  We study the $q$-hypergeometric difference operator $L$ on a particular
Hilbert space. In this setting $L$ can be considered as an extension of the
Jacobi operator for $q^{-1}$-Al-Salam--Chihara polynomials. Spectral analysis
leads to unitarity and an explicit inverse of a $q$-analog of the Jacobi
function transform. As a consequence a solution of the Al-Salam--Chihara
indeterminate moment problem is obtained.
