Norm inequalities for a class of elementary operators generated by analytic functions with non-negative Taylor coefficients in ideals of compact operators related to p-modified unitarily invariant norms

Abstract

Let ∑n=1∞(‖Anh‖2+‖An⁎h‖2+‖Bnh‖2+‖Bn⁎h‖2)<+∞ for all h in a Hilbert space H, for some families {An}n=1∞ and {Bn}n=1∞ of bounded operators on H, where at least one of them consists of mutually commuting normal operators. If p⩾2, Φ is a symmetrically normed (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[Figure presented], associated to the s.n. function Φ(p)⁎, then for all[Figure presented] ‖∑n=1∞AnXBn‖Φ(p)⁎⩽‖(∑n=1∞An⁎An)1/2X(∑n=1∞BnBn⁎)1/2‖Φ(p)⁎. Amongst other applications, this new Cauchy–Schwarz type norm inequality was used to explore a class of elementary operators induced by an analytic functions with non-negative Taylor coefficients to prove that, under conditions required for (1), ‖f(∑n=1∞An⊗Bn)X‖Φ(p)⁎⩽‖f(∑n=1∞An⁎⊗An)(I)Xf(∑n=1∞Bn⊗Bn⁎)(I)‖Φ(p)⁎, whenever ‖∑n=1∞An⁎An‖, ‖∑n=1∞AnAn⁎‖, ‖∑n=1∞Bn⁎Bn‖ and ‖∑n=1∞BnBn⁎‖ are smaller then the radius of convergence of an analytic function f, where An⊗Bn stands for the bilateral multipliers[Figure presented]. Different applications and examples for the obtained norm inequalities are also provided.