Singular value estimates of oblique projections

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