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 $C_B(f)=f\circ B$ are considered, where $B$ is a finite Blaschke product. $C_B^*$ is obtained as a finite linear combination of operators of the form $T_gA_BT_h,$ where $g$ and $h$ are rational functions, $T_g,T_h$ are associated Toeplitz operators and $A_B$ is defined by