Oblique projections and Schur complements
Abstract
Let H be a Hilbert space, L(H) the algebra of all bounded linear operators on H and <, >_A : H \times H \to C the bounded sesquilinear form induced by a selfadjoint A in L(H), < \xi, \eta >_A = < A \xi, \eta >, \xi, \eta in H. Given T in L(H), T is Aselfadjoint if AT = T^*A. If S \subseteq H is a closed subspace, we study the set of Aselfadjoint projections onto S, P(A, S) = {Q in L(H): Q^2 = Q, R(Q) = S, AQ = Q*A} for different choices of A, mainly under the hypothesis that A\geq 0. There is a closed relationship between the Aselfadjoint projections onto S and the shorted operator (also called Schur complement) of A to S^\perp. Using this relation we find several conditions which are equivalent to the fact that P(A, S) \neq \emptyset, in particular in the case of A\geq 0 with A injective or with R(A) closed. If A is itself a projection, we relate the set P(A, S) with the existence of a projection with fixed kernel and range and we determine its norm.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2000
 arXiv:
 arXiv:math/0006120
 Bibcode:
 2000math......6120C
 Keywords:

 Mathematics  Operator Algebras;
 47A64;
 47A07;
 46C99
 EPrint:
 20 pages, Latex