Abstract
In this paper, we discuss the isometries of composition operators on the holomorphic general family function spaces F(p, q, s). First, we classify the isometric composition operators acting on a general Banach spaces. For 1 < p < 2, we display that an isometry of C-phi is caused only by a rotation of the disk. We scrutinize the previous work on the case for p >= 2. Also, we characterize many of the foregoing results about all alpha-Besov-type spaces F(p, alpha p-2, s), alpha > 0. We exhibit that in every classes F(p, alpha p-2, s) except for the Dirichlet space D = F(2, 0, 0), rotations are the only that produce isometries.