Adjoints of a class of composition operators

Adjoints of certain operators of composition type are calculated. Specifically, on the classical Hardy space H-2 (D) of the open unit disk D operators of the form C-B (f) = f circle B are considered, where B is a finite Blaschke product. C-B* is obtained as a finite linear combination of operators of the form T-g A(B) T-h, where g and h are rational functions, T-g, T-h are associated Toeplitz operators and A(B) is defined by [GRAPHICS]