This issue of "Note di Matematica" is dedicated to the memory of Bruno Moscatelli, Editor in Chief of "Note di Matematica", until his death in 2008. Several authors and friends contributed with papers about various subjects of Mathematical Analysis.

We study the analyticity of the semigroups generated by some classes of degenerate second order differential operators in the space of continuous function on a domain with corners. These semigroups arise from the theory of dynamics of populations.

We solve a long standing open problem, by proving the analyticity of the semigroups generated by a class of degenerate second order differential operators in the space $C(S_d)$, where $S_d$ is the canonical simplex of $R^d$. The semigroups arise from the theory of Fleming--Viot processes in population genetics.

We give characterizations of chaos for $C_0$-semigroups induced by semiflows on $L^p_rho(Omega)$ for open $OmegasubseteqR$ similar to the characterizations of hypercyclicity and mixing of such $C_0$-semigroups proved in cite{kalmes2009hypercyclic}. Moreover, we characterize hypercyclicity, mixing, and chaos for these classes of $C_0$-semigroups on $W^{1,p}_*(I)$ for a bounded interval $IsubsetR$ and prove that these $C_0$-semigroups are never hypercyclic on $W^{1,p}(I)$. We apply our results to concrete first order partial differential equations, such as the von Foerster-Lasota equation.

Distributional chaos for strongly continuous semigroups is studied and characterized. It is shown to be equivalent to the existence of a distributionally irregular vector. Finally, a sufficient condition for distributional chaos on the point spectrum of the generator of the semigroup is presented. An application to the semigroup generated in $L^2(mathbb{R})$ by a translation of the Ornstein-Uhlenbeck operator is also given.

Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is studied. The results are achieved by establishing an analogy between frequent hypercyclicity for translation semigroups and for weighted pseudo-shifts and by characterizing frequently hypercyclic weighted pseudo-shifts on spaces of vanishing sequences. Frequently hypercyclic translation semigroups on weighted $L^p$-spaces are also characterized.

Abstract. We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be veried in certain cases in terms of the infinitesimal generator of the semigroup. Applications are given for semigroups generated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of p-integrable functions, or continuous functions that, multiplied by the weight, vanish at infinity.

The chaotic and hypercyclic behavior of the C0-semigroups of operators generated by a perturbation of the Ornstein-Uhlenbeck operator with a multiple of the identity in L2(RN) is investigated. Negative and positive results are presented, depending on the signs of the real parts of the eigenvalues of the matrix appearing in the drift of the operator.

We show the existence of solutions for a second order ordinary differential equation coupled with a boundary value condition and an integral condition.

The aim of this paper is to present some results about generation, sectoriality and gradient estimates both for the semigroup and for the resolvent of suitable realizations of the operators $A^{gamma,b}u(x) = gamma xu''(x) + bu'(x)$, with constants $ gamma > 0$ and $ bgeq 0$, in the space $C([0,infty])$.

We prove some permanence results with respect to quotient spaces and to projective and injective tensor products of the Dunford--Pettis and Grothendieck properties in the setting of locally convex Hausdorff spaces.