A noncompletely continuous operator on 𝐿₁(𝐺) whose random Fourier transform is in 𝑐₀(𝐺̂)

Abstract

Let T T be a bounded linear operator from L 1 ( G , λ ) {L_1}(G,\lambda ) into L 1 ( Ω , F , P ) {L_1}(\Omega ,\mathcal {F},P) , where ( G , λ ) (G,\lambda ) is a compact abelian metric group with its Haar measure, and ( Ω , F , P ) (\Omega ,\mathcal {F},P) is a probability space. Let ( μ ω ) ({\mu _\omega }) be the random measure on G G associated to T T ; that is, T f ( ω ) = ∫ G f ( t ) d μ ω ( t ) Tf(\omega ) = \int _G {f(t)d{\mu _\omega }(t)} for each f f in L 1 ( G ) {L_1}(G) . We show that, unlike the ideals of representable and Kalton operators, there is no subideal B B of M ( G ) \mathcal {M}(G) such that T T is completely continuous if and only if μ ω ∈ B {\mu _\omega } \in B for almost ω \omega in Ω \Omega . We actually exhibit a noncompletely continuous operator T T such that μ ^ ω ∈ l 2 + ε ( G ^ ) {\hat \mu _\omega } \in {l_{2 + \varepsilon }}(\hat G) for each ε > 0 \varepsilon > 0 .