Distributionally chaotic families of operators on Fréchet spaces

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September 2016, 15(5): 1915-1939. doi: 10.3934/cpaa.2016022

Distributionally chaotic families of operators on Fréchet spaces

J. Alberto Conejero ^1, , Marko Kostić ^2, , Pedro J. Miana ^3, and Marina Murillo-Arcila ^4,

1.	Dept. Matemàtica Aplicada and IUMPA, Universitat Politècnica de València, València, 46022
2.	Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia
3.	Departamento Matemáticas e I.U.M.A., Universidad de Zaragoza, Zaragoza, Spain
4.	Instituto Universitario de Matemática Pura y Aplicada, Universitat Politécnica de València, València, Spain

Received June 2015 Revised May 2016 Published July 2016

Abstract
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The 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 $\alpha$-times integrated semigroups on Fréchet spaces. We also add a study of rescaled distributionally chaotic $C_0$-semigroups. Some examples are provided to illustrate all these results.

Keywords: $C$-regularized semigroups, hypercyclicity, well-posedness, Distributional chaos, Fréchet spaces, cosine functions, abstract time-fractional equations, integrated semigroups.

Mathematics Subject Classification: 47A16, 47D03, 47D06, 47D9.

Citation: J. Alberto Conejero, Marko Kostić, Pedro J. Miana, Marina Murillo-Arcila. Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1915-1939. doi: 10.3934/cpaa.2016022

References:

[1]	A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2069. doi: 10.3934/cpaa.2013.12.2069.
[2]	A. A. Albanese, J. Bonet and W. J. Ricker, $C_0$-semigroups and mean ergodic operators in a class of Fréchet spaces,, \emph{J. Math. Anal. Appl.}, 365 (2010), 142. doi: 10.1016/j.jmaa.2009.10.014.
[3]	R. M. Aron, J. B. Seoane-Sepúlveda and A. Weber, Chaos on function spaces,, \emph{Bull. Austral. Math. Soc.}, 71 (2005), 411. doi: 10.1017/S0004972700038417.
[4]	J. Aroza and A. Peris, Chaotic behaviour of birth-and-death models with proliferation,, \emph{J. Difference Equ. Appl.}, 18 (2012), 647. doi: 10.1080/10236198.2011.631535.
[5]	J. Banasiak, Birth-and-death type systems with parameter and chaotic dynamics of some linear kinetic models,, \emph{Z. Anal. Anwendungen}, 24 (2005), 675. doi: 10.4171/ZAA/1262.
[6]	J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 959. doi: 10.3934/dcds.2005.12.959.
[7]	J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos,, \emph{Discrete Contin. Dyn. Syst.}, 29 (2011), 67. doi: 10.3934/dcds.2011.29.67.
[8]	X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations,, \emph{Abstr. Appl. Anal.}, (4570).
[9]	X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Distributional chaos for the forward and backward control traffic model,, \emph{Linear Algebra Appl.}, 479 (2015), 202. doi: 10.1016/j.laa.2015.04.010.
[10]	X. Barrachina and A. Peris, Distributionally chaotic translation semigroups,, \emph{J. Difference Equ. Appl.}, 18 (2012), 751. doi: 10.1080/10236198.2011.625945.
[11]	F. Bayart and T. Bermúdez, Semigroups of chaotic operators,, \emph{Bull. Lond. Math. Soc.}, 41 (2009), 823. doi: 10.1112/blms/bdp055.
[12]	F. Bayart and É. Matheron, Dynamics of Linear Operators, vol. 179 of Cambridge Tracts in Mathematics,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511581113.
[13]	T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators,, \emph{J. Math. Anal. Appl.}, 373 (2011), 83.
[14]	T. Bermúdez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria,, \emph{Bull. Austral. Math. Soc.}, 70 (2004), 45. doi: 10.1017/S0004972700035802.
[15]	T. Bermúdez and V. G. Miller, On operators $T$ such that $f(T)$ is hypercyclic,, \emph{Extracta Math.}, 15 (2000), 237.
[16]	L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators,, \emph{Studia Math.}, 157 (2003), 17. doi: 10.4064/sm157-1-2.
[17]	L. Bernal-González, D. Pellegrino and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 51 (2014), 71. doi: 10.1090/S0273-0979-2013-01421-6.
[18]	N. C. Bernardes Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators,, \emph{J. Funct. Anal.}, 265 (2013), 2143. doi: 10.1016/j.jfa.2013.06.019.
[19]	J. Bès, K. C. Chan and S. M. Seubert, Chaotic unbounded differentiation operators,, \emph{Integral Equations Operator Theory}, 40 (2001), 257. doi: 10.1007/BF01299846.
[20]	J. Bès and A. Peris, Hereditarily hypercyclic operators,, \emph{J. Funct. Anal.}, 167 (1999), 94. doi: 10.1006/jfan.1999.3437.
[21]	J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Linear chaos for the quick-thinking-driver model,, \emph{Semigroup Forum}, (). doi: 10.1007/s00233-015-9704-6.
[22]	J. A. Conejero and F. Martínez-Giménez, Chaotic differential operators,, \emph{Rev. R. Acad. Cienc. Exactas F\'\i s. Nat. Ser. A Math. RACSAM}, 105 (2011), 423. doi: 10.1007/s13398-011-0026-6.
[23]	J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators,, \emph{Mediterr. J. Math.}, 7 (2010), 101. doi: 10.1007/s00009-010-0030-7.
[24]	J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup,, \emph{J. Funct. Anal.}, 244 (2007), 342. doi: 10.1016/j.jfa.2006.12.008.
[25]	J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 1195. doi: 10.3934/dcds.2009.25.1195.
[26]	R. deLaubenfels, H. Emamirad and K.-G. Grosse-Erdmann, Chaos for semigroups of unbounded operators,, \emph{Math. Nachr.}, 261/262 (2003), 47. doi: 10.1002/mana.200310112.
[27]	R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, vol. 1570 of Lecture Notes in Mathematics,, Springer-Verlag, (1994).
[28]	B. Dembart, On the theory of semigroups of operators on locally convex spaces,, \emph{J. Functional Analysis}, 16 (1974), 123.
[29]	W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators,, \emph{Ergodic Theory Dynam. Systems}, 17 (1997), 793. doi: 10.1017/S0143385797084976.
[30]	J. Duan, X.-C. Fu, P.-D. Liu and A. Manning, A linear chaotic quantum harmonic oscillator,, \emph{Appl. Math. Lett.}, 12 (1999), 15. doi: 10.1016/S0893-9659(98)00119-0.
[31]	S. El Mourchid, The imaginary point spectrum and hypercyclicity,, \emph{Semigroup Forum}, 73 (2006), 313. doi: 10.1007/s00233-005-0533-x.
[32]	H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 2043.
[33]	K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics,, Springer-Verlag, (2000).
[34]	L. Frerick, E. Jordá, T. Kalmes and J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces,, \emph{J. Math. Anal. Appl.}, 412 (2014), 121. doi: 10.1016/j.jmaa.2013.10.053.
[35]	G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds,, \emph{J. Funct. Anal.}, 98 (1991), 229. doi: 10.1016/0022-1236(91)90078-J.
[36]	J. A. Goldstein, Some remarks on infinitesimal generators of analytic semigroups,, \emph{Proc. Amer. Math. Soc.}, 22 (1969), 91.
[37]	M. González and F. León-Saavedra, Hypercyclicity for the elements of the commutant of an operator,, \emph{Integral Equations Operator Theory}, 80 (2014), 265. doi: 10.1007/s00020-014-2129-x.
[38]	K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos,, Universitext, (2011). doi: 10.1007/978-1-4471-2170-1.
[39]	K.-G. Grosse-Erdmann, Universal families and hypercyclic operators,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 36 (1999), 345. doi: 10.1090/S0273-0979-99-00788-0.
[40]	G. Herzog, On a universality of the heat equation,, \emph{Math. Nachr.}, 188 (1997), 169. doi: 10.1002/mana.19971880110.
[41]	L. Ji and A. Weber, Dynamics of the heat semigroup on symmetric spaces,, \emph{Ergodic Theory Dynam. Systems}, 30 (2010), 457. doi: 10.1017/S0143385709000133.
[42]	T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators,, \emph{Proc. Amer. Math. Soc.}, 137 (2009), 3833. doi: 10.1090/S0002-9939-09-09955-9.
[43]	T. Kōmura, Semigroups of operators in locally convex spaces,, \emph{Journal of Functional Analysis}, 2 (1968), 258.
[44]	M. Kostić, Generalized Semigroups and Cosine Functions, vol. 23 of Posebna Izdanja [Special Editions],, Matemati\v cki Institut SANU, (2011).
[45]	M. Kostić, Some contributions to the theory of abstract Volterra equations,, \emph{Int. J. Math. Anal. (Ruse)}, 5 (2011), 1529.
[46]	M. Kostić, Abstract Volterra equations in locally convex spaces,, \emph{Sci. China Math.}, 55 (2012), 1797. doi: 10.1007/s11425-012-4477-9.
[47]	M. Kostić, Hypercyclic and chaotic integrated $C$-cosine functions,, \emph{Filomat}, 26 (2012), 1. doi: 10.2298/FIL1201001K.
[48]	M. Kostić, Abstract Volterra integro-differential equations: approximation and convergence of resolvent operator families,, \emph{Numer. Funct. Anal. Optim.}, 35 (2014), 1579. doi: 10.1080/01630563.2014.908211.
[49]	F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts,, \emph{J. Math. Anal. Appl.}, 351 (2009), 607. doi: 10.1016/j.jmaa.2008.10.049.
[50]	F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for operators with full scrambled sets,, \emph{Math. Z.}, 274 (2013), 603. doi: 10.1007/s00209-012-1087-8.
[51]	Q. Menet, Linear chaos and frequent hypercyclicity,, \emph{Trans. Amer. Math. Soc.}, ().
[52]	G. Metafune, $L_p$-spectrum of Ornstein-Uhlenbeck operators,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 30 (2001), 97.
[53]	V. Müller, On the Salas theorem and hypercyclicity of $f(T)$,, \emph{Integral Equations Operator Theory}, 67 (2010), 439. doi: 10.1007/s00020-010-1791-x.
[54]	P. Oprocha, A quantum harmonic oscillator and strong chaos,, \emph{J. Phys. A}, 39 (2006), 14559. doi: 10.1088/0305-4470/39/47/003.
[55]	B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval,, \emph{Trans. Amer. Math. Soc.}, 344 (1994), 737. doi: 10.2307/2154504.
[56]	T. Xiao and J. Liang, Laplace transforms and integrated, regularized semigroups in locally convex spaces,, \emph{J. Funct. Anal.}, 148 (1997), 448. doi: 10.1006/jfan.1996.3096.
[57]	K. Yosida, Functional Analysis, vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, Sixth edition, (1980).

show all references

References:

[1]	A. A. Albanese, X. Barrachina, E. M. Mangino and A. Peris, Distributional chaos for strongly continuous semigroups of operators,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2069. doi: 10.3934/cpaa.2013.12.2069.
[2]	A. A. Albanese, J. Bonet and W. J. Ricker, $C_0$-semigroups and mean ergodic operators in a class of Fréchet spaces,, \emph{J. Math. Anal. Appl.}, 365 (2010), 142. doi: 10.1016/j.jmaa.2009.10.014.
[3]	R. M. Aron, J. B. Seoane-Sepúlveda and A. Weber, Chaos on function spaces,, \emph{Bull. Austral. Math. Soc.}, 71 (2005), 411. doi: 10.1017/S0004972700038417.
[4]	J. Aroza and A. Peris, Chaotic behaviour of birth-and-death models with proliferation,, \emph{J. Difference Equ. Appl.}, 18 (2012), 647. doi: 10.1080/10236198.2011.631535.
[5]	J. Banasiak, Birth-and-death type systems with parameter and chaotic dynamics of some linear kinetic models,, \emph{Z. Anal. Anwendungen}, 24 (2005), 675. doi: 10.4171/ZAA/1262.
[6]	J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos,, \emph{Discrete Contin. Dyn. Syst.}, 12 (2005), 959. doi: 10.3934/dcds.2005.12.959.
[7]	J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos,, \emph{Discrete Contin. Dyn. Syst.}, 29 (2011), 67. doi: 10.3934/dcds.2011.29.67.
[8]	X. Barrachina and J. A. Conejero, Devaney chaos and distributional chaos in the solution of certain partial differential equations,, \emph{Abstr. Appl. Anal.}, (4570).
[9]	X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Distributional chaos for the forward and backward control traffic model,, \emph{Linear Algebra Appl.}, 479 (2015), 202. doi: 10.1016/j.laa.2015.04.010.
[10]	X. Barrachina and A. Peris, Distributionally chaotic translation semigroups,, \emph{J. Difference Equ. Appl.}, 18 (2012), 751. doi: 10.1080/10236198.2011.625945.
[11]	F. Bayart and T. Bermúdez, Semigroups of chaotic operators,, \emph{Bull. Lond. Math. Soc.}, 41 (2009), 823. doi: 10.1112/blms/bdp055.
[12]	F. Bayart and É. Matheron, Dynamics of Linear Operators, vol. 179 of Cambridge Tracts in Mathematics,, Cambridge University Press, (2009). doi: 10.1017/CBO9780511581113.
[13]	T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators,, \emph{J. Math. Anal. Appl.}, 373 (2011), 83.
[14]	T. Bermúdez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria,, \emph{Bull. Austral. Math. Soc.}, 70 (2004), 45. doi: 10.1017/S0004972700035802.
[15]	T. Bermúdez and V. G. Miller, On operators $T$ such that $f(T)$ is hypercyclic,, \emph{Extracta Math.}, 15 (2000), 237.
[16]	L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators,, \emph{Studia Math.}, 157 (2003), 17. doi: 10.4064/sm157-1-2.
[17]	L. Bernal-González, D. Pellegrino and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 51 (2014), 71. doi: 10.1090/S0273-0979-2013-01421-6.
[18]	N. C. Bernardes Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators,, \emph{J. Funct. Anal.}, 265 (2013), 2143. doi: 10.1016/j.jfa.2013.06.019.
[19]	J. Bès, K. C. Chan and S. M. Seubert, Chaotic unbounded differentiation operators,, \emph{Integral Equations Operator Theory}, 40 (2001), 257. doi: 10.1007/BF01299846.
[20]	J. Bès and A. Peris, Hereditarily hypercyclic operators,, \emph{J. Funct. Anal.}, 167 (1999), 94. doi: 10.1006/jfan.1999.3437.
[21]	J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Linear chaos for the quick-thinking-driver model,, \emph{Semigroup Forum}, (). doi: 10.1007/s00233-015-9704-6.
[22]	J. A. Conejero and F. Martínez-Giménez, Chaotic differential operators,, \emph{Rev. R. Acad. Cienc. Exactas F\'\i s. Nat. Ser. A Math. RACSAM}, 105 (2011), 423. doi: 10.1007/s13398-011-0026-6.
[23]	J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators,, \emph{Mediterr. J. Math.}, 7 (2010), 101. doi: 10.1007/s00009-010-0030-7.
[24]	J. A. Conejero, V. Müller and A. Peris, Hypercyclic behaviour of operators in a hypercyclic $C_0$-semigroup,, \emph{J. Funct. Anal.}, 244 (2007), 342. doi: 10.1016/j.jfa.2006.12.008.
[25]	J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors,, \emph{Discrete Contin. Dyn. Syst.}, 25 (2009), 1195. doi: 10.3934/dcds.2009.25.1195.
[26]	R. deLaubenfels, H. Emamirad and K.-G. Grosse-Erdmann, Chaos for semigroups of unbounded operators,, \emph{Math. Nachr.}, 261/262 (2003), 47. doi: 10.1002/mana.200310112.
[27]	R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, vol. 1570 of Lecture Notes in Mathematics,, Springer-Verlag, (1994).
[28]	B. Dembart, On the theory of semigroups of operators on locally convex spaces,, \emph{J. Functional Analysis}, 16 (1974), 123.
[29]	W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators,, \emph{Ergodic Theory Dynam. Systems}, 17 (1997), 793. doi: 10.1017/S0143385797084976.
[30]	J. Duan, X.-C. Fu, P.-D. Liu and A. Manning, A linear chaotic quantum harmonic oscillator,, \emph{Appl. Math. Lett.}, 12 (1999), 15. doi: 10.1016/S0893-9659(98)00119-0.
[31]	S. El Mourchid, The imaginary point spectrum and hypercyclicity,, \emph{Semigroup Forum}, 73 (2006), 313. doi: 10.1007/s00233-005-0533-x.
[32]	H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation,, \emph{Proc. Amer. Math. Soc.}, 140 (2012), 2043.
[33]	K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics,, Springer-Verlag, (2000).
[34]	L. Frerick, E. Jordá, T. Kalmes and J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces,, \emph{J. Math. Anal. Appl.}, 412 (2014), 121. doi: 10.1016/j.jmaa.2013.10.053.
[35]	G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds,, \emph{J. Funct. Anal.}, 98 (1991), 229. doi: 10.1016/0022-1236(91)90078-J.
[36]	J. A. Goldstein, Some remarks on infinitesimal generators of analytic semigroups,, \emph{Proc. Amer. Math. Soc.}, 22 (1969), 91.
[37]	M. González and F. León-Saavedra, Hypercyclicity for the elements of the commutant of an operator,, \emph{Integral Equations Operator Theory}, 80 (2014), 265. doi: 10.1007/s00020-014-2129-x.
[38]	K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos,, Universitext, (2011). doi: 10.1007/978-1-4471-2170-1.
[39]	K.-G. Grosse-Erdmann, Universal families and hypercyclic operators,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 36 (1999), 345. doi: 10.1090/S0273-0979-99-00788-0.
[40]	G. Herzog, On a universality of the heat equation,, \emph{Math. Nachr.}, 188 (1997), 169. doi: 10.1002/mana.19971880110.
[41]	L. Ji and A. Weber, Dynamics of the heat semigroup on symmetric spaces,, \emph{Ergodic Theory Dynam. Systems}, 30 (2010), 457. doi: 10.1017/S0143385709000133.
[42]	T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators,, \emph{Proc. Amer. Math. Soc.}, 137 (2009), 3833. doi: 10.1090/S0002-9939-09-09955-9.
[43]	T. Kōmura, Semigroups of operators in locally convex spaces,, \emph{Journal of Functional Analysis}, 2 (1968), 258.
[44]	M. Kostić, Generalized Semigroups and Cosine Functions, vol. 23 of Posebna Izdanja [Special Editions],, Matemati\v cki Institut SANU, (2011).
[45]	M. Kostić, Some contributions to the theory of abstract Volterra equations,, \emph{Int. J. Math. Anal. (Ruse)}, 5 (2011), 1529.
[46]	M. Kostić, Abstract Volterra equations in locally convex spaces,, \emph{Sci. China Math.}, 55 (2012), 1797. doi: 10.1007/s11425-012-4477-9.
[47]	M. Kostić, Hypercyclic and chaotic integrated $C$-cosine functions,, \emph{Filomat}, 26 (2012), 1. doi: 10.2298/FIL1201001K.
[48]	M. Kostić, Abstract Volterra integro-differential equations: approximation and convergence of resolvent operator families,, \emph{Numer. Funct. Anal. Optim.}, 35 (2014), 1579. doi: 10.1080/01630563.2014.908211.
[49]	F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts,, \emph{J. Math. Anal. Appl.}, 351 (2009), 607. doi: 10.1016/j.jmaa.2008.10.049.
[50]	F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for operators with full scrambled sets,, \emph{Math. Z.}, 274 (2013), 603. doi: 10.1007/s00209-012-1087-8.
[51]	Q. Menet, Linear chaos and frequent hypercyclicity,, \emph{Trans. Amer. Math. Soc.}, ().
[52]	G. Metafune, $L_p$-spectrum of Ornstein-Uhlenbeck operators,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 30 (2001), 97.
[53]	V. Müller, On the Salas theorem and hypercyclicity of $f(T)$,, \emph{Integral Equations Operator Theory}, 67 (2010), 439. doi: 10.1007/s00020-010-1791-x.
[54]	P. Oprocha, A quantum harmonic oscillator and strong chaos,, \emph{J. Phys. A}, 39 (2006), 14559. doi: 10.1088/0305-4470/39/47/003.
[55]	B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval,, \emph{Trans. Amer. Math. Soc.}, 344 (1994), 737. doi: 10.2307/2154504.
[56]	T. Xiao and J. Liang, Laplace transforms and integrated, regularized semigroups in locally convex spaces,, \emph{J. Funct. Anal.}, 148 (1997), 448. doi: 10.1006/jfan.1996.3096.
[57]	K. Yosida, Functional Analysis, vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, Sixth edition, (1980).

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