Determinacy of bounded complex perturbations of Jacobi matrices (Q1577926)

Let \(G\) be a semi-infinite tridiagonal complex matrix such that the nonzero off-diagonal elements satisfy \(a_j:=g_{j+1,j}=g_{j,j+1} \neq 0\) for \(j=1,2,\dots\), while the diagonal elements \(b_1, b_2, \dots\) are any complex numbers. Such a matrix is called a Jacobi matrix. \(G\) is said to be determinate if there exists \(z \in \mathbb{C}\) such that \[ \sum_{n \geq 1} |p_n(z)|^2 = \infty \quad \text{ or} \quad \sum_{n \geq 1} |q_n(z)|^2 = \infty, \] where \(p_n(z)\) and \(q_n(z)\) are the solutions of the recurrence relation \[ a_{n+1}y_{n+1}+b_ny_n+a_ny_{n-1}=zy_n, \quad n=0,1,2,\dots \quad (a_0=1), \] satisfying the initial conditions \[ p_{-1}(z)=0, p_0(z)=1 \quad \text{ and} \quad q_0(z)=0, q_1(z)=1/a_1, \] respectively. Let \(\mathbf{G}\) denote the closure of the operator defined by the matrix \(G\) in the sequence space \(\ell^2\) with domain consisting of all sequences which have only finitely many nonzero components. Assume that \(G\) has a decomposition \(G=J+C\), where \(J\) is a real Jacobi matrix and the elements of \(C\) are uniformly bounded. It is proved that the determinacy of \(G\) is equivalent to \(D({\mathbf{G}})=D({\mathbf{G}}^{*})\). This result was known only in the case \(C=0\). The determinacy has applications in the study of continued fractions.