Indefinite inner product space

In mathematics, in the field of functional analysis, an indefinite inner product space

is an infinite dimensional complex vector space equipped with both an indefinite inner product

and a positive semi definite inner product

where the metric operator is an endomorphism of obeying

The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi definite inner product on 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 space) if is positive definite and possesses a majorant topology. Krein spaces are named in honor of the Soviet mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989).

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 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 and compatible with the same indefinite inner product on result in Hilbert spaces and whose decompositions and 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 and into which a given if they are equal. All topological notions in a Krein space, like continuity, closed ness of sets, and the spectrum of an operator on , are understood with respect to this Hilbert space topology.

Let , , be subspaces of The subspace for all is called the orthogonal companion of , and is the isotropic part of If , is called non degenerate; otherwise it is degenerate. If for all , then the two subspaces are said to be orthogonal, and we write If where , we write If, in addition, this is a direct sum, we write .

If , the Krein space is called a Pontryagin space or space. (Conventionally, the indefinite inner product is given the sign that makes finite.) In this case is known as the number of positive squares of Pontrjagin spaces are named after Lev Semenovich Pontryagin.

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.