Self-adjoint difference operators and symmetric Al-Salam--Chihara polynomials

The symmetric Al-Salam--Chihara polynomials for $q>1$ are associated with an indeterminate moment problem. There is a self-adjoint second order difference operator on $\ell^2(\Z)$ to which these polynomials are eigenfunctions. We determine the spectral decomposition of this self-adjoint operator. This leads to a class of discrete orthogonality measures, which have been obtained previously by Christiansen and Ismail using a different method, and we give an explicit orthogonal basis for the corresponding weighted $\ell^2$-space. In particular, the orthocomplement of the polynomials is described explicitly. Taking a limit we obtain all the $N$-extremal solutions to the $q^{-1}$-Hermite moment problem, a result originally obtained by Ismail and Masson in a different way. Some applications of the results are discussed.