Indefinite inner product space

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

(K, \langle \cdot,\,\cdot \rangle, J)

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

\langle \cdot,\,\cdot \rangle \,

and a positive semi-definite inner product

(x,\,y) \ \stackrel{\mathrm{def}}{=}\  \langle x,\,Jy \rangle,

where the metric operator J is an endomorphism of K obeying

J^3 = J. \,

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

Inner products and the metric operator[edit]

Consider a complex vector space K equipped with an indefinite hermitian form \langle \cdot ,\, \cdot \rangle. 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:

K_{0}  \ \stackrel{\mathrm{def}}{=}\  \{ x \in K : \langle x,\,x \rangle = 0 \} ("neutral")
K_{++} \ \stackrel{\mathrm{def}}{=}\  \{ x \in K : \langle x,\,x \rangle > 0 \} ("positive")
K_{--} \ \stackrel{\mathrm{def}}{=}\  \{ x \in K : \langle x,\,x \rangle < 0 \} ("negative")
K_{+0} \ \stackrel{\mathrm{def}}{=}\  K_{++} \cup K_{0} ("non-negative")
K_{-0} \ \stackrel{\mathrm{def}}{=}\  K_{--} \cup K_{0} ("non-positive")

A subspace L \subset K lying within K_{0} is called a neutral subspace. Similarly, a subspace lying within K_{+0} (K_{-0}) is called positive (negative) semi-definite, and a subspace lying within K_{++} \cup \{0\} (K_{--} \cup \{0\}) 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 K = K_+ \oplus K_-, called the fundamental decomposition, which respects the complex structure on K. Hence the corresponding linear projection operators P_\pm coincide with the identity on K_\pm and annihilate K_\mp, and they commute with multiplication by the i of the complex structure. If this decomposition is such that K_+ \subset K_{+0} and K_- \subset K_{-0}, then K is called an indefinite inner product space; if K_\pm \subset K_{\pm\pm} \cup \{0\}, then K is called a Krein space, subject to the existence of a majorant topology on K.

The operator J \ \stackrel{\mathrm{def}}{=}\  P_+ - P_- is called the (real phase) metric operator or fundamental symmetry, and may be used to define the Hilbert inner product (\cdot,\,\cdot):

(x,\,y) \ \stackrel{\mathrm{def}}{=}\  \langle x,\,Jy \rangle = \langle x,\,P_+ y \rangle - \langle x,\,P_- y \rangle

On a Krein space, the Hilbert inner product is positive definite, giving K the structure of a Hilbert space (under a suitable topology). Under the weaker constraint K_pm \subset K_{\pm0}, some elements of the neutral subspace K_0 may still be neutral in the Hilbert inner product, but many are not. For instance, the subspaces K_0 \cap K_\pm are part of the neutral subspace of the Hilbert inner product, because an element k \in K_0 \cap K_\pm obeys (k,\,k) \ \stackrel{\mathrm{def}}{=}\  \langle k,\,Jk \rangle = \pm \langle k,\,k \rangle = 0. But an element k = k_+ + k_- (k_\pm \in K_\pm) which happens to lie in K_0 because \langle k_-,\,k_- \rangle = - \langle k_+,\,k_+ \rangle 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:

\langle x,\,y \rangle = \frac{1}{4} (\langle x+y,\,x+y \rangle - \langle x-y,\,x-y \rangle)

Therefore the indefinite inner product of any two elements x,\,y \in K which differ only by an element x-y \in K_0 is equal to the square norm of their average \frac{x+y}{2}. Consequently, the inner product of any non-zero element k_0 \in (K_0 \cap K_\pm) with any other element k_\pm \in K_\pm must be zero, lest we should be able to construct some k_\pm + 2 \lambda k_0 whose inner product with k_\pm has the wrong sign to be the square norm of k_\pm + \lambda k_0 \in K_\pm.

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 K_{00} = (K_0 \cap K_+) \oplus (K_0 \cap K_-), that elements of this neutral space have zero Hilbert inner product with any element of K, and that the Hilbert inner product is positive semi-definite. It therefore induces a positive definite inner product (also denoted (\cdot,\,\cdot)) on the quotient space \tilde{K} \ \stackrel{\mathrm{def}}{=}\  K / K_{00}, which is the direct sum of \tilde{K}_\pm \ \stackrel{\mathrm{def}}{=}\  K_\pm / (K_0 \cap K_\pm). Thus (\tilde{K},\,(\cdot,\,\cdot)) is a Hilbert space (given a suitable topology).

Properties and applications[edit]

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 (\cdot,\,\cdot) 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 J and J^\prime compatible with the same indefinite inner product on K result in Hilbert spaces \tilde{K} and \tilde{K}^\prime whose decompositions \tilde{K}_\pm and \tilde{K}^\prime_\pm 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 \tilde{k} \in \tilde{K} and \tilde{k}^\prime \in \tilde{K}^\prime into which a given k \in K falls are equal. All topological notions in a Krein space, like continuity, closed-ness of sets, and the spectrum of an operator on \tilde{K}, are understood with respect to this Hilbert space topology.

Isotropic part and degenerate subspaces[edit]

Let L, L_{1}, L_{2} be subspaces of K. The subspace L^{[\perp]} \ \stackrel{\mathrm{def}}{=}\  \{ x \in K : \langle x,\,y \rangle = 0 for all y \in L \} is called the orthogonal companion of L, and L^{0} \ \stackrel{\mathrm{def}}{=}\  L \cap L^{[\perp]} is the isotropic part of L. If L^{0} = \{0\}, L is called non-degenerate; otherwise it is degenerate. If \langle x,\,y \rangle = 0 for all x \in L_{1},\,\, y \in L_{2}, then the two subspaces are said to be orthogonal, and we write L_{1} [\perp] L_{2}. If L = L_{1} + L_{2} where L_{1} [\perp] L_{2}, we write L = L_{1} [+] L_{2}. If, in addition, this is a direct sum, we write L= L_{1} [\dot{+}] L_{2}.

Pontrjagin space[edit]

If \kappa := \min \{ \dim K_{+}, \dim K_{-} \} < \infty, the Krein space (K, \langle \cdot,\,\cdot \rangle, J) is called a Pontrjagin space or \Pi_{\kappa}-space. (Conventionally, the indefinite inner product is given the sign that makes \dim K_{+} finite.) In this case \dim K_{+} is known as the number of positive squares of \langle \cdot,\,\cdot \rangle. Pontrjagin spaces are named after Lev Semenovich Pontryagin.

Pesonen operator[edit]

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.

References[edit]