Pontryagin space

A Hilbert space with an indefinite metric that has a finite rank of indefiniteness . Basic facts concerning the geometry of these spaces were established by L.S. Pontryagin [1]. Besides the facts common for spaces with an indefinite metric, the following properties hold.

If is an arbitrary non-negative linear manifold in , then ; if is a positive linear manifold and , then its -orthogonal complement is a negative linear manifold and . Moreover, is a complete space with respect to the norm . If the linear manifold is non-degenerate, then its -orthogonal complement is non-degenerate as well and .

The spectrum (in particular, the discrete spectrum) of a -unitary (-self-adjoint) operator is symmetric with respect to the unit circle (real line), all elementary divisors corresponding to eigen values , , are of finite order , , . The sum of the dimensions of the root subspaces of a -unitary (-self-adjoint) operator corresponding to eigen values , (), does not exceed .

The following theorem [1] is fundamental in the theory of -self-adjoint operators on a Pontryagin space : For each -self-adjoint operator () there exists a -dimensional (maximal) non-negative invariant subspace in which all eigen values of have non-negative imaginary parts, and a -dimensional non-negative invariant subspace in which all eigen values have non-positive imaginary parts. A similar statement in which the upper (lower) half-plane is replaced by the exterior (interior) of the unit disc is also valid for -unitary operators, and under certain additional conditions — even for operators on the space .

If is a -unitary operator, then its maximal invariant subspaces , can be chosen so that the elementary divisors of the operator , are of minimal order. In order that a polynomial with no roots inside the unit disc has the property: , , it is necessary and sufficient that it can be divided by the minimal annihilating polynomial of the operator . If is a cyclic operator, then its non-negative invariant subspaces of dimension are uniquely determined. In this case the above-mentioned property of the polynomial with roots outside the unit disc, , is equivalent to the divisibility of by the characteristic polynomial of .

Each completely-continuous -self-adjoint operator on a Pontryagin space such that zero belongs to its continuous spectrum does not have a residual spectrum. The root vectors of such an operator form a Riesz basis in with respect to the (definite) norm .

Many facts concerning invariant subspaces and the spectrum can be generalized to a case of -isometric and -non-expanding operators. Thus, if is an arbitrary set of eigen values of a -isometric operator, , , and if is the order of the elementary divisor at the point , then . Any -non-expanding boundedly-invertible operator has a -dimensional invariant non-negative subspace such that all eigen values of the restriction lie in the unit disc [2]. A similar fact holds for maximal -dissipative operators. In general, a -dissipative operator , , has at most eigen values in the upper half-plane. -isometric and -symmetric (and more generally, -non-expanding and -dissipative) operators are related by the Cayley transformation (cf. Cayley transform), which has on all natural properties [2]. This fact allows one to develop the extension theory simultaneously for -isometric and -symmetric operators. In particular, every -isometric (-symmetric) operator can be extended to a maximal one. If its deficiency indices are different, then it has no -unitary (-self-adjoint) extensions. If these indices are equal and finite, then any maximal extension is -unitary (-self-adjoint).

For completely-continuous operators on , a number of statements on the completeness of the system of root vectors, analogous to the corresponding facts from the theory of dissipative operators on spaces with a definite metric, is valid.

References

 [1] L.S. Pontryagin, "Hermitian operators in a space with indefinite metric" Izv. Akad. Nauk. SSSR Ser. Mat. , 8 (1944) pp. 243–280 (In Russian) [2] I.S. Iokhvidov, M.G. Krein, "Spectral theory of operators in a space with indefinite metric I" Transl. Amer. Math. Soc. (2) , 13 (1960) pp. 105–175 Trudy Moskov. Mat. Obshch. , 5 (1956) pp. 367–432 [3] I.S. Iokhvidov, M.G. Krein, "Spectral theory of operators in a space with indefinite metric II" Trudy Moskov. Mat. Obshch. , 8 (1959) pp. 413–496 (In Russian) [4] 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 [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] M.A. Naimark, R.S. Ismagilov, "Representations of groups and algebras in a space with indefinite metric" Itogi Nauk. i Tekhn. Mat. Anal. (1969) pp. 73–105 (In Russian) [7] L. Nagy, "State vector spaces with indefinite metric in quantum field theory" , Noordhoff (1966)