Orthogonal projector

orthoprojector

A mapping $ P _ {L} $ of a Hilbert space $ H $ onto a subspace $ L $ of it such that $ x- P _ {L} x $ is orthogonal to $ P _ {L} x $: $ x- P _ {L} x \perp P _ {L} x $. An orthogonal projector is a bounded self-adjoint operator, acting on a Hilbert space $ H $, such that $ P _ {L} ^ {2} = P _ {L} $ and $ \| P _ {L} \| = 1 $. On the other hand, if a bounded self-adjoint operator acting on a Hilbert space $ H $ such that $ P ^ {2} = P $ is given, then $ L _ {P} = \{ {Px } : {x \in H } \} $ is a subspace, and $ P $ is an orthogonal projector onto $ L _ {P} $. Two orthogonal projectors $ P _ { L _ 1 } , P _ { L _ 2 } $ are called orthogonal if $ P _ { L _ 1 } P _ { L _ 2 } = P _ { L _ 2 } P _ { L _ 1 } = 0 $; this is equivalent to the condition that $ L _ {1} \perp L _ {2} $.

Properties of an orthogonal projector. 1) In order that the sum $ P _ { L _ 1 } + P _ { L _ 2 } $ of two orthogonal projectors is itself an orthogonal projector, it is necessary and sufficient that $ P _ { L _ 1 } P _ { L _ 2 } = 0 $, in this case $ P _ { L _ 1 } + P _ { L _ 2 } = P _ {L _ {1} \oplus L _ {2} } $; 2) in order that the composite $ P _ { L _ 1 } P _ { L _ 2 } $ is an orthogonal projector, it is necessary and sufficient that $ P _ { L _ 1 } P _ { L _ 2 } = P _ { L _ 2 } P _ { L _ 1 } $, in this case $ P _ { L _ 1 } P _ { L _ 2 } = P _ {L _ {1} \cap L _ {2} } $.

An orthogonal projector $ P _ {L ^ \prime } $ is called a part of an orthogonal projector $ P _ {L} $ if $ L ^ \prime $ is a subspace of $ L $. Under this condition $ P _ {L} - P _ {L ^ \prime } $ is an orthogonal projector on $ L \ominus L ^ \prime $— the orthogonal complement to $ L ^ \prime $ in $ L $. In particular, $ I - P _ {L} $ is an orthogonal projector on $ H \ominus L $.

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.