We will state a connection between the adjoints of a vast variety of bounded operators on 2 different weighted Hardy spaces. We will apply it to determine the adjoints of rationally induced composition operators on Dirichlet and Bergman spaces. | Turkish Journal of Mathematics Turk J Math (2014) 38: 862 – 871 ¨ ITAK ˙ c TUB ⃝ doi: Research Article Adjoints of rationally induced composition operators on Bergman and Dirichlet spaces Aliakbar GOSHABULAGHI∗, Hamid VAEZI Department of Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Received: • • Accepted: Published Online: • Printed: Abstract: We will state a connection between the adjoints of a vast variety of bounded operators on 2 different weighted Hardy spaces. We will apply it to determine the adjoints of rationally induced composition operators on Dirichlet and Bergman spaces. Key words: Weighted composition operator, adjoint, weighted Hardy space 1. Introduction Let U denote the open unit disk of the complex plane. For each sequence β = {βn } of positive numbers, the ∑∞ weighted Hardy space H 2 (β) consists of analytic functions f (z) = n=0 an z n on U for which the norm ( ∥f ∥β = ) 12 ∞ ∑ |an |2 βn2 n=0 is finite. Notice that the above norm is induced by the following inner product: ⟨ ∞ ∑ n=0 n an z , ∞ ∑ n=0 ⟩ bn z n = β ∞ ∑ an b¯n βn2 , n=0 and that the monomials z n form a complete orthogonal system for H 2 (β) . Consequently, the polynomials are dense in H 2 (β) (see [4, Section ]). Observe that particular instances of the sequence β = {βn } yield well-known Hilbert spaces of analytic functions. Indeed, βn = 1 corresponds to the Hardy space H 2 (U) . If β0 = 1 and βn = n1/2 for n ≥ 1, the resulting space is the classical Dirichlet space D , and if βn = (n + 1)−1/2 , we have the Bergman space A2 (U). If u is analytic on the open unit disk U and φ is an analytic map of the unit disk into itself, the weighted composition operator on H 2 (β) with symbols u and φ is the operator (Wu,φ f )(z) = u(z)f (φ(z)) for f in H 2 (β). When u(z) ≡ 1 we call the operator a composition operator and denote .
