For a s-finite measures m on O and m-weakly à -measurable families fA t g tAO and fB t g tAO of Hilbert space operators we have the non-commutative Cauchy-Schwarz inequalities ins p for all X A p ðHÞ and for all p; q; rX1 such that 1 q þ 1 r ¼ 2 p : If both fA t g tAO and fB t g tAO consists of commuting normal operators, then

No abstract

No abstract

No abstract

Abstract. For a probability measure μ and for square integrable fields (A t ) and (B t ) (t ∈ Ω ) of commuting normal operators we prove Landau type inequalityfor all X ∈ B(H ) and for all unitarily invariant norms ||| · ||| .For Schatten p -norms similar inequalities are given for arbitrary double square integrable fields. Also, for all bounded self-adjoint fields satisfying C A t D and E B t F for all t ∈ Ω and some bounded self-adjoint operators C,D,E and F , and for all X ∈ C |||·||| (H ) we prove Grüss type inequalityMore general results for arbitrary bounded fields are also given.Mathematics subject classification (2010): Primary 47A63; Secondary 46L05, 47B10, 47A30, 47B15.