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Hörmander Type Functional Calculus and Square Function Estimates

We investigate Hörmander spectral multiplier theorems as they hold on $X = L^p(Ω),\: 1 < p < \infty,$ for many self-adjoint elliptic differential operators $A$ including the standard Laplacian on $\R^d.$ A strengthened matricial extension is considered, which coincides with a completely bounded map between operator spaces in the case that $X$ is a Hilbert space. We show that the validity of the matricial Hörmander theorem can be characterized in terms of square function estimates for imaginary powers $A^{it}$, for resolvents $R(λ,A),$ and for the analytic semigroup $\exp(-zA).$ We deduce Hörmander spectral multiplier theorems for semigroups satisfying generalized Gaussian estimates.