UDC 517.948.35:513.88

MATHEMATICS

M. A. NAIMARK

AN ANALOGUE OF STONE’S THEOREM IN A SPACE WITH AN INDEFINITE METRIC

(Presented by Academician L. S. Pontryagin on 18 I 1966)

1. Let \(\mathfrak H\) be a Hilbert space; let \(P\) be an orthoprojector in \(\mathfrak H\), different from 0 and 1; let \(Q=1-P\), \(J=P-Q\), \(\varkappa=\min\{\dim P,\dim Q\}\). Put
   \([x,y]=(Jx,y)\), where \((x,y)\) is the scalar product in \(\mathfrak H\). The space \(\mathfrak H\), with the indefinite form \([x,y]\) thus defined, is called a space \(\Pi_{\varkappa}\).* In a natural way one introduces the notions of an isometric, unitary, Hermitian, and self-adjoint operator with respect to \([x,y]\); they are called, respectively, a \(J\)-isometric, \(J\)-unitary, \(J\)-Hermitian, and \(J\)-self-adjoint operator in \(\Pi_{\varkappa}\).

Let \(A\) be a \(J\)-Hermitian operator. If there exists a non-real number \(\lambda\) such that \(\lambda\) and \(\bar\lambda\) belong to the resolvent set of \(A\), then, as is not hard to show, \(A\) is a \(J\)-self-adjoint operator.

1. A family \(U(t)\), \(-\infty<t<\infty\), of operators in \(\Pi_{\varkappa}\) is called a one-parameter group of \(J\)-unitary operators in \(\Pi_{\varkappa}\) if 1) \(U(t)\) is a \(J\)-unitary operator for each \(t\in(-\infty,\infty)\); 2) \(U(t_1+t_2)=U(t_1)U(t_2)\) for all \(t_1,t_2\in(-\infty,\infty)\); 3) \(U(t)x\) is a strongly continuous (in the sense of the norm in \(\Pi_{\varkappa}=\mathfrak H\)) function of \(t\) on \((-\infty,\infty)\) for every \(x\in\Pi_{\varkappa}\).

An analogue of the well-known Stone theorem is the following

Theorem 1. The generating operator \(A\) of any one-parameter group of \(J\)-unitary operators in \(\Pi_{\varkappa}\) has the form \(A=iH\), where \(H\) is a \(J\)-self-adjoint operator in \(\Pi_{\varkappa}\) such that, for all sufficiently large in modulus integers \(n\) \((|n|\ge n_0)\) and some constants \(M>0\), \(\beta>0\): 1) \((1-in^{-1}H)^{-1}\) exists and is bounded; 2)
\[ \left\|(1-in^{-1}H)^{-m}\right\| \le M(1-|n^{-1}|\beta)^{-m},\quad m=1,2,3,\ldots . \]
Conversely, for every \(J\)-self-adjoint operator \(H\) in \(\Pi_{\varkappa}\) satisfying conditions 1) and 2), the operator \(A=iH\) is the generating operator of a one-parameter group of \(J\)-unitary operators \(U(t)\) in \(\Pi_{\varkappa}\). The correspondence \(U(t)\leftrightarrow H\), thus established, is one-to-one.

Proof. Let \(U(t)\) be a one-parameter group of \(J\)-unitary operators in \(\Pi_{\varkappa}\). Considering \(\Pi_{\varkappa}\) as a Banach space with the usual Hilbert norm in \(\mathfrak H\), we conclude that (see, for example, \((^4)\), Ch. IX, Secs. 1, 2 and 9): a) \(\|U(t)\|\le Me^{\beta|t|}\) for some constants \(M>0\), \(\beta>0\); b) the generating operator \(A\) of the one-parameter group \(U(t)\) satisfies the conditions: b\(_1\)) the domain of definition \(D(A)\) of the operator \(A\) is dense in \(\Pi_{\varkappa}\); b\(_2\)) \((1-n^{-1}A)^{-1}\) exists and is bounded and
\[ \left\|(1-n^{-1}A)^{-m}\right\| \le M(1-|n^{-1}|\beta)^{-m},\quad m=1,2,\ldots, \]
for all sufficiently large in modulus integers \(n\).

Put \(A=iH\) and prove that \(H\) is a \(J\)-self-adjoint operator in \(\Pi_{\varkappa}\). Since \(U(t)\) is a \(J\)-unitary operator, we have \([U(t)x,U(t)y]=[x,y]\) for all \(x,y\in\Pi_{\varkappa}\). Taking here \(x,y\in D(H)=D(A)\) and differentiating with respect to \(t\) at the point \(t=0\), we obtain
\[ [Ax,y]+[x,Ay]=0,\quad \text{i.e. } i[Hx,y]- \]

* For \(\varkappa<\infty\) the basic properties of the space \(\Pi_{\varkappa}\) and of operators in it are considered in \((^1,^2)\); for more general spaces with indefinite metric see \((^3)\).

\(-i[x, Hy] = 0\) for all \(x, y \in D(H)\). This means that \(H\) is \(J\)-Hermitian. Setting \(A = iH\) in b\(_2\)), we conclude that \(H\) satisfies conditions 1) and 2). But then, by the remark at the end of Sec. 1, \(H\) is a \(J\)-self-adjoint operator in \(\Pi_\varkappa\). Conversely, let \(H\) be a \(J\)-self-adjoint operator in \(\Pi_\varkappa\) satisfying conditions 1) and 2). Put \(A = iH\). Then the operator \(A\) satisfies conditions b\(_1\)) and b\(_2\)), and therefore (see, for example, (4), Ch. IX, Sec. 9) \(A\) is the generating operator of a strongly continuous group of bounded operators \(U(t)\) in \(\Pi_\varkappa\). We shall prove that \(U(t)\) is \(J\)-unitary. Since \(\frac{d}{dt}U(t)x = U(t)Ax = AU(t)x\) for all \(x \in D(A)\), we have

\[ \frac{d}{dt}[U(t)x,\ U(t)y] = [AU(t)x,\ U(t)y] + [U(t)x,\ Au(t)y] = i\{[HU(t)x,\ U(t)y] - [U(t)x,\ HU(t)y]\} = 0; \]

hence

\[ [U(t)x,\ U(t)y] = \operatorname{const} = [U(0)x,\ U(0)y] = [x,y]. \]

Consequently, \(U(t)\) is \(J\)-isometric; its \(J\)-unitarity now follows from the equality

\[ U(t)U(-t) = U(-t)U(t) = U(0) = 1. \]

Finally, the correspondence \(U(t) \to H\) is one-to-one, since the correspondence \(U(t) \to A\) is one-to-one.

If \(iH\) is the generating operator of the group \(U(t)\), we shall write \(U(t)=\exp(itH)\). This notation is justified by the fact that, for some set \(D\) dense in \(\Pi_\varkappa\),

\[ U(t)x = x + \frac{1}{1!}\,itHx + \frac{1}{2!}(itH)^2x + \cdots \quad \text{for } x \in D, \]

where the series on the right-hand side converges absolutely (see, for example, (5)).

Remark 1. If \(A=iH\) is the generating operator of a one-parameter group of \(J\)-unitary operators in \(\Pi_\varkappa\), then conditions 1) and 2) of Theorem 1 are also fulfilled for all nonzero \(n\) for which \(|\operatorname{Im} n|\) is greater than some constant.

Remark 2. If \(H = H_1 + iH_2\), where \(H_1, H_2\) are ordinary self-adjoint operators in \(\mathfrak H = \Pi_\varkappa\), and \(H_2\) is bounded, while \(H\) is a \(J\)-self-adjoint operator in \(\Pi_\varkappa\), then \(H\) satisfies conditions 1) and 2) of Theorem 1.

Indeed,

\[ 1 - in^{-1}H = 1 - in^{-1}H_1 + n^{-1}H_2 = \]

\[ = (1 - in^{-1}H_1)\,[1 + (1 - in^{-1}H_1)^{-1}n^{-1}H_2], \tag{1} \]

where \((1 - in^{-1}H_1)^{-1}\) exists and
\(\|(1 - in^{-1}H_1)^{-1}\| \leq 1\). Therefore

\[ \|(1 - in^{-1}H_1)^{-1}n^{-1}H_2\| \leq |n|^{-1}\|H_2\| < 1 \quad \text{when } |n| > \|H_2\|. \tag{2} \]

Consequently, for \(|n| > \|H_2\|\) there exists
\((1 + (1 - n^{-1}H_1)^{-1}n^{-1}H_2)^{-1}\), and

\[ \|(1 + (1 - n^{-1}H_1)^{-1}n^{-1}H_2)^{-1}\| \leq (1 - |n|^{-1}\|H_2\|)^{-1}. \]

But then from (1) and (2) we conclude that for \(|n| > \|H_2\|\) there also exists

\[ (1 - in^{-1}H)^{-1} = [1 + (1 - in^{-1}H_1)^{-1}n^{-1}H_2]^{-1}(1 - in^{-1}H_2)^{-1} \]

and

\[ \|(1 - in^{-1}H)^{-m}\| \leq (1 - |n|^{-1}\|H_2\|)^{-m},\quad m=1,2,\ldots . \]

From Remark 2 and Theorem 1 there follows

Theorem 2. In the space \(\Pi_\varkappa\), \(\varkappa < \infty\), every one-parameter group of \(J\)-unitary operators \(U(t)\) is represented in the form \(U(t)=\exp(itH)\), where \(H\) is a \(J\)-self-adjoint operator; conversely, for every \(J\)-self-adjoint \(H\) there exists such a one-parameter group of \(J\)-unitary operators \(U(t)\) that \(U(t)=\exp(itH)\).

Indeed, every \(J\)-self-adjoint operator in \(\Pi_\varkappa\), \(\varkappa < \infty\), can be represented in the form \(H = H_1 + iH_2\), where \(H_1, H_2\) are ordinary self-adjoint operators and \(H_2\) is bounded.

1. Theorem 1 carries over, with the appropriate changes, to strongly continuous one-parameter semigroups of \(J\)-isometric bounded operators \(U(t)\), \(t \geq 0\). In this case, in conditions 1 and 2 of Theorem 1, \(n\) must be regarded as positive, and \(H\) as a maximal \(J\)-Hermitian operator*.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
14 I 1966

REFERENCES

1. L. S. Pontryagin, Izv. Akad. Nauk SSSR, Ser. Mat., 8, 243 (1944).
2. I. S. Iokhvidov, M. G. Krein, Tr. Mosk. Mat. Obshch., 5, 367 (1956).
3. Yu. P. Ginzburg, I. S. Iokhvidov, UMN, 17, 3 (1962).
4. K. Yosida, Functional Analysis, Berlin, 1965.
5. I. M. Gelfand, DAN, 25, 711 (1939).
6. M. G. Krein, Proceedings of the IV All-Union Mathematical Congress, Leningrad, 1961, p. 1333.
7. Shah Tao-shing, Sci. Sinica, 11, No. 9, 1147 (1962).

* Note added in proof. As the author has learned, Theorem 2 was formulated by M. G. Krein in his lecture at the IV All-Union Mathematical Congress (6) and proved by Shah Tao-shing (7).