Singular value estimates of oblique projections

Abstract

Let W and M be two finite dimensional subspaces of a Hilbert space H such that H = W ⊕ M⊥, and let PW {norm of matrix} M⊥ denote the oblique projection with range W and nullspace M⊥. In this article we get the following formula for the singular values of PW {norm of matrix} M⊥2 (sk (PW {norm of matrix} M⊥) - 1) = under(min, (F, H) ∈ X (W, M))2,where the minimum is taken over the set of all operator pairs (F, H) on H such that R (F) = W, R (H) = M and FH* = PW {norm of matrix} M⊥, and k ∈ {1, ..., dim W}. We also characterize all the pairs where the minimum is attained.