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Functional calculus for generators of symmetric contraction semigroups  [PDF]
Andrea Carbonaro,Oliver Dragi?evi?
Mathematics , 2013,
Abstract: We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1
Isometric dilations and $H^\infty$ calculus for bounded analytic semigroups and Ritt operators  [PDF]
Cédric Arhancet,Stephan Fackler,Christian Le Merdy
Mathematics , 2015,
Abstract: We show that any bounded analytic semigroup on $L^p$ (with $1
On the structure of semigroups on $L_p$ with a bounded $H{^\infty}$-calculus  [PDF]
Stephan Fackler
Mathematics , 2013, DOI: 10.1112/blms/bdu062
Abstract: We show that a bounded analytic semigroup on an $L_p$-space has a bounded $H^{\infty}(\Sigma_{\varphi})$-calculus for some $\varphi < \frac{\pi}{2}$ if and only if the semigroup can be obtained, after restricting to invariant subspaces, factorizing through invariant subspaces and similarity transforms, from a bounded analytic semigroup on some bigger $L_p$-space which is positive and contractive on the real line.
Functional calculus for semigroup generators via transference  [PDF]
Markus Haase,Jan Rozendaal
Mathematics , 2013, DOI: 10.1016/j.jfa.2013.08.019
Abstract: In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if $-A$ generates a $C_0$-semigroup on a Hilbert space, then for each $\tau>0$ the operator $A$ has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy $f(z)=O(e^{-\tau\textrm{Re}(z)})$ as $|z|\rightarrow \infty$. The bound of this calculus grows at most logarithmically as $\tau\searrow 0$. As a consequence, $f(A)$ is a bounded operator for each holomorphic function $f$ (on a right half-plane) with polynomial decay at $\infty$. Then we show that each semigroup generator has a so-called (strong) $m$-bounded calculus for all $m\in\mathbb{N}$, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called $\gamma$-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.
$H^\infty$ functional calculus and maximal inequalities for semigroups of contractions on vector-valued $L_p$-spaces  [PDF]
Quanhua Xu
Mathematics , 2014,
Abstract: Let $\{T_t\}_{t>0}$ be a strongly continuous semigroup of positive contractions on $L_p(X,\mu)$ with $10}\frac1t\Big|\int_0^tT_s(f(\cdot,\omega))(x)ds\Big|,\quad (x,\omega)\in X\times\Omega.$$ Then the following maximal ergodic inequality holds $$\big\|\mathcal M(f)\big\|_{L_p(X; E)}\lesssim \big\|f\big\|_{L_p(X; E)},\quad f\in L_p(X; E).$$ If the semigroup $\{T_t\}_{t>0}$ is additionally assumed to be analytic, then $\{T_t\}_{t>0}$ extends to an analytic semigroup on $L_p(X; E)$ and $\mathcal M(f)$ in the above inequality can be replaced by the following sectorial maximal function $$\mathcal T_\theta(f)(x, \omega)=\sup_{|{\rm arg}(z)|<\theta}\big|T_z(f(\cdot,\omega))(x)\big|$$ for some $\theta>0$. Under the latter analyticity assumption and if $E$ is a complex interpolation space between a Hilbert space and a UMD Banach space, then $\{T_t\}_{t>0}$ extends to an analytic semigroup on $L_p(X; E)$ and its negative generator has a bounded $H^\infty(\Sigma_\sigma)$ calculus for some $\sigma<\pi/2$.
Generators of Quantum Markov Semigroups  [PDF]
George Androulakis,Matthew Ziemke
Mathematics , 2014, DOI: 10.1063/1.4928936
Abstract: Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to Lindblad \cite{lindblad}, Stinespring \cite{stinespring}, and Kraus \cite{kraus} imply that the generator of the semigroup has the form $$L(A)=\sum_{n=1}^{\infty}V_n^*AV_n +GA+AG^*$$ where $V_n$ and G are elements of the underlying operator algebra. In the present paper we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus' result. Lastly we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.
Analytic semigroups on vector valued noncommutative $L^p$-spaces  [PDF]
Cédric Arhancet
Mathematics , 2012,
Abstract: We give sufficient conditions on an operator space $E$ and on a semigroup of operators on a von Neumann algebra $M$ to obtain a bounded analytic or a $R$-analytic semigroup $(T_t \otimes Id_E)_{t \geq 0}$ on the vector valued noncommutative $L^p$-space $L^p(M,E)$. Moreover, we give applications to the $H^\infty(\Sigma_\theta)$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.
Quantum Stochastic Semigroups and Their Generators  [PDF]
V. P. Belavkin
Mathematics , 2005,
Abstract: A rigged Hilbert space characterisation of the unbounded generators of quantum completely positive (CP) stochastic semigroups is given. The general form and the dilation of the stochastic completely dissipative (CD) equation over the algebra L(H) is described, as well as the unitary quantum stochastic dilation of the subfiltering and contractive flows with unbounded generators is constructed.
$H^{\infty}$ functional calculus and square functions on noncommutative $L^p$-spaces  [PDF]
Marius Junge,Christian Le Merdy,Quanhua Xu
Mathematics , 2006,
Abstract: In this work we investigate semigroups of operators acting on noncommutative $L^p$-spaces. We introduce noncommutative square functions and their connection to sectoriality, variants of Rademacher sectoriality, and $H^\infty$ functional calculus. We discuss several examples of noncommutative diffusion semigroups. This includes Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups, and the noncommutative Poisson semigroup on free groups.
Functional calculus for $C_0$-semigroups using infinite-dimensional systems theory  [PDF]
Felix Schwenninger,Hans Zwart
Mathematics , 2015,
Abstract: In this short note we use ideas from systems theory to define a functional calculus for infinitesimal generators of strongly continuous semigroups on a Hilbert space. Among others, we show how this leads to new proofs of (known) results in functional calculus.
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