If $\{\mathcal A_t\}_{t\in\Omega}$ and $\{\mathcal B_t\}_{t\in\Omega}$ are weakly*-measurable families of bounded Hilbert space operators such that transformers $X\mapsto\int_\Omega\mathcal A^*_tX\mathcal A_td\mu(t)$ and $X\mapsto\int_\Omega\mathcal B_t^*X\mathcal B_td\mu(t)$ on $\boldsymbol{\mathcal{B}}(\boldsymbol{\mathcal{H}})$ have their spectra contained in the unit disc, then for all bounded operators $X$ \begin{equation} \|\Delta_{\mathcal A}X\Delta_{\mathcal B}\|eq\Big\|X-ıt_mega\mathcal A^*_tXB_td\mu(t)\Big\|, \end{equation} where $\Delta_{\mathcal A}\stackrel{def}{=}s-\lim_{r\nearrow1}\big(I+\sum^\infty_{n-1}r^{2n}\int_\Omega\dots\int_\Omega|\mathcal A_{t_1}\dots\mathcal A_{t_n}|^2d\mu^n(t_1,\dots,t_n)\big)^{-1/2}$ and $\Delta_\mathcal B$ by analogy. If additionally $\sum_{n=1}^\infty\int_{\Omega^n}|\mathcal A^*_{t_1}\dots\mathcal A^*_{t_n}|^2d\mu^n(t_1,\dots,t_n)$ and $\sum_{n=1}^\infty\int_{\Omega^n}|\mathcal B^*_{t_1}\dots\mathcal B^*_{t_n}|^2d\mu^n(t_1,\dots,t_n)$ both represent bounded operators, then for all $p,q,s\geq1$ such that $\frac1q+\frac1s=\frac2p$ and for all Schatten $p$ trace class operators $X$ \begin{equation} \Big\|\Delta^{1-\frac1q}_\mathcal AX\Delta^{1-\frac1s}_\mathcal B\Big\|_peq\Big\|\Delta^{-\frac1q}_{\mathcal A^*}\Big(X-ıt_mega\mathcal{A^*}_tX\mathcal B_td\mu(t)\Big)\Delta^{-\frac1s}_{\mathcal B^*}\Big\|_p. \end{equation} 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.