Algebras of Bounded Analytic Functions containing the Disk Algebra

Abstract

Let D be the open unit disk and let ∂D be its boundary. We denote by C the algebra of continuous functions on ∂d, and by L∞ the algebra of essentially bounded measurable functions with respect to the normalized Lebesgue measure m on ∂D. Let H∞ be the algebra of bounded analytic functions on D. Identifying with their boundary functions, we regard H∞ as a closed subalgebra of L∞. Let A = H∞ Pi C, which is called the disk algebra. The algebras A and H∞ have been studied extensively [5, 6, 7]. In these fifteen years, norm closed subalgebras between H∞ and L∞, called Douglas algebras, have received considerable attention in connection with Toeplitz operators [12]. A norm closed subalgebra between A and H∞ is called an analytic subalgebra. In [2], Dawson studied analytic subalgebras and he remarked that there are many different types of analytic subalgebras. One problem is to study which analytic subalgebras are backward shift invariant. Here, a subset E of H∞ is called backward shift invariant if