Difference between revisions of "Toeplitz form, indefinite"
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| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/T092/T.0902940 Toeplitz form, indefinite | ||
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| − | + | A quadratic form, defined on the space $ \Phi $ | |
| + | of infinite sequences $ x = \{ \xi _ {p} \} _ {- \infty } ^ \infty $ | ||
| + | of finite support by the expression | ||
| − | + | $$ | |
| + | ( x, x) = \sum _ {- \infty } ^ { {+ } \infty } | ||
| + | c _ {p - q } \xi _ {p} \overline \xi \; _ {q} , | ||
| + | $$ | ||
| − | An indefinite scalar product may be introduced in | + | where the sequence $ c = \{ c _ {p} \} _ {- \infty } ^ \infty $, |
| + | $ c _ {0} = \overline{ {c _ {0} }}\; $, | ||
| + | is such that, from some dimension $ N $ | ||
| + | onwards, the form $ ( x, x) $ | ||
| + | reduces to canonical form as a sum of $ \kappa $ | ||
| + | squares on each subspace | ||
| + | |||
| + | $$ | ||
| + | \Phi ^ {N} \subset \Phi ,\ \ | ||
| + | \Phi ^ {N} = \{ {\xi _ {p} } : {\xi _ {p} = 0, | p | > N } \} | ||
| + | . | ||
| + | $$ | ||
| + | |||
| + | An indefinite scalar product may be introduced in $ \Phi $ | ||
| + | by means of the Toeplitz form; after factorization over the isotropic subspace and completion, $ \Phi $ | ||
| + | becomes a [[Pontryagin space|Pontryagin space]]. | ||
Revision as of 08:25, 6 June 2020
A quadratic form, defined on the space $ \Phi $
of infinite sequences $ x = \{ \xi _ {p} \} _ {- \infty } ^ \infty $
of finite support by the expression
$$ ( x, x) = \sum _ {- \infty } ^ { {+ } \infty } c _ {p - q } \xi _ {p} \overline \xi \; _ {q} , $$
where the sequence $ c = \{ c _ {p} \} _ {- \infty } ^ \infty $, $ c _ {0} = \overline{ {c _ {0} }}\; $, is such that, from some dimension $ N $ onwards, the form $ ( x, x) $ reduces to canonical form as a sum of $ \kappa $ squares on each subspace
$$ \Phi ^ {N} \subset \Phi ,\ \ \Phi ^ {N} = \{ {\xi _ {p} } : {\xi _ {p} = 0, | p | > N } \} . $$
An indefinite scalar product may be introduced in $ \Phi $ by means of the Toeplitz form; after factorization over the isotropic subspace and completion, $ \Phi $ becomes a Pontryagin space.
How to Cite This Entry:
Toeplitz form, indefinite. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toeplitz_form,_indefinite&oldid=48981
Toeplitz form, indefinite. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Toeplitz_form,_indefinite&oldid=48981
This article was adapted from an original article by N.K. Nikol'skiiB.S. Pavlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article