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Register a new userGraph Laplacians do not generate strongly continuous semigroups
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We show that for graph Laplacians $Delta_G$ on a connected locally finite simplicial undirected graph $G$ with countable infinite vertex set $V$ none of the operators $alpha,mathrm{Id}+betaDelta_G, alpha,betainmathbb{K},beta e 0$, generate a strongly continuous semigroup on $mathbb{K}^V$ when the latter is equipped with the product topology.
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In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.
In this paper we study the main properties of the Ces`aro means of bi-continuous semigroups, introduced and studied by K{u}hnemund in [24]. We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in $C_b(R^N)$ and to evolution operators associated with nonautonomous second-order differential operators in $C_b(R^N)$ with time-periodic coefficients.
In finite dimensions, we provide characterizations of the quantum dynamical semigroups that do not decrease the von Neumann, the Tsallis and the Renyi entropies, as well as a family of functions of density operators strictly related to the Schatten norms. A few remarkable consequences --- in particular, a description of the associated infinitesimal generators --- are derived, and some significant examples are discussed. Extensions of these results to semigroups of trace-preserving positive (i.e., not necessarily completely positive) maps and to a more general class of quantum entropies are also considered.
In [B1, Theorem 2.36] we proved the equivalence of six conditions on a continuous function f on an interval. These conditions define a subset of the set of operator convex functions, whose elements are called strongly operator convex. Two of the six conditions involve operator-algebraic semicontinuity theory, as given by C. Akemann and G. Pedersen in [AP], and the other four conditions do not involve operator algebras at all. Two of these conditions are operator inequalities, one is a global condition on f, and the fourth is an integral representation of f stronger than the usual integral representation for operator convex functions. The purpose of this paper is to make the equivalence of these four conditions accessible to people who do not know operator algebra theory as well as to operator algebraists who do not know the semicontinuity theory. We also provide a similar treatment of one theorem from [B1] concerning (usual) operator convex functions. And in two final sections we give a somewhat tentative treatment of some other operator inequalities for strongly operator convex functions, and we give a differential criterion for strong operator convexity.
We consider families of E_0-semigroups continuously parametrized by a compact Hausdorff space, which are cocycle-equivalent to a given E_0-semigroup beta. When the gauge group of $beta$ is a Lie group, we establish a correspondence between such families and principal bundles whose structure group is the gauge group of beta.
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