Toeplitz form, indefinite

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.