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

Abstract

Let ${f(z){\mathop {=}\limits ^{\tiny \mathrm{def}}}\sum _{m=0}^\infty c_mz^m}$ be an analytic function with positive Taylor coefficients $c_m\geqslant 0,$ with $R_f>0$ as its radius of convergence, let $T,S:{\mathbb R}\rightarrow {{\mathbb {C}}}$ be trigonometrical polynomials and $f_{TS,t},f_{{\bar{T}}T,t},f_{{\bar{S}}S,t}$ their associated analytic functions. Let also $\{{{A}_n}\}_{n=1}^\infty ,\{{{B}_n}\}_{n=1}^\infty ,\{{{C}_n}\}_{n=1}^\infty$ and $\{{{D}_n}\}_{n=1}^\infty$ be strongly square summable families in each of them consisting of mutually commuting normal operators, satisfying also that $A_mC_n=C_nA_m$ and $B_mD_n=D_nB_m$ for all $m,n\in {{\mathbb {N}}}.$ Then for any symmetrically norming function $\Phi ,$ for any and $t\in {{\mathbb {R}}},$ such that

$$\begin{aligned}&\Biggl \vert \!\Biggl \vert \!\,\!\biggl \vert {\!\,\!\sum _{n=1}^\infty \!(A_n^*A_n^{}\!\;\!\!-C_n^*C_n^{}\;\!\!)\!}\biggr \vert ^{\!\,\!\frac{1}{2}}\!\!\biggl ({\!\;\!\!f_{TS,t}\!\Biggl ({\!\,\!\sum _{n=1}^\infty \!A_n{\otimes } B_n\!\;\!\!}\Biggr )\;\!\!X\!\,\!-\!\,\!f_{TS,t}\!\Biggl ({\!\,\!\sum _{n=1}^\infty \!C_n{\otimes } D_n\!\;\!\!}\Biggr )\;\!\!X\!\!\,\!}\biggr )\nonumber \\&\qquad \times \biggl \vert {\sum _{n=1}^\infty (B_n^{}B_n^*\!\,\!-D_n^{}D_n^*)}\biggr \vert ^\frac{1}{2}\Biggr \vert \!\Biggr \vert _\Phi \nonumber \\&\quad \leqslant \Biggl \vert \!\Biggl \vert \;\!\!\biggl \vert {f_{{\bar{T}}T\!\,\!,t}\!\,\!\Biggl ({\sum _{n=1}^\infty \!\,\!A_n^*A_n^{}\!\;\!\!}\Biggr ) -\!\,\!f_{{\bar{T}}T\!\,\!,t}\!\,\!\Biggl ({\sum _{n=1}^\infty C_n^*C_n^{}\!\;\!\!}\Biggr )\;\!\!}\biggr \vert ^\frac{1}{2}\!\!\sum _{n=1}^\infty (A_nXB_n\!\,\!-C_nXD_n\!\,\!)\nonumber \\&\qquad \times \biggl \vert {f_{{\bar{S}}S,t}\!\,\!\Biggl ({\sum _{n=1}^\infty \!\,\!B_n^{}B_n^*\!}\Biggr ) -f_{{\bar{S}}S,t}\!\,\!\Biggl ({\sum _{n=1}^\infty \!\,\!D_n^{}D_n^*\!}\Biggr )\;\!\!}\biggr \vert ^\frac{1}{2}\Biggr \vert \!\Biggr \vert _\Phi , \!\!\!\;\!\! \end{aligned}$$

(1)

if $\max \bigl \{{\bigl \vert {\!\bigl \vert {\!\,\!\sum \nolimits _{n=1}^\infty \!\,\!A_n^*A_n^{}\!\,\!}\bigr \vert \!}\bigr \vert ,\bigl \vert {\!\bigl \vert {\!\,\!\sum \nolimits _{n=1}^\infty \!\,\!B_n^{}B_n^*\!\,\!}\bigr \vert \!}\bigr \vert , \bigl \vert {\!\bigl \vert {\!\,\!\sum \nolimits _{n=1}^\infty \!\,\!C_n^*C_n^{}\!\,\!}\bigr \vert \!}\bigr \vert ,\bigl \vert {\!\bigl \vert {\!\,\!\sum \nolimits _{n=1}^\infty \!\,\!D_n^{}D_n^*\!\,\!}\bigr \vert \!}\bigr \vert }\bigr \}< R_f.$ 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 $\sum _{n=1}^\infty (A_n^{}A_n^*-C_n^{}C_n^*)$ or $\sum _{n=1}^\infty (B_n^*B_n^{}-D_n^*D_n^{})$ is a semi-definite operator.