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

Abstract

For a probability measure μ and for square integrable fields (A t) and (Bt) (t ∈ Ω) of commuting normal operators we prove Landau type inequality |||∫Ω A tXBtdμ(t) - ∫ΩA tdμ(t)X ∫Ω Btdμ(t)||| ≤ |||√∫Ω |At|2 dμ(t)- | ∫Ω At dμ(t) |2 X√ ∫Ω |Bt dμ(t) |2 ||| for 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 ≤ Bt ≤ 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 inequality |||∫Ω AtXBt dμ(t) - ∫Ω At dμ(t)X ∫Ω B t dμ(t) ||| ≤ ||D - C|| · ||F - E||/4 · |||X|||. More general results for arbitrary bounded fields are also given. © ELEMEN.