Norms of Hankel operators and uniform algebras, II

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. 0. Introduction. Let X be a compact Hausdorff space, let C(X) be the algebra of complex-valued continuous functions on X, and let A be a uniform algebra on X. For r E MA, the maximal ideal space of A, set Ao = {f E A: r(f) O}. Let m be a representing measure for r on X. The abstract Hardy space HP = HP(m), 1 < p < oo, determined by A is defined to be the closure of A in LP = LP(m) when p is finite and to be the weak*-closure of A in L?' = L?'(m) when p = x. Suppose Ho = {f f HP: fxfdm =O KO = {f E LP: fxfgdm = 0 for all g in A} and KP = {f E LP: fxfgdm = 0 for all g in Ao}. Then Ho C KP and HP C KP. Moreover put KO = KO n c(x) and K = KP nc(X). Then Ao c KO and A c K. Let Q(') be the orthogonal projection from L2 to Ko and Q(2) the orthogonal projection from L2 to Ho. For a function q in L?? we denote by MO, the multiplication operator on L2 that it determines. The two generalizations of the classical Hankel operators that we consider in this paper are defined as follows. For q E L? and f E H2 H (,f = Q(i)M f (j = 1, 2). If A is a disc algebra and r(f) = f(O) where f denotes the holomorphic extension of f E A, then r is in MA. Let m be a normalized Lebesgue measure on the unit circle; then m is a representing measure for r. Then H2 is the classical Hardy space and H -= K . Hence H(1) = H(2) It is well known that IIH(1) IH(2) =I+H where 11 + H??ll = inf{j11 + gIIjo: g E H?"?}. This is Nehari's theorem (cf. [11, Theorem 1.3]). However generalizations to uniform algebras are unknown except for Corollary 2.1.1 in [4]. This appears to be due to the lack of a factorization theorem of H1, that is, if h E H1 and fx IhI dm < 1, then h = fg, f E H2 and g E H2 where fIf 2dm < 1 and fX jg2 dm < 1. Received by the editors August 23, 1985 and, in revised form, February 12, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46J15, 47B35, 30D55.