Indefinite inner product space

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

[math]\displaystyle{ (K, \langle \cdot,\,\cdot \rangle, J) }[/math]

is an infinite-dimensional complex vector space [math]\displaystyle{ K }[/math] equipped with both an indefinite inner product

[math]\displaystyle{ \langle \cdot,\,\cdot \rangle \, }[/math]

and a positive semi-definite inner product

[math]\displaystyle{ (x,\,y) \ \stackrel{\mathrm{def}}{=}\ \langle x,\,Jy \rangle, }[/math]

where the metric operator [math]\displaystyle{ J }[/math] is an endomorphism of [math]\displaystyle{ K }[/math] obeying

[math]\displaystyle{ J^3 = J. \, }[/math]

The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on [math]\displaystyle{ K }[/math] 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 [math]\displaystyle{ J }[/math]-space) if [math]\displaystyle{ (x,\,y) }[/math] is positive definite and [math]\displaystyle{ K }[/math] 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 [math]\displaystyle{ K }[/math] equipped with an indefinite hermitian form [math]\displaystyle{ \langle \cdot ,\, \cdot \rangle }[/math]. 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:

[math]\displaystyle{ K_{0} \ \stackrel{\mathrm{def}}{=}\ \{ x \in K : \langle x,\,x \rangle = 0 \} }[/math] ("neutral")
[math]\displaystyle{ K_{++} \ \stackrel{\mathrm{def}}{=}\ \{ x \in K : \langle x,\,x \rangle \gt 0 \} }[/math] ("positive")
[math]\displaystyle{ K_{--} \ \stackrel{\mathrm{def}}{=}\ \{ x \in K : \langle x,\,x \rangle \lt 0 \} }[/math] ("negative")
[math]\displaystyle{ K_{+0} \ \stackrel{\mathrm{def}}{=}\ K_{++} \cup K_{0} }[/math] ("non-negative")
[math]\displaystyle{ K_{-0} \ \stackrel{\mathrm{def}}{=}\ K_{--} \cup K_{0} }[/math] ("non-positive")

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

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

[math]\displaystyle{ (x,\,y) \ \stackrel{\mathrm{def}}{=}\ \langle x,\,Jy \rangle = \langle x,\,P_+ y \rangle - \langle x,\,P_- y \rangle }[/math]

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

[math]\displaystyle{ \langle x,\,y \rangle = \frac{1}{4} (\langle x+y,\,x+y \rangle - \langle x-y,\,x-y \rangle) }[/math]

(Note: This is not correct for complex-valued Hermitian forms. It only gives the real part.) Therefore the indefinite inner product of any two elements [math]\displaystyle{ x,\,y \in K }[/math] which differ only by an element [math]\displaystyle{ x-y \in K_0 }[/math] is equal to the square norm of their average [math]\displaystyle{ \frac{x+y}{2} }[/math]. Consequently, the inner product of any non-zero element [math]\displaystyle{ k_0 \in (K_0 \cap K_\pm) }[/math] with any other element [math]\displaystyle{ k_\pm \in K_\pm }[/math] must be zero, lest we should be able to construct some [math]\displaystyle{ k_\pm + 2 \lambda k_0 }[/math] whose inner product with [math]\displaystyle{ k_\pm }[/math] has the wrong sign to be the square norm of [math]\displaystyle{ k_\pm + \lambda k_0 \in K_\pm }[/math].

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 [math]\displaystyle{ K_{00} = (K_0 \cap K_+) \oplus (K_0 \cap K_-) }[/math], that elements of this neutral space have zero Hilbert inner product with any element of [math]\displaystyle{ K }[/math], and that the Hilbert inner product is positive semi-definite. It therefore induces a positive definite inner product (also denoted [math]\displaystyle{ (\cdot,\,\cdot) }[/math]) on the quotient space [math]\displaystyle{ \tilde{K} \ \stackrel{\mathrm{def}}{=}\ K / K_{00} }[/math], which is the direct sum of [math]\displaystyle{ \tilde{K}_\pm \ \stackrel{\mathrm{def}}{=}\ K_\pm / (K_0 \cap K_\pm) }[/math]. Thus [math]\displaystyle{ (\tilde{K},\,(\cdot,\,\cdot)) }[/math] 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 [math]\displaystyle{ (\cdot,\,\cdot) }[/math] 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 [math]\displaystyle{ J }[/math] and [math]\displaystyle{ J^\prime }[/math] compatible with the same indefinite inner product on [math]\displaystyle{ K }[/math] result in Hilbert spaces [math]\displaystyle{ \tilde{K} }[/math] and [math]\displaystyle{ \tilde{K}^\prime }[/math] whose decompositions [math]\displaystyle{ \tilde{K}_\pm }[/math] and [math]\displaystyle{ \tilde{K}^\prime_\pm }[/math] 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 [math]\displaystyle{ \tilde{k} \in \tilde{K} }[/math] and [math]\displaystyle{ \tilde{k}^\prime \in \tilde{K}^\prime }[/math] into which a given [math]\displaystyle{ k \in K }[/math] if they are equal. All topological notions in a Krein space, like continuity, closed-ness of sets, and the spectrum of an operator on [math]\displaystyle{ \tilde{K} }[/math], are understood with respect to this Hilbert space topology.

Isotropic part and degenerate subspaces

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

Pontryagin space

If [math]\displaystyle{ \kappa := \min \{ \dim K_{+}, \dim K_{-} \} \lt \infty }[/math], the Krein space [math]\displaystyle{ (K, \langle \cdot,\,\cdot \rangle, J) }[/math] is called a Pontryagin space or [math]\displaystyle{ \Pi_{\kappa} }[/math]-space. (Conventionally, the indefinite inner product is given the sign that makes [math]\displaystyle{ \dim K_{+} }[/math] finite.) In this case [math]\displaystyle{ \dim K_{+} }[/math] is known as the number of positive squares of [math]\displaystyle{ \langle \cdot,\,\cdot \rangle }[/math]. 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.

References



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