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MATHEMATICS
E. M. DZHABRAILOVA
ON THE COMPLETENESS OF THE SYSTEM OF FINITE-DIMENSIONAL INVARIANT SUBSPACES OF A NONSELFADJOINT SINGULAR DIFFERENTIAL OPERATOR OF SECOND ORDER WITH RESPECT TO A REAL WEIGHT FUNCTION
(Presented by Academician L. S. Pontryagin, 17 III 1964)
In the present paper we consider the question of the completeness of the linear span of the finite-dimensional invariant subspaces of a pair of operators \((L,Q)\), connected with the equation
\[ -y''+p(x)y=\lambda q(x)y \tag{1} \]
and acting in \(L^2(-\infty,+\infty)\). In equation (1), \(p(x)=p_1(x)+ip_2(x)\) is a complex-valued function of the real parameter \(x\), \(p_1(x)\) and \(p_2(x)\) are summable on every finite interval of the real axis, and \(q(x)\) is a complex essentially bounded function. We shall assume that the condition
\[ \operatorname{Re} p(x)=p_1(x)\geqslant 1 \]
is satisfied.
Under this condition the operator \(L_0\), generated by the differential expression
\[ l(y)=-y''+(p_1(x)+ip_2(x))y \]
and defined on the manifold \(D_0\) of finite functions \(\mathcal L(x)\), each of which is absolutely continuous together with its derivative and for which \(l(z)\in L^2(-\infty,+\infty)\), admits the closure \(\overline{L}_0=L\).
The domain of definition \(D(L)\) of the operator \(L\) is determined by the conditions:
a) \(y(x)\in L^2(-\infty,+\infty)\), \(y(x)\) and \(y'(x)\) are absolutely continuous on every finite interval of the real axis;
b) the function \(-y''+p(x)y\) belongs to \(L^2(-\infty,+\infty)\).
The following proposition holds (see ([1])):
\(1^\circ\). Under the conditions \(p_1(x)\to\infty\) and \(p_1(x)\geqslant 1\), the operator \(L\) possesses a completely continuous inverse.
We denote by \(Q\) the operator of multiplication by the essentially bounded function \(q(x)\), bounded and defined everywhere in \(L^2(-\infty,+\infty)\).
The number \(\lambda\) will be called an eigenvalue of the pair of operators \((L,Q)\) if in \(D(L)\) there exists a function \(y\ne 0\) such that \(Ly=\lambda Qy\); in this case \(y\) is called an eigenfunction of \((L,Q)\) corresponding to the eigenvalue \(\lambda\).
The value \(\lambda\) is called a regular point of the pair of operators \((L,Q)\) if the operator \((L-\lambda Q)^{-1}\) exists, is defined on all of \(L^2(-\infty,+\infty)\), and is bounded. All nonregular points are called points of the spectrum of the pair of operators \((L,Q)\).
It is not difficult to show that the spectrum of the pair of operators \((L,Q)\) consists of no more than a countable number of eigenvalues \(\lambda_1,\lambda_2,\ldots,\lambda_s,\ldots\) and is connected with the spectrum of the completely continuous operator \(QL^{-1}\), \(\lambda'_1,\lambda'_2,\ldots,\lambda'_s,\ldots\), by the equalities \(\lambda_s=1/\lambda'_s\).
A subspace \(\mathfrak N\) of the space \(L^2\), belonging to \(D(L)\), is called an invariant subspace of the pair of opera-
\[
\operatorname{tor}(L,Q),
\]
if \(Q\mathfrak N \subset L\mathfrak N\), i.e., if for every element \(f\in\mathfrak N\) there is an element \(g\in\mathfrak N\) such that
\[
Qf=Lg.
\]
One says that \(\mathfrak N\) is an invariant subspace corresponding to the eigenvalue \(\lambda\) if \(\mathfrak N\) contains no eigenfunctions of the pair of operators \((L,Q)\) corresponding to eigenvalues different from \(\lambda\).
If there is no invariant subspace \(\mathfrak N_1\) of the pair of operators \((L,Q)\) such that \(\mathfrak N_1\supset\mathfrak N\), and corresponding to the same eigenvalue as \(\mathfrak N_1\), then \(\mathfrak N\) is called a maximal invariant subspace of the pair of operators \((L,Q)\), corresponding to the eigenvalue \(\lambda\) (5).
Consider the completely continuous operator \(S=L^{-1}Q\). Any invariant subspace of the pair of operators \((L,Q)\) corresponding to the eigenvalue \(\lambda_k\) is an invariant subspace of the operator \(S\), corresponding to the eigenvalue \(\lambda'_k=1/\lambda_k\). Therefore, to each eigenvalue of the pair of operators \((L,Q)\) there corresponds a maximal finite-dimensional invariant subspace. Moreover, the maximal invariant subspace of the pair of operators \((L,Q)\) corresponding to \(\lambda_k\) coincides with the maximal invariant subspace of the operator \(S\), corresponding to the eigenvalue \(\lambda'_k=1/\lambda_k\).
Indeed, let \(\mathfrak N_k\) be the maximal invariant subspace of the operator \(S\), corresponding to the eigenvalue \(\lambda'_k\). As is known, in \(\mathfrak N_k\) one can choose a basis in which the transformation is written as a Jordan matrix. This basis consists of several vectors
\[ f_{11}, f_{21}, \ldots, f_{s_1 1};\ f_{12}, f_{22}, \ldots, f_{s_2 2};\ \ldots;\ f_{1q}, f_{2q}, \ldots, f_{s_q q}. \tag{2} \]
Each chain forms a basis in one of the invariant subspaces into which the given maximal invariant subspace \(\mathfrak N_k\) decomposes. For the elements of each chain the equalities hold
\[ Sf_{1\tau}=\lambda'_k f_{1\tau},\quad Sf_{2\tau}=\lambda'_k f_{2\tau}+f_{1\tau},\ldots,\quad Sf_{s_\tau\tau}=\lambda'_k f_{s_\tau\tau}+f_{s_\tau-1,\tau}. \]
If we recall that \(S=L^{-1}Q\), we see that all elements of the basis (2) belong to \(D\); consequently, \(\mathfrak N_k\subset D\). Since \(S\mathfrak N_k\subset\mathfrak N_k\), we have \(Q\mathfrak N_k\subset L\mathfrak N_k\), i.e., \(\mathfrak N_k\) is an invariant subspace of the pair of operators \((L,Q)\). And since the eigenspace of the pair of operators \((L,Q)\) corresponding to the eigenvalue \(\lambda_k\) coincides with the eigenspace of the operator \(S\) corresponding to the eigenvalue \(\lambda'_k=1/\lambda_k\), and any invariant subspace of the pair of operators \((L,Q)\) is an invariant subspace of the operator \(S\), the maximal invariant subspace of the pair of operators \((L,Q)\), corresponding to \(\lambda_k\), coincides with \(\mathfrak N_k\).
Thus, the question of the completeness of the linear span of the finite-dimensional invariant subspaces of the pair of operators \((L,Q)\) reduces to the question of the completeness of the linear span of the invariant subspaces of the completely continuous operator \(S\), or, equivalently, to the question of the completeness of the system of eigen- and associated elements of the operator \(S\).
The latter, by arguments close to those of V. B. Lidskii \((^1)\), under certain restrictions on the functions \(p(x)\) and \(q(x)\), can be reduced to the question of the completeness of the system of eigen- and associated elements of the equation
\[ g=Pg+\lambda Hg, \]
where \(H=A^{-1/2}QA^{-1/2}\) is a complete self-adjoint operator with finite absolute norm, \(P=A^{-1/2}p_2A^{-1/2}\) is a completely continuous operator; here \(A^{-1/2}\) is the square root of a self-adjoint positive operator—
operator \(A^{-1}\), inverse to the operator \(A=-d^2/dx^2+p_1(x)\). By the theorem of M. V. Keldysh (³), the system of eigen- and associated elements of such an equation is complete.
Thus, the following holds.
Theorem. Let the function \(p_1(x)\), for some \(\alpha>0\), satisfy the condition
\[ \lim_{|x|\to\infty}\frac{p_1(x)}{|x|^\alpha}\ge C>0 \]
and let the function \(q(x)\) be real, essentially bounded, and vanish only on a set of measure zero.
Then, for completeness in \(L^2(-\infty,+\infty)\) of the system of finite-dimensional invariant subspaces of the pair of operators \((L,Q)\), it is sufficient that
\[ \lim_{|x|\to\infty}\frac{|p_2(x)|}{p_1(x)}=0 \tag{3} \]
and, for all \(x\),
\[ \frac{|p_2(x)|}{p_1(x)}<1. \]
Remark. Completeness of the system of finite-dimensional invariant subspaces of the pair of operators \((L,Q)\) also holds in the case where (3) is replaced by the condition
\[ \lim_{|x|\to\infty}\frac{|p_2(x)|}{p_1(x)}=C, \]
where \(C<\sin\psi_0,\ \psi_0=\min\left(\frac{\pi}{2},\frac{\alpha\pi}{\alpha+2}\right)\).
Indeed, in this case the following theorem of J. E. Allakhverdiev (⁴) is applicable.
Let \(H\) be a complete, completely continuous normal operator of order \(\rho\), whose eigenvalues are situated on a finite number of rays (let their number be \(k\), and let the arguments of the rays be denoted by \(\varphi_1,\varphi_2,\ldots,\varphi_k\) in increasing order); let \(A\) be a bounded operator with strictly bounded part \(B_0,\ \|B_0\|<1\); denote by \(\varepsilon_0\) the smallest angle for which \(\sin\varepsilon=\|B_0\|\). If
\[ 2\varepsilon_0<\min\left(|\varphi_{i+1}-\varphi_i|,\frac{\pi}{\rho}\right) \quad (i=1,\ldots,k)\;(\varphi_{k+1}=\varphi_1), \]
then the system of eigen- and associated elements of the equation
\[ y=(A+\lambda H)y \]
is complete in the space \(\mathcal H\).
For any \(\varepsilon>\varepsilon_0\), outside the angles \(\varphi_i-\varepsilon\le \arg\lambda\le \varphi_i+\varepsilon\) there can be only a finite number of eigenvalues of the operator \(A+\lambda H\).
I express my gratitude to Prof. M. A. Naimark for valuable advice and comments.
Received
4 III 1964
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