On the spectrum of tridiagonal operators and the support of orthogonalization measures (Q1945789)

The authors study orthogonal polynomials \(\{Q_n\}_{n\geq0}\) which satisfy the three-term recurrence relations \[ xQ_n(x)=a_nQ_{n+1}(x)+b_nQ_n(x)+c_nQ_{n-1}(x), \qquad n=0,1,2,\dots, \] with the normalization \(Q_n(1)=a_n+b_n+c_n=1\). Here, \[ a_0>0, \;c_0=0; \quad a_n,c_n>0, \;n=1,2,\dots; \quad b_n\in\mathbb{R}, \;|b_n|\leq M. \] The orthogonality relations for \(Q_n\) are of the form \[ \int_S Q_n(x)Q_m(x)\,\pi(dx)=\frac{\delta_{m,n}}{h(n)}, \quad h(0)=1, \;\;h(n)=\prod_{j=1}^n \frac{a_{j-1}}{c_j}. \] An operator \(T=J(\{a_n\},\{b_n\},\{c_n\})\), given by the tridiagonal matrix, arises naturally. The authors show that it is bounded and self-adjoint in the weighted Hilbert space \(\ell^2(h)\), \(h=\{h(n)\}_{n\geq0}\), and its spectrum \(\sigma(T)=S\subset[-(2M+1), 1]\). The main result provides a necessary and sufficient condition for \(1\in S\): \[ \frac{\sum_{k=0}^n h(k)}{a_nh(n)}\in\ell^\infty(\mathbb(Z)_+). \]