Abstract:
Consider --- for the generator \({-}A\) of a symmetric contraction semigroup over some measure space $\mathrm{X}$, $1\le p < \infty$, $q$ the dual exponent and given measurable functions $F_j,\: G_j : \mathbb{C}^d \to \mathbb{C}$ --- the statement: $$ \mathrm{Re}\, \sum_{j=1}^m \int_{\mathrm{X}} A F_j(\mathbf{f}) \cdot G_j(\mathbf{f}) \,\,\ge \,\,0 $$ {\em for all $\mathbb{C}^d$-valued measurable functions $\mathbf{f}$ on $\mathrm{X}$ such that $F_j(\mathbf{f}) \in \mathrm{dom}(A_p)$ and $G_j(\mathbf{f}) \in \mathrm{L}^q(\mathrm{X})$ for all $j$.} It is shown that this statement is valid in general if it is valid for $\mathrm{X}$ being a two-point Bernoulli $(\frac{1}{2}, \frac{1}{2})$-space and $A$ being of a special form. As a consequence we obtain a new proof for the optimal angle of $\mathrm{L}^{p}$-analyticity for such semigroups, which is essentially the same as in the well-known sub-Markovian case. The proof of the main theorem is a combination of well-known reduction techniques and some representation results about operators on $\mathrm{C}(K)$-spaces. One focus of the paper lies on presenting these auxiliary techniques and results in great detail.

Abstract:
We provide necessary and sufficient conditions for a Hilbert space-valued Ornstein-Uhlenbeck process to be reversible with respect to its invariant measure $\mu$. For a reversible process the domain of its generator in $L^p(\mu )$ is characterized in terms of appropriate Sobolev spaces thus extending the Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We provide also a formula for the size of the spectral gap of the generator. Those results are applied to study the Ornstein-Uhlenbeck process in a chaotic environment. Necessary and sufficient conditions for a transition semigroup $(R_t)$ to be compact, Hilbert-Schmidt and strong Feller are given in terms of the coefficients of the Ornstein-Uhlenbeck operator. We show also that the existence of spectral gap implies a smoothing property of $R_t$ and provide an estimate for the (appropriately defined) gradient of $R_t\phi$. Finally, in the Hilbert-Schmidt case, we show that for any $\phi\in L^p(\mu)$ the function $R_t\phi$ is an (almost) classical solution of a version of the Kolmogorov equation.

Abstract:
We investigate the boundedness of the $H^\infty$-calculus by estimating the bound $b(\varepsilon)$ of the mapping $H^{\infty}\rightarrow \mathcal{B}(X)$: $f\mapsto f(A)T(\varepsilon)$ for $\varepsilon$ near zero. Here, $-A$ generates the analytic semigroup $T$ and $H^{\infty}$ is the space of bounded analytic functions on a domain strictly containing the spectrum of $A$. We show that $b(\varepsilon)=\mathcal{O}(|\log\varepsilon|)$ in general, whereas $b(\varepsilon)=\mathcal{O}(1)$ for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield $b(\varepsilon)=\mathcal{O}(\sqrt{|\log\varepsilon|})$.

Abstract:
The main objective of the present work is to study contraction semigroups generated by Laplace operators on metric graphs, which are not necessarily self-adjoint. We prove criteria for such semigroups to be continuity and positivity preserving. Also we provide a characterization of generators of Feller semigroups on metric graphs.

Abstract:
We find the structure of generators of norm continuous quantum Markov semigroups on B(h) that are symmetric with respect to the scalar product tr(\rho^{1/2}x\rho^{1/2}y) induced by a faithful normal invariant state invariant state \rho and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the so-called standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.

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.

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.

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.

Abstract:
An alternative presentation of the π-calculus is given. This version of the π-calculus is symmetric in the sense that communications are symmetric and there is no difference between input and output prefixes. The point of the symmetric π-calculus is that it has no abstract names. The set of closed names is therefore homogeneous. The π-calculus can be fully embedded into the symmetric π-calculus. The symmetry changes the emphasis of the communication mechanism of the π-calculus and opens up possibility for further variations. Supported by NNSFC, grant number 69503006. Fu Yuxi is an Associate Professor in the Department of Computer Science, Shanghai Jiao Tong University. He received his Ph.D. degree in computer science in 1992 from Manchester University, England. His current research interests include type theory, semantics and concurrency theory.

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.