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''orthoprojector''
 
''orthoprojector''
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703601.png" /> of a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703602.png" /> onto a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703603.png" /> of it such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703604.png" /> is orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703605.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703606.png" />. An orthogonal projector is a bounded self-adjoint operator, acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703607.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o0703609.png" />. On the other hand, if a bounded [[Self-adjoint operator|self-adjoint operator]] acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036011.png" /> is given, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036012.png" /> is a subspace, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036013.png" /> is an orthogonal projector onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036014.png" />. Two orthogonal projectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036015.png" /> are called orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036016.png" />; this is equivalent to the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036017.png" />.
+
A mapping $  P _ {L} $
 +
of a [[Hilbert space|Hilbert space]] $  H $
 +
onto a subspace $  L $
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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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036018.png" /> of two orthogonal projectors is itself an orthogonal projector, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036019.png" />, in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036020.png" />; 2) in order that the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036021.png" /> is an orthogonal projector, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036022.png" />, in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036023.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036024.png" /> is called a part of an orthogonal projector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036025.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036026.png" /> is a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036027.png" />. Under this condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036028.png" /> is an orthogonal projector on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036029.png" /> — the orthogonal complement to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036031.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036032.png" /> is an orthogonal projector on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070360/o07036033.png" />.
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elements of functional analysis" , Wiley &amp; Hindustan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  V.I. Sobolev,  "Elements of functional analysis" , Wiley &amp; Hindustan Publ. Comp.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Cf. also [[Projector|Projector]].
 
Cf. also [[Projector|Projector]].

Latest revision as of 08:04, 6 June 2020


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.

How to Cite This Entry:
Orthogonal projector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_projector&oldid=48078
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article