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 begin{multline*} 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-\\left|\\int_\\Omega\\mathscr{A}_t\\dt\\right|^2}X \\sqrt{\\,\\int_\\Omega|\\mathscr{B}_t|^2 \\dt-\\left|\\int_\\Omega\\mathscr{B}_t\\dt\\right|^2} \\rru \\end{multline*} 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 \\begin{equation} \\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.\\nonumber \\end{equation} More general results for arbitrary bounded fields are also given.

Keywords