Cauchy–Schwarz norm inequalities for elementary operators and inner product type transformers generated by families of subnormal operators

Abstract

Recently obtained Cauchy–Schwarz norm inequalities for normal inner product type (i.p.t.) transformers are generalized to i.p.t. transformers generated by some subnormal families of operators. If L2(Ω , μ) is separable, Φ is a symmetrically norming (s.n.) function and at least one of square integrable families {At}t∈Ω or {Bt∗}t∈Ω is uniformly extendable to (strongly) square integrable families of commuting normal operators, then for all p⩾ 2 and X∈CΦ(p)∗(H)||∫ΩAtXBtdμ(t)||Φ(p)∗⩽||(∫ΩAt∗Atdμ(t))1/2X(∫ΩBtBt∗dμ(t))1/2||Φ(p)∗.If f(z)=∑n=0∞cnzn is analytic function with non-negative Taylor coefficients cn≥ 0 and A and B∗ are bounded quasinormal operators, such that max{||A||2,||B||2} is less than the radius of convergence for f, then for all X∈ CΦ(H) ||f(A⊗B)X||Φ=||∑n=0∞cnAnXBn||Φ⩽||f(A∗A)Xf(BB∗)||Φ.If, in addition, A, B∗ are contractions and 0<∑n=0∞cn<+∞, then ||I-A∗A(f(A)X-Xf(B))I-BB∗||Φ⩽||f(1)I-f(A∗A)(AX-XB)f(1)I-f(BB∗)||Φ⩽||f(1)I-|f(A)|2/f(1)(AX-XB)f(1)I-|f(B∗)|2/f(1)||Φ.