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We present a general method to extend results on Hilbert space operators tothe Banach space setting by representing certain sets of Banach space operators$\Gamma$ on a Hilbert space. Our assumption on $\Gamma$ is expressed in termsof $\alpha$-boundedness for a Euclidean structure $\alpha$ on the underlyingBanach space $X$. This notion is originally motivated by $\mathcal{R}$- or$\gamma$-boundedness of sets of operators, but, for example, any operator idealfrom 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 aHilbert space. By choosing the Euclidean structure $\alpha$ accordingly, we geta unified and more general approach to classical factorization and extensiontheorems. Furthermore we use these Euclidean structures to build vector-valuedfunction spaces and define an interpolation method based on these spaces, whichhas formulations modelled after both the real and the complex interpolationmethod. Using our representation theorem we prove a transference principle forsectorial operators on a Banach space, enabling us to extend Hilbert spaceresults for sectorial operators to the Banach space setting. We definegeneralizations of the classical square function estimates in $L^p$-spaces andestablish, via the $H^\infty$-calculus, a version of Littlewood-Paley theoryand associated spaces of fractional smoothness for a rather large class ofsectorial operators. Our results for sectorial operators lead to somesophisticated counterexamples.

Euclidean structures and operator theory in Banach spaces