Norm Inequalities for Elementary Operators and Other Inner Product Type Integral Transformers with the Spectra Contained in the Unit Disc

Abstract

If  $\{\mathscr A_t\}_{t\in\Omega}$ and $\{\mathscr B_t\}_{t\in\Omega}$ are
 weakly*-measurable families of bounded Hilbert space operators such that  transformers
$ X\mapsto \int_{\Omega}\mathscr A_t^* X \mathscr A_t d\mu(t)$
   and $ X\mapsto \int_{\Omega}\mathscr B_t^* X \mathscr B_t d\mu(t)$ on $\mathcal B(\mathcal{H})$ have their spectra
contained in the unit disc, then for all bounded operators $X$
\begin{equation}
\bigl\vert\!\!\;\bigl\vert \Delta_{\mathscr A} X \Delta_{\mathscr B} \bigr\vert\!\!\; \bigr\vert
\leqslant \biggl\vert \!\!\; \biggr\vert \, X -\! \int_{\Omega}\mathscr A_t^* X \mathscr B_t d\mu(t) \biggr\vert \!\!\; \biggr\vert , \label{zvezda}
\end{equation}
where $\Delta_{\mathscr A} \stackrel{def}{=} {s\!-\!\lim_{r\nearrow 1} \Bigl( I +
   \sum_{n=1}^{\infty}r^{2n}\!\! \int_{\Omega} \cdots \int_{\Omega} \bigl\vert \mathscr A_{t_1} \cdots \mathscr A_{t_n} \bigr\vert^2 d\mu^n(t_1,\!\cdots\!,t_n) \:\!\! \Bigr)^{-1/2}}$
 and $\Delta_{\mathscr B}$ by analogy.

If additionally
$\sum_{n=1}^\infty\int_{{\Omega}^n} \bigl\vert \mathscr A_{t_1}^*\cdots\mathscr A_{t_n}^* \bigr\vert^2 d\mu^n(t_1,\!\cdots\!,t_n)$
and
$\sum_{n=1}^\infty\int_{{\Omega}^n} \bigl\vert \mathscr B_{t_1}^*\cdots\mathscr B_{t_n}^* \bigr\vert^2 d\mu^n(t_1,\!\cdots\!,t_n)$
both represent bounded operators, then
for all  $p,q,s\geqslant 1$  such that $\frac 1{q}+\frac1{s}=\frac 2{p} $
and for all Schatten $p$ trace class operators $X $
\begin{equation}
\left\vert \!\!\; \left\vert \Delta_{\mathscr A}^{1-\frac1{q}}X \Delta_{\mathscr B}^{1-\frac1{s}} \right\vert \!\!\; \right\vert_p
\leqslant \biggl\vert \!\!\; \biggl\vert \:\! \Delta_{\mathscr A^*}^{-\frac1{q}}\biggl( X-\!\int_{\Omega}\mathscr A_t^* X \mathscr B_t d\mu(t)\:\!\! \biggr)\,\Delta_{\mathscr B^*}^{-\frac1{s}}
\:\! \biggr\vert \!\!\; \biggr\vert_p.
\label{hag2}
\end{equation}

If at least one of those families consists of bounded commuting normal operators,
then (\ref{zvezda}) holds for all unitarily invariant Q-norms.
Applications to the shift operators are also given.