RESUMEN

In this paper we investigate the following question: when a bounded analytic function ' on the unit disk D, fixing 0, is such that the family {'n : n = 0, 1, 2, . . . } is orthogonal in the Dirichlet space D?. We also consider the problem of characterizing the univalent, full self-maps ' of D in terms of the norm of the induced composition operator C' : D → D. The first problem is analogous to a celebrated question asked byW. Rudin on the Hardy space setting that was answered recently ([3] and [14]). The second problem resembles a problem investigated by J. Shapiro in [13] about characterization of inner functions _ in the terms of kC_kH2 .