Spectral multiplicity of certain contractions

One computes the spectral multiplicity of the direct sum of operators of the following form: a unitary operator U=U_aU_s, a contraction T of class C₀, a one-sided shift S, and its conjugate S*. Theorem: where _mo=1 if m=0 and _mo=0 it m>0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 23–29, 1987.