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Register a new user$H^infty$-calculus for semigroup generators on BMO
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We prove that the negative infinitesimal generator $L$ of a semigroup of positive contractions on $L^infty$ has a bounded $H^infty(S_eta^0)$-calculus on the associated Poisson semigroup-BMO space for any angle $eta>pi/2$, provided the semigroup satisfies Bakry-Emrys $Gamma_2 $ criterion. Our arguments only rely on the properties of the underlying semigroup and works well in the noncommutative setting. A key ingredient of our argument is a quasi monotone property for the subordinated semigroup $T_{t,alpha}=e^{-tL^alpha},0<alpha<1$, that is proved in the first half of the article.
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In this paper we extend the $H^infty$ functional calculus to quaternionic operators and to $n$-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called $S$-functional calculus. The $S$-functional calculus has t
The main result is a Paleys theory for lacunary Fourier series using semigroup-BMO and $H^1$ spaces. This interpretation allows an extension of Paleys theory to general discrete groups, complementing the work of Rudin for abelian groups with a total order, and Lust-Piquard and Pisiers work for lacunary Fourier series with operator-valued coefficients.
Generally-unbounded infinitesimal generators are studied in the context of operator topology. Beginning with the definition of seminorm, the concept of locally convex topological vector space is introduced as well as the concept of Fr{e}chet space. These are the basic concepts for defining an operator topology. Consequently, by associating the topological concepts with the convergence of sequence, a suitable mathematical framework for obtaining the logarithmic representation of infinitesimal generators is presented.
Let $1leq p,q < infty$ and $1leq r leq infty$. We show that the direct sum of mixed norm Hardy spaces $big(sum_n H^p_n(H^q_n)big)_r$ and the sum of their dual spaces $big(sum_n H^p_n(H^q_n)^*big)_r$ are both primary. We do so by using Bourgains localization method and solving the finite dimensional factorization problem. In particular, we obtain that the spaces $big(sum_{nin mathbb N} H_n^1(H_n^s)big)_r$, $big(sum_{nin mathbb N} H_n^s(H_n^1)big)_r$, as well as $big(sum_{nin mathbb N} BMO_n(H_n^s)big)_r$ and $big(sum_{nin mathbb N} H^s_n(BMO_n)big)_r$, $1 < s < infty$, $1leq r leq infty$, are all primary.
We show that the non-separable Banach space $SL^infty$ is primary. This is achieved by directly solving the infinite dimensional factorization problem in $SL^infty$. In particular, we bypass Bourgains localization method.
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