Hypercyclic algebras for convolution and composition operators

[EN] We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator D of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as , , or , where . In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras.This work is supported in part by MEC, Project MTM 2016-7963-P. We also thank Angeles Prieto for comments and suggestions.Bès, J.; Conejero, JA.; Papathanasiou, D. (2018). Hypercyclic algebras for convolution and composition operators. Journal of Functional Analysis. 274(10):2884-2905. https://doi.org/10.1016/j.jfa.2018.02.003S288429052741

Convolution operators supporting hypercyclic algebras

[EN] We show that any convolution operator induced by a non-constant polynomial that vanishes at zero supports a hypercyclic algebra. This partially solves a question raised by R. AronThis work is supported in part by MICINN and FEDER, Project MTM2013-47093-P, and by GVA, Project ACOMP/2015/005.Bès, JP.; Conejero, JA.; Papathanasiou, D. (2017). Convolution operators supporting hypercyclic algebras. Journal of Mathematical Analysis and Applications. 445(2):1232-1238. https://doi.org/10.1016/j.jmaa.2016.01.029S12321238445

Dipolar plasma source modeling: a first approach

International audienceThe scaling up of conventional plasmas presents limitations in terms of plasma density, limited to the critical density, and of uniformity, due to the difficulty of achieving constant amplitude standing wave patterns along linear microwave applicators in the meter range. An alternative solution lies in the concept of distribution from one- to two-dimensional networks of elementary plasma. Each elementary plasma source consists in a permanent magnet on which microwaves are applied via an independent coaxial line [1]. The plasma is produced by the electrons accelerated at ECR (Electron Cyclotron Resonance) and trapped in the dipolar magnetic field. Large-size uniform plasmas can be obtained by assembling as many such elementary plasma sources as necessary, without any physical or technical limitations [2]. Simulation of the plasma produced by a dipolar source requires a global, self consistent, modeling of its functioning. In order to obtain results to lead a first optimization of the dipolar source, magnetostatics, microwave propagation and fast electrons trajectories (Particles in Cell (PIC) and Monte-Carlo hybrid method [3]) have been performed with Comsol Multiphysics and MatLab

The sextic oscillator as a γ\gamma-independent potential

The sextic oscillator is proposed as a two-parameter solvable $\gamma$-independent potential in the Bohr Hamiltonian. It is shown that closed analytical expressions can be derived for the energies and wavefunctions of the first few levels and for the strength of electric quadrupole transitions between them. Depending on the parameters this potential has a minimum at $\beta=0$ or at $\beta>0$, and might also have a local maximum before reaching its minimum. A comparison with the spectral properties of the infinite square well and the $\beta^4$ potential is presented, together with a brief analysis of the experimental spectrum and E2 transitions of the $^{134}$Ba nucleus.Comment: 15 pages, 5 figures; to appear in Phys. Rev.

The low-lying quadrupole collective excitations of Ru and Pd isotopes

Quadrupole excitations of even-even Ru and Pd isotopes are described within microscopic approach based on the general collective Bohr model which includes the effect of coupling with the pairing vibrations. The excitation energies and E2 transition probabilities observed in 104-114Ru and 106-110Pd are reproduced in the frame of the calculation containing no free parameters.Comment: 11 pages, 18 figures in EPS forma

Recurrence properties of hypercyclic operators

[EN] We generalize the notions of hypercyclic operators, U-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics, namely A-hypercyclicity. We then state an A-hypercyclicity criterion, inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion, and we show that this criterion characterizes the A-hypercyclicity for weighted shifts. We also investigate which density properties can the sets N(x, U) = {n is an element of N; T-n x is an element of U} have for a given hypercyclic operator, and we study the new notion of reiteratively hypercyclic operators.This work is supported in part by MEC and FEDER, Project MTM2013-47093-P, and by GVA, Projects PROMETEOII/2013/013 and ACOMP/2015/005. The second author was a postdoctoral researcher of the Belgian FNRS.Bès, JP.; Menet, Q.; Peris Manguillot, A.; Puig-De Dios, Y. (2016). Recurrence properties of hypercyclic operators. Mathematische Annalen. 366(1):545-572. https://doi.org/10.1007/s00208-015-1336-3S5455723661Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211, 766–793 (2007)Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358, 5083–5117 (2006)Bayart, F., Grivaux, S.: Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. 94, 181–210 (2007)Bayart, F., Matheron, É.: Dynamics of linear operators, Cambridge Tracts in Mathematics, 179. Cambridge University Press, Cambridge (2009)Bayart, F., Matheron, É.: (Non-)weakly mixing operators and hypercyclicity sets. Ann. Inst. Fourier 59, 1–35 (2009)Bayart, F., Ruzsa, I.: Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory Dynam. Syst. 35, 691–709 (2015)Bergelson, V.: Ergodic Ramsey Theory- an update, Ergodic Theory of $$\mathbb{Z}^d$$ Z d -actions. Lond. Math. Soc. Lecture Note Ser. 28, 1–61 (1996)Bernal-González, L., Grosse-Erdmann, K.-G.: The Hypercyclicity Criterion for sequences of operators. Studia Math. 157, 17–32 (2003)Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergodic Theory Dynam. Syst. 27, 383–404 (2007)Bonilla, A., Grosse-Erdmann, K.-G.: Erratum: Ergodic Theory Dynam. Systems 29, 1993–1994 (2009)Chan, K., Seceleanu, I.: Hypercyclicity of shifts as a zero-one law of orbital limit points. J. Oper. Theory 67, 257–277 (2012)Costakis, G., Sambarino, M.: Topologically mixing hypercyclic operators. Proc. Amer. Math. Soc. 132, 385–389 (2004)Furstenberg, H.: Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton (1981)Giuliano, R., Grekos, G., Mišík, L.: Open problems on densities II, Diophantine Analysis and Related Fields 2010. AIP Conf. Proc. 1264, 114–128 (2010)Grosse-Erdmann, K.-G.: Hypercyclic and chaotic weighted shifts. Studia Math. 139, 47–68 (2000)Grosse-Erdmann, K.-G., Peris, A.: Frequently dense orbits. C. R. Math. Acad. Sci. Paris 341, 123–128 (2005)Grosse-Erdmann, K.G., Peris, A.: Weakly mixing operators on topological vector spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 104, 413–426 (2010)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear chaos, Universitext. Springer, London (2011)Menet, Q.: Linear chaos and frequent hypercyclicity. Trans. Amer. Math. Soc. arXiv:1410.7173Puig, Y.: Linear dynamics and recurrence properties defined via essential idempotents of $$\beta {\mathbb{N}}$$ β N (2014) arXiv:1411.7729 (preprint)Salas, H.N.: Hypercyclic weighted shifts. Trans. Amer. Math. Soc. 347, 993–1004 (1995)Salat, T., Toma, V.: A classical Olivier’s theorem and statistical convergence. Ann. Math. Blaise Pascal 10, 305–313 (2003)Shkarin, S.: On the spectrum of frequently hypercyclic operators. Proc. Am. Math. Soc. 137, 123–134 (2009

Collective quadrupole excitations in the 50<Z,N<82 nuclei with the generalized Bohr Hamiltonian

The generalized Bohr Hamiltonian is applied to a description of low-lying collective excitations in even-even isotopes of Te, Xe, Ba, Ce, Nd and Sm. The collective potential and inertial functions are determined by means of the Strutinsky method and the cranking model, respectively. A shell-dependent parametrization of the Nilsson potential is used. An approximate particle-number projection is performed in treatment of pairing correlations. The effect of coupling with the pairing vibrations is taken into account approximately when determining the inertial functions. The calculation does not contain any free parameter.Comment: Latex2e source, 20 pages, 14 figures in EPS format, tar gzipped fil

Finite Intersection Property and Dynamical Compactness

[EN] Dynamical compactness with respect to a family as a new concept of chaoticity of a dynamical system was introduced and discussed in Huang et al. (J Differ Equ 260(9):6800-6827, 2016). In this paper we continue to investigate this notion. In particular, we prove that all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. We investigate weak mixing and weak disjointness by using the concept of dynamical compactness. We also explore further difference between transitive compactness and weak mixing. As a byproduct, we show that the -limit and the -limit sets of a point may have quite different topological structure. Moreover, the equivalence between multi-sensitivity, sensitive compactness and transitive sensitivity is established for a minimal system. Finally, these notions are also explored in the context of linear dynamics.Wen Huang and Sergii Kolyada acknowledge the hospitality of the School of Mathematical Sciences of the Fudan University, Shanghai. Sergii Kolyada also acknowledges the hospitality of the Max-Planck-Institute fur Mathematik (MPIM) in Bonn, the Departament de Matematica Aplicada of the Universitat Politecnica de Valencia, the partial support of Project MTM2013-47093-P, and the Department of Mathematics of the Chinese University of Hong Kong. We thank the referees for careful reading and constructive comments that have resulted in substantial improvements to this paper. Wen Huang was supported by NNSF of China (11225105, 11431012); Alfred Peris was supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P, and by GVA, Project PROMETEOII/2013/013; and Guohua Zhang was supported by NNSF of China (11671094).Huang, W.; Khilko, D.; Kolyada, S.; Peris Manguillot, A.; Zhang, G. (2018). 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Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces

[EN] Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.J. Bonet was partially supported by the research projects MTM2016-76647-P and GV Prometeo 2017/102 (Spain). M. Worku is supported by ISP project, Addis Ababa University, Ethiopia.Bonet Solves, JA.; Mengestie, T.; Worku, M. (2019). Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces. Results in Mathematics. 74(4):1-15. https://doi.org/10.1007/s00025-019-1123-7S115744Abanin, A.V., Tien, P.T.: Differentiation and integration operators on weighted Banach spaces of holomorphic functions. Math. Nachr. 290(8–9), 1144–1162 (2017)Atzmon, A., Brive, B.: Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions, Bergman spaces and related topics in complex analysis, Contemp. Math., vol. 404, Amer. Math. Soc., Providence, RI, pp. 27–39 (2006)Bayart, F., Matheron, E.: Dynamics of Linear Operators, Cambridge Tracts in Math, vol. 179. Cambridge Univ. Press, Cambridge (2009)Bermúdez, T., Bonilla, A., Peris, A.: On hypercyclicity and supercyclicity criteria. Bull. Austral. Math. Soc. 70, 45–54 (2004)Beltrán, M.J.: Dynamics of differentiation and integration operators on weighted space of entire functions. Studia Math. 221, 35–60 (2014)Beltrán, M.J., Bonet, J., Fernández, C.: Classical operators on weighted Banach spaces of entire functions. Proc. Am. Math. Soc. 141, 4293–4303 (2013)Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)Bonet, J.: Dynamics of the differentiation operator on weighted spaces of entire functions. Math. Z. 26, 649–657 (2009)Bonet, J.: The spectrum of Volterra operators on weighted Banach spaces of entire functions. Q. J. Math. 66, 799–807 (2015)Bonet, J., Bonilla, A.: Chaos of the differentiation operator on weighted Banach spaces of entire functions. Complex Anal. Oper. Theory 7, 33–42 (2013)Bonet, J., Taskinen, J.: A note about Volterra operators on weighted Banach spaces of entire functions. Math. Nachr. 288, 1216–1225 (2015)Constantin, O., Persson, A.-M.: The spectrum of Volterra-type integration operators on generalized Fock spaces. Bull. Lond. Math. Soc. 47, 958–963 (2015)Constantin, O., Peláez, J.-Á.: Integral operators, embedding theorems and a Littlewood–Paley formula on weighted Fock spaces. J. Geom. Anal. 26, 1109–1154 (2016)De La Rosa, M., Read, C.: A hypercyclic operator whose direct sum is not hypercyclic. J. Oper. Theory 61, 369–380 (2009)Dunford, N.: Spectral theory. I. Convergence to projections. Trans. Am. Math. Soc. 54, 185–217 (1943)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear Chaos. Springer, New York (2011)Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)Lyubich, Yu.: Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition. Studia Mathematica 143(2), 153–167 (1999)Mengestie, T.: A note on the differential operator on generalized Fock spaces. J. Math. Anal. Appl. 458(2), 937–948 (2018)Mengestie, T.: Spectral properties of Volterra-type integral operators on Fock–Sobolev spaces. J. Kor. Math. Soc. 54(6), 1801–1816 (2017)Mengestie, T.: On the spectrum of volterra-type integral operators on Fock–Sobolev spaces. Complex Anal. Oper. Theory 11(6), 1451–1461 (2017)Mengestie, T., Ueki, S.: Integral, differential and multiplication operators on weighted Fock spaces. Complex Anal. Oper. Theory 13, 935–95 (2019)Mengestie, T., Worku, M.: Isolated and essentially isolated Volterra-type integral operators on generalized Fock spaces. Integr. Transf. Spec. Funct. 30, 41–54 (2019)Nagy, B., Zemanek, J.A.: A resolvent condition implying power boundedness. Studia Math. 134, 143–151 (1999)Nevanlinna, O.: Convergence of iterations for linear equations. Lecture Notes in Mathematics. ETH Zürich, Birkhäuser, Basel (1993)Ritt, R.K.: A condition that $$\lim _{n\rightarrow \infty } n^{-1}T^n =0$$. Proc. Am. Math. Soc. 4, 898–899 (1953)Ueki, S.: Characterization for Fock-type space via higher order derivatives and its application. Complex Anal. Oper. Theory 8, 1475–1486 (2014)Yosida, K.: Functional Analysis. Springer, Berlin (1978)Yosida, K., Kakutani, S.: Operator-theoretical treatment of Marko’s process and mean ergodic theorem. Ann. Math. 42(1), 188–228 (1941

Mean ergodicity of weighted composition operators on spaces of holomorphic functions

[EN] Let phi be a self-map of the unit disc D of the complex plane C and let psi be a holomorphic function on D. We investigate the mean ergodicity and power boundedness of the weighted composition operator C-phi,C-psi(f) = psi(f o phi) with symbol phi and multiplier psi on the space H(D). We obtain necessary and sufficient conditions on the symbol phi and on the multiplier psi which characterize when the weighted composition operator is power bounded and (uniformly) mean ergodic. One necessary condition is that the symbol phi has a fixed point in D. If phi is not a rational rotation, the sufficient conditions are related to the modulus of the multiplier on the fixed point of phi. Some of our results are valid in an open connected set U of the complex plane.This research was partially supported by MINECO, Project MTM2013-43540-P. The second and third authors were partially supported by GVA, Project AICO/2016/054.Beltrán Meneu, MJ.; Gómez Collado, MDC.; Jorda Mora, E.; Jornet Casanova, D. (2016). Mean ergodicity of weighted composition operators on spaces of holomorphic functions. Journal of Mathematical Analysis and Applications. 444(2):1640-1651. https://doi.org/10.1016/j.jmaa.2016.07.039S16401651444