CC-admissibility and analytic CC-semigroups

We introduce the notion of CC-admissible subspaces and obtain various conditions of CC-admissibility, generalizing well known results of Vu and Schuler. Moreover, we show the uniqueness of solutions for the operator equation AX−XB=CDAX−XB=CD with AA generating an analytic CC-semigroup which generalize results of Vu.