Landau and Gruss type inequalities for inner product type integral transformers in norm ideals

TL;DR

This paper establishes Landau and Grüss type inequalities for operator-valued fields in norm ideals, extending classical inequalities to the setting of normal operators and unitarily invariant norms.

Contribution

It introduces new Landau and Grüss inequalities for operator fields in norm ideals, including Schatten norms, with conditions on bounded self-adjoint fields.

Findings

Proved Landau type inequalities for commuting normal operator fields.

Derived Grüss type inequalities for bounded self-adjoint operator fields.

Extended inequalities to arbitrary bounded fields and various norms.

Abstract

For a probability measure $\mu$ and for square integrable fields $(\mathscr{A}_t)$ and $(\mathscr{B}_t)$ ($t\in\Omega$) of commuting normal operators we prove Landau type inequality \llu\int_\Omega\mathscr{A}_tX\mathscr{B}_td\mu(t)- \int_\Omega\mathscr{A}_t\,d\mu(t)X \int_\Omega\mathscr{B}_t\,d\mu(t) \rru \le \llu \sqrt{\,\int_\Omega|\mathscr{A}_t|^2\dt-|\int_\Omega\mathscr{A}_t\dt|^2}X \sqrt{\,\int_\Omega|\mathscr{B}_t|^2 \dt-|\int_\Omega\mathscr{B}_t\dt|^2} \rru for all $X\in\mathcalb{B}(\mathcal{H})$ and for all unitarily invariant norms $\lluo\cdot\rruo$. For Schatten $p$-norms similar inequalities are given for arbitrary double square integrable fields. Also, for all bounded self-adjoint fields satisfying $C\le\mathscr{A}_t\le D$ and $E\le\mathscr{B}_t\le F$ for all $t\in\Omega $ and some bounded self-adjoint operators $C,D,E$ and $F$, then for all $X\in\ccu$ we prove Gr\"uss type inequality \llu\int_\Omega\mathscr{A}_tX\mathscr{B}_t \dt- \int_\Omega \mathscr{A}_t\,d\mu(t)X \int_\Omega\mathscr{B}_t\,d\mu(t) \rru\leq \frac{\|D-C\|\cdot\|F-E\|}4\cdot\lluo X\rruo. More general results for arbitrary bounded fields are also given.