Weak null maximum and integration of families of multiplication operators

Let X be a reflexive Hardy space or weighted Bergman space on the unit disk in the complex plane. For a bounded linear operator S on X, let wem(S):= (fₙ) ₙ \|Sfₙ\|, that is, the supremum of cluster points of n↦ \|S fₙ\|, where (fₙ) is any unit norm weakly null sequence. This quantity coincides with the essential norm on the reflexive weighted Bergman spaces. For a suitable family \ gₜ : t∈]0,1[ \ of bounded analytic functions on the unit disk, we characterize when one can exchange wem(·) and integration over t of the multiplication operators Mgₜ, that is, when wem( ∫ Mgₜ\, dt ) = ∫ wem( Mgₜ ) \, dt; when the functions gₜ,t∈]0,1[ can be continuously extended to the unit circle, we obtain a neat function-theoretic characterization.