  We present a general method to extend results on Hilbert space operators to
the Banach space setting by representing certain sets of Banach space operators
$\Gamma$ on a Hilbert space. Our assumption on $\Gamma$ is expressed in terms
of $\alpha$-boundedness for a Euclidean structure $\alpha$ on the underlying
Banach space $X$. This notion is originally motivated by $\mathcal{R}$- or
$\gamma$-boundedness of sets of operators, but, for example, any operator ideal
from the Euclidean space $\ell^2_n$ to $X$ defines such a structure. Therefore,
our method is quite flexible. Conversely we show that $\Gamma$ has to be
$\alpha$-bounded for some Euclidean structure $\alpha$ to be representable on a
Hilbert space. By choosing the Euclidean structure $\alpha$ accordingly, we get
a unified and more general approach to classical factorization and extension
theorems. Furthermore we use these Euclidean structures to build vector-valued
function spaces and define an interpolation method based on these spaces, which
has formulations modelled after both the real and the complex interpolation
method. Using our representation theorem we prove a transference principle for
sectorial operators on a Banach space, enabling us to extend Hilbert space
results for sectorial operators to the Banach space setting. We define
generalizations of the classical square function estimates in $L^p$-spaces and
establish, via the $H^\infty$-calculus, a version of Littlewood-Paley theory
and associated spaces of fractional smoothness for a rather large class of
sectorial operators. Our results for sectorial operators lead to some
sophisticated counterexamples.
