Orthogonal projector
orthoprojector
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
.
References
[1] | L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Wiley & Hindustan Publ. Comp. (1974) (Translated from Russian) |
[2] | 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) |
[3] | F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French) |
Comments
Cf. also Projector.
Orthogonal projector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_projector&oldid=14998