We show that any bounded analytic semigroup on L p (with 1 < p < â) whose negative generator admits a bounded H â (ÎŁ Ξ ) functional calculus for some Ξ â (0, Ï 2 ) can be dilated into a bounded analytic semigroup (R t ) t 0 on a bigger L p -space in such a way that R t is a positive contraction for any t 0. We also establish a discrete analogue for Ritt operators and consider the case when L p -spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier's unitarization theorem.2010 Mathematics Subject Classification. Primary 47A60; Secondary 47D06, 47A20, 22D12.
We present a completely new structure theoretic approach to the dilation theory of linear operators. Our main result is the following theorem: if X is a superâreflexive Banach space and T is contained in the weakly closed convex hull of all invertible isometries on X, then T admits a dilation to an invertible isometry on a Banach space Y with the same regularity as X. The classical dilation theorems of Sz.âNagy and AkcogluâSucheston are easy consequences of our general theory.
Abstract. This is a survey on recent progress concerning maximal regularity of non-autonomous equations governed by time-dependent forms on a Hilbert space. It also contains two new results showing the limits of the theory.
We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions. 2010 Mathematics Subject Classification. Primary 42B15; Secondary 42B25, 42B20.
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