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
Related Papers

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 and 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, Commun. Pure Appl. Anal., 12 (2013), 2069-2082. 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, J. Math. Anal. Appl., 365 (2010), 142-157. doi: 10.1016/j.jmaa.2009.10.014.
[3]	R. M. Aron, J. B. Seoane-Sepúlveda and A. Weber, Chaos on function spaces, Bull. Austral. Math. Soc., 71 (2005), 411-415. doi: 10.1017/S0004972700038417.
[4]	J. Aroza and A. Peris, Chaotic behaviour of birth-and-death models with proliferation, J. Difference Equ. Appl., 18 (2012), 647-655. doi: 10.1080/10236198.2011.631535.
[5]	J. Banasiak, Birth-and-death type systems with parameter and chaotic dynamics of some linear kinetic models, Z. Anal. Anwendungen, 24 (2005), 675-690. doi: 10.4171/ZAA/1262.
[6]	J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discrete Contin. Dyn. Syst., 12 (2005), 959-972. doi: 10.3934/dcds.2005.12.959.
[7]	J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011), 67-79. 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, Abstr. Appl. Anal., Art. ID 457019, 11.
[9]	X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Distributional chaos for the forward and backward control traffic model, Linear Algebra Appl., 479 (2015), 202-215. doi: 10.1016/j.laa.2015.04.010.
[10]	X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Difference Equ. Appl., 18 (2012), 751-761. doi: 10.1080/10236198.2011.625945.
[11]	F. Bayart and T. Bermúdez, Semigroups of chaotic operators, Bull. Lond. Math. Soc., 41 (2009), 823-830. 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, Cambridge, 2009. doi: 10.1017/CBO9780511581113.
[13]	T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373 (2011), 83-93.
[14]	T. Bermúdez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria, Bull. Austral. Math. Soc., 70 (2004), 45-54. doi: 10.1017/S0004972700035802.
[15]	T. Bermúdez and V. G. Miller, On operators $T$ such that $f(T)$ is hypercyclic, Extracta Math., 15 (2000), 237-241.
[16]	L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math., 157 (2003), 17-32. 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, Bull. Amer. Math. Soc. (N.S.), 51 (2014), 71-130. 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, J. Funct. Anal., 265 (2013), 2143-2163. doi: 10.1016/j.jfa.2013.06.019.
[19]	J. Bès, K. C. Chan and S. M. Seubert, Chaotic unbounded differentiation operators, Integral Equations Operator Theory, 40 (2001), 257-267. doi: 10.1007/BF01299846.
[20]	J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal., 167 (1999), 94-112. 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, Semigroup Forum, To appear. doi: 10.1007/s00233-015-9704-6.
[22]	J. A. Conejero and F. Martínez-Giménez, Chaotic differential operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 105 (2011), 423-431. doi: 10.1007/s13398-011-0026-6.
[23]	J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators, Mediterr. J. Math., 7 (2010), 101-109. 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, J. Funct. Anal., 244 (2007), 342-348. doi: 10.1016/j.jfa.2006.12.008.
[25]	J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors, Discrete Contin. Dyn. Syst., 25 (2009), 1195-1208. doi: 10.3934/dcds.2009.25.1195.
[26]	R. deLaubenfels, H. Emamirad and K.-G. Grosse-Erdmann, Chaos for semigroups of unbounded operators, Math. Nachr., 261/262 (2003), 47-59. doi: 10.1002/mana.200310112.
[27]	R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, vol. 1570 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994.
[28]	B. Dembart, On the theory of semigroups of operators on locally convex spaces, J. Functional Analysis, 16 (1974), 123-160.
[29]	W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17 (1997), 793-819. doi: 10.1017/S0143385797084976.
[30]	J. Duan, X.-C. Fu, P.-D. Liu and A. Manning, A linear chaotic quantum harmonic oscillator, Appl. Math. Lett., 12 (1999), 15-19. doi: 10.1016/S0893-9659(98)00119-0.
[31]	S. El Mourchid, The imaginary point spectrum and hypercyclicity, Semigroup Forum, 73 (2006), 313-316. doi: 10.1007/s00233-005-0533-x.
[32]	H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc., 140 (2012), 2043-2052.
[33]	K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
[34]	L. Frerick, E. Jordá, T. Kalmes and J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces, J. Math. Anal. Appl., 412 (2014), 121-124. doi: 10.1016/j.jmaa.2013.10.053.
[35]	G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98 (1991), 229-269. doi: 10.1016/0022-1236(91)90078-J.
[36]	J. A. Goldstein, Some remarks on infinitesimal generators of analytic semigroups, Proc. Amer. Math. Soc., 22 (1969), 91-93.
[37]	M. González and F. León-Saavedra, Hypercyclicity for the elements of the commutant of an operator, Integral Equations Operator Theory, 80 (2014), 265-274. doi: 10.1007/s00020-014-2129-x.
[38]	K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011. doi: 10.1007/978-1-4471-2170-1.
[39]	K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 345-381. doi: 10.1090/S0273-0979-99-00788-0.
[40]	G. Herzog, On a universality of the heat equation, Math. Nachr., 188 (1997), 169-171. doi: 10.1002/mana.19971880110.
[41]	L. Ji and A. Weber, Dynamics of the heat semigroup on symmetric spaces, Ergodic Theory Dynam. Systems, 30 (2010), 457-468. doi: 10.1017/S0143385709000133.
[42]	T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators, Proc. Amer. Math. Soc., 137 (2009), 3833-3848. doi: 10.1090/S0002-9939-09-09955-9.
[43]	T. Kōmura, Semigroups of operators in locally convex spaces, Journal of Functional Analysis, 2 (1968), 258-296.
[44]	M. Kostić, Generalized Semigroups and Cosine Functions, vol. 23 of Posebna Izdanja [Special Editions], Matematički Institut SANU, Belgrade, 2011.
[45]	M. Kostić, Some contributions to the theory of abstract Volterra equations, Int. J. Math. Anal. (Ruse), 5 (2011), 1529-1551.
[46]	M. Kostić, Abstract Volterra equations in locally convex spaces, Sci. China Math., 55 (2012), 1797-1825. doi: 10.1007/s11425-012-4477-9.
[47]	M. Kostić, Hypercyclic and chaotic integrated $C$-cosine functions, Filomat, 26 (2012), 1-44. doi: 10.2298/FIL1201001K.
[48]	M. Kostić, Abstract Volterra integro-differential equations: approximation and convergence of resolvent operator families, Numer. Funct. Anal. Optim., 35 (2014), 1579-1606. doi: 10.1080/01630563.2014.908211.
[49]	F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts, J. Math. Anal. Appl., 351 (2009), 607-615. 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, Math. Z., 274 (2013), 603-612. doi: 10.1007/s00209-012-1087-8.
[51]	Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc., to appear.
[52]	G. Metafune, $L_p$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 97-124.
[53]	V. Müller, On the Salas theorem and hypercyclicity of $f(T)$, Integral Equations Operator Theory, 67 (2010), 439-448. doi: 10.1007/s00020-010-1791-x.
[54]	P. Oprocha, A quantum harmonic oscillator and strong chaos, J. Phys. A, 39 (2006), 14559-14565. 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, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.2307/2154504.
[56]	T. Xiao and J. Liang, Laplace transforms and integrated, regularized semigroups in locally convex spaces, J. Funct. Anal., 148 (1997), 448-479. 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, Springer-Verlag, Berlin-New York, 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, Commun. Pure Appl. Anal., 12 (2013), 2069-2082. 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, J. Math. Anal. Appl., 365 (2010), 142-157. doi: 10.1016/j.jmaa.2009.10.014.
[3]	R. M. Aron, J. B. Seoane-Sepúlveda and A. Weber, Chaos on function spaces, Bull. Austral. Math. Soc., 71 (2005), 411-415. doi: 10.1017/S0004972700038417.
[4]	J. Aroza and A. Peris, Chaotic behaviour of birth-and-death models with proliferation, J. Difference Equ. Appl., 18 (2012), 647-655. doi: 10.1080/10236198.2011.631535.
[5]	J. Banasiak, Birth-and-death type systems with parameter and chaotic dynamics of some linear kinetic models, Z. Anal. Anwendungen, 24 (2005), 675-690. doi: 10.4171/ZAA/1262.
[6]	J. Banasiak and M. Moszyński, A generalization of Desch-Schappacher-Webb criteria for chaos, Discrete Contin. Dyn. Syst., 12 (2005), 959-972. doi: 10.3934/dcds.2005.12.959.
[7]	J. Banasiak and M. Moszyński, Dynamics of birth-and-death processes with proliferation-stability and chaos, Discrete Contin. Dyn. Syst., 29 (2011), 67-79. 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, Abstr. Appl. Anal., Art. ID 457019, 11.
[9]	X. Barrachina, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Distributional chaos for the forward and backward control traffic model, Linear Algebra Appl., 479 (2015), 202-215. doi: 10.1016/j.laa.2015.04.010.
[10]	X. Barrachina and A. Peris, Distributionally chaotic translation semigroups, J. Difference Equ. Appl., 18 (2012), 751-761. doi: 10.1080/10236198.2011.625945.
[11]	F. Bayart and T. Bermúdez, Semigroups of chaotic operators, Bull. Lond. Math. Soc., 41 (2009), 823-830. 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, Cambridge, 2009. doi: 10.1017/CBO9780511581113.
[13]	T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li-Yorke and distributionally chaotic operators, J. Math. Anal. Appl., 373 (2011), 83-93.
[14]	T. Bermúdez, A. Bonilla and A. Peris, On hypercyclicity and supercyclicity criteria, Bull. Austral. Math. Soc., 70 (2004), 45-54. doi: 10.1017/S0004972700035802.
[15]	T. Bermúdez and V. G. Miller, On operators $T$ such that $f(T)$ is hypercyclic, Extracta Math., 15 (2000), 237-241.
[16]	L. Bernal-González and K.-G. Grosse-Erdmann, The hypercyclicity criterion for sequences of operators, Studia Math., 157 (2003), 17-32. 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, Bull. Amer. Math. Soc. (N.S.), 51 (2014), 71-130. 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, J. Funct. Anal., 265 (2013), 2143-2163. doi: 10.1016/j.jfa.2013.06.019.
[19]	J. Bès, K. C. Chan and S. M. Seubert, Chaotic unbounded differentiation operators, Integral Equations Operator Theory, 40 (2001), 257-267. doi: 10.1007/BF01299846.
[20]	J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal., 167 (1999), 94-112. 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, Semigroup Forum, To appear. doi: 10.1007/s00233-015-9704-6.
[22]	J. A. Conejero and F. Martínez-Giménez, Chaotic differential operators, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 105 (2011), 423-431. doi: 10.1007/s13398-011-0026-6.
[23]	J. A. Conejero and E. M. Mangino, Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators, Mediterr. J. Math., 7 (2010), 101-109. 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, J. Funct. Anal., 244 (2007), 342-348. doi: 10.1016/j.jfa.2006.12.008.
[25]	J. A. Conejero and A. Peris, Hypercyclic translation $C_0$-semigroups on complex sectors, Discrete Contin. Dyn. Syst., 25 (2009), 1195-1208. doi: 10.3934/dcds.2009.25.1195.
[26]	R. deLaubenfels, H. Emamirad and K.-G. Grosse-Erdmann, Chaos for semigroups of unbounded operators, Math. Nachr., 261/262 (2003), 47-59. doi: 10.1002/mana.200310112.
[27]	R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, vol. 1570 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994.
[28]	B. Dembart, On the theory of semigroups of operators on locally convex spaces, J. Functional Analysis, 16 (1974), 123-160.
[29]	W. Desch, W. Schappacher and G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems, 17 (1997), 793-819. doi: 10.1017/S0143385797084976.
[30]	J. Duan, X.-C. Fu, P.-D. Liu and A. Manning, A linear chaotic quantum harmonic oscillator, Appl. Math. Lett., 12 (1999), 15-19. doi: 10.1016/S0893-9659(98)00119-0.
[31]	S. El Mourchid, The imaginary point spectrum and hypercyclicity, Semigroup Forum, 73 (2006), 313-316. doi: 10.1007/s00233-005-0533-x.
[32]	H. Emamirad, G. R. Goldstein and J. A. Goldstein, Chaotic solution for the Black-Scholes equation, Proc. Amer. Math. Soc., 140 (2012), 2043-2052.
[33]	K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
[34]	L. Frerick, E. Jordá, T. Kalmes and J. Wengenroth, Strongly continuous semigroups on some Fréchet spaces, J. Math. Anal. Appl., 412 (2014), 121-124. doi: 10.1016/j.jmaa.2013.10.053.
[35]	G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98 (1991), 229-269. doi: 10.1016/0022-1236(91)90078-J.
[36]	J. A. Goldstein, Some remarks on infinitesimal generators of analytic semigroups, Proc. Amer. Math. Soc., 22 (1969), 91-93.
[37]	M. González and F. León-Saavedra, Hypercyclicity for the elements of the commutant of an operator, Integral Equations Operator Theory, 80 (2014), 265-274. doi: 10.1007/s00020-014-2129-x.
[38]	K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011. doi: 10.1007/978-1-4471-2170-1.
[39]	K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 345-381. doi: 10.1090/S0273-0979-99-00788-0.
[40]	G. Herzog, On a universality of the heat equation, Math. Nachr., 188 (1997), 169-171. doi: 10.1002/mana.19971880110.
[41]	L. Ji and A. Weber, Dynamics of the heat semigroup on symmetric spaces, Ergodic Theory Dynam. Systems, 30 (2010), 457-468. doi: 10.1017/S0143385709000133.
[42]	T. Kalmes, Hypercyclic $C_0$-semigroups and evolution families generated by first order differential operators, Proc. Amer. Math. Soc., 137 (2009), 3833-3848. doi: 10.1090/S0002-9939-09-09955-9.
[43]	T. Kōmura, Semigroups of operators in locally convex spaces, Journal of Functional Analysis, 2 (1968), 258-296.
[44]	M. Kostić, Generalized Semigroups and Cosine Functions, vol. 23 of Posebna Izdanja [Special Editions], Matematički Institut SANU, Belgrade, 2011.
[45]	M. Kostić, Some contributions to the theory of abstract Volterra equations, Int. J. Math. Anal. (Ruse), 5 (2011), 1529-1551.
[46]	M. Kostić, Abstract Volterra equations in locally convex spaces, Sci. China Math., 55 (2012), 1797-1825. doi: 10.1007/s11425-012-4477-9.
[47]	M. Kostić, Hypercyclic and chaotic integrated $C$-cosine functions, Filomat, 26 (2012), 1-44. doi: 10.2298/FIL1201001K.
[48]	M. Kostić, Abstract Volterra integro-differential equations: approximation and convergence of resolvent operator families, Numer. Funct. Anal. Optim., 35 (2014), 1579-1606. doi: 10.1080/01630563.2014.908211.
[49]	F. Martínez-Giménez, P. Oprocha and A. Peris, Distributional chaos for backward shifts, J. Math. Anal. Appl., 351 (2009), 607-615. 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, Math. Z., 274 (2013), 603-612. doi: 10.1007/s00209-012-1087-8.
[51]	Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc., to appear.
[52]	G. Metafune, $L_p$-spectrum of Ornstein-Uhlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 97-124.
[53]	V. Müller, On the Salas theorem and hypercyclicity of $f(T)$, Integral Equations Operator Theory, 67 (2010), 439-448. doi: 10.1007/s00020-010-1791-x.
[54]	P. Oprocha, A quantum harmonic oscillator and strong chaos, J. Phys. A, 39 (2006), 14559-14565. 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, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.2307/2154504.
[56]	T. Xiao and J. Liang, Laplace transforms and integrated, regularized semigroups in locally convex spaces, J. Funct. Anal., 148 (1997), 448-479. 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, Springer-Verlag, Berlin-New York, 1980.

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