Generalized orthogonal projections and shorted operators

G. CORACH, A. MAESTRIPIERI Y D. STOJANOFF

Margarita mathematica: en memoria de José Javier (Chicho) Guadalupe Hernández

Servicio de Publicaciones, Univ. de La Rioja,

Lugar: Logroño; Año: 2001; p. 607 - 626

Abstract. Let H be a Hilbert space, L(H)the algebra of all bounded linearoperators on H and , A : H×H→C the bounded sesquilinear form inducedby a selfadjoint A ∈ L(H), ξ, ηA = Aξ, η, ξ, η ∈ H. Given T ∈ L(H), T isA-selfadjoint if AT = T ∗A. If S ⊆H is a closed subspace, we study the set ofA-selfadjoint projections onto S,P(A,S) = {Q ∈ L(H) : Q2 = Q, R(Q) = S, AQ= Q∗A}for different choices of A, mainly under the hypothesis that A ≥ 0. In thispaper we study the close relationship between the existence and properties ofA-selfadjoint projections onto S and the shorted operator (also called Schurcomplement) A/S of A to S and the S-compression AS = A − A/S .