  We introduce and study the notion of generating operators as those norm-one
operators $G\colon X\longrightarrow Y$ such that for every $0<\delta<1$, the
set $\{x\in X\colon \|x\|\leq 1,\ \|Gx\|>1-\delta\}$ generates the unit ball of
$X$ by closed convex hull. This class of operators includes isometric
embeddings, spear operators (actually, operators with the alternative Daugavet
property), and other examples like the natural inclusions of $\ell_1$ into
$c_0$ and of $L_\infty[0,1]$ into $L_1[0,1]$. We first present a
characterization in terms of the adjoint operator, make a discussion on the
behaviour of diagonal generating operators on $c_0$-, $\ell_1$-, and
$\ell_\infty$-sums, and present examples in some classical Banach spaces. Even
though rank-one generating operators always attain their norm, there are
generating operators, even of rank-two, which do not attain their norm. We
discuss when a Banach space can be the domain of a generating operator which
does not attain its norm in terms of the behaviour of some spear sets of the
dual space. Finally, we study when the set of all generating operators between
two Banach spaces $X$ and $Y$ generates all non-expansive operators by closed
convex hull. We show that this is the case when $X=L_1(\mu)$ and $Y$ has the
Radon-Nikod\'ym property with respect to $\mu$. Therefore, when
$X=\ell_1(\Gamma)$, this is the case for every target space $Y$. Conversely, we
also show that a real finite-dimensional space $X$ satisfies that generating
operators from $X$ to $Y$ generate all non-expansive operators by closed convex
hull only in the case that $X$ is an $\ell_1$-space.
