On the support of the measure of orthogonality of a class of orthogonal polynomials. (Q1855675)

It is a well-known fact that an orthonormal polynomial sequence \((p_n)_n\) satisfies a three-term recurrence relation. Let \(T\) be the tridiagonal operator associated to \((p_n)_n\) and defined by \[ Te_n=\alpha_n e_{n+1} \beta_n e_n+\alpha_{n-1}e_{n-1},\quad \alpha_n>0,\quad \] \((e_n)_n\) being an orthonormal basis of the Hilbert space. The main aim of the present paper is to study the spectrum of \(T\) that coincides with the support of the orthogonality measure of the family \((p_n)_n\). In particular, for the polynomials in the Nevai class \(M(1/2,0)\) the authors find a general sufficient condition such that the support of the orthogonality measure is the entire interval \([-1,1]\). Finally, they apply the results to several families of orthogonal polynomials such as Pollaczek and random walk polynomials.