Indefinite inner product space

In mathematics, in the field of functional analysis, an indefinite inner product space 
is an infinite-dimensional complex vector space formula_2 equipped with both an indefinite inner product 
and a positive semi-definite inner product 
where the metric operator formula_5 is an endomorphism of formula_2 obeying 
The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on formula_2 implies that one can form a quotient space on which there is a positive definite inner product. Given a strong enough topology on this quotient space, it has the structure of a Hilbert space, and many objects of interest in typical applications fall into this quotient space.
An indefinite inner product space is called a Krein space (or formula_5"-space") if formula_10 is positive definite and formula_2 possesses a majorant topology. Krein spaces are named in honor of the Soviet mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989).
Inner products and the metric operator.
Consider a complex vector space formula_2 equipped with an indefinite hermitian form formula_13. In the theory of Krein spaces it is common to call such an hermitian form an indefinite inner product. The following subsets are defined in terms of the square norm induced by the indefinite inner product:
A subspace formula_19 lying within formula_20 is called a "neutral subspace". Similarly, a subspace lying within formula_21 (formula_22) is called "positive" ("negative") "semi-definite", and a subspace lying within formula_23 (formula_24) is called "positive" ("negative") "definite". A subspace in any of the above categories may be called "semi-definite", and any subspace that is not semi-definite is called "indefinite".
Let our indefinite inner product space also be equipped with a decomposition into a pair of subspaces formula_25, called the "fundamental decomposition", which respects the complex structure on formula_2. Hence the corresponding linear projection operators formula_27 coincide with the identity on formula_28 and annihilate formula_29, and they commute with multiplication by the formula_30 of the complex structure. If this decomposition is such that formula_31 and formula_32, then formula_2 is called an indefinite inner product space; if formula_34, then formula_2 is called a Krein space, subject to the existence of a majorant topology on formula_2 (a locally convex topology where the inner product is jointly continuous).
The operator formula_37 is called the (real phase) "metric operator" or "fundamental symmetry", and may be used to define the "Hilbert inner product" formula_38:
On a Krein space, the Hilbert inner product is positive definite, giving formula_2 the structure of a Hilbert space (under a suitable topology). Under the weaker constraint formula_41, some elements of the neutral subspace formula_42 may still be neutral in the Hilbert inner product, but many are not. For instance, the subspaces formula_43 are part of the neutral subspace of the Hilbert inner product, because an element formula_44 obeys formula_45. But an element formula_46 (formula_47) which happens to lie in formula_42 because formula_49 will have a positive square norm under the Hilbert inner product.
We note that the definition of the indefinite inner product as a Hermitian form implies that:
Therefore the indefinite inner product of any two elements formula_51 which differ only by an element formula_52 is equal to the square norm of their average formula_53. Consequently, the inner product of any non-zero element formula_54 with any other element formula_47 must be zero, lest we should be able to construct some formula_56 whose inner product with formula_57 has the wrong sign to be the square norm of formula_58.
Similar arguments about the Hilbert inner product (which can be demonstrated to be a Hermitian form, therefore justifying the name "inner product") lead to the conclusion that its neutral space is precisely formula_59, that elements of this neutral space have zero Hilbert inner product with any element of formula_2, and that the Hilbert inner product is positive semi-definite. It therefore induces a positive definite inner product (also denoted formula_38) on the quotient space formula_62, which is the direct sum of formula_63. Thus formula_64 is a Hilbert space (given a suitable topology).
Properties and applications.
Krein spaces arise naturally in situations where the indefinite inner product has an analytically useful property (such as Lorentz invariance) which the Hilbert inner product lacks. It is also common for one of the two inner products, usually the indefinite one, to be globally defined on a manifold and the other to be coordinate-dependent and therefore defined only on a local section.
In many applications the positive semi-definite inner product formula_38 depends on the chosen fundamental decomposition, which is, in general, not unique. But it may be demonstrated (e. g., cf. Proposition 1.1 and 1.2 in the paper of H. Langer below) that any two metric operators formula_5 and formula_67 compatible with the same indefinite inner product on formula_2 result in Hilbert spaces formula_69 and formula_70 whose decompositions formula_71 and formula_72 have equal dimensions. Although the Hilbert inner products on these quotient spaces do not generally coincide, they induce identical square norms, in the sense that the square norms of the equivalence classes formula_73 and formula_74 into which a given formula_75 falls are equal. All topological notions in a Krein space, like continuity, closed-ness of sets, and the spectrum of an operator on formula_69, are understood with respect to this Hilbert space topology.
Isotropic part and degenerate subspaces.
Let formula_77, formula_78, formula_79 be subspaces of formula_2. The subspace formula_81 for all formula_82 is called the orthogonal companion of formula_77, and formula_84 is the isotropic part of formula_77. If formula_86, formula_77 is called non-degenerate; otherwise it is degenerate. If formula_88 for all formula_89, then the two subspaces are said to be orthogonal, and we write formula_90. If formula_91 where formula_90, we write formula_93. If, in addition, this is a direct sum, we write formula_94.
Pontryagin space.
If formula_95, the Krein space formula_1 is called a Pontryagin space or formula_97-space. (Conventionally, the indefinite inner product is given the sign that makes formula_98 finite.) In this case formula_98 is known as the "number of positive squares" of formula_100. Pontrjagin spaces are named after Lev Semenovich Pontryagin.
Pesonen operator.
A symmetric operator "A" on an indefinite inner product space "K" with domain "K" is called a Pesonen operator if ("x","x") = 0 = ("x","Ax") implies "x" = 0.