Singular value estimates of oblique projections

JORGE ANTEZANA; GUSTAVO CORACH

LINEAR ALGEBRA AND ITS APPLICATIONS

ELSEVIER SCIENCE INC

Año: 2009 vol. 430 p. 386 - 395

0024-3795

Let $mathcal{W}$ and $mathcal{M}$ be two finite dimensional subspaces of a Hilbert space $mathcal{H}$ such that $mathcal{H}=mathcal{W}oplusmathcal{M}^ot$, and let $P_{_{mathcal{W}||mathcal{M}^ot}}$ denote the oblique projection with range $mathcal{W}$ and nullspace $mathcal{M}^ot$. In this article we get the following formula for the singular values of $P_{_{mathcal{W}||mathcal{M}^ot}}$$$ 2(s_k(sub{P}{ewe||eme^ot})-1)=min_{(F,H)inconjunto} s_k(F-H)^2,,$$where   the minimum is taken over the set of all  operator pairs $(F,H)$ on $hil$ such that $R(F)=ewe$, $R(H)=eme$  and $FH^*=sub{P}{ewe||eme^ot}$, and $kin{1,ldots,dim ewe}$. We also characterize all the pairs where the minimum is attained.