Banach-compact operators, $\mathcal A$-precompactness, and frames in Hilbert $C^*$-modules
For a couple $\mathcal M$, $\mathcal N$ of Hilbert $C^*$-modules over a $C^*$-algebra $\mathcal A$, one has two notions of ``$\mathcal A$-rank 1 operators'': $\theta_{x,y}:\mathcal M\to\mathcal N$, $\theta_{x,y}(z)=x\langle y,z\rangle$, where $y,z\in\mathcal M$, $x\in\mathcal N$, (called elementary $\mathcal A$-compact, or elementary Kasparov, operators) and $\theta_{x,f}:\mathcal M\to\mathcal N$, $\theta_{x,f}(z)=xf(z)$, where $z\in\mathcal M$, $x\in\mathcal N$, and $f$ is a bounded $\mathcal A$-functional on $\mathcal M$ (introduced by Manuilov). They generate a $C^*$-bimodule ${\mathbf{K}}(\mathcal M,\mathcal N)$ ($\mathcal A$-compact operators) over the $C^*$-algebras of adjointable operators and a Banach bimodule ${\mathbf{BK}}(\mathcal M,\mathcal N)$ (Banach-compact operators) over the algebras of all bounded morphisms, respectively. In order to give a geometrical characterization of these classes of operators, we introduce the notion of $\mathcal A$-compactness (developing the one introduced by Manuilov). Banach-compact operators can be characterized as those with $\mathcal A$-precompact image of the unit ball. Another obtained characterization is in terms of total boundedness of this set relatively the uniform structure introduced by one of us previously. The constructions and proofs turn out to be closely related to the concept of frame in a Hilbert $C^*$-module.