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)

Comments

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)