Paper detail

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

For a probability measure $μ$ and for square integrable fields $(\mathscr{A}_t)$ and $(\mathscr{B}_t)$ ($t\inΩ$) of commuting normal operators we prove Landau type inequality \llu\int_Ω\mathscr{A}_tX\mathscr{B}_tdμ(t)- \int_Ω\mathscr{A}_t\,dμ(t)X \int_Ω\mathscr{B}_t\,dμ(t) \rru \le \llu \sqrt{\,\int_Ω|\mathscr{A}_t|^2\dt-|\int_Ω\mathscr{A}_t\dt|^2}X \sqrt{\,\int_Ω|\mathscr{B}_t|^2 \dt-|\int_Ω\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 $t\inΩ$ and some bounded self-adjoint operators $C,D,E$ and $F$, then for all $X\in\ccu$ we prove Grüss type inequality \llu\int_Ω\mathscr{A}_tX\mathscr{B}_t \dt- \int_Ω\mathscr{A}_t\,dμ(t)X \int_Ω\mathscr{B}_t\,dμ(t) \rru\leq \frac{\|D-C\|\cdot\|F-E\|}4\cdot\lluo X\rruo. More general results for arbitrary bounded fields are also given.