Analyticity of a class of degenerate evolution equations on the canonical simplex of Rd\R^d arising from Fleming--Viot processes

We study 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.Comment: 32 page

A note on supercyclic operators in locally convex spaces

We treat some questions related to supercyclicity of continuous linear operators when acting in locally convex spaces. We extend results of Ansari and Bourdon and consider doubly power bounded operators in this general setting. Some examples are given

Convolutors on Sω(RN)\mathcal{S}_\omega(\mathbb{R}^N)

In this paper we continue the study of the spaces $\mathcal{O}_{M,\omega}(\mathbb{R}^N)$ and $\mathcal{O}_{C,\omega}(\mathbb{R}^N)$ undertaken in [1]. We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $\mathcal{O}'_{C,\omega}(\mathbb{R}^N)$ is the space of convolutors of the space $\mathcal{S}_\omega(\mathbb{R}^N)$ of the $\omega$-ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $\mathcal{S}'_\omega(\mathbb{R}^N)$. We also establish that the Fourier transform is an isomorphism from $\mathcal{O}'_{C,\omega}(\mathbb{R}^N)$ onto $\mathcal{O}_{M,\omega}(\mathbb{R}^N)$. In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $\mathcal{L}_b(\mathcal{S}_\omega(\mathbb{R}^N))$ and the last space is endowed with its natural lc-topology.Comment: arXiv admin note: text overlap with arXiv:2011.0396

Topological Properties of Weighted Composition Operators in Sequence Spaces

For fixed sequences u=(ui)i∈N,φ=(φi)i∈N , we consider the weighted composition operator Wu,φ with symbols u, φ defined by x=(xi)i∈N↦u(x∘φ)=(uixφi)i∈N . We characterize the continuity and the compactness of the operator Wu,φ when it acts on the weighted Banach spaces lp(v) , 1 ≤ p≤ ∞ , and c(v) , with v=(vi)i∈N a weight sequence on N . We extend these results to the case in which the operator Wu,φ acts on sequence (LF)-spaces of type lp(V) and on sequence (PLB)-spaces of type ap(V) , with p∈ [1 , ∞] ∪ { 0 } and V a system of weights on N . We also characterize other topological properties of Wu,φ acting on lp(V) and on ap(V) , such as boundedness, reflexivity and to being Montel.

Dissipative operators and additive perturbations in locally convex spaces

"This is the peer reviewed version of the following article: Albanese, Angela A., and David Jornet. 2015. Dissipative Operators and Additive Perturbations in Locally Convex Spaces. Mathematische Nachrichten 289 (8 9). Wiley: 920 49. doi:10.1002/mana.201500150, which has been published in final form at https://doi.org/10.1002/mana.201500150. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."[EN] Let (A, D(A)) be a densely defined operator on a Banach space X. Characterizations of when (A, D(A)) generates a C-0-semigroup on X are known. The famous result of Lumer and Phillips states that it is so if and only if (A, D(A)) is dissipative and rg(lambda I - A) subset of X is dense in X for some lambda > 0. There exists also a rich amount of Banach space results concerning perturbations of dissipative operators. In a recent paper Tyran-Kaminska provides perturbation criteria of dissipative operators in terms of ergodic properties. These results, and others, are shown to remain valid in the setting of general non-normable locally convex spaces. Applications of the results to concrete examples of operators on function spaces are also presented. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimThe research of the second author was partially supported by MINECO of Spain, Project MTM2013-43540-P, by Programa de Apoyo a la Investigacion y Desarrollo de la UPV, PAID-06-12 and by Generalitat Valenciana ACOMP/2015/186.Albanese, AA.; Jornet Casanova, D. (2016). Dissipative operators and additive perturbations in locally convex spaces. Mathematische Nachrichten. 289(8-9):920-949. https://doi.org/10.1002/mana.201500150S9209492898-9Albanese, A. A., Bonet, J., & Ricker, W. J. (2010). C0-semigroups and mean ergodic operators in a class of Fréchet spaces. Journal of Mathematical Analysis and Applications, 365(1), 142-157. doi:10.1016/j.jmaa.2009.10.014Albanese, A. A., Bonet, J., & Ricker, W. J. (2011). Mean ergodic semigroups of operators. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 106(2), 299-319. doi:10.1007/s13398-011-0054-2Albanese, A. A., Bonet, J., & Ricker, W. J. (2013). Montel resolvents and uniformly mean ergodic semigroups of linear operators. Quaestiones Mathematicae, 36(2), 253-290. doi:10.2989/16073606.2013.779978Albanese, A. A., Bonet, J., & Ricker, W. J. (2013). Convergence of arithmetic means of operators in Fréchet spaces. Journal of Mathematical Analysis and Applications, 401(1), 160-173. doi:10.1016/j.jmaa.2012.11.060Albanese, A. A., Bonet, J., & Ricker, W. J. (2014). Uniform mean ergodicity of $C_0$-semigroups\newline in a class of Fréchet spaces. Functiones et Approximatio Commentarii Mathematici, 50(2), 307-349. doi:10.7169/facm/2014.50.2.8Domański, P., & Langenbruch, M. (2011). On the abstract Cauchy problem for operators in locally convex spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 106(2), 247-273. doi:10.1007/s13398-011-0052-4Frerick, L., Jordá, E., Kalmes, T., & Wengenroth, J. (2014). Strongly continuous semigroups on some Fréchet spaces. Journal of Mathematical Analysis and Applications, 412(1), 121-124. doi:10.1016/j.jmaa.2013.10.053B. Jacob S.-A. Wegner Asymptotics of solution equations beyond Banach spaces Semigroup ForumJacob, B., Wegner, S.-A., & Wintermayr, J. (2015). Desch-Schappacher perturbation of one-parameter semigroups on locally convex spaces. Mathematische Nachrichten, 288(8-9), 925-934. doi:10.1002/mana.201400116Köthe, G. (1983). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. doi:10.1007/978-3-642-64988-2Köthe, G. (1979). Topological Vector Spaces II. Grundlehren der mathematischen Wissenschaften. doi:10.1007/978-1-4684-9409-9KOMATSU, H. (1964). Semi-groups of operators in locally convex spaces. Journal of the Mathematical Society of Japan, 16(3), 230-262. doi:10.2969/jmsj/01630230Kōmura, T. (1968). Semigroups of operators in locally convex spaces. Journal of Functional Analysis, 2(3), 258-296. doi:10.1016/0022-1236(68)90008-6Lumer, G., & Phillips, R. S. (1961). Dissipative operators in a Banach space. Pacific Journal of Mathematics, 11(2), 679-698. doi:10.2140/pjm.1961.11.679Miyadera, I. (1959). Semi-groups of operators in Fréchet space amd applications to partial differential equations. Tohoku Mathematical Journal, 11(2), 162-183. doi:10.2748/tmj/1178244580Moscatelli, V. B. (1980). Fréchet Spaces without continuous Norms and without Bases. Bulletin of the London Mathematical Society, 12(1), 63-66. doi:10.1112/blms/12.1.63Tyran-Kamińska, M. (2009). Ergodic theorems and perturbations of contraction semigroups. Studia Mathematica, 195(2), 147-155. doi:10.4064/sm195-2-4S.-A. Wegner The growth bound for strongly continuous semigroups on Fréchet spaces Proc. Edinb. Math. Soc. (2) (to appear) 10.1017/S001309151500031

A Note on Supercyclic Operators in Locally Convex Spaces

[EN] We treat some questions related to supercyclicity of continuous linear operators when acting in locally convex spaces. We extend results of Ansari and Bourdon and consider doubly power bounded operators in this general setting. Some examples are given.We are indebted to Prof. Jose Bonet for his helpful suggestions on the topic of this paper. The authors were partially supported by the project MTM2016-76647-P.Albanese, AA.; Jornet Casanova, D. (2019). A Note on Supercyclic Operators in Locally Convex Spaces. Mediterranean Journal of Mathematics. 16(5):1-10. https://doi.org/10.1007/s00009-019-1386-yS11016