Spectra, eigenvectors and overlap functions for representation operators of q-deformed algebras

Operators of representations corresponding to symmetric elements of the q-deformed algebras U-q(su(1,1)), U-q(So(2,1)), U-q(So(3,1)), U-q(SOn) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebras U-q(Su(1,1)) and U-q(So(2,1)) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms of q-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory of q-deformed algebras.