Hilbert space idempotents and involutions

Norms of idempotents, involutions, and the Hermitian and skewHermitian parts of involutions are shown to be elementary trigonometric functions of an angle between two subspaces of Hilbert space. When the spaces involved are nontrivial, the norm of a linear idempotent is the cosecant of the angle between its range and kernel; the norm of a linear involution is the cotangent of half the angle between the involution’s eigenspaces. A bounded linear idempotent M on a Hilbert space H induces a splitting of H into the direct sum of two subspaces: the range R of M and the kernel K. (Throughout this paper the term “subspace” always means a closed linear manifold. That R is a subspace follows from the fact that it is the kernel of the bounded idempotent I − M.) The eigenspaces of a linear involution V belonging to the eigenvalues 1 and −1 are the range and kernel of the associated idempotent, M = (I+V )/2. In this paper we study relations between these operators, the orthogonal projections PR and PK of H onto R and K, and the angle between the subspaces R and K. Our principal results are contained in the following theorems. Theorem 1. Let R and K be subspaces of a Hilbert space H, and let PR and PK denote the orthogonal projections of H onto these subspaces. The following are equivalent : (i) H is the direct sum of R and K. (ii) There exists a bounded linear idempotent M with range R and kernel K. (iii) The operator PR − PK is invertible. (iv) ‖PR + PK − I‖ < 1. The angle between two nontrivial subspaces R and K whose intersection is {0} is the number θ in [0, π/2] whose cosine equals sup{|(r, k)| : r ∈ R, k ∈ K, ‖r‖ = ‖k‖ = 1}. In view of (r, k) = (PRr, PKk) = (r, PRPKk), we have cos(θ) = ‖PRPK‖ = ‖PKPR‖. See [2] for a discussion of the angle between subspaces whose intersection is nontrivial. Throughout this paper the term idempotent is used to mean a linear transformation M satisfying M = M , and involution is used to mean a linear transformation V satisfying V 2 = I. Neither is necessarily bounded, and unbounded examples exist on every infinite dimensional Hilbert space. A self-adjoint idempotent is an orthogonal projection. Received by the editors October 25, 1996 and, in revised form, July 2, 1998. 1991 Mathematics Subject Classification. Primary 46C05; Secondary 47A05, 47A30. c ©2000 American Mathematical Society