MATHEMATICS

I. S. IOKHVIDOV

REGULAR AND PROJECTION-COMPLETE LINEALS IN SPACES WITH A GENERAL HERMITIAN-BILINEAR METRIC

(Presented by Academician L. S. Pontryagin on 21 III 1961)

1. Consider an infinite-dimensional complex linear system (space) (\mathfrak C), in which a Hermitian bilinear functional (G(x,y)=[x,y]), or, briefly, a (G)-metric, is given. Since this (G)-metric is, generally speaking, indefinite, vectors (x\in\mathfrak C) may be positive, null, and negative (([x,x]\gtreqless 0), respectively, (x\ne0)). A lineal (\mathfrak L(\subseteq\mathfrak C)) will be called positive (negative) if all nonzero vectors (x\in\mathfrak L) are positive (negative). Positive and negative lineals are called definite.

Let (\mathfrak L) be some lineal in (\mathfrak C), and let (y_0\in\mathfrak C). A vector (x_0\in\mathfrak L) is called the (G)-projection of the vector (y_0) onto (\mathfrak L), if ([x,y_0-x_0]=0) for all (x\in\mathfrak L) (i.e. the vector (y_0-x_0) is “(G)-orthogonal” to (\mathfrak L)).

The question of the existence of (G)-projections was studied in the literature ((^{1-3})) for the special case in which (\mathfrak C=\mathfrak H) is a Hilbert space with scalar product ((x,y)) ((x,y\in\mathfrak H)), and the (G)-metric is given by means of the form

[
[x,y]=(Gx,y)\qquad (x,y\in\mathfrak H),
]

where (G) is a bounded Hermitian* operator in (\mathfrak H) (the so-called Gram operator of the form ([x,y]) in (\mathfrak H) ((^3))). Still earlier (and, apparently, for the first time) (G)-projections were studied by L. S. Pontryagin ((^4)) for the case in which (G=P^+-P^-), where (P^+), (P^-) are orthoprojectors in (\mathfrak H), (P^+P^-=P^-P^+=0), (P^+ + P^- = I), and (\min(\dim P^+\mathfrak H,\dim P^-\mathfrak H)=\varkappa<\infty). In papers ((^5,^6)) a space with such a (G)-metric was denoted by (\Pi_\varkappa). We shall adhere to this notation here as well.

In all the papers mentioned, the natural requirement of closedness in the norm

[
|x|=(x,x)^{1/2}\qquad (x\in\mathfrak H),
]

was imposed on the lineal (\mathfrak L), i.e. (\mathfrak L) was a subspace of the Hilbert space (\mathfrak H). With the aid of the so-called canonical decomposition of the subspace (\mathfrak L) ((^1,^2)), the question of (G)-projection onto an arbitrary subspace was reduced to (G)-projection onto definite subspaces.

In the general case, i.e. for an arbitrary linear system (\mathfrak C), not endowed with any topology, the methods used in ((^{1-4})) become inapplicable, even if one restricts (G)-projection only to definite lineals. Meanwhile, such a problem has recently arisen more than once in a number of works on quantum field theory (see, for example, ((^7,^8))). The present note is devoted to the solution of this problem and to some related questions.

* In paper ((^3)), (G) is an arbitrary bounded operator.

1. Let (\mathfrak L\;(\subset \mathfrak E)) be a definite lineal, and let (y) be some vector of (\mathfrak E). Consider the linear form (functional) (f_y(x)=[x,y]). If in (\mathfrak L) one introduces the norm

[
|x|=|[x,x]|^{1/2}\qquad (x\in \mathfrak L),
]

then, with respect to it, the functional (f_y(x)) may turn out to be continuous (bounded) or discontinuous. If, for any vector (y\in\mathfrak E), the functional (f_y(x)) is continuous with respect to (|x|) ((x\in\mathfrak L)), then the lineal (\mathfrak L) will be called regular. If, however, for some (y_0\in\mathfrak E) the functional (f_{y_0}(x)) is discontinuous with respect to (|x|), then the lineal (\mathfrak L) will be called singular.

Theorem 1. Let (\mathfrak L\;(\subset \mathfrak E)) be a definite lineal, and let (y_0\in\mathfrak E). In order that the functional (f_{y_0}(x)=[x,y_0]) be bounded with respect to the norm (|x|), ((x\in\mathfrak L)), it is necessary and sufficient that

[
\inf_{x\in\mathfrak L}[x-y_0,x-y_0]=m(y_0)>-\infty,
\tag{1}
]

if the lineal (\mathfrak L) is positive, and

[
\sup_{x\in\mathfrak L}[x-y_0,x-y_0]=M(y_0)<+\infty,
\tag{1'}
]

if the lineal (\mathfrak L) is negative.

Thus, a definite lineal (\mathfrak L) is regular if and only if condition (1) (respectively (1′)) is fulfilled for all (y_0\in\mathfrak E).

We shall formulate another criterion for the regularity of a definite lineal in terms of singular lineals. We shall call a (G)-metric ([x,y]) nondegenerate on the lineal (\mathfrak L) if from the equality ([x_0,y]=0), valid for some (x_0\in\mathfrak L) and all (y\in\mathfrak L), it follows that (x_0=0). In the opposite case ((x_0\ne0)) one says that the (G)-metric degenerates on (\mathfrak L), and (x_0) is called an isotropic vector of the lineal (\mathfrak L) ((^5)).

Theorem 2. Let (\mathfrak L) be a definite lineal in the space (\mathfrak E) with a nondegenerate (G)-metric. In order that the lineal (\mathfrak L) be singular, it is necessary and sufficient that (\mathfrak L\subset \mathfrak E_1\subset \mathfrak E), where (\mathfrak E_1) is a linear system admitting a completion (see ((^5)), Theorem 1.4) to the space (\Pi_1), after which the closure (\overline{\mathfrak L}) of the lineal (\mathfrak L) in the norm of the space (\Pi_1) contains vectors isotropic (in (\mathfrak L)).

In the work ((^8)), A. Wightman proceeds from the assumption that in the case of a complete definite lineal (\mathfrak L), with respect to the norm (|x|=|[x,x]|^{1/2}) ((x\in\mathfrak L)), this lineal is regular. This assertion was used by A. Wightman as the basis of the scheme of quantum mechanics in a space with an indefinite metric developed by him.* Meanwhile, it is easy to give examples refuting A. Wightman’s hypothesis. In essence, such an example of a singular lineal complete in the norm (|x|) had already been constructed in ((^6)) on p. 421; moreover, as the following theorem shows, this example exhausts the entire class of similar lineals.

Theorem 3. In order that a singular lineal (\mathfrak L\;(\subset\mathfrak E)) be complete with respect to the norm (|x|=|[x,x]|^{1/2}), it is necessary and sufficient that, after the isometric embedding of (\mathfrak L) into the space (\Pi_1) (according to Theorem 2), its closure (\overline{\mathfrak L}) admit a decomposition into the direct sum (\overline{\mathfrak L}=\mathfrak L\dotplus{x_0}), where ({x_0}) is a one-dimensional subspace spanned by an isotropic vector (x_0) of the subspace (\overline{\mathfrak L}).

1. The question of the regularity of a definite lineal is closely connected with the problem of (G)-projection. We shall call a definite lineal (\mathfrak L\;(\subset\mathfrak E)) projection-complete if, for every vector (y_0\in\mathfrak E), there exists a (G)-projection (x_0) onto the lineal (\mathfrak L). It is obvious that a necessary condition for projection-completeness is the regularity of the lineal. The following

* Our attention was drawn to work ((^8)) by the Hungarian mathematician János Bognár, who expressed the first considerations calling A. Wightman’s hypothesis into question and gave examples of (incomplete) singular lineals in (\mathfrak E).

Theorem 4 establishes a criterion for the projection completeness of a definite lineal.

Theorem 4. In order that a vector (y_0\in \mathfrak E) have a (G)-projection (x_0) onto a definite lineal (\mathfrak L\subset \mathfrak E), it is necessary and sufficient that the following two conditions be satisfied:

(1^\circ). The functional (f_{y_0}(x)=[x,y_0]) ((x\in \mathfrak L)) is bounded with respect to the norm (|x|). We denote the norm of this functional by (|f_{y_0}|).

(2^\circ). For at least one sequence ({x_n}\subset \mathfrak L) having the property

[
\lim_{n\to\infty}\frac{[x_n,y_0]}{|x_n|}=|f_{y_0}|,
\tag{2}
]

and normalized by the conditions (|x_n|=|f_{y_0}|) ((n=1,2,\ldots)), there exists a limit (x_0\in \mathfrak L): (\lim_{n\to\infty}|x_n-x_0|=0). The vector ({\operatorname{sign}[x_0,x_0]}\cdot x_0) is then the (G)-projection of the vector (y_0) onto the lineal (\mathfrak L).

Thus, the lineal (\mathfrak L) will be projection complete if and only if it is regular and condition (2^\circ) is satisfied for all vectors (y_0\in \mathfrak E).

Let us note that if condition (2^\circ) is satisfied for a given (y_0) for at least one normalized sequence ({x_n}) having property (2), then it will also be satisfied for any such sequence, and the vector (x_0) is determined uniquely. It is easy to see that any sequence having property (2) is fundamental with respect to (|x|) (cf. (1)). Therefore condition (2^\circ) will in any case be satisfied for all vectors (y_0\in \mathfrak E), if the lineal (\mathfrak L) is complete with respect to the norm (|x|).

Thus, regularity and completeness of a definite lineal (\mathfrak L) entail its projection completeness. At the same time, by Theorems 3 and 4, completeness of the lineal (\mathfrak L) alone is not sufficient for its projection completeness.*

1. Returning to Theorem 1, we point out that condition (1) (respectively, ((1'))) was first established by R. Nevanlinna ({}^{(1)}) as necessary for the existence of the (G)-projection of a vector (y_0) onto a definite lineal (\mathfrak L) under the additional restrictions:

1) (\mathfrak E=\mathfrak H) is a Hilbert space with (G)-metric (see item 1).
2) (\mathfrak L) is a (closed) subspace of (\mathfrak H).

There it is also shown ({}^{(1)}) that condition (1) (respectively, ((1'))) is sufficient if (\mathfrak L) is complete (with respect to the norm (|x|)).

Comparison of Theorems 1 and 4 formulated by us shows that the restrictions 1) and 2) imposed by R. Nevanlinna are superfluous.

In the work ({}^{(3)}) of Yu. P. Ginzburg, under the same restrictions 1) and 2), there was first obtained a necessary and sufficient condition for the existence of the (G)-projection of a vector (y_0\in \mathfrak H) onto a subspace (\mathfrak L) (even for the case of a non-Hermitian operator (G)). If (P_{\mathfrak L}) is the orthoprojector from (\mathfrak H) onto (\mathfrak L), then (G_{\mathfrak L}=P_{\mathfrak L}GP_{\mathfrak L}) is the Gram operator of the form ([x,y]) on the subspace (\mathfrak L) (see item 1). It turns out (({}^{(3)}), Theorem 2) that in the case of a Hermitian operator (G) of interest to us, for the existence of the (G)-projection of a vector (y_0) onto (\mathfrak L), it is necessary and sufficient that for every sequence ({x_n}\subset \mathfrak L) having the property

[
\lim_{n\to\infty}|G_{\mathfrak L}x_n|=0,
]

the equality

[
\lim_{n\to\infty}[x_n,y_0]=0
]

hold.

[
\text{* Already after submitting the present article for publication, we became acquainted with the work }{}^{(9)},\text{ in which other criteria for the projection completeness of arbitrary lineals were obtained.}
]

The necessity of this condition is directly evident from the relation

[
[x,y_0]=[x,x_0]=(G_{\Omega}x,x_0)\qquad (x\in\Omega),
]

in which (x_0) denotes the (G)-projection of the vector (y_0) onto (\Omega). Using the closedness of the operator (G_{\Omega}^{-1}) with respect to the norm (|x|), it is not difficult to show that, in its sufficient part, the criterion under consideration follows from our Theorem 4.

Odessa Civil Engineering Institute

Received
17 III 1961

REFERENCES

1. R. Nevanlinna, Ann. Acad. Sci. Fennicae, AI, 163 (1954).
2. J. S. Louhivaara, Ann. Acad. Sci. Fennicae, AI, 252 (1958).
3. Yu. P. Ginzburg, DAN, 139, No. 4 (1961).
4. L. S. Pontryagin, Izv. AN SSSR, ser. matem., 8, 243 (1944).
5. I. S. Iokhvidov, M. G. Krein, Tr. Mosk. matem. obshch., 5, 367 (1956).
6. I. S. Iokhvidov, M. G. Krein, Tr. Mosk. matem. obshch., 8, 413 (1959).
7. R. Ascoli, E. Minardi, Nuclear Phys., 9, No. 2, 249 (1958).
8. A. Uhlmann, Nuclear Phys., 9, No. 4, 429 (1959).
9. E. Scheibe, Ann. Acad. Sci. Fennicae, AI, 294 (1960).