Norms of Hankel operators and uniform algebras

Abstract

Two generalizations of the classical Hankel operators are defined on an abstract Hardy space that is associated with a uniform algebra. In this paper the norms of Hankel operators are studied. This has applications to weighted norm inequalities for conjugation operators, and invertible Topelitz operators. The results in this paper have applications to concrete uniform algebras, for example, a polydisc algebra and a uniform algebra which consists of rational functions.

A Lifting Theorem and Uniform Algebras

“…Then S is a subgroup of 7É'. ê is defined by the first author [12]. is positive on A X A0 for all v in S, then it is positive on A X A0 for all v in SC.…”

“…Hence the supremum is attained with v = exp(T."=xSjUj) in S. This completes the proof. If /l is a disc algebra, then ||y//J| = \\(j> + /V°°|| (see [12]). This is a theorem of Nehari.…”

“…This is a theorem of Nehari. The first author [12] proved the following theorem. We shall prove it by Corollary 2.1.…”

“…For weights W such that W = \h\2 for some outer function « in H2, the following theorems are obtained by the first author (see [12,13]).…”

“…But it does not seem to be true for general uniform algebras, even for uniform algebras on finitely connected regions. More recently, the first author [12,13] has given a new approach to this kind of problem for uniform algebras and he has studied the norms of Hankel operators, Helson-Szegö type theorems, and the left invertibility of Toeplitz operators for uniform algebras. The second author [15] has given another proof of the lifting theorem for a disc algebra and he has studied Helson-Szegö type theorems and Koosis type theorems using the lifting theorem for the disc algebra.…”