UDC 513.88:517.948.35

MATHEMATICS

I. S. IOKHVIDOV

BANACH SPACES WITH A \(J\)-METRIC.

\(J\)-NONNEGATIVE OPERATORS

(Presented by Academician L. S. Pontryagin on 5 XI 1965)

1. Let a Banach space \(\mathfrak B\) be the direct sum of two of its subspaces: \(\mathfrak B=\mathfrak B_+\dotplus \mathfrak B_-\), and let \(P_\pm\) be the bounded projectors corresponding to this decomposition \((\mathfrak B_\pm=P_\pm\mathfrak B)\). Introduce in \(\mathfrak B\) the so-called \(J\)-metric by putting

\[ J(x)=\|P_+x\|^2-\|P_-x\|^2\qquad (x\in\mathfrak B). \]

(We note that the squares here could be replaced by any other positive power; cf. \((^{1,2})\).) The function \(J(x)\) is, obviously, strongly continuous and, moreover, uniformly (strongly) continuous in every ball \(\|x\|\le C\). Depending on the sign assumed by this function on vectors and sets of vectors from \(\mathfrak B\), these vectors and sets are naturally classified as \(J\)-positive (\(J\)-nonnegative), \(J\)-negative (\(J\)-nonpositive), and \(J\)-neutral.

For the case in which \(\mathfrak B\) is a Hilbert space and \(P_\pm\) are orthoprojectors \((\mathfrak B=\mathfrak B_+\oplus\mathfrak B_-)\), the geometry and the theory of operators in spaces \(\mathfrak B\) with a \(J\)-metric were first studied by L. S. Pontryagin, M. G. Krein, and the author under the assumption \(\min\{\dim\mathfrak B_+,\dim\mathfrak B_-\}=\varkappa<\infty\) \((^{3-8})\), and then (already without the restriction \(\varkappa<\infty\), partially removed earlier in \((^6), \S 13\)) this theory received further development and application in a number of works by Soviet and foreign authors (for a complete bibliography see \((^{9,10})\)). The passage to a Banach space, apparently, was first undertaken by F. Bonsall \((^1)\) \((\varkappa=1)\), then by M. L. Brodskii \((^{11})\) and K. Fan \((^2)\) \((\varkappa<\infty)\), and for \(\varkappa=\infty\) by M. G. Krein \((^{12})\), the author \((^{13})\), and K. Fan \((^{14})\).

2. A \(J\)-nonnegative (\(J\)-nonpositive) subspace is called maximal if it is not a proper part of any other \(J\)-nonnegative (\(J\)-nonpositive) subspace of \(\mathfrak B\). From Zorn’s lemma one obtains (cf. \((^9), \S 3,10\)) the so-called

Maximality principle. Every \(J\)-nonnegative (\(J\)-nonpositive) subspace is contained in some maximal \(J\)-nonnegative (\(J\)-nonpositive) subspace.

The following facts hold (cf. \((^9), \S 6\)):

\(1^\circ\). The projector \(P_+\) \((P_-)\) maps every \(J\)-nonnegative (\(J\)-nonpositive) linear manifold \(\mathfrak L\) from \(\mathfrak B\) homeomorphically onto the linear manifold \(P_+\mathfrak L\subset\mathfrak B_+\) \((P_-\mathfrak L\subset\mathfrak B_-)\). In particular, \(\dim\mathfrak L\le \dim\mathfrak B_+\) \((\dim\mathfrak L\le \dim\mathfrak B_-)\).

\(2^\circ\). If, under the conditions of assertion \(1^\circ\), \(P_+\mathfrak L=\mathfrak B_+\) \((P_-\mathfrak L=\mathfrak B_-)\), then \(\mathfrak L\) is a maximal \(J\)-nonnegative (\(J\)-nonpositive) subspace.

Denote by \(\mathfrak T_+\) \((\mathfrak T_-)\) the class of all subspaces \(\mathfrak L\) of \(\mathfrak B\) that are mapped by the projector \(P_+\) \((P_-)\) one-to-one onto all of \(\mathfrak B_+\) \((\mathfrak B_-)\). With the aid of the Hahn–Banach theorem the following assertion is proved:

\(3^\circ\). Every one-dimensional \(J\)-nonnegative (\(J\)-nonpositive) subspace is contained in some maximal \(J\)-nonnegative (\(J\)-nonpositive) subspace \(\mathfrak L\in\mathfrak T_+\) \((\mathfrak L\in\mathfrak T_-)\).

For subspaces of larger dimension this assertion is, generally speaking, no longer true.

Lemma 1. In order that every \(J\)-nonnegative subspace be contained in some maximal \(J\)-nonnegative subspace of the class \(\mathfrak{T}_{+}\), it is necessary and sufficient that every bounded linear operator acting from an arbitrary subspace \(\mathfrak{B}_{+}^{0}\subset \mathfrak{B}_{+}\) into the subspace \(\mathfrak{B}_{-}\) admit an extension, with the same norm, to an operator acting from all of \(\mathfrak{B}_{+}\) into \(\mathfrak{B}_{-}\).

Remark. As is known \((^{15,16})\), the condition of Lemma 1 will be satisfied if, for example, any one of the following requirements is satisfied:
\(\alpha)\ \mathfrak{B}_{+}\) is a unitary space; \(\beta)\ \mathfrak{B}_{-}\) is a space of type \(\mathfrak{M}\).

An assertion analogous to Lemma 1 holds, of course, also for \(J\)-nonpositive subspaces.

A special role is played by the particular case in which
\[ \min\{\dim \mathfrak{B}_{+},\dim \mathfrak{B}_{-}\}=\varkappa<\infty. \]
For definiteness we shall assume \(\varkappa=\dim \mathfrak{B}_{+}\). In the following lemma \(0\leq \varkappa<\infty\).

Lemma 2. If the linear manifold \(\mathfrak{L}(\subset \mathfrak{B})\) contains a \(\varkappa\)-dimensional \(J\)-positive subspace \(*\), then the same property is possessed by every linear manifold \(\mathfrak{D}(\subset \mathfrak{L})\) dense in \(\mathfrak{L}\) (i.e. such that \(\overline{\mathfrak{D}}\supset \mathfrak{L}\)).

1. A linear operator \(V\), defined on some linear manifold \(\mathfrak{D}_{V}\) in \(\mathfrak{B}=\mathfrak{B}_{+}\dot{+}\mathfrak{B}_{-}\), will be called \(J\)-nonnegative if from \(x\in\mathfrak{D}_{V}\) and \(J(x)\geq 0\) it always follows that \(J(Vx)\geq 0\). \(J\)-nonpositive operators are defined analogously; for them in what follows all assertions are easily obtained from those given in the text.

Theorem 1. Let \(V\) be a \(J\)-nonnegative operator defined on some linear manifold \(\mathfrak{D}_{V}\) in \(\mathfrak{B}\), and let \(\mathfrak{D}_{V}\cap \mathfrak{B}_{+}\ne\{0\}\). The operator \(V\) is bounded if and only if the operator \(P_{+}V\) is bounded.

From this theorem and Lemma 2 one can obtain the following generalization and strengthening of one result of M. L. Brodskii (see \((^{11}), 1^\circ\)):

Theorem 2. Let \(V\) be a \(J\)-nonnegative operator in \(\mathfrak{B}=\mathfrak{B}_{+}\dot{+}\mathfrak{B}_{-}\), and let \(\dim \mathfrak{B}_{+}=\varkappa\ (1\leq \varkappa<\infty)\). If \(\mathfrak{D}_{V}\cap \mathfrak{B}_{+}\ne\{0\}\) and \(\mathfrak{D}_{V}\) contains a \(\varkappa\)-dimensional \(J\)-positive subspace, while the operator \(V\) annihilates no \(J\)-positive vector, then this operator is bounded.

In the particular case when \(\mathfrak{B}\) is a Hilbert space and \(\mathfrak{B}_{\pm}\) are its mutually orthogonal subspaces, the conditions of Theorem 2 imposed on \(\mathfrak{D}_{V}\) are satisfied, for example, when \(\overline{\mathfrak{D}}_{V}=\mathfrak{B}\). The condition requiring that the operator \(V\) not annihilate \(J\)-positive vectors is essential, since even in a Hilbert space \(\mathfrak{B}\), for \(\mathfrak{D}_{V}=\mathfrak{B}\), one can give examples of one-dimensional \(J\)-nonnegative operators which, when this condition is violated, turn out to be unbounded.

Theorem 3. If, in a Banach space \(\mathfrak{B}=\mathfrak{B}_{+}\dot{+}\mathfrak{B}_{-}\), a \(J\)-nonnegative operator \(V\) annihilates some \(J\)-positive vector, then the linear manifold \(\mathfrak{R}_{V}=V\mathfrak{D}_{V}\) is \(J\)-nonnegative (and, consequently, by \(1^\circ\), \(\dim \mathfrak{R}_{V}\leq \dim \mathfrak{B}_{+}\)) (cf. \((^{11}), 1^\circ\)).

Corollary. If, under the conditions of Theorem 3, \(\mathfrak{D}_{V}=\mathfrak{B}\), then the linear manifold \(\mathfrak{R}_{V}\) is a \(J\)-nonnegative invariant linear manifold for the operator \(V\).

1. The question touched upon in the corollary to Theorem 3, concerning the existence for a \(J\)-nonnegative operator of a \(J\)-nonnegative invariant subspace, is connected with a well-known problem in operator theory which has given rise to an extensive literature \((^{1-8,10-14,17})\). The central point in these works is the proof of the existence of maximal \(J\)-nonnegative invariant subspaces of the class \(\mathfrak{T}_{+}\) for various classes of operators in spaces with a \(J\)-metric. The most fruitful idea here proved to be that of M. G. Krein \((^{4-6,12})\) [[unclear: continuation cut off at bottom]]

* A linear manifold (subspace) will be called \(J\)-positive if all its nonzero vectors are \(J\)-positive. \(J\)-negative linear manifolds are defined analogously.

...of obtaining such proofs by means of various fixed-point principles.

The greatest progress in the question under consideration has been achieved in the paper \({}^{(12)}\) (see also \({}^{(13)}\))*, where, in proving the existence of \(J\)-nonnegative invariant subspaces of class \(\mathfrak J_+\) for a \(J\)-nonnegative operator \(V\) with \(\mathfrak D_V=\mathfrak B\), the following requirements are imposed: 1) \(V\) is bounded; 2) the operator \(P_+VP_-\) admits approximation in norm by finite-dimensional operators (in the case when \(\mathfrak B\) is a Hilbert space or a Banach space with a basis, this is equivalent to the requirement that \(P_+VP_-\) be a completely continuous operator).

The assertion contained in a recent paper of K. Fan (\({}^{(14)}\), Corollary 2 of Theorem 2), if valid, would constitute, in the case of its applicability, a further step toward the generalization of the results named in \({}^{(9,12,13,17)}\), since condition 2) is absent from the formulation of this assertion of K. Fan. However, unfortunately, the proof of Corollary 2 in \({}^{(14)}\), as its analysis shows, contains a fundamental error.

Thus, even for \(J\)-unitary operators \(V\) \((\mathfrak D_V=\mathfrak R_V=\mathfrak B,\ J(Vx)=J(x),\ x\in\mathfrak B)\) in a Hilbert space \(\mathfrak B=\mathfrak B_+\oplus\mathfrak B_-\), the possibility of dispensing with condition 2) remains problematic (condition 1) is satisfied automatically in this case \({}^{(18)}\)). Closely connected with this question is another unsolved problem, already for families of commuting \(J\)-unitary operators, posed by R. S. Phillips \({}^{(19)}\) (see the commentary to the translation \({}^{(20)}\) of this article).

Odessa
Civil Engineering Institute

Received
3 XI 1965

CITED LITERATURE

\({}^{1}\) F. F. Bonsall, Quart. J. Math., Oxford, 6 (2), 175 (1955).
\({}^{2}\) K. Fan, Bull. Am. Math Soc., 69, 6, 773 (1963).
\({}^{3}\) L. S. Pontryagin, Izv. AN SSSR, ser. matem., 8, 243 (1944).
\({}^{4}\) M. G. Krein, M. A. Rutman, UMN, 3, 1 (23), 3 (1948).
\({}^{5}\) M. G. Krein, UMN, 5, 2 (36), 180 (1950).
\({}^{6}\) I. S. Iokhvidov, M. G. Krein, Tr. Mosk. matem. obshch., 5, 367 (1956); 8, 413 (1959).
\({}^{7}\) I. S. Iokhvidov, Zap. Kharkovsk. matem. obshch., 21, 4, 79 (1949).
\({}^{8}\) I. S. Iokhvidov, Tr. III Vsesoyuzn. matem. s"ezda, 3, 254 (1958).
\({}^{9}\) Yu. P. Ginzburg, I. S. Iokhvidov, UMN, 17, 4 (106), 3 (1962).
\({}^{10}\) M. G. Krein, Second Summer Mathematical School, Inst. of Mathematics, Academy of Sciences of the Ukrainian SSR, 1965, p. 15.
\({}^{11}\) M. L. Brodskii, UMN, 14, 1 (85), 147 (1959).
\({}^{12}\) M. G. Krein, DAN, 154, No. 5, 1023 (1964).
\({}^{13}\) I. S. Iokhvidov, DAN, 159, No. 3, 501 (1954).
\({}^{14}\) K. Fan, Israel J. Math., 2, No. 1, 19 (1964).
\({}^{15}\) L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semiordered Spaces, 1950.
\({}^{16}\) L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, Moscow, 1959.
\({}^{17}\) H. Langer, Math. Ann., 146, 60 (1962); 152, 434 (1963).
\({}^{18}\) I. S. Iokhvidov, UMN, 16, 4 (100), 167 (1961); 20, 3 (123), 175 (1965).
\({}^{19}\) R. S. Phillips, Proc. Intern. Symp. Lin. Spaces, Jerusalem, 1960, p. 366, 1961.
\({}^{20}\) R. S. Phillips, Sborn. per. Matematika, 8, 6, 81 (1964).

* We take this opportunity to make the following clarification: in Theorem 3 of \({}^{(13)}\), if \(\mathfrak B=\mathfrak B_+\dot{+}\mathfrak B_-\) is a Banach space, then one must additionally require of the operator \(A\) that it send every \(J\)-nonnegative subspace of class \(\mathfrak J_+\) into a subspace of class \(\mathfrak J_+\). In a Hilbert space \(\mathfrak B=\mathfrak B_+\oplus\mathfrak B_-\) this requirement is satisfied automatically by virtue of the maximality principle (see item 2) and item a) of the remark to Lemma 1 of the present note.