# Space with an indefinite metric

$G$- space

A pair of objects $( E , G )$, the first of which is a vector space $E$ over the field of complex numbers, while the second is a bilinear (more precisely, sesquilinear) form $G$ on $E$; this form is also called a $G$- metric. If $G$ is a positive-definite (a so-called definite) form, then it is a scalar product in $E$, and one can use it to canonically introduce (cf., e.g., Hilbert space with an indefinite metric) a norm and a distance (i.e. an ordinary metric) for the elements of $E$. In the case of a general sesquilinear form $G$ there is neither a norm nor a metric canonically related to $G$, and the phrase "G-metric" only recalls the close relation of definite sesquilinear forms with certain metrics in vector spaces.

The theory of finite-dimensional spaces with an indefinite metric, more often called bilinear metric spaces, or spaces with a bilinear metric, was developed already by G. Frobenius, and is expounded in courses on linear algebra (cf. [1]).

The main purpose of the general theory of spaces with an indefinite metric is the separation and study of relatively simple, but for applications important, classes of non-self-adjoint operators in a Hilbert space (cf. Non-self-adjoint operator). Spaces with an indefinite metric were for the first time introduced by L.S. Pontryagin [2] (for more detail, see Pontryagin space).

The theory of spaces with an indefinite metric has been developed in two directions — their geometry and linear operations on them.

In the geometry of spaces with an indefinite metric one basically studies: a) the relation between the $G$- metric and various topologies on $E$; b) the classification of vector subspaces (linear manifolds) in $E$ relative to the $G$- metric (especially, the so-called definite subspaces, see below); c) the properties of $G$- projections; and d) bases of $G$- spaces.

In the case of a Hermitian $G$- metric (a $G ^ {H}$- metric), i.e. such that $G ( x , y ) = \overline{ {G ( y , x ) }}\;$ for all $x , y \in E$, the most important results and concepts in the geometry of spaces with an indefinite metric are as follows. Suppose that each vector $y \in E$ is put in correspondence with a linear functional $G _ {y} : x \rightarrow G ( x , y )$, $x \in E$. A topology $\tau$ on $E$ is called subordinate to the $G$- metric if $G _ {y}$ is continuous in $\tau$ for all $y \in E$; $\tau$ is called compatible with the $G$- metric if it is subordinate to $G$ and if every $\tau$- continuous functional has the form $G _ {y}$, $y \in E$. In a space $E$ with an indefinite metric one cannot specify more than one Fréchet topology subordinate to $G$, and not every $G$- metric allows such a topology (cf. [4]). If a topology, subordinate to the $G$- metric, is a pre-Hilbert topology on $E$ and is given by a scalar product $H ( \cdot , \cdot )$ in $E$, then $H$ is called a Hermitian non-negative majorant of $G$; in this case

$$| G ( x , y ) | ^ {2} \leq CH ( x , x ) H ( y , y ) ,\ \ C = \textrm{ const } ,\ \ x , y \in E .$$

After completing in the $H$- norm one obtains a Hilbert space with indefinite metric $( \widetilde{E} , \widetilde{G} )$, where $\widetilde{G}$ is the continuous extension of $G$ to the entire space $\widetilde{E}$. Here, $\widetilde{G}$ may turn out to be a degenerate metric, even if $G$ is non-degenerate. This degeneration does not occur if $G$ is a non-degenerate metric and if the largest of the dimensions $\kappa$ of the positive subspaces of $E$ is finite. In the latter case one obtains the Pontryagin space $\Pi _ \kappa$.

A subspace $L$ in a space $( E , G )$ with an indefinite metric is called a positive subspace, a negative subspace (a more general name is: a definite subspace) or a neutral subspace, depending on whether $G ( x , x ) > 0$, $G ( x , x ) < 0$ or $G ( x , x ) = 0$ for all $x \in L$. A subspace is called maximally positive if it is positive and cannot be extended with preservation of this property. Every subspace of the type indicated above is contained in a maximal subspace of the same type.

An important part in the classification of subspaces in spaces with an indefinite metric is played by the notions of a canonical decomposition and a $G$- orthogonal projection.

A vector $x \in E$ is called $G$- orthogonal to a subspace $L \subset E$( is isotropic with respect to $L$) if $G ( x , y ) = 0$ for all $y \in L$. A subspace $L$ is called degenerate if it contains at least one non-zero vector that is isotropic with respect to $L$.

If $L$ is a subspace in a space $E$ with an indefinite metric, then $L ^ \prime = \{ {y } : {G ( x , y ) = 0 \textrm{ for all } x \in L } \}$ is its $G$- orthogonal complement. Always $L ^ {\prime\prime} = L ^ \tau$, where $\tau$ is any topology compatible with $G$. The $G$- orthogonal complement $L ^ \prime$ of a degenerate vector subspace $L$ is a degenerate vector subspace that is closed in a topology $\tau$ compatible with $G$, and $L \cap L ^ \prime$ is the vector subspace of isotropic elements. A subspace $L$ is called projection complete if each $y \in E$ has a $G$- projection on $L$, i.e. if there is an $y _ {0} \in L$ for which $G ( x , y - y _ {0} ) = 0$ for every $x \in L$. Uniqueness of a $G$- projection on $L$ is equivalent with $L$ being a non-degenerate subspace, while its existence depends on the continuity of the functional $G _ {y}$ in topologies on $L$ compatible with $G$. If $N$ and $M$ are $G$- orthogonal subspaces and $M + N = E$, then $M$ and $N$ are projection complete; if $L$ is a projection-complete subspace, then $L + L ^ \prime = E$; the sum is the direct sum if $E$ is a non-degenerate space with an indefinite metric.

Suppose that $L$ is a definite subspace in a space with an indefinite metric $E$. It is called regular if every functional $G _ {y}$, $y \in E$, is continuous on $E$ in the norm $\| x \| _ {G} = | G ( x , x ) | ^ {1/2 }$. Otherwise it is called singular. Every non-degenerate infinite-dimensional space with an indefinite metric contains singular subspaces. A definite subspace $L$ is projection complete if and only if it is regular and if for every $y \in E$ there is an $x \in L$ such that

$$\| x \| _ {G} ^ {2} = \| G _ {y} \| _ {G} ^ {2} = G ( x , y ) .$$

Linear operators in spaces with an indefinite metric have been studied mainly in Hilbert spaces with an indefinite metric; for Banach analogues there is a survey in [8].

As in the case of Hilbert spaces with an indefinite metric, an important tool in the study of the geometry of spaces with an indefinite metric and of linear operators in spaces $( E , G )$ endowed with some topology compatible with $G$, are the so-called $G$- orthogonal bases in $E$, i.e. bases $\{ e _ {n} \}$ of the topological vector space $E$ for which $G( e _ {k} , e _ {n} ) = \pm \delta _ {kn}$, $k , n = 1 , 2 , . . .$( cf. [4]).

#### References

 [1] A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian) [2] L.S. Pontryagin, "Hermitian operators in spaces with indefinite metric" Izv. Akad. Nauk SSSR Ser. Mat. , 8 (1944) pp. 243–280 (In Russian) [3] I.S. Iokhvidov, M.G. Krein, "Spectral theory in spaces with an indefinite metric I" Transl. Amer. Math. Soc. , 13 (1960) pp. 105–176 Trudy Moskov. Mat. Obshch. , 5 (1956) pp. 367–432 [4] Yu.P. Ginzburg, I.S. Iokhvidov, "The geometry of infinite-dimensional spaces with a bilinear metric" Russian Math. Surveys , 17 : 4 (1962) pp. 1–51 Uspekhi Mat. Nauk , 17 : 4 (1962) pp. 3–56 [5] M.G. Krein, "Introduction to the geometry of indefinite -spaces and the theory of operators in these spaces" , Second Math. Summer School , 1 , Kiev (1965) pp. 15–92 (In Russian) [6] T.Ya. Azizov, I.S. Iokhvidov, "Linear operators in Hilbert spaces with -metric" Russian Math. Surveys , 26 : 4 (1971) pp. 45–97 Uspekhi Mat. Nauk , 26 : 4 (1971) pp. 43–92 [7] K.L. Nagy, "State vector spaces with indefinite metric in quantum field theory" , Noordhoff (1966) [8] I.S. Iokhvidov, "Banach spaces with a -metric and certain classes of linear operators in these spaces" Izv. Akad. Nauk MoldavSSR , 1 (1968) pp. 60–80 (In Russian)

A topology $\tau$ that is compatible with the $G$- metric is also called an admissible topology. For an admissible topology, $L ^ \tau$ denotes the $\tau$- closure of $L$ in $E$. The $G$- orthogonal complement is called the orthogonal companion in [a2]. The weak topology on the $G$- space $E$ is the locally convex topology defined by the family of semi-norms (cf. Semi-norm)

$$p _ {y} ( x) = | G ( x , y) | .$$

It is an admissible topology, and the weakest such. As a consequence of the double orthogonal complement theorem, $L ^ {\prime\prime} = L ^ \tau$, one thus has that $L ^ {\prime\prime} = L$ if and only if $L$ is weakly closed.

For additional information about spaces with an indefinite metric see Krein space and [a1][a4].

#### References

 [a1] T.Ya. Azizov, I.S. [I.S. Iokhvidov] Iohidov, "Linear operators in spaces with an indefinite metric" , Wiley (1989) (Translated from Russian) [a2] J. Bognár, "Indefinite inner product spaces" , Springer (1974) [a3] I. [I. Gokhberg] Gohberg, P. Lancaster, L. Rodman, "Matrices and indefinite scalar products" , Birkhäuser (1983) [a4] I.S. [I.S. Iokhvidov] Iohidov, M.G. Krein, H. Langer, "Introduction to the spectral theory of operators in spaces with an indefinite metric" , Akademie Verlag (1982)
How to Cite This Entry:
Space with an indefinite metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Space_with_an_indefinite_metric&oldid=48753
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