Abstract We study the interchange of essential norm and integration of certain families of weighted composition operators acting on the standard weighted Bergman spaces Aᵖ_ A α p, where p>1 p > 1 and 0 α ≥ 0. To be more precise, we give a sufficient condition for \| uₜC 䂻\, dt\| ₑ = \| uₜC 䂻\| ₑ \, dt ∫ u t C ϕ t d t e = ∫ u t C ϕ t e d t to hold in terms of geometric properties of uₜ u t and ₜ ϕ t. We also provide some necessary conditions for the equality to hold and calculate the essential norm of some integral operators such as some Volterra operators.
David Norrbo (Fri,) studied this question.