Approximation in operator algebras on bounded analytic functions

Abstract

Let B denote the algebra of bounded analytic functions on the open unit disc in the complex plane. Let (B, 3) denote B endowed with the strict topology 83. In 1956, R. C. Buck introduced [,8:gI, the algebra of all continuous linear operators from (B, 3) into (B, 3). This paper studies the algebra [/3:,3] and some of its subalgebras, in the norm topology and in the topology of uniform convergence on bounded subsets. We also study a special class of operators, the translation operators. For k an analytic map of the open unit disc into itself, the translation operator U , is defined on B by Uof(x) = f(Q x). In particular we obtain an expression for the norm of the difference of two translation operators.