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Dergiler / Constructive Mathematical Analysis / 2026 / Volume: 7 Issue: 1

Extensions of the operator Bellman and operator Holder type inequalities

Mojtaba Bakherad0000-0003-0323-6310
Fuad Kıttaneh0000-0003-0308-365X
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Yayın Tarihi
2024-03-15
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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.

📌 3 Maddede Bu Makale

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Amaç & Hipotez: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.
📊
Temel Bulgular & Veri: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.
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Sonuç & Katkı: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.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Bellman inequality · Cauchy-Schwarz inequality · H\"{o}lder inequality

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.

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Bu Makaleye Atıf Yap

Bakherad, M., Kıttaneh, F. (2024). Extensions of the operator Bellman and operator Holder type inequalities. Constructive Mathematical Analysis, Volume: 7 Issue: 1. https://doi.org/10.33205/cma.1435944

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