Oblique projections and Schur complements
By Gustavo Corach, Alejandra Maestripieri, D. D. Matematica, FCEN-UBA, Buenos Aires, Argentina., Instituto de Ciencias, Ungs, San Miguel, FCE-UNLP and 1 more. 2000
60 citations recorded by the source corpus.
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), _A = , \xi, \eta in H. Given T in L(H), T is A-selfadjoint if AT = T^*A. If S \subseteq H is a closed subspace, we study the set of A-selfadjoint 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 A-selfadjoint 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.
Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/ffff02b2387903672c86dfc3743c307404f78935
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Citations in the corpus
- Generalized Schur complements and oblique projections (2002, methodology): “This paper can be seen as a continuation of [10].”
- Ergodic properties of operators in some semi-Hilbertian spaces (2013, background): “For more details concerning A-bounded operators, see [3,7,22,23].”