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arXiv · 2306.06685

An extension of the weighted geometric mean in unital $JB$-algebras

Amir Ghasem Ghazanfari·Somayeh Malekinejad
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Abstract

Let $\mathcal{A}$ be a unital $JB$-algebra and $A,~B\in\mathcal{A}$, we extend the weighted geometric mean $A\sharp_r B$, from $r\in [0,1]$ to $r\in (-1, 0)\cup(1, 2)$. We will notice that many results will be reversed when the domain of $r$ change from $[0,1]$ to $(-1,0)$ or $(1,2)$. We investigate some property of $A\sharp_r B$ for such quantities of $r$, such as we show that $A\sharp_r B$ is separately operator convex with respect to $A, B$. We also introduce the Heinz and Heron means for unital $JB$-algebras and give some famous inequalities involving them.

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Amir Ghasem Ghazanfari, Somayeh Malekinejad. 2023-06-11. An extension of the weighted geometric mean in unital $JB$-algebras. https://arxiv.org/abs/2306.06685

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