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 $(B,\beta )$ denote $B$ endowed with the strict topology $\beta$. In 1956, R. C. Buck introduced $[\beta :\beta ]$, the algebra of all continuous linear operators from $(B,\beta )$ into $(B,\beta )$. This paper studies the algebra $[\beta :\beta ]$ 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 $\phi$ an analytic map of the open unit disc into itself, the translation operator ${U_\phi }$ is defined on $B$ by ${U_\phi }f(x) = f(\phi x)$. In particular we obtain an expression for the norm of the difference of two translation operators.