Orthogonal projector

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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)