Cauchy–Schwarz inequalities for inner product type transformers in Q ^∗ norm ideals of compact operators

Abstract

Let ∫Ω||Ath||2+||Bt∗h||2dμ(t)<+∞ for all h in a Hilbert space H, for some weakly ∗-measurable families {At}t∈Ω and {Bt}t∈Ω of bounded operators on H, where at least one of them consists of mutually commuting normal operators. If p⩾ 2 , Φ is a symmetrically norming (s.n.) function, Φ(p) is its p-modification, Φ(p)∗ is a s.n. function adjoint to Φ(p) and ||·||Φ(p)∗ is a norm on the ideal [InlineEquation not available: see fulltext.] associated to s.n. function Φ(p)∗, then for all [InlineEquation not available: see fulltext.]||∫ΩAtXBtdμ(t)||Φ(p)∗⩽||(∫ΩAt∗Atdμ(t))1/2X(∫ΩBtBt∗dμ(t))1/2||Φ(p)∗.This enable us to prove that if μ is a complex Borel measure on R+, with its total variation | μ| (R+) ⩽ 1 and [InlineEquation not available: see fulltext.] are such that A, B are dissipative and at least one of them is normal, such that [InlineEquation not available: see fulltext.] then ||iA∗-iA(μ^(A)X-Xμ^(B))iB∗-iB||Φ(p)∗⩽||I-|μ^(A)|2(AX-XB)I-|μ^(B)∗|2||Φ(p)∗.Some others norm inequalities for operator valued and transformer valued Fourier transformations are also provided.