101 research outputs found
C-spectrality of the Schrödinger operator in Lp spaces
AbstractIn [1], the notions of C-regularized functional calculus and C-regularized scalar operator are defined and their mutual relationship with temperate C-regularized groups is given. In this note, we apply these notions in two ways: first we consider the Schrödinger operator in Lp(Ω) with Dirichlet boundary condition, when Ω is a bounded domain in Rn. The second application will be the operator −Δ + V in Lp(Rn), when V belongs to the Kato's class of potentials
C-Semigroups and the Cauchy Problem
AbstractWe extend the definition of generator to C-semigroups that may not be exponentially bounded, where the range of C may not be dense. We then characterize linear operators, A, for which the associated abstract Cauchy problem has a unique solution, for every initial value in the domain of another operator, B, without assuming that the domain of A is dense, or the solutions are exponentially bounded. We also give Hille-Yosida type characterizations of generators that may not be densely defined
Convoluted -cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines
Convoluted -cosine functions and semigroups in a Banach space setting
extending the classes of fractionally integrated -cosine functions and
semigroups are systematically analyzed. Structural properties of such operator
families are obtained. Relations between convoluted -cosine functions and
analytic convoluted -semigroups, introduced and investigated in this paper
are given through the convoluted version of the abstract Weierstrass formula
which is also proved in the paper. Ultradistribution and hyperfunction sines
are connected with analytic convoluted semigroups and ultradistribution
semigroups. Several examples of operators generating convoluted cosine
functions, (analytic) convoluted semigroups as well as hyperfunction and
ultradistribution sines illustrate the abstract approach of the authors. As an
application, it is proved that the polyharmonic operator
acting on with appropriate boundary
conditions, generates an exponentially bounded -convoluted cosine
function, and consequently, an exponentially bounded analytic
-convoluted semigroup of angle for suitable
exponentially bounded kernels and $K_{n+1}.
Optimal estimates for the semigroup generated by the classical Volterra operator on L p -spaces
Chaos for linear fractional transformations of shifts
[EN] We characterize chaos for \phi;(B) on Banach sequence spaces, where \phi; is a Linear Fractional Transformation and B is the usual backward shift operator. Characterizations are computable since they involve only the four complex numbers defining \phi.Supported by MICINN and FEDER, Project MTM2013-47093-P, and by GVA, Projects PROMETEOII/2013/013 and ACOMP/2015/005Jimenez-Munguia, RR.; Galán-Céspedes, VJ.; MartÃnez Jiménez, F.; Peris Manguillot, A. (2016). Chaos for linear fractional transformations of shifts. Topology and its Applications. 203:84-90. https://doi.org/10.1016/j.topol.2015.12.077S849020
Chaotic differential operators
We give sufficient conditions for chaos of (differential) operators on Hilbert spaces of entire functions. To this aim we establish conditions on the coefficients of a polynomial P(z) such that P(B) is chaotic on the space lp, where B is the backward shift operator. © 2011 Springer-Verlag.This work was partially supported by the MEC and FEDER Projects MTM2007-64222, MTM2010-14909, and by GVA Project GV/2010/091, and by UPV Project PAID-06-09-2932. The authors would like to thank A. Peris for helpful comments and ideas that produced a great improvement of the paper's presentation. We also thank the referees for their helpful comments and for reporting to us a gap in Theorem 1.Conejero Casares, JA.; MartÃnez Jiménez, F. (2011). Chaotic differential operators. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 105(2):423-431. https://doi.org/10.1007/s13398-011-0026-6S4234311052Bayart, F., Matheron, É.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bermúdez T., Miller V.G.: On operators T such that f(T) is hypercyclic. Integr. Equ. Oper. Theory 37(3), 332–340 (2000)Bonet J., MartÃnez-Giménez F., Peris A.: Linear chaos on Fréchet spaces. Int. J. Bifur. Chaos Appl. Sci. Eng. 13(7), 1649–1655 (2003)Chan K.C., Shapiro J.H.: The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40(4), 1421–1449 (1991)Conejero J.A., Müller V.: On the universality of multipliers on . J. Approx. Theory. 162(5), 1025–1032 (2010)deLaubenfels R., Emamirad H.: Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory Dyn. Syst. 21(5), 1411–1427 (2001)Devaney, R.L.: An introduction to chaotic dynamical systems, 2nd edn. In: Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989)Godefroy G., Shapiro J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2), 229–269 (1991)Grosse-Erdmann K.-G.: Hypercyclic and chaotic weighted shifts. Stud. Math. 139(1), 47–68 (2000)Grosse-Erdmann, K.-G., Peris, A.,: Linear chaos. Universitext, Springer, New York (to appear, 2011)Herzog G., Schmoeger C.: On operators T such that f(T) is hypercyclic. Stud. Math. 108(3), 209–216 (1994)Kahane, J.-P.: Some random series of functions, 2nd edn. In: Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge University Press, Cambridge (1985)MartÃnez-Giménez F., Peris A.: Chaos for backward shift operators. Int. J. Bifur. Chaos Appl. Sci. Eng. 12(8), 1703–1715 (2002)MartÃnez-Giménez F.: Chaos for power series of backward shift operators. Proc. Am. Math. Soc. 135, 1741–1752 (2007)Müller V.: On the Salas theorem and hypercyclicity of f(T). Integr. Equ. Oper. Theory 67(3), 439–448 (2010)Protopopescu V., Azmy Y.Y.: Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2(1), 79–90 (1992)Rolewicz S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)Salas H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347(3), 93–1004 (1995)Shapiro, J.H.: Simple connectivity and linear chaos. Rend. Circ. Mat. Palermo. (2) Suppl. 56, 27–48 (1998
Chaotic semigroups from second order partial differential equations
[EN] We give general conditions on given parameters to ensure Devaney and distributional chaos for the solution C-0-semigroup corresponding to a class of second-order partial differential equations. We also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups. In the case of chaos, we prove that the Co-semigroup admits a strongly mixing measure with full support. We also give concrete examples of partial differential equations, such as the telegraph equation, whose solutions satisfy these properties. (C) 2017 Elsevier Inc. All rights reserved.The first and third authors are supported in part by MINECO and FEDER, grant MTM2016-75963-P. The second author is partially supported by CONICYT, under Fondecyt Grant number 1140258 and CONICYT-PIA, Anillo ACT1416.Conejero, JA.; Lizama, C.; Murillo Arcila, M. (2017). Chaotic semigroups from second order partial differential equations. Journal of Mathematical Analysis and Applications. 456(1):402-411. https://doi.org/10.1016/j.jmaa.2017.07.013S402411456
Distributional chaos for the Forward and Backward Control traffic model
The interest in car-following models has increased in the
last years due to its connection with vehicle-to-vehicle
communications and the development of driverless cars. Some
non-linear models such as the Gazes–Herman–Rothery model
were already known to be chaotic. We consider the linear
Forward and Backward Control traffic model for an infinite
number of cars on a track. We show the existence of solutions
with a chaotic behaviour by using some results of linear
dynamics of C0-semigroups. In contrast, we also analyse which
initial configurations lead to stable solutions.
© 2015 Elsevier Inc. All rights reserved.The first three authors were supported by MTM2013-47093-P. The first author was supported by the ERC grant HEVO no. 2177691. The third author was also supported by MTM2013-47093-P and by a grant from the FPU program of MEC (AP 2010-4361).Barrachina Civera, X.; Conejero, JA.; Murillo Arcila, M.; Seoane-Sepulveda, JB. (2015). Distributional chaos for the Forward and Backward Control traffic model. Linear Algebra and its Applications. 479:202-215. https://doi.org/10.1016/j.laa.2015.04.010S20221547
Distributionally chaotic families of operators on Fréchet spaces
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis (CPAA) following peer review. The definitive publisher-authenticated version Conejero, J. A., Kostić, M., Miana, P. J., & Murillo-Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces.Communications on Pure and Applied Analysis, 2016, vol. 15, no 5, p. 1915-1939, is available online at: http://dx.doi.org/10.3934/cpaa.2016022The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C-0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C-0-semigroups, C-regularized semigroups, and alpha-timesintegrated semigroups on Frechet spaces. We also add a study of rescaled distributionally chaotic C-0-semigroups. Some examples are provided to illustrate all these results.The first and fourth authors are supported in part by MEC Project MTM2010-14909, MTM2013-47093-P, and Programa de Investigacion y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia. The third author has been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragon, and Project UZCUD2014-CIE-09, Universidad de Zaragoza. The fourth author is supported by a grant of the FPU Program of Ministry of education of Spain.Conejero, JA.; Kostic, M.; Miana Sanz, PJ.; Murillo Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure and Applied Analysis. 15(5):1915-1939. https://doi.org/10.3934/cpaa.2016022S1915193915
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