  We investigate structure of the optimal domains for the Hardy-type operators
including, for example, the classical Ces\`aro, Copson and Volterra operators
as well as for some of their generalizations. We prove that, in some sense, the
abstract Ces\`aro and Copson function spaces are closely related to the space
$L^1$, namely, they contain "in the middle" a complemented copy of $L^1[0,1]$,
asymptotically isometric copy of $\ell^1$ and also can be renormed to contain
an isometric copy of $L^1[0,1]$. Moreover, the generalized Tandori function
spaces are quite similar to $L^\infty$ because they contain an isometric copy
of $\ell^\infty$ and can be renormed to contain an isometric copy of
$L^\infty[0,1]$. Several applications to the metric fixed point theory will be
given. Next, we prove that the Ces\`aro construction $X \mapsto CX$ does not
commutate with the truncation operation of the measure space support. We also
study whether a given property transfers between a Banach function space $X$
and the space $TX$, where $T$ is the Ces\`aro or the Copson operator. In
particular, we find a large class of properties which do not lift from $TX$
into $X$ and prove that the abstract Ces\`aro and Copson function spaces are
never reflexive, are not isomorphic to a dual space and do not have the
Radon--Nikodym property in general.
