# Hilbert space with an indefinite metric

A Hilbert space $E$ over the field of complex numbers endowed with a continuous bilinear (more exactly, sesquilinear) form $G$ that is not, generally speaking, positive definite. The form $G$ is often referred to as the $G$- metric. The most important example of a Hilbert space with an indefinite metric is a so-called $J$- space — a Hilbert space with an indefinite metric in which $G$ is defined by a certain Hermitian involution $J$ in $E$ by the formula $G( x, y) = ( Jx, y)$. The form $G$ is then also denoted by the letter $J$ and is called a $J$- metric. The involution $J$ may be represented as $J = P _ {+} - P _ {-}$, where $P _ {+}$ and $P _ {-}$ are orthogonal projections in $E$, and $P _ {+} + P _ {-} = I$; the number $\kappa = \min ( \mathop{\rm dim} P _ {+} , \mathop{\rm dim} P _ {-} )$ is called the rank of indefiniteness of the $J$- metric or of the $J$- space. If $\kappa < + \infty$, the Hilbert space with the indefinite metric $( E, J)$ is called a Pontryagin space $\Pi _ \kappa$; see also Space with an indefinite metric.

Two Hilbert spaces $( E, G)$ and $( E _ {1} , G _ {1} )$ with indefinite metrics are said to be metrically equivalent if there exists a linear homeomorphism $U$ of $E$ onto $E _ {1}$ which transforms $G$ to $G _ {1}$. A $G$- metric generated by an invertible Hermitian operator $G$ by the formula $G( x, y) = ( Gx, y)$ is said to be regular; after the introduction of a new scalar product that is metrically equivalent to the old one a regular $G$- metric becomes a $J$- metric. Any Hilbert space with an indefinite metric with a Hermitian form $G$ may be $G$- isometrically (i.e. with preservation of $G$) imbedded in some $J$- space , .

The principal trends in the theory of Hilbert spaces with an indefinite metric are the same as those in general spaces with an indefinite metric, but with a considerable stress on the spectral theory. The geometry of Hilbert spaces with an indefinite metric is much richer than that of ordinary spaces with an indefinite metric. Thus, in the case of $J$- spaces there is an effective description of the maximal subspaces $L$ among all the non-negative (non-positive, neutral) ones: these are the $L$' s for which $P _ {+} L = P _ {+} E$( or, correspondingly, $P _ {-} L = P _ {-} E$; at least one of these equalities must be valid). Hence the analogue of the law of inertia of quadratic forms: If $E = L _ {+} \dot{+} L _ {-}$ is the canonical decomposition of the $J$- space into a sum of semi-definite subspaces, then $\mathop{\rm dim} L _ \pm = \mathop{\rm dim} P _ \pm E$. The subspace $L$ is maximal and non-negative if and only if $L$ has an angular operator $K$ with respect to $E _ {+}$, i.e. if $L = \{ {x + Kx } : {x \in E _ {+} } \}$ and $\| K \| \leq 1$.

A theory of bases has been developed in $J$- spaces; this theory helps in the study of the geometry of Hilbert spaces with an indefinite metric as well as of the operators on them. A $J$- orthonormal basis of a $J$- space $( E, J)$ is a basis in the Hilbert space $E$ satisfying the conditions $( Je _ {k} , e _ {n} ) = \delta _ {kn}$; $k$, $n = 1, 2 ,\dots$. For a $J$- orthonormal sequence ${\mathcal E}$ to be a Riesz basis of $E$ it is necessary and sufficient that $E = M _ {+} \dot{+} M _ {-}$, where $M _ \pm$ is the closed linear hull of the vectors $\{ {e _ {k} } : {( Je _ {k} , e _ {k} ) = \pm 1 } \}$. If ${\mathcal E}$ is a $J$- orthonormal basis in $E$, then the decomposition $E = M _ {+} \dot{+} M _ {-}$ is the canonical decomposition of the $J$- space $E$. A large group of geometrical problems in Hilbert spaces with an indefinite metric are connected with the structure and properties of so-called dual pairs of subspaces of a Hilbert space with an indefinite metric $( E, J)$, i.e. with pairs $N, P$ of subspaces in $E$ such that $N$ and $P$ are mutually orthogonal, while $N$ is a non-positive and $P$ is a non-negative space. A dual pair is said to be maximal if $N$ and $P$ are maximal semi-definite subspaces.

## Theory of operators in a Hilbert space with an indefinite metric.

The metric $G$ is considered to be Hermitian and non-degenerate, while the operators that are considered are densely defined. For an operator $T$ with domain of definition $D _ {T}$ let there be defined a $G$- adjoint operator $T ^ {c}$ by the equation

$$G ( Tx, y) = \ G ( x, T ^ {c} y),\ \ y \in D _ {T} ,\ \ y \in D _ {T ^ {c} } .$$

where $T ^ {c} = G ^ {-} 1 T ^ {*} G$ and

$$D _ {T ^ {c} } = \ G ^ {-} 1 \{ GE \cap T ^ {*- 1 } ( TE \cap GE) \} .$$

An operator $T$ is said to be $G$- self-adjoint if $T = T ^ {c}$, and is said to be $G$- symmetric if $G( Tx, y) = G( x, Ty)$, $x, y \in D _ {T}$. Root subspaces $L _ \lambda ( T)$ and $L _ \mu ( T)$, $\lambda \neq \overline \mu \;$, of a $G$- symmetric operator $T$ are $G$- orthogonal; in particular, if $\lambda \neq \overline \lambda \;$, then $L _ \lambda ( T)$ is a neutral subspace.

If $G$ is a regular metric, then the spectrum $\sigma ( T)$ of the $G$- self-adjoint operator $T$ is symmetric with respect to the real axis; if it is not regular, this is usually not the case. The $J$- self-adjointness of an operator $T$ is equivalent to the self-adjointness of $JT$. If $\zeta , \overline \zeta \; \in \sigma ( T)$, then the Cayley transform $U = ( T - \overline \zeta \; I) ( T - \zeta I ) ^ {-} 1$ is a $J$- unitary operator, i.e. is such that $UJU ^ {*} = U ^ {*} JU = J$. The spectrum of $U$ is symmetric with respect to the circle $S = \{ {\lambda \in \mathbf C } : {| \lambda | = 1 } \}$.

Beginning with the study of L.S. Pontryagin , the principal problem of the theory is the existence of semi-definite invariant subspaces. Let $T$ be a bounded operator in a $J$- space $E$ and let $( JTx, Tx) \geq 0$ for $( Jx, x) \geq 0$, $x \in E$( the so-called plus-operator); if $P _ {+} TP _ {-}$ is a completely-continuous operator, then there exists a maximal non-negative $T$- invariant subspace $L$. This result is applicable, in particular, to $J$- unitary operators $U$ on the spaces $\Pi _ \kappa$, in which it is the base of the so-called definization method — a construction of an operator polynomial $p( U)$ that maps $E$ into a semi-definite subspace. This method makes it possible to obtain, e.g., analogues of the ordinary spectral expansion for $J$- unitary and $J$- self-adjoint operators on $\Pi _ \kappa$.

The theory of operators in Hilbert spaces with an indefinite metric is used in an essential way in the theory of canonical systems of ordinary differential equations; for example, the criterion of stability for such equations may be written as follows in terms of the monodromy operator $U$: Stability holds if and only if a maximal $U$- invariant dual pair of subspaces exists. Another important use of this theory is in the spectral theory of quadratic operator pencils, which is important in many problems of mathematical physics.

For the theory of representations in Hilbert spaces with an indefinite metric see .

How to Cite This Entry:
Hilbert space with an indefinite metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_space_with_an_indefinite_metric&oldid=47234
This article was adapted from an original article by N.K. Nikol'skiiB.S. Pavlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article