Norm inequalities for elementary operators and other inner product type integral transformers with the spectra contained in the unit disc

Abstract

If {At}t∈Ω and {Bt}t∈Ω are weakly*-measurable families of bounded Hilbert space operators such that transformers X ↦ ∫Ω A*t XAt dμ (t) and X ↦ ∫Ω B*t XBt dμ(t) B(H) have their spectra contained in the unit disc, then for all bounded operators (equation found) where (equation found) by analogy. If additionally (equation found) both represent bounded operators, then for all p, q, s ≥ 1 such that 1/q + 1/s = 2/p and for all Schatten p trace class operators X (equation found) If at least one of those families consists of bounded commuting normal operators, then (1) holds for all unitarily invariant Q-norms. Applications to the shift operators are also given.