A Lifting Theorem and Uniform Algebras

Abstract

Abstract. In this paper we discuss the possible generalizations of a lifting theorem of a 2 X 2 matrix to uniform algebras. These have applications to Hankel operators, weighted norm inequalities for conjugation operators and Toeplitz operators on uniform algebras. For example, the Helson-Szegö theorems for general uniform algebras follow.1. Introduction. We will consider a fixed uniform algebra A on a compact Hausdorff space X, and a fixed homomorphism t in MA, the maximal ideal space of A. The set of representing measures for t will be denoted by #T. The kernel of t will be denoted by A0. That is, A0 consists of the functions f in A such that t(/) = 0. We study only 2x2 measure matrices u. = (u, ) whose elements are finite complex regular Borel measures, and satisfy the conditions:

Weighted norm inequalities and uniform algebras

“…The following theorem is known in the case when I = A0 or I = K0 [9, Theorems 14 and 14']. For a general / we can show a lifting theorem on A x / as in the proof of [9,Theorem 2]. Then the following theorem follows as in the proof of [9,Theorem 14].…”

“…For a general / we can show a lifting theorem on A x / as in the proof of [9,Theorem 2]. Then the following theorem follows as in the proof of [9,Theorem 14]. However, we will sketch of a direct proof, not using a lifting theorem.…”

A Lifting Theorem and Analytic Operator Algebras

“…This problem is related with a lifting theorem of Cotlar and Sadosky [4] for the disk algebra and a generalized lifting theorem of the author and Yamamoto [10] for a general uniform algebra. In the special case, they consider the problem above when 38 is commutative.…”