Weighted shift operators, orthogonal polynomials and chain sequences

The main purpose of this paper is to use chain sequences to study spectral properties of weighted shift operators A and of tridiagonal operators Re A. Characterizations of chain sequences and relations to Haar sequences are derived. We use these results to compare the spectral radius, the numerical radius and the norm of A and Re A. As an example we study orthogonal polynomials defined by a recursion formula with almost constant coefficients.