Space with an indefinite metric

-space

A pair of objects , the first of which is a vector space over the field of complex numbers, while the second is a bilinear (more precisely, sesquilinear) form on ; this form is also called a -metric. If is a positive-definite (a so-called definite) form, then it is a scalar product in , 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 . In the case of a general sesquilinear form there is neither a norm nor a metric canonically related to , 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 -metric and various topologies on ; b) the classification of vector subspaces (linear manifolds) in relative to the -metric (especially, the so-called definite subspaces, see below); c) the properties of -projections; and d) bases of -spaces.

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

After completing in the -norm one obtains a Hilbert space with indefinite metric , where is the continuous extension of to the entire space . Here, may turn out to be a degenerate metric, even if is non-degenerate. This degeneration does not occur if is a non-degenerate metric and if the largest of the dimensions of the positive subspaces of is finite. In the latter case one obtains the Pontryagin space .

A subspace in a space 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 , or for all . 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 -orthogonal projection.

A vector is called -orthogonal to a subspace (is isotropic with respect to ) if for all . A subspace is called degenerate if it contains at least one non-zero vector that is isotropic with respect to .

If is a subspace in a space with an indefinite metric, then is its -orthogonal complement. Always , where is any topology compatible with . The -orthogonal complement of a degenerate vector subspace is a degenerate vector subspace that is closed in a topology compatible with , and is the vector subspace of isotropic elements. A subspace is called projection complete if each has a -projection on , i.e. if there is an for which for every . Uniqueness of a -projection on is equivalent with being a non-degenerate subspace, while its existence depends on the continuity of the functional in topologies on compatible with . If and are -orthogonal subspaces and , then and are projection complete; if is a projection-complete subspace, then ; the sum is the direct sum if is a non-degenerate space with an indefinite metric.

Suppose that is a definite subspace in a space with an indefinite metric . It is called regular if every functional , , is continuous on in the norm . Otherwise it is called singular. Every non-degenerate infinite-dimensional space with an indefinite metric contains singular subspaces. A definite subspace is projection complete if and only if it is regular and if for every there is an such that

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 endowed with some topology compatible with , are the so-called -orthogonal bases in , i.e. bases of the topological vector space for which , (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 that is compatible with the -metric is also called an admissible topology. For an admissible topology, denotes the -closure of in . The -orthogonal complement is called the orthogonal companion in [a2]. The weak topology on the -space is the locally convex topology defined by the family of semi-norms (cf. Semi-norm)

It is an admissible topology, and the weakest such. As a consequence of the double orthogonal complement theorem, , one thus has that if and only if 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)