Full Text
A. V. KUZHEL
ON NONSELFADJOINT OPERATORS GENERATED BY JACOBI MATRICES
(Presented by Academician V. I. Smirnov on 17 X 1963)
In this note an equation is found for the nonreal eigenvalues of the nonselfadjoint operator \(A_\theta\) generated by a Jacobi matrix; a completeness condition is established for the system of eigenvectors of the operator \(A_\theta\), and a condition is also obtained for the unitary equivalence of certain (simple) parts of nonselfadjoint operators generated by different Jacobi matrices.
- Let \(A'\) be the linear operator defined in the separable Hilbert space \(H\) by means of the Jacobi matrix
\[ \left\| \begin{array}{ccccc} a_0 & b_0 & 0 & 0 & \cdots\\ b_0 & a_1 & b_1 & 0 & \cdots\\ 0 & b_1 & a_2 & b_2 & \cdots\\ \cdot & \cdot & \cdot & \cdot & \cdots\\ \cdot & \cdot & \cdot & \cdot & \cdots \end{array} \right\| \qquad (b_k>0,\ \bar a_k=a_k,\ k=0,1,2,\ldots) \tag{1} \]
by the relation
\[ A'e_k=b_{k-1}e_{k-1}+a_ke_k+b_ke_{k+1} \quad (k=0,1,2,\ldots;\ b_{-1}=0), \]
where \(\{e_k\}_{k=0}^{\infty}\) is an orthonormal basis in \(H\). As is known, the operator \(A'^*\) adjoint to \(A'\) (which in what follows we shall denote by \(A\)) exists and is either selfadjoint, or a symmetric operator with defect index \((1,1)\). In what follows the latter case is considered.
Then (see, for example, \((^1)\), p. 639, or \((^2)\), p. 175) the polynomials \(P_k(\lambda)\) \((k=0,1,2,\ldots)\), defined by the relations
\[ b_{k-1}P_{k-1}(\lambda)+a_kP_k(\lambda)+b_kP_{k+1}(\lambda)=\lambda P_k(\lambda) \]
\[ \left(k=1,2,3,\ldots;\quad P_0(\lambda)=1,\ P_1(\lambda)=\frac{\lambda-a_0}{b_0}\right), \]
satisfy the condition
\[ \sum_{k=0}^{\infty}\left|P_k(\lambda)\right|^2<\infty \qquad (\operatorname{Im}\lambda\ne0). \]
Here the defect subspace \(\mathfrak N_\lambda\) of the operator \(A\) is spanned by the vector
\[ g_\lambda=\sum_{k=0}^{\infty}P_k(\bar\lambda)e_k. \]
Let us note that the polynomial \(P_k(\lambda)\) is related to the characteristic polynomial of the “truncated” Jacobi matrix
\[ J_k= \left\| \begin{array}{ccccc} a_0 & b_0 & 0 & \cdots & 0\\ b_0 & a_1 & b_1 & \cdots & 0\\ 0 & b_1 & a_2 & \cdots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & & a_{k-1} \end{array} \right\| \]
by the relation
\[ P_k(\lambda)=\frac{\det(\lambda E-J_k)}{b_0 b_1 \ldots b_{k-1}}, \]
where \(E\) is the identity matrix of order \(k\).
In what follows an important role will be played by the function
\[ h(\lambda,\mu)=\sum_{k=0}^{\infty} P_k(\lambda)P_k(\mu), \]
which we shall call the polynomial kernel of the system \(\{P_k(\lambda)\}_{k=0}^{\infty}\).
- Let \(D_A\) be the domain of definition of the operator \(A\) \((=A^*)\). Consider the linear manifold \(D_{A_\theta}\), which, in addition to the manifold \(D_A\), also contains the vector \(\psi=\theta g_i+g_{-i}\), where \(\theta\) is an arbitrary fixed complex number. Define on \(D_{A_\theta}\) the operator \(A_\theta\) by the relation
\[ A_\theta f=A^*f \qquad (f\in D_{A_\theta}). \]
Then
\[ A_\theta^*=A_{1/\bar{\theta}}, \]
whence, in particular, it follows that the operator \(A_\theta\) is self-adjoint if and only if \(|\theta|=1\).
In what follows we assume that \(|\theta|\ne 1\). Then the operator \(A_\theta\) is a \(K_{\Pi}^{1}\)-operator \({}^{(3)}\), and, consequently, the results of \({}^{(3,4)}\) may be applied to it. In particular, the \(\alpha\)-basis of the operator \(A_\theta\) consists of the vector \(g=\tau g_{-i}\), where
\[ \tau=\left|1-|\theta|^2\right|^{1/2} \left(2\sum_{k=0}^{\infty}|P_k(i)|^2\right)^{-1/2}, \]
and the corresponding coefficient \(J\) is determined by the relation \(J=\operatorname{sign}(1-|\theta|^2)\). This makes it possible to compute the characteristic function \(\chi_{A_\theta}(\lambda)\) of the operator \(A_\theta\), which in our case is determined by the relation \({}^{(3)}\)
\[ \chi_{A_\theta}(\lambda)\chi_{A_\theta}(i) =1+i(\lambda+i)\bigl((A_\theta^*-iI)(A_\theta^*-\lambda I)^{-1}g,g\bigr)J. \]
As a result we obtain that
\[ \chi_{A_\theta}(\lambda)=\frac{\omega(\lambda,\theta)}{\theta\,\omega(\lambda,1/\bar{\theta})}, \]
where
\[ \omega(\lambda,t)=(\lambda+i)\,t\,h(\lambda,-i)+(\lambda-i)\,h(\lambda,i) \]
(\(h(\lambda,\mu)\) is the polynomial kernel of the system \(\{P_k(\lambda)\}_{k=0}^{\infty}\)).
Using now the results of \({}^{(3)}\), we obtain the following assertion:
Theorem 1. The non-real spectrum of the operator \(A_\theta\) coincides with the set of non-real zeros of the function \(\omega(\lambda,\theta)\). Moreover, the multiplicity of an arbitrary non-real eigenvalue of the operator \(A_\theta\) is equal to 1.
- Let \(|\theta|<1\). Then the operator \(A_\theta\) is dissipative (i.e., for every \(f\) from \(D_{A_\theta}\), \(\operatorname{Im}(A_\theta f,f)\ge 0\)). If \(|\theta|>1\), instead of the operator \(A_\theta\) one may consider the operator \(A_\theta^*=A_{1/\bar{\theta}}\), which, by the preceding, will be dissipative. Consequently, without loss of generality, we may consider only the case \(|\theta|<1\).
Theorem 2. The non-real spectrum \(\{\lambda_k\}_{k=1}^{N}\) of the dissipative operator \(A_\theta\) satisfies the condition
\[ \prod_{k=1}^{N} \left|\frac{\bar{\lambda}_k+i}{\lambda_k+i}\right| \ge |\theta| \qquad (N\le \infty). \tag{2} \]
Moreover, if the operator \(A_\theta\) is simple \((^3)\), then the system of eigenvectors of this operator will be complete in the space \(H\) if and only if equality holds in relation (2).
Let us note that in the case when the system of eigenvectors of the operator \(A_\theta\) is complete in the space \(H\) (i.e., when equality holds in (2) and \(N=\infty\)), the discrete spectrum \(\{\lambda_k\}_{k=1}^{\infty}\) of the operator \(A_\theta\) satisfies the condition
\[ \sum_{k=1}^{\infty} \operatorname{Im}\lambda_k=\infty . \]
In this case the set \(\{\lambda_k\}_{k=1}^{\infty}\) is unbounded.
In the same case, when \(\{\lambda_k\}_{k=1}^{\infty}\) is a bounded set, the series
\[ \sum_{k=1}^{\infty} \operatorname{Im}\lambda_k \]
converges. Consequently, in this case, on the basis of the preceding, the system of eigenvectors of the operator \(A_\theta\) cannot be complete in \(H\).
- Let us consider, in addition to the operator \(A_\theta\), the operator \(\widetilde A_\theta\), which is generated by the matrix
\[ \left\| \begin{matrix} \widetilde a_0 & \widetilde b_0 & 0 & 0 & \ldots \\ \widetilde b_0 & \widetilde a_1 & \widetilde b_1 & 0 & \ldots \\ 0 & \widetilde b_1 & \widetilde a_2 & \widetilde b_2 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{matrix} \right\| \qquad (\widetilde b_k>0,\ \widetilde a_k=a_k,\ k=0,1,2,\ldots), \]
analogously to how the operator \(A_\theta\) is generated by the matrix (1). Let, further,
\[ \widetilde h(\lambda,\mu)=\sum_{k=0}^{\infty}\widetilde P_k(\lambda)\widetilde P_k(\mu) \]
be the polynomial kernel of the system \(\{\widetilde P_k(\lambda)\}_{k=0}^{\infty}\). Then, using the preceding results and the results of paper \((^3)\), we obtain the following assertion:
Theorem 3. Let the polynomial kernels \(h(\lambda,i)\) and \(\widetilde h(\lambda,i)\) of the systems \(\{P_k(\lambda)\}_{k=0}^{\infty}\) and \(\{\widetilde P_k(\lambda)\}_{k=0}^{\infty}\) coincide. Then the simple parts of the operators \(A_\theta\) and \(\widetilde A_\theta\) are isomorphic.
Thus, by virtue of the preceding theorem, the simple part of the operator \(A_\theta\) is determined by the polynomial kernel up to isomorphism. Moreover, the simple part of the operator \(A_\theta\) can be constructively recovered (up to isomorphism) if the polynomial kernel is given.
Uman State
Pedagogical Institute
Received
27 VII 1963
CITED LITERATURE
\(^1\) V. I. Smirnov, A Course of Higher Mathematics, 5, Moscow, 1959.
\(^2\) N. I. Akhiezer, The Classical Moment Problem, Moscow, 1961.
\(^3\) A. V. Kuzhel, DAN, 119, No. 5 (1958).
\(^4\) A. V. Kuzhel, DAN, 125, No. 1 (1959).