Abstract:In this article, we prove the proximinality of closed unit ball of $M$-ideals of compact operators on Banach spaces. We show that every positive (self-adjoint) operator on a Hilbert space has a positive (self-adjoint) compact approximant from the closed unit ball of space of compact operators. We also show that $\mathcal {K}(\ell _1)$, the space of compact operators on $\ell _1$, is ball proximinal in $\mathcal {B}(\ell _1)$, the space of bounded operators on $\ell _1$, even though $\mathcal {K}(\ell _1)$ is not an $M$-ideal in $\mathcal {B}(\ell _1)$. Moreover, we prove the ball proximinality of $M$-embedded spaces in their biduals.

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中文翻译：

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紧算子的𝑀-理想的球邻近性

摘要：在本文中，我们证明了在 Banach 空间上紧算子的 $M$-理想闭单位球的邻近性。我们表明，希尔伯特空间上的每个正（自伴随）算子都有一个正（自伴随）紧致近似于紧凑算子空间的封闭单位球。我们还表明 $\mathcal {K}(\ell _1)$，$\ell _1$ 上的紧算子空间，在 $\mathcal {B}(\ell _1)$，有界空间$\ell _1$ 上的运算符，即使 $\mathcal {K}(\ell _1)$ 在 $\mathcal {B}(\ell _1)$ 中不是 $M$ 理想。此外，我们证明了在他们的双人中的 $M$ 嵌入空间的球邻近性。

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