Extensions of the operator Bellman and operator Holder type inequalities

Year 2024, Volume: 7 Issue: 1, 12 - 29, 15.03.2024

Mojtaba Bakherad , Fuad Kıttaneh

https://doi.org/10.33205/cma.1435944

Cited By: 1

Abstract

In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator H\"{o}lder type inequalities. Among other results, it is shown that if $\mathbf{A}=(A_t)_{t\in \Omega}$ and $\mathbf{B}=(B_t)_{t\in \Omega}$ are continuous fields of positive invertible operators in a unital $C^*$-algebra ${\mathscr A}$ such that $\int_{\Omega}A_t\,d\mu(t)\leq I_{\mathscr A}$ and $\int_{\Omega}B_t\,d\mu(t)\leq I_{\mathscr A}$, and if $\omega_f$ is an arbitrary operator mean with the representing function $f$, then
\begin{align*}
\left(I_{\mathscr A}-\int_{\Omega}(A_t \omega_f B_t)\,d\mu(t)\right)^p
\geq\left(I_{\mathscr A}-\int_{\Omega}A_t\,d\mu(t)\right) \omega_{f^p}\left(I_{\mathscr A}-\int_{\Omega}B_t\,d\mu(t)\right)
\end{align*}
for all $0 < p \leq 1$, which is an extension of the operator Bellman inequality.

Keywords

Bellman inequality, Cauchy-Schwarz inequality, H\"{o}lder inequality, operator mean, Hadamard product, continuous field of operators, $C^*$-algebra.

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