This is an announcement for the paper “Euclidean structures and operator theory in Banach spaces” by Nigel J. Kaltonhttps://arxiv.org/search/math?searchtype=author&query=Kalton%2C+N+J, Emiel Loristhttps://arxiv.org/search/math?searchtype=author&query=Lorist%2C+E, Lutz Weishttps://arxiv.org/search/math?searchtype=author&query=Weis%2C+L.

Abstract: 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 $X$ (like the $\gamma$-radonifying or the $2$-summing operator ideal) defines such a structure. Therefore our method is quite general and flexible and allows to unify the approach to seemingly unrelated theorems. Conversely we show that $\Gamma$ has to be $\alpha$-bounded for some Euclidean structure $\alpha$ for it 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 quite general 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.

https://arxiv.org/abs/1912.09347