Landau and Gruss type inequalities for inner product type integral transformers in norm ideals

For a probability measure $\mu$ and for square integrable fields $(\mathscr{A}_t)$ and $(\mathscr{B}_t)$ ($t\in\Omega$) of commuting normal operators we prove Landau type inequality \llu\int_\Omega\mathscr{A}_tX\mathscr{B}_td\mu(t)- \int_\Omega\mathscr{A}_t\,d\mu(t)X \int_\Omega\mathscr{B}_t\,d\mu(t) \rru \le \llu \sqrt{\,\int_\Omega|\mathscr{A}_t|^2\dt-|\int_\Omega\mathscr{A}_t\dt|^2}X \sqrt{\,\int_\Omega|\mathscr{B}_t|^2 \dt-|\int_\Omega\mathscr{B}_t\dt|^2} \rru for all $X\in\mathcalb{B}(\mathcal{H})$ and for all unitarily invariant norms $\lluo\cdot\rruo$. For Schatten $p$-norms similar inequalities are given for arbitrary double square integrable fields. Also, for all bounded self-adjoint fields satisfying $C\le\mathscr{A}_t\le D$ and $E\le\mathscr{B}_t\le F$ for all $t\in\Omega $ and some bounded self-adjoint operators $C,D,E$ and $F$, then for all $X\in\ccu$ we prove Gr\"uss type inequality \llu\int_\Omega\mathscr{A}_tX\mathscr{B}_t \dt- \int_\Omega \mathscr{A}_t\,d\mu(t)X \int_\Omega\mathscr{B}_t\,d\mu(t) \rru\leq \frac{\|D-C\|\cdot\|F-E\|}4\cdot\lluo X\rruo. More general results for arbitrary bounded fields are also given.