The Complete Continuity Property in Banach Spaces

Abstract

Let X be a complex Banach space. We show that the following are equivalent: (i) X has the complete continuity property, (ii) for every (or equivalently for some) 1 < p < ∞, for f ∈ h(D, X) and rn ↑ 1, the sequence frn is p-Pettis-Cauchy, where frn is defined by frn(t) = f(rne ) for t ∈ [0, 2π], (iii) for every (or equivalently for some) 1 < p < ∞, for every μ ∈ V (X), the bounded linear operator T : L(0, 2π) → X defined by Tφ = ∫ 2π 0 φdμ is compact, where 1/q+1/p = 1, (iv) for every (or equivalently for some) 1 < p <∞, each μ ∈ V (X) has a relatively compact range. Through this note (X, ‖ · ‖) denotes a complex Banach space, D denotes the open unit disc in the complex plane. λ will be the normalized Lebesgue measure on [0, 2π]. For a Banach space Y , we denote by BY the closed unit ball of Y . Given 1 < p <∞ we denote by hp(D, X) the space of all X-valued harmonic functions f on D such that ‖f‖p = sup 0<r<1 ( ∫ 2π 0 ‖f(re)‖dλ(t)) <∞. h∞(D, X) will be the space of all X-valued bounded harmonic functions on D equipped with the norm ‖f‖∞ = supz∈D ‖f(z)‖. For f ∈ hp(D, X) and n ∈ Z, we define the Fourier coefficient f(n) by