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>SUP>⊥</SUP>, and let PW ‖ M⊥ denote the oblique projection with range W and nullspace M⊥. In this article we get the following formula for the singular values of P W‖M⊥: 2 (sk (P W ‖ M⊥) - 1) = min, (F, H) ∈ X (W, M) Sk (F - H)<SUP>2</SUP>,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* = P W ‖ M⊥, and k ∈ {1, ..., dim W}. We also characterize all the pairs where the minimum is attained.