We introduce a uniform structure on any Hilbert $C^*$-module $\mathcal N$ and prove the following theorem: suppose, $F:{\mathcal M}\to {\mathcal N}$ is a bounded adjointable morphism of Hilbert $C^*$-modules over $\mathcal A$ and $\mathcal N$ is countably generated. Then $F$ belongs to the Banach space generated by operators $\theta_{x,y}$, $\theta_{x,y}(z):=x\langle y,z\rangle$, $x\in {\mathcal N}$, $y,z\in {\mathcal M}$ (i.e. $F$ is ${\mathcal A}$-compact, or "compact") if and only if $F$ maps the unit ball of ${\mathcal M}$ to a totally bounded set with respect to this uniform structure (i.e. $F$ is a compact operator).

中文翻译：

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Hilbert C⁎-模上“紧凑”算子的几何本质

我们在任何 Hilbert $C^*$-module $\mathcal N$ 上引入统一结构并证明以下定理：假设，$F:{\mathcal M}\to {\mathcal N}$ 是一个有界伴随态射Hilbert $C^*$-$\mathcal A$ 和 $\mathcal N$ 上的模块是可数生成的。那么$F$属于算子$\theta_{x,y}$, $\theta_{x,y}(z):=x\langle y,z\rangle$, $x\in { \mathcal N}$, $y,z\in {\mathcal M}$（即 $F$ 是 ${\mathcal A}$-compact，或“compact”）当且仅当 $F$ 映射单位球${\mathcal M}$ 到关于这个统一结构的完全有界集（即 $F$ 是一个紧凑的运算符）。