Isometries among composition operators on Besov type spaces

Given p > 1, alpha > -1, let B-p,B-alpha denote the Besov type space of analytic functions on the unit disk D. Allen, Heller and Pons have shown that the isometries among composition operators on certain Besov spaces, B-p,B- p-2, are induced by rotations. We extend this to all Besov spaces and in fact to all Besov type spaces B-p,B-alpha. We show that in every Besov type space, except on B-2,B-0, rotations are the only symbols inducing isometries. We show that this is the case in every weighted Dirichlet space D-alpha, alpha not equal 0 as well.