Derived space of L p (G, H)

Abstract

Let G denote a locally compact abelian group and H a separable Hilbert space. Let Lp(G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual Lp space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space Lp(G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then Lp0(G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let Sp(G, H) be the L1(G) Banach module of functions in Lp(G, H) having unconditionally convergent Fourier series in Lp-norm. We show that Sp(G, H) coincides with the derived space Lp0(G, H), as in the scalar valued case. We also show that if G is compact and abelian, then Lp0(G, H) = L2(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ Lp(G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in Lp-norm, then F ∈ L2(G, H).