On the spectrum of tridiagonal operators in the context of orthogonal polynomials

Abstract

The basis for our studies is a large class of orthogonal polynomial sequences \((P_n)_{n\in {{\mathbb {N}}}_0}\), which is normalized by \(P_n(x_0)=1\) for all \(n\in {\mathbb {N}}_0\) where the coefficients in the three-term recurrence relation are bounded. The goal is to check if \(x_0 \in {\mathbb {R}}\) is in the support of the orthogonalization measure \(\mu \). For this purpose, we use, among other things, a result of G. H. Hardy concerning Cesàro operators on weighted \(l^2\)-spaces. These investigations generalize ideas from Lasser et al. (Arch Math 100:289–299, 2013).