UDC 513.88:517.948.35

MATHEMATICS

I. S. IOKHVIDOV

\(J\)-NONEXPANDING OPERATORS IN A BANACH SPACE

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

The present note adjoins our previous communication \((^1)\) and uses the notation, definitions, and results from \((^1)\).

1. In the Banach space \(\mathfrak{B}=\mathfrak{B}_{+}\dot{+}\mathfrak{B}_{-}\) \((\mathfrak{B}_{\pm}=P_{\pm}\mathfrak{B})\) one considers the \(J\)-metric

\[ J(x)=\|P_{+}x\|^{2}-\|P_{-}x\|^{2}\quad (x\in\mathfrak{B}). \tag{1} \]

In \((^1)\) certain properties were established of linear operators \(V\) \((\mathfrak{D}_{V}\subset\mathfrak{B})\) possessing the property of \(J\)-nonnegativity: \(J(Vx)\geqslant 0\) for \(x\in\mathfrak{D}_{V}\) with \(J(x)\geqslant 0\), or of \(J\)-nonpositivity: \(J(Vx)\leqslant 0\) for \(x\in\mathfrak{D}_{V}\) with \(J(x)\leqslant 0\). We shall now consider in more detail one subclass of the class of \(J\)-nonpositive operators.

A linear operator \(V\) with \(\mathfrak{D}_{V}\subset\mathfrak{B}=\mathfrak{B}_{+}\dot{+}\mathfrak{B}_{-}\) will be called \(J^{-}\)-nonexpanding if

\[ J(Vx)\leqslant J(x) \tag{2} \]

for vectors \(x\in\mathfrak{D}_{V}\) having the property \(J(x)\leqslant 0\). If, however, inequality (2) is valid for all \(x\in\mathfrak{D}_{V}\), then the operator \(V\) will be called simply \(J\)-nonexpanding.

In the case when \(\mathfrak{B}\) is a Hilbert (or finite-dimensional unitary) space, and \(\mathfrak{B}_{+}\perp\mathfrak{B}_{-}\), i.e. \(\mathfrak{B}=\mathfrak{B}_{+}\oplus\mathfrak{B}_{-}\), \(J\)-nonexpanding operators have found wide application in the theory of stability of solutions of canonical systems of differential equations (see the works of M. G. Krein \((^{2-6})\) and subsequent works developing these ideas— a detailed exposition of the question and a bibliography are given in \((^7)\)). In turn, these works, in combination with the theory of characteristic matrix functions of M. S. Livshits \((^8)\), gave rise to a series of investigations in the theory of non-self-adjoint operators in Hilbert space, in which the notions of analytic \(J\)-nonexpanding matrix- and operator-functions began to play an essential role \((^{9-11})\); for details and bibliography see \((^{12})\).

On the other hand, in the theory of operators in spaces with an indefinite metric, M. G. Krein \((^{13})\) introduced and studied a more general class of \(J\)-nonexpanding operators*, though with the restriction that \(\dim \mathfrak{B}_{-}=\varkappa<\infty\). Yu. P. Ginzburg \((^{10,14})\) began to develop the general theory of \(J\)-nonexpanding operators in a Hilbert space \(\mathfrak{B}=\mathfrak{B}_{+}\oplus\mathfrak{B}_{-}\) (without restrictions on \(\dim \mathfrak{B}_{\pm}\)). In all these works the \(J\)-nonexpanding (and more general) operators were assumed a priori to be bounded and defined on the whole space.

* More precisely, in \((^{13})\) the discussion concerned operators which in our terminology should be called \(J^{+}\)-noncontracting \((\dim \mathfrak{B}_{+}<\infty)\).

In the present note we consider \(J\)- and \(J\)-nonexpanding operators defined on arbitrary lineals of a Banach space
\[ \mathfrak B=\mathfrak B_+\dot{+}\mathfrak B_-, \]
and are interested chiefly in questions of their closedness and boundedness. In doing so, some facts are found that had previously gone unnoticed even for \(J\)-isometric operators \(V\) \((J(Vx)=J(x),\ x\in\mathfrak D_V)\) in a Hilbert space.

1. We begin by setting out a number of simple properties of the operators of interest to us, for which purpose we introduce one more definition.

A lineal \(\mathfrak L\) in \(\mathfrak B\) will be called (cf. \((^{15})\)) uniformly \(J\)-positive (uniformly \(J\)-negative) if for all \(x\in\mathfrak L\) we have
\[ J(x)\ge c\|x\|^2 \]
(respectively
\[ J(x)\le -c\|x\|^2), \qquad c=\operatorname{const}>0. \]
The simplest examples of such lineals are \(\mathfrak B_+\) (respectively \(\mathfrak B_-\)).

\(1^\circ\). A \(J\)-nonexpanding operator \(V\) carries every \(J\)-negative lineal \(\mathfrak L\subset\mathfrak D_V\) into a \(J\)-negative lineal \(V\mathfrak L\). Moreover, if the operator \(V\) is bounded, then every uniformly \(J\)-negative lineal \(\mathfrak L\) is mapped by the operator \(V\) homeomorphically onto a uniformly \(J\)-negative lineal \(V\mathfrak L\).

\(2^\circ\). The nullifying lineal
\[ \mathfrak Z=\{x:\ x\in\mathfrak D_V,\ Vx=0\} \]
of a \(J\)-nonexpanding operator \(V\) is \(J\)-nonpositive.

It is curious that, as examples show, this latter assertion (which follows directly from \(1^\circ\)) cannot, when \(\mathfrak D_V\ne\mathfrak B\), be strengthened (i.e. one cannot assert that \(\mathfrak Z\) is a \(J\)-positive lineal) either by imposing continuity of the operator \(V\), or by passing from \(J\)-nonexpanding operators to \(J\)-isometric ones (even in a finite-dimensional unitary space). At the same time, for \(\mathfrak D_V=\mathfrak B\) the following proposition of Yu. P. Ginzburg holds \((^{14,15})\):

\(3^\circ\). The nullifying subspace \(\mathfrak Z\) of a bounded \(J\)-nonexpanding operator defined on the whole Hilbert space
\[ \mathfrak B=\mathfrak B_+\oplus\mathfrak B_- \]
is uniformly \(J\)-positive.

From \(2^\circ\) it follows that the nullifying lineal \(\mathfrak Z\) of a \(J\)-isometric operator is \(J\)-neutral.

We shall call a \(J\)-isometric operator \(V\) \(J\)-semiunitary if \(\mathfrak D_V=\mathfrak B\), and \(J\)-unitary if \(\mathfrak D_V=\mathfrak B\) and \(\mathfrak R_V\equiv V\mathfrak D_V=\mathfrak B\).

\(4^\circ\). If the domain \(\mathfrak D_V\) of a \(J\)-isometric operator \(V\) is “decomposable,” i.e. is representable as a direct sum
\[ \mathfrak D_V=\mathfrak D_+\dot{+}\mathfrak D_- \]
of a \(J\)-positive \((\mathfrak D_+)\) and a \(J\)-negative \((\mathfrak D_-)\) lineal, then \(\mathfrak Z=\{0\}\), i.e. the operator \(V\) is one-to-one on its (also decomposable) range \(\mathfrak R_V\), and the inverse operator \(V^{-1}\) is again \(J\)-isometric. In particular, every \(J\)-semiunitary operator \(V\) is one-to-one, and if \(V\) is \(J\)-unitary, then \(V^{-1}\) is also \(J\)-unitary.

The last assertion immediately implies

\(5^\circ\). A \(J\)-unitary operator carries every maximal \((^1)\) \(J\)-nonnegative (\(J\)-nonpositive) subspace into a maximal subspace of the same “sign.”

1. Inequality (2), proceeding from (1), can be rewritten in a form more convenient in many cases:
   \[ \|P_+Vx\|^2+\|P_-x\|^2\le \|P_-Vx\|^2+\|P_+x\|^2. \tag{3} \]

Using this inequality, it is not difficult to obtain the following strengthening of the general theorem 1 from \((^1)\) for the special case of \(J\)-nonexpanding operators:

Theorem 1. For the boundedness of a \(J\)-nonexpanding operator \(V\), defined on an arbitrary lineal \(\mathfrak D_V(\subset\mathfrak B=\mathfrak B_+\dot{+}\mathfrak B_-)\), it is necessary and sufficient that the operator \(P_-V\) be bounded.

In turn, theorem 2 from \((^1)\) gives the following result for \(J\)-nonexpanding operators:

Theorem 2. If a \(J\)-nonexpanding operator \(V\) is defined on the whole Banach space \(\mathfrak B=\mathfrak B_+ \dotplus \mathfrak B_-\) and \((1\le)\varkappa=\dim\mathfrak B_-<\infty\), then the operator \(V\) is bounded.

As the examples show (see \((^{16})\)), the condition \(\varkappa<\infty\) is essential in Theorem 2 even in the case of a Hilbert space \(\mathfrak B=\mathfrak B_+\oplus\mathfrak B_-\). As for the condition \(\mathfrak D_V=\mathfrak B\), for a \(J\)-nonexpanding operator \(V\) it can be weakened by requiring only that \(\mathfrak D_V\) contain a \(\varkappa\)-dimensional \(J\)-negative subspace. This is not difficult to verify by comparing Theorem 1 of the present note with Theorem 2 of \((^1)\).

1. Our further results rest on the following two lemmas, of which the second has a purely algebraic character, while the first contains, in addition to algebraic assertions, other assertions obtained with the aid of inequality (3).

Lemma 1. If \(V\) is a \(J\)-nonexpanding operator, then the operators \(P_+\dotplus P_-V\) and \(P_+\dotplus VP_-\) are one-to-one invertible. Moreover, if \(V\) is \(J\)-noncontracting, then the operators \((P_+\dotplus P_-V)^{-1}\) and \((P_+\dotplus VP_-)^{-1}\) are bounded.

Lemma 2. Let \(\mathfrak L=\mathfrak L_+\dotplus\mathfrak L_-\) be an arbitrary (not necessarily normed) linear space, \(\mathfrak L_\pm\) two of its subspaces, and \(P_\pm\) the projectors from \(\mathfrak L\) onto \(\mathfrak L_\pm\), respectively, defined by the decomposition \(\mathfrak L=\mathfrak L_+\dotplus\mathfrak L_-\). Let \(T\) be a linear operator given everywhere in \(\mathfrak L\). The following three assertions are equivalent:

1) the operator \(P_+\dotplus TP_-\) maps \(\mathfrak L\) one-to-one onto all of \(\mathfrak L\);

2) the operator \(P_+\dotplus P_-T\) maps \(\mathfrak L\) one-to-one onto all of \(\mathfrak L\);

3) the operator \(P_-TP_-\) maps \(\mathfrak L_-\) one-to-one onto all of \(\mathfrak L_-\).

Of course, the analogue of Lemma 2 is also valid in which the operators \(P_+\) and \(P_-\) and the subspaces \(\mathfrak L_+\) and \(\mathfrak L_-\) are interchanged. In addition, in assertions 1) and 2) the sums of operators may be replaced by the differences of the same operators.

Theorem 3. In order that a \(J\)-nonexpanding operator \(V\) with \(\mathfrak D_V\subset\mathfrak B=\mathfrak B_+\dotplus\mathfrak B_-\) be closed, it is necessary and sufficient that the operator \(P_-V\) be closed.

From this theorem and Lemma 1 one obtains

Corollary 1. For a \(J\)-nonexpanding operator \(V\) to be closed, it is necessary that both lineals \((P_+\dotplus P_-V)\mathfrak D_V\) be closed, and sufficient that at least one of them be closed.

In combination with Theorem 1 this implies

Corollary 2. In the case where at least one of the two lineals \(\mathfrak D_V\) and \((P_+-P_-V)\mathfrak D_V\) is closed, for the boundedness of a \(J\)-nonexpanding operator \(V\) it is necessary and sufficient that the second lineal also be closed. In particular, every \(J\)-nonexpanding operator \(V\) defined everywhere in \(\mathfrak B\) and such that \((P_+-P_-V)\mathfrak B=\mathfrak B\) is bounded.

The last assertion of Corollary 2 is immediately connected with Lemma 2, from which it is seen that the condition \((P_+-P_-V)\mathfrak B=\mathfrak B\) may be replaced by the equivalent condition (taking Lemma 1 into account) \(P_-V\mathfrak B_-=\mathfrak B_-\), where the mapping \(P_-V:\mathfrak B_-\to\mathfrak B_-\) is one-to-one. This fact plays a role in the theory of the so-called bi-nonexpanding \(J\)-nonexpanding operators in Hilbert space \((^{10,14})\).

Using the terminology from \((^1)\), the last result may be formulated also as follows:

\(6^\circ\). If \(V\) is a \(J\)-nonexpanding operator and \(\mathfrak D_V=\mathfrak B=\mathfrak B_+\dotplus\mathfrak B_-\), then for the boundedness of the operator \(V\) it is necessary and sufficient that the space \(V\mathfrak B_-\) belong to the class \(\mathfrak T_-\). This condition is always fulfilled, for example, for \(J\)-unitary operators (see \(5^\circ\)) in Hilbert space \((^1)\), whence there follows a new proof of the known fact that all such operators are bounded \((^{16,17})\).

In conclusion, we state a theorem, one special case of which was already reported by us in the form of a footnote in paper (16):

Theorem 4. In a Banach space $\mathfrak{B}=\mathfrak{B}_{+}+\mathfrak{B}_{-}$, where $\dim \mathfrak{B}_{-}=\chi<\infty$, every $J$-nonexpanding operator $V$ ($\mathfrak{D}_{V}\subset \mathfrak{B}$) is bounded if and only if it admits a closure.

Odessa Civil Engineering Institute

Received
3 XI 1965

REFERENCES

1. I. S. Iokhvidov, DAN, 169, No. 2 (1966).
2. M. G. Krein, UMN, 3, 3 (25), 166 (1948).
3. M. G. Krein, DAN, 73, 445 (1950).
4. M. G. Krein, UMN, 6, 1 (41), 171 (1951).
5. M. G. Krein, Ukr. Mat. Zh., 3, No. 2, 164 (1951).
6. M. G. Krein, Collection in Memory of A. A. Andronov, Moscow, 1955, p. 413.
7. M. G. Krein, Lectures on the Theory of Stability of Solutions of Differential Equations in Banach Space, Kiev, 1964.
8. M. S. Livshits, Mat. sbornik, 34 (76), 1, 145 (1954).
9. V. P. Potapov, Tr. Mosk. Mat. Obshch., 4, 125 (1955).
10. Yu. P. Ginzburg, DAN, 117, No. 2, 171 (1957).
11. Yu. P. Ginzburg, Izv. Vyssh. Uchebn. Zaved., Mathematics, No. 1 (32), 42 (1962).
12. M. S. Brodskii, M. S. Livshits, UMN, 13, 1 (79), 3 (1958).
13. M. G. Krein, UMN, 5, 2 (36), 180 (1950).
14. Yu. P. Ginzburg, Scientific Notes of the Physics and Mathematics Faculty of the Odessa Pedagogical Institute, 22, 1, 13 (1958).
15. M. G. Krein, Second Summer Mathematical School, Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, Kiev, 1965, p. 15.
16. I. S. Iokhvidov, UMN, 20, 3 (123), 175 (1965).
17. I. S. Iokhvidov, UMN, 16, 4 (100), 167 (1961).