Interpolation norms between row and column spaces and the norm problem for elementary operators

For the class of transformers acting as X→∫ΩAtXBtdμ(t) on the space of bounded Hilbert space operators we give formulae for its norm on the Hilbert–Schmidt classequation(1)X→∫ΩAtXBtdμ(t)B(C2(H))=limn→∞∫Ω2ntr∏k=1nAtn+1-k∗Asn+1-ktr∏k=1nBskBtk∗∏k=1ndμ(sk)dμ(tk)2n,whenever ∫Ω‖At‖p‖Bt‖pdμ(t)<∞ for some p>0p>0. We also estimate from below its norm on the other Schatten classes. This answers a question of characterizing (θ=)12 interpolation norm between column and row space norm for operator valued functions, with the discrete case providing the solution of the norm problem for elementary operators acting on the Hilbert–Schmidt class.