1 and PW2, onto closed subspaces W1 and W2, to attain its maximum, namely ∥PW1+PW2∥=2. These conditions are expressed in terms of the geometric relationship and symmetry between the ranges of the projections. We apply these results to orthogonal projections associated with a closed-range operator via its Moore–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 and their algebraic structure governs norm extremality and extends a recent finite-dimensional result to the general Hilbert space setting.