Grüss-Landau inequalities for elementary operators and inner product type transformers in Q and Q* norm ideals of compact operators

Abstract

\DeclareMathAlphabet\EuScript{U}{eus}{m}{n}
\DeclareMathAlphabet\EuScriptBold{U}{eus}{b}{n}
\DeclareMathAlphabet\Eurm{U}{eur}{m}{n}
\DeclareMathAlphabet\Eurb{U}{eur}{b}{n}
\newcommand{\mathrom}{\Eurm}%
\newcommand{\mathcalb}{\EuScriptBold}

\begin{document}

\newcommand{\zzat}[1]{\biggl\vert{#1}\biggr\vert}
\newcommand{\zzac}[1]{\Biggl\vert{#1}\Biggr\vert}
\newcommand{\nnot}[1]{\zzat{\!\!\;\zzat{\:\!#1\:\!}\!\!\;}}
\newcommand{\nnoc}[1]{\zzac{\!\!\;\zzac{\:\!#1\:\!}\!\!\;}}
\newcommand{\zzmc}[1]{\Biggl({#1}\Biggr)}
\newcommand{\HH}{{\mathcal H}}
\newcommand{\BH}{ {\mathcalb B}({\mathcal H})}
\newcommand\Phip{{\Phi^\malieksp{p}}}
\newcommand{\malieksp}[1]{{^(\;\!\!^{#1}\;\!\!^)}}
\newcommand\Phipz{{\Phi^{\malieksp{p}^{_*}}}}
\newcommand{\nnoo}[1]{\zzao{\!\!\;\zzao{\:\!#1\!\:}\!\!\;}}
\newcommand{\zzao}[1]{\vert{#1}\vert}
\newcommand{\ccphipz}{ {\mathcalb C}_{\Phi^{\malieksp{p}^{_*}}}\!({\mathcal H}) }

\noindent For probability measure $\mu$ on $\Omega$ and square integrable (Hilbert space)
operator valued functions
$\{A_t^*\}_{t\in\Omega}$, $\{B_t\}_{t\in\Omega},$ we prove Gr\"uss-Landau type
operator inequality for inner product type transformers
\begin{align*}
&\zzat{\int_\Omega A_tXB_t\,d\mu(t)-\int_\Omega A_t\,d\mu(t) X \int_\Omega B_t\,d\mu(t)}^{2\eta}\\
&\qquad\qquad\qquad\qquad\leqslant \nnoc{\int_\Omega A_t A_t^*\,d\mu(t)-\zzat{\int_\Omega A_t^*\,d\mu(t)}^2}^\eta
\zzmc{\int_\Omega B_t^*X^*XB_t \,d\mu(t)-\zzat{X\int_\Omega B_t\,d\mu(t)}^2}^\eta, \label{1}
\end{align*}
for all $X\in\BH$ and for all $\eta\in[0,1].$

Let $p\geqslant 2,$ $\Phi$ to be a symmetrically norming (s.n.) function, $\Phip$ to be its $p$-modification,
$\Phipz$ is a s.n. function adjoint to $\Phip$ and $\nnoo{\cdot}_\Phipz$
to be a norm on its associated ideal $\ccphipz$ of compact operators. %associated to the s.n. function $\Phipz,$then
If $X\in\ccphipz$ and $\{\alpha_n\}_{n=1}^\infty$ is a sequence in $(0,1],$ such that $\sum_{n=1}^\infty \alpha_n = 1$
and $\sum_{n=1}^\infty \nnoo{{\alpha_n^{-1/2}A_n} f}^2 + \nnoo{{\alpha_n^{-1/2}B_n^*}f}^2<+\infty$
for some families $\{A_n\}_{n=1}^\infty$ and
$\{B_n\}_{n=1}^\infty$ of bounded operators on Hilbert space $\mathcal{H}$
and for all $f\in \HH,$ then
\begin{equation}\notag
\nnot{\sum_{n=1}^\infty \alpha_n^{-1} A_n XB_n - \sum_{n=1}^\infty A_n X \sum_{n=1}^\infty B_n}_\Phipz
\leqslant \nnoc{\sqrt{\sum_{n=1}^\infty \alpha_n^{-1}|A_n|^2 -\zzat{\sum_{n=1}^\infty A_n}^2}X
\sqrt{\sum_{n=1}^\infty \alpha_n^{-1}|B_n^*|^2 -\zzat{\sum_{n=1}^\infty B_n^*}^2}
}_\Phipz,
\end{equation}
if at least one of those operator families consists of mutually commuting normal operators.

The related Gr\"uss-Landau type $\nnoo{\cdot}_\Phip$ norm inequalities for
inner product type transformes are also provided.

\end{document}