Abstract
Full Text
V. I. Derguzov
On the Stability of Solutions of Hamiltonian Equations in Hilbert Space with Unbounded Periodic Operator Coefficients
(Presented by Academician V. I. Smirnov on 10 V 1963)
Let us consider the Hamiltonian equation
\[ J \frac{dx}{dt} = H(t)x \tag{1} \]
in the complete separable complex Hilbert space \(\mathfrak{H}\). Here \(J\) is a bounded operator with bounded inverse, anti-Hermitian \((J^*=-J)\); \(H(t)\) is an unbounded symmetric \(T\)-periodic operator, subject to certain general conditions formulated below. Equation (1) often occurs in applications \((^1)\). In the case of a finite-dimensional space \(\mathfrak{H}\), M. G. Krein found sufficient conditions for strong stability of the solutions of equation (1) in terms of the type of points of the spectrum of the monodromy operator \((^2)\). I. M. Gelfand and V. B. Lidskii \((^3)\) showed that these conditions are also necessary (for another proof see \((^4)\)). In the present paper these results, with natural modifications, are carried over to equation (1).
\(1^\circ\). We shall assume that \(H(t)\) is equal to the sum of two operators
\[ H(t)=H_0(t)+H_1(t), \tag{2} \]
where \(H_0(t)\) is a self-adjoint positive-definite operator, \(H_0^*(t)=H_0(t)\geq \beta I\) \((\beta=\mathrm{const}>0)\), and the following condition is satisfied:
A. The domain of definition \(D(H_0^{1/2})\) of the positive square root \(H_0^{1/2}(t)\) of the operator \(H_0(t)\) is constant, and for \(t,t'\in[0,T]\), for any elements \(x,y\in D(H_0^{1/2})\), the estimate
\[ \left| \left(H_0^{1/2}(t)x,H_0^{1/2}(t)y\right) - \left(H_0^{1/2}(t')x,H_0^{1/2}(t')y\right) \right| \leq \]
\[ \leq \mathrm{const}\cdot |t-t'|\, \|H_0^{1/2}(t)x\|\cdot \|H_0^{1/2}(t)y\| \]
holds.
The operator \(H_1(t)\) is, in the following sense, subordinated to the operators \(J\) and \(H_0(t)\):
B. For almost all \(t\in[0,T]\), the symmetric operator \(H_1(t)\) is defined on the set \(D(H_0^{1/2})\); the operator
\[ A(t)=H_0^{1/2}(0)J^{-1}H_1(t)H_0^{-1/2}(0), \tag{3} \]
is meaningful, is bounded for almost all \(t\in[0,T]\), is strongly measurable on \([0,T]\), and \(\|A(t)\|\in \mathcal{L}(0,T)\).
Definition 1. We shall call a function \(x(t)\) a generalized solution of equation (1) if it has the following properties:
1) \(x(t)\in D(H_0^{1/2})\) for all \(t\), and the function \(H_0^{1/2}(0)x(t)\) is weakly continuous;
2) for almost all \(t\), the function \(H_0^{-1/2}(0)Jx(t)\) is strongly differentiable, and \(x(t)\) satisfies, for these \(t\), the equation
\[ \frac{d}{dt}\left[H_0^{-1/2}(0)J(x,t)\right] = \left[H_0^{1/2}(t)H_0^{-1/2}(0)\right]^* H_0^{1/2}(t)x(t) + H_0^{-1/2}(0)H_1(t)x(t) \]
and, by weak continuity, the initial condition.
Theorem 1. If conditions A and B are satisfied for the operator (2), then equation (1) with the initial condition \(x(0)\in D(H_0^{1/2})\) has a unique generalized solution \(x(t)\). The resolving operator \(X(t)\), defined by the formula \(X(t)x(0)=x(t)\), is representable in the form \(X(t)=H_0^{-1/2}(0)z(t)H_0^{1/2}(0)\).
The operator \(Z(t)\) is bounded together with its inverse uniformly in \(t \in [0,T]\) and satisfies the relation \(Z^*(t)FZ(t)=F\), where
\[ F=iH_0^{-1/2}(0)JH_0^{-1/2}(0). \tag{4} \]
By means of the resolution of the identity \(E_\lambda\) of the self-adjoint operator \(F\), define the bounded operator
\[ |F|^{1/2}=\int_{-\infty}^{+\infty}|\lambda|^{1/2}\,dE_\lambda, \tag{5} \]
whose inverse is unbounded.
Let the operator \(H_0(t)\) in formula (2) be fixed, and let \(M\) be the set of all operators \(H_1(t)\) satisfying condition B. Introduce in \(M\) the distance between the operators \(H_1=H_1(t)\) and \(\widetilde H_1=\widetilde H_1(t)\) by the formula
\[ \rho(H_1,\widetilde H_1) = \int_0^T \left\| |F|^{1/2}\bigl(A(t)-\widetilde A(t)\bigr)|F|^{-1/2} \right\|\,dt, \tag{6} \]
where the operator \(A(t)\) is defined by formula (3), and
\[ \widetilde A(t)=H_0^{1/2}(0)J^{-1}\widetilde H_1(t)H_0^{-1/2}(0). \]
We note that for any pair of operators \(H_1(t)\) and \(\widetilde H_1(t)\) in \(M\), \(\rho(H_1,\widetilde H_1)<\infty\). The latter assertion follows from the estimate
\[ \left\| |F|^{1/2}\bigl(A(t)-\widetilde A(t)\bigr)|F|^{-1/2} \right\| \le \|A(t)-\widetilde A(t)\|, \]
valid for almost all \(t\).
Definition 2. Equation (1) is called stable if, for every generalized solution \(x(t)\) of it, the estimate
\[ \bigl\||F|^{1/2}H_0^{1/2}(0)x(t)\bigr\| \le \mathrm{const}\cdot \bigl\||F|^{1/2}H_0^{1/2}(0)x(0)\bigr\| \tag{7} \]
holds.
If, in addition, all equations (1) are stable under small changes of the operator \(H_1(t)\) in the metric (6), then equation (1) is called strongly stable.
The main result is formulated in the following theorem.
Theorem 2. Suppose that on the period \([0,T]\) the coefficients of equation (1) satisfy the conditions of Theorem 1; \(X(t)\) and \(F\) are the operators defined in Theorem 1. In order that equation (1) be strongly stable, it is sufficient that the spectrum of the operator \(Y(T)=|F|^{1/2}Z(T)|F|^{-1/2}\) have no points of mixed kind.
The definition of the kind of points of the spectrum of the operator \(Y(T)\) is analogous to the finite-dimensional case and will be given below.
There exists an example showing that if, in inequality (7), the operator \(|F|^{1/2}\) is replaced by the identity operator \(I\), then the assertion of Theorem 2 loses its force.
\(2^\circ\). Definition 3. If the operator \(Z\) is bounded together with its inverse and satisfies the relation \(Z^*FZ=F\), then it is called \(F\)-unitary.
Definition 4. A bounded projector \(P\) satisfying the relation \(FP=P^*F\) is called a projector of the first kind (a projector of the second kind) with respect to the operator \(F\), if \((FPx,x)>0\) \(((FPx,x)<0)\) for every \(x=Px\ne0\).
Definition 5. Suppose the entire spectrum of the operator \(Z\) can be surrounded by a finite number of closed contours \(\Gamma_j\), pairwise nonintersecting and not intersecting the spectrum of \(Z\), each of which is located symmetrically with respect to the unit circle. Suppose that each of the projectors
\[ P_j=\frac{1}{2\pi i}\int_{\Gamma_j}(\xi I-Z)^{-1}\,d\xi, \]
is a projector of the first or second kind with respect to \(F\). Then one says that the \(F\)-unitary operator \(Z\) has no points of spectrum of mixed kind.
Definition 6. The operator \(Z\) is called stable if
\[ \left\|\,|F|^{1/2} Z^n |F|^{-1/2} x\,\right\| \leqslant \mathrm{const}\cdot \|x\| \qquad \left(x \in D\left(|F|^{-1/2}\right)\right) \]
for \(n=\pm 1,\pm 2,\ldots\). If, in addition, all \(F\)-unitary operators from some neighborhood of \(Z\) in the uniform operator topology are stable, then \(Z\) is called strongly stable.
One can give an example of an \(F\)-unitary operator \(Z\), having no spectral points of mixed type, for which the set of norms of integral powers is unbounded. However, the following holds:
Theorem 3. In order that an \(F\)-unitary operator \(Z\) be strongly stable, it is sufficient that it have no spectral points of mixed type.
Theorem 4. If \(Z\) is an \(F\)-unitary operator, then the operators
\[ Y=|F|^{1/2}Z|F|^{-1/2},\qquad Y^{-1}=|F|^{1/2}Z^{-1}|F|^{-1/2}, \tag{8} \]
defined on the set \(D(|F|)^{-1/2}\), are bounded. If \(\lambda\) and \((\overline{\lambda})^{-1}\) are regular points of the operator \(Z\), then they will also be regular points for the operator \(Y\).
It is now clear that Theorem 3 asserts the boundedness of the powers of the operators \(Y\) and \(Y^{-1}\), obtained from the operator \(Z\) by formulas (8).
By means of the resolution of the identity \(E_\lambda\) of the operator \(F\), introduce the bounded symmetric operator, together with its inverse,
\[ G=\int_0^{+\infty} dE_\lambda-\int_{-\infty}^{0} dE_\lambda . \tag{9} \]
From Theorem 4 there follows
Corollary. If an \(F\)-unitary operator \(Z\) has no spectral points of mixed type, then the \(G\)-unitary operator \(Y\), obtained by formulas (8) from the operator \(Z\), has no spectral points of mixed type.
Examples show the possibility of a case in which an \(F\)-unitary operator \(Z\) has spectral points of mixed type, while the \(G\)-unitary operator \(Y\), defined from \(Z\) by formulas (8), has no spectral points of mixed type. In particular, this indicates that Theorem 3, applied to the \(G\)-unitary operator \(Y\), has a wider range of application than the same theorem applied to the \(F\)-unitary operator \(Z\). Since the operator \(G\) is bounded together with its inverse, Theorem 3, applied to the \(G\)-unitary operator \(Y\), asserts the boundedness of the powers of the operator \(Y\).
It can be shown that small changes of the operator \(H_1(t)\) in the metric (6) correspond to small changes, in the uniform operator topology, of the \(G\)-unitary operator \(Y(t)\) defined in Theorem 2. Using Theorem 4 and Theorem 3 applied to the \(G\)-unitary operator \(Y(t)\), it is easy to prove Theorem 2.
Received
2 V 1963
REFERENCES
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