COMPOSITION OPERATORS ON THE DIRICHLET SPACE AND RELATED PROBLEMS

Abstract. In this paper we investigate the following problem: when a bounded analytic function ϕ on the unit disk D, fixing 0, is such that {ϕn: n = 0, 1, 2,...} is orthogonal in D?, and consider the problem of characterizing the univalent, full self-maps of D in terms of the norm of the composition operator induced. The first problem is analogous to a celebrated question asked by W. Rudin on the Hardy space setting that was answered recently ([3] and [15]). The second problem is analogous to a problem investigated by J. Shapiro in [14] about characterization of inner functions in the setting of H2. Let D denote the unit disk in the complex plane. By a self- map of D we mean an analytic map such that ϕ(D) ⊂ D. The composition operator induced by ϕ is the linear transformation Cϕ defined by Cϕ(f) = f ◦ϕ in the space of the holomorphic functions on D. The composition operators have been studied in many settings, and in particular in functional Banach spaces (cf. the books [4], [13], the survey of recent develop-ments [8], and the references therein). Recall that a functional Banach space is a Banach space of analytic functions (on the disk or other domains of C or Cn) where the evaluation functionals are continuous. The goal of this theory is to obtain char-acterizations of operator-theoretic properties of Cϕ by function-theoretic properties of the symbol ϕ. Conversely, operator-theoretic properties of Cϕ could suggest, or help to understand certain phenomena about function-theoretic properties of ϕ. Particular instances of functional Banach spaces are the Hardy spaceH2, and the Bergman space A2 of the unit disk. In these spaces, as a consequence of Littlewood’s Subordination Principle, every self-map of D induces a bounded composition oper-ator. A very interesting setting for studying such operators is the Dirichlet space. Recall that if dA(z) = 1pidx dy = 1 pi r dr dθ, (z = x + iy = re iθ) denotes the nor-malized area Lebesgue measure on D, the Dirichlet space D is the Hilbert space of analytic functions in D with a square integrable derivative, with the norm given b