Perturbation norm inequalities for elementary operators generated by analytic functions with positive Taylor coefficients

Abstract

Let f(z)=def∑m=0∞cmzm be an analytic function with positive Taylor coefficients cm⩾ 0 , with Rf> 0 as its radius of convergence, let T, S: R→ C be trigonometrical polynomials and fTS,t, fT¯T,t, fS¯S,t their associated analytic functions. Let also {An}n=1∞,{Bn}n=1∞,{Cn}n=1∞ and {Dn}n=1∞ be strongly square summable families in [InlineEquation not available: see fulltext.] each of them consisting of mutually commuting normal operators, satisfying also that AmCn= CnAm and BmDn= DnBm for all m, n∈ N. Then for any symmetrically norming function Φ , for any [InlineEquation not available: see fulltext.] and t∈ R, such that [InlineEquation not available: see fulltext.]|||∑n=1∞(An∗An-Cn∗Cn)|12(fTS,t(∑n=1∞An⊗Bn)X-fTS,t(∑n=1∞Cn⊗Dn)X)×|∑n=1∞(BnBn∗-DnDn∗)|12||Φ⩽|||fT¯T,t(∑n=1∞An∗An)-fT¯T,t(∑n=1∞Cn∗Cn)|12∑n=1∞(AnXBn-CnXDn)×|fS¯S,t(∑n=1∞BnBn∗)-fS¯S,t(∑n=1∞DnDn∗)|12||Φ,if max{||∑n=1∞An∗An||,||∑n=1∞BnBn∗||,||∑n=1∞Cn∗Cn||,||∑n=1∞DnDn∗||}<Rf. We also provide versions of the inequality (1) for Q, Q∗ and the Schatten-von Neumann ideals of compact operators, with reduced requirements for the normality and commutativity for the observed families, if ∑n=1∞(AnAn∗-CnCn∗) or ∑n=1∞(Bn∗Bn-Dn∗Dn) is a semi-definite operator.