If {A t } t∈Ω and {B t } t∈Ω are weakly*-measurable families of bounded Hilbert space operators such that transformers X → Ω A * t XA t dµ(t) and X → Ω B * t XB t dµ(t) on B(H) have their spectra contained in the unit disc, then for all bounded operators X If at least one of those families consists of bounded commuting normal operators, then (1) holds for all unitarily invariant Q-norms. Applications to the shift operators are also given.
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