On the weighted contraharmonic means
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916285575593984 |
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| author | Zamani, Ali |
| author_facet | Zamani, Ali |
| contents | Let $\mathscr{A}$ be a unital $C^*$-algebra with unit $e$ and let $ν\in(0, 1)$. We introduce the concept of the $ν$-weighted contraharmonic of two positive definite elements $a$ and $b$ of $\mathscr{A}$ by \begin{align*} {C}_ν(a, b):= (1-ν)ν^{-1}b + ν(1-ν)^{-1}a - \left((1-ν)a^{-1}+νb^{-1}\right)^{-1}. \end{align*} We show that \begin{align*} {C}_ν(a, b)= \displaystyle{\max_{x+y=e}}\left\{(1-ν)^{-1}\left(νa - x^*ax\right) + ν^{-1}\left((1-ν)b - y^*by\right)\right\}, \end{align*} and then apply it to present some properties of this weighted mean. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the weighted contraharmonic means Zamani, Ali Functional Analysis Operator Algebras 46L05, 47A63, 47A64, 47B65 Let $\mathscr{A}$ be a unital $C^*$-algebra with unit $e$ and let $ν\in(0, 1)$. We introduce the concept of the $ν$-weighted contraharmonic of two positive definite elements $a$ and $b$ of $\mathscr{A}$ by \begin{align*} {C}_ν(a, b):= (1-ν)ν^{-1}b + ν(1-ν)^{-1}a - \left((1-ν)a^{-1}+νb^{-1}\right)^{-1}. \end{align*} We show that \begin{align*} {C}_ν(a, b)= \displaystyle{\max_{x+y=e}}\left\{(1-ν)^{-1}\left(νa - x^*ax\right) + ν^{-1}\left((1-ν)b - y^*by\right)\right\}, \end{align*} and then apply it to present some properties of this weighted mean. |
| title | On the weighted contraharmonic means |
| topic | Functional Analysis Operator Algebras 46L05, 47A63, 47A64, 47B65 |
| url | https://arxiv.org/abs/2406.08986 |