Pontryagin space

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A Hilbert space with an indefinite metric $\Pi _ \kappa$ that has a finite rank of indefiniteness $\kappa$. 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 ${\mathcal P}$ is an arbitrary non-negative linear manifold in $\Pi _ \kappa$, then $\mathop{\rm dim} {\mathcal P} \leq \kappa$; if ${\mathcal P}$ is a positive linear manifold and $\mathop{\rm dim} {\mathcal P} = \kappa$, then its $J$- orthogonal complement $N$ is a negative linear manifold and $\Pi _ \kappa = {\mathcal P} \oplus N$. Moreover, $N$ is a complete space with respect to the norm $| x | = \sqrt {- J ( x , x ) }$. If the linear manifold $L \subset \Pi _ \kappa$ is non-degenerate, then its $J$- orthogonal complement $M$ is non-degenerate as well and $\Pi _ \kappa = M \oplus L$.

The spectrum (in particular, the discrete spectrum) of a $J$- unitary ( $J$- self-adjoint) operator is symmetric with respect to the unit circle (real line), all elementary divisors corresponding to eigen values $\lambda$, $| \lambda | > 1$, are of finite order $\rho _ \lambda$, $\rho _ \lambda \leq \kappa$, $\rho _ \lambda = - \rho _ {\lambda ^ {-} 1 }$. The sum of the dimensions of the root subspaces of a $J$- unitary ( $J$- self-adjoint) operator corresponding to eigen values $\lambda$, $| \lambda | > 1$( $\mathop{\rm lm} \lambda > 0$), does not exceed $\kappa$.

The following theorem [1] is fundamental in the theory of $J$- self-adjoint operators on a Pontryagin space $\Pi _ \kappa$: For each $J$- self-adjoint operator $A$( $\overline{ {D ( A) }}\; = \Pi _ \kappa$) there exists a $\kappa$- dimensional (maximal) non-negative invariant subspace ${\mathcal T}$ in which all eigen values of $A$ have non-negative imaginary parts, and a $\kappa$- dimensional non-negative invariant subspace ${\mathcal T} ^ \prime$ 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 $J$- unitary operators, and under certain additional conditions — even for operators on the space $\Pi _ \infty$.

If $U$ is a $J$- unitary operator, then its maximal invariant subspaces ${\mathcal T}$, ${\mathcal T} ^ \prime$ can be chosen so that the elementary divisors of the operator $U _ {\mathcal T} = U \mid _ {\mathcal T}$, $U _ { {\mathcal T} ^ \prime } = U \mid _ { {\mathcal T} ^ \prime }$ are of minimal order. In order that a polynomial $P ( \lambda )$ with no roots inside the unit disc has the property: $( P ( U) x , P ( U) x ) \leq 0$, $x \in \Pi _ \kappa$, it is necessary and sufficient that it can be divided by the minimal annihilating polynomial of the operator $U _ {\mathcal T}$. If $U$ is a cyclic operator, then its non-negative invariant subspaces of dimension $\kappa$ are uniquely determined. In this case the above-mentioned property of the polynomial $P$ with roots $\{ \lambda _ {i} \}$ outside the unit disc, $| \lambda _ {i} | > 1$, is equivalent to the divisibility of $P ( \lambda )$ by the characteristic polynomial of $U _ {\mathcal T}$.

Each completely-continuous $J$- self-adjoint operator $A$ on a Pontryagin space $\Pi _ \kappa$ 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 $\Pi _ \kappa$ with respect to the (definite) norm $( | J | x , x )$.

Many facts concerning invariant subspaces and the spectrum can be generalized to a case of $J$- isometric and $J$- non-expanding operators. Thus, if $\lambda _ {1} \dots \lambda _ {n}$ is an arbitrary set of eigen values of a $J$- isometric operator, $\lambda _ {i} \overline \lambda \; _ {k} \neq 1$, $i , k = 1 \dots n$, and if $\rho _ {i}$ is the order of the elementary divisor at the point $\lambda _ {i}$, then $\sum _ {1} ^ {n} \rho _ {i} \leq \kappa$. Any $J$- non-expanding boundedly-invertible operator $T$ has a $\kappa$- dimensional invariant non-negative subspace ${\mathcal T}$ such that all eigen values of the restriction $T \mid _ {\mathcal T}$ lie in the unit disc [2]. A similar fact holds for maximal $J$- dissipative operators. In general, a $J$- dissipative operator $A$, $D ( A) \subset D ( A ^ {*} )$, has at most $\kappa$ eigen values in the upper half-plane. $J$- isometric and $J$- symmetric (and more generally, $J$- non-expanding and $J$- dissipative) operators are related by the Cayley transformation (cf. Cayley transform), which has on $\Pi _ \kappa$ all natural properties [2]. This fact allows one to develop the extension theory simultaneously for $J$- isometric and $J$- symmetric operators. In particular, every $J$- isometric ( $J$- symmetric) operator can be extended to a maximal one. If its deficiency indices are different, then it has no $J$- unitary ( $J$- self-adjoint) extensions. If these indices are equal and finite, then any maximal extension is $J$- unitary ( $J$- self-adjoint).

For completely-continuous operators on $\Pi _ \kappa$, 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)

Pontryagin spaces form a subclass of the class of Krein spaces (cf. Krein space and also Hilbert space with an indefinite metric). The operator $J$ appearing in the beginning of the main article above is the fundamental symmetry (see Krein space), which defines the indefinite inner product via the formula $[ x, y] = ( Jx, y)$.