A mapping of a Hilbert space onto a subspace of it such that is orthogonal to : . An orthogonal projector is a bounded self-adjoint operator, acting on a Hilbert space , such that and . On the other hand, if a bounded self-adjoint operator acting on a Hilbert space such that is given, then is a subspace, and is an orthogonal projector onto . Two orthogonal projectors are called orthogonal if ; this is equivalent to the condition that .
Properties of an orthogonal projector. 1) In order that the sum of two orthogonal projectors is itself an orthogonal projector, it is necessary and sufficient that , in this case ; 2) in order that the composite is an orthogonal projector, it is necessary and sufficient that , in this case .
An orthogonal projector is called a part of an orthogonal projector if is a subspace of . Under this condition is an orthogonal projector on — the orthogonal complement to in . In particular, is an orthogonal projector on .
|||L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Wiley & Hindustan Publ. Comp. (1974) (Translated from Russian)|
|||N.I. [N.I. Akhiezer] Achieser, I.M. [I.M. Glaz'man] Glasman, "Theorie der linearen Operatoren im Hilbert Raum" , Akademie Verlag (1958) (Translated from Russian)|
|||F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)|
Cf. also Projector.
Orthogonal projector. V.I. Sobolev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Orthogonal_projector&oldid=14998