Formes bilinéaires hankéliennes compactes sur H2 (X) X H2 (Y).
We give new results on square functions associated to a sectorial operator on for . Under the assumption that is actually -sectorial, we prove equivalences of the form for suitable functions . We also show that has a bounded functional calculus with respect to . Then we apply our results to the study of conditions under which we have an estimate , when generates a bounded semigroup on and is a linear mapping.
Given a completely bounded map from an operator space into a von Neumann algebra (or merely a unital dual algebra) , we define to be -semidiscrete if for any operator algebra , the tensor operator is bounded from into , with norm less than . We investigate this property and characterize it by suitable approximation properties, thus generalizing the Choi-Effros characterization of semidiscrete von Neumann algebras. Our work is an extension of some recent work of Pisier on an analogous...
To any bounded analytic semigroup on Hilbert space or on -space, one may associate natural ’square functions’. In this survey paper, we review old and recent results on these square functions, as well as some extensions to various classes of Banach spaces, including noncommutative -spaces, Banach lattices, and their subspaces. We give some applications to functional calculus, similarity problems, multiplier theory, and control theory.
Let Y be a Banach space and let be a subspace of an space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and is a positive contraction on for any...
Let be a positive contraction, with . Assume that is analytic, that is, there exists a constant such that for any integer . Let and let be the space of all complex sequences with a finite strong -variation. We show that for any , the sequence belongs to for almost every , with an estimate . If we remove the analyticity assumption, we obtain an estimate , where denotes the ergodic average of . We also obtain similar results for strongly continuous semigroups of positive...
Let X be a closed subspace of B(H) for some Hilbert space H. In [9], Pisier introduced Sp [X] (1 ≤ p ≤ +∞) by setting Sp [X] = (S∞ [X] , S1 [X])θ , (where θ =1/p , S∞ [X] = S∞ ⊗min X and S1 [X] = S1 ⊗∧ X) and showed that there are p−matricially normed spaces. In this paper we prove that conversely, if X is a p−matricially normed space with p = 1, then there is an operator structure on X, such that M1,n (X) = S1 [X] where Sn,1 [X] is the finite dimentional version of S1 [X]. For p...
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