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February  2015, 35(2): 637-652. doi: 10.3934/dcds.2015.35.637

Classical operators on the Hörmander algebras

María José Beltrán 1, , José Bonet 2, and  Carmen Fernández 3,

1. 

Facultad de Magisterio, Universitat de València, Avda. Tarongers, 4, E-46022-Valencia, Spain
2. 

Instituto Universitario de Matemática Pura y Aplicada, IUMPA Universitat Politència, Camino de Vera, s/n., E-46022 Valencia, Spain
3. 

Departamento de Análisis Matemático, Universitat de València, C/ Dr. Moliner, 50, E-46100-Burjassot, Spain

Received  July 2013 Revised  October 2013 Published  September 2014

We study the integration operator, the differentiation operator and more general differential operators on radial Fréchet or (LB) Hörmander algebras of entire functions. We analyze when these operators are power bounded, hypercyclic and (uniformly) mean ergodic.
Keywords: growth conditions, chaotic, power bounded and (uniformly) mean ergodic operators, Hypercyclic, weighted spaces of entire functions, entire functions, Hörmander algebras., differential operators.
Mathematics Subject Classification: Primary: 47B38; Secondary: 47A16, 46E1.
Citation: María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637
References:
[1]

A. Albanese, J. Bonet and W. J. Ricker, Mean ergodic operators in Fréchet spaces, Ann. Acad. Sci. Fenn. Math., 34 (2009), 401-436.  Google Scholar

[2]

A. Albanese, J. Bonet and W. J. Ricker, On mean ergodic operators, Oper. Theory Adv. Appl., 201 (2010), 1-20.  Google Scholar

[3]

F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Mathematics 179, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511581113.  Google Scholar

[4]

M. J. Beltrán, Dynamics of differentiation and integration operators on weighted spaces of entire functions, Studia Math., 221 (2014), 35-60. doi: 10.4064/sm221-1-3.  Google Scholar

[5]

M. J. Beltrán, J. Bonet and C. Fernández, Classical operators on weighted Banach spaces of entire functions, Proc. Amer. Math. Soc., 141 (2013), 4293-4303. doi: 10.1090/S0002-9939-2013-11685-0.  Google Scholar

[6]

C. Berenstein and R. Gay, Complex Analysis and Special Topics in Harmonic Analysis, Springer, New York, 1995. doi: 10.1007/978-1-4613-8445-8.  Google Scholar

[7]

C. Berenstein and B. A. Taylor, A new look at interpolation theory for entire functions of one variable, Adv. in Math., 33 (1979), 109-143. doi: 10.1016/S0001-8708(79)80002-X.  Google Scholar

[8]

L. Bernal-González and A. Bonilla, Exponential type of hypercyclic entire functions, Arch. Math. (Basel), 78 (2002), 283-290. doi: 10.1007/s00013-002-8248-7.  Google Scholar

[9]

K. D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions, Stud. Math., 127 (1998), 137-168.  Google Scholar

[10]

K. D. Bierstedt, R. Meise and W. H. Summers, A projective description of weighted inductive limits, Trans. Am. Math. Soc., 272 (1982), 107-160. doi: 10.1090/S0002-9947-1982-0656483-9.  Google Scholar

[11]

K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A, 54 (1993), 70-79. doi: 10.1017/S1446788700036983.  Google Scholar

[12]

C. Blair and L. A. Rubel, A triply universal entire function, Enseign. Math. (2), 30 (1984), 269-274.  Google Scholar

[13]

J. Bonet, Hypercyclic and chaotic convolution operators, J. London Math. Soc., 62 (2000), 253-262. doi: 10.1112/S0024610700001174.  Google Scholar

[14]

J. Bonet, Dynamics of the differentiation operator on weighted spaces of entire functions, Math. Z., 261 (2009), 649-677. doi: 10.1007/s00209-008-0347-0.  Google Scholar

[15]

J. Bonet and A. Bonilla, Chaos of the differentiation operator on weighted Banach spaces of entire functions, Complex Anal. Oper. Theory, 7 (2013), 33-42. doi: 10.1007/s11785-011-0134-5.  Google Scholar

[16]

R. Braun, Weighted algebras of entire functions in which each closed ideal admits two algebraic generators, Michigan Math. J., 34 (1987), 441-450. doi: 10.1307/mmj/1029003623.  Google Scholar

[17]

K. G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math., 139 (2000), 47-68.  Google Scholar

[18]

K. G. Grosse-Erdmann and A. Peris, Linear Chaos, Springer, London, 2011. doi: 10.1007/978-1-4471-2170-1.  Google Scholar

[19]

A. Harutyunyan and W. Lusky, On the boundedness of the differentiation operator between weighted spaces of holomorphic functions, Studia Math., 184 (2008), 233-247. doi: 10.4064/sm184-3-3.  Google Scholar

[20]

W. Lusky, On the Fourier series of unbounded harmonic functions, J. London Math. Soc., 61 (2000), 568-580. doi: 10.1112/S0024610799008443.  Google Scholar

[21]

F. Martínez-Giménez and A. Peris, Chaos for backward shift operators, Int. J. Bifurcation and Chaos, 12 (2002), 1703-1715. doi: 10.1142/S0218127402005418.  Google Scholar

[22]

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.  Google Scholar

[23]

R. Meise, Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals, J. Reine Angew. Math., 363 (1985), 59-95. doi: 10.1515/crll.1985.363.59.  Google Scholar

[24]

R. Meise and B. A. Taylor, Sequence space representations for (FN)-algebras of entire functions modulo closed ideals, Studia Math., 85 (1987), 203-227.  Google Scholar

[25]

R. Meise and D. Vogt, Introduction to Functional Analysis, The Clarendon Press, Oxford University Press, New York, 1997.  Google Scholar

[26]

H. Neus, Über die Regularitätsbegriffe induktiver lokalkonvexer Sequenzen, Manuscripta Math., 25 (1978), 135-145. doi: 10.1007/BF01168605.  Google Scholar

[27]

E. Wolf, Weighted Fréchet spaces of holomorphic functions, Stud. Math., 174 (2006), 255-275. doi: 10.4064/sm174-3-3.  Google Scholar

show all references

References:
[1]

A. Albanese, J. Bonet and W. J. Ricker, Mean ergodic operators in Fréchet spaces, Ann. Acad. Sci. Fenn. Math., 34 (2009), 401-436.  Google Scholar

[2]

A. Albanese, J. Bonet and W. J. Ricker, On mean ergodic operators, Oper. Theory Adv. Appl., 201 (2010), 1-20.  Google Scholar

[3]

F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Mathematics 179, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511581113.  Google Scholar

[4]

M. J. Beltrán, Dynamics of differentiation and integration operators on weighted spaces of entire functions, Studia Math., 221 (2014), 35-60. doi: 10.4064/sm221-1-3.  Google Scholar

[5]

M. J. Beltrán, J. Bonet and C. Fernández, Classical operators on weighted Banach spaces of entire functions, Proc. Amer. Math. Soc., 141 (2013), 4293-4303. doi: 10.1090/S0002-9939-2013-11685-0.  Google Scholar

[6]

C. Berenstein and R. Gay, Complex Analysis and Special Topics in Harmonic Analysis, Springer, New York, 1995. doi: 10.1007/978-1-4613-8445-8.  Google Scholar

[7]

C. Berenstein and B. A. Taylor, A new look at interpolation theory for entire functions of one variable, Adv. in Math., 33 (1979), 109-143. doi: 10.1016/S0001-8708(79)80002-X.  Google Scholar

[8]

L. Bernal-González and A. Bonilla, Exponential type of hypercyclic entire functions, Arch. Math. (Basel), 78 (2002), 283-290. doi: 10.1007/s00013-002-8248-7.  Google Scholar

[9]

K. D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions, Stud. Math., 127 (1998), 137-168.  Google Scholar

[10]

K. D. Bierstedt, R. Meise and W. H. Summers, A projective description of weighted inductive limits, Trans. Am. Math. Soc., 272 (1982), 107-160. doi: 10.1090/S0002-9947-1982-0656483-9.  Google Scholar

[11]

K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A, 54 (1993), 70-79. doi: 10.1017/S1446788700036983.  Google Scholar

[12]

C. Blair and L. A. Rubel, A triply universal entire function, Enseign. Math. (2), 30 (1984), 269-274.  Google Scholar

[13]

J. Bonet, Hypercyclic and chaotic convolution operators, J. London Math. Soc., 62 (2000), 253-262. doi: 10.1112/S0024610700001174.  Google Scholar

[14]

J. Bonet, Dynamics of the differentiation operator on weighted spaces of entire functions, Math. Z., 261 (2009), 649-677. doi: 10.1007/s00209-008-0347-0.  Google Scholar

[15]

J. Bonet and A. Bonilla, Chaos of the differentiation operator on weighted Banach spaces of entire functions, Complex Anal. Oper. Theory, 7 (2013), 33-42. doi: 10.1007/s11785-011-0134-5.  Google Scholar

[16]

R. Braun, Weighted algebras of entire functions in which each closed ideal admits two algebraic generators, Michigan Math. J., 34 (1987), 441-450. doi: 10.1307/mmj/1029003623.  Google Scholar

[17]

K. G. Grosse-Erdmann, Hypercyclic and chaotic weighted shifts, Studia Math., 139 (2000), 47-68.  Google Scholar

[18]

K. G. Grosse-Erdmann and A. Peris, Linear Chaos, Springer, London, 2011. doi: 10.1007/978-1-4471-2170-1.  Google Scholar

[19]

A. Harutyunyan and W. Lusky, On the boundedness of the differentiation operator between weighted spaces of holomorphic functions, Studia Math., 184 (2008), 233-247. doi: 10.4064/sm184-3-3.  Google Scholar

[20]

W. Lusky, On the Fourier series of unbounded harmonic functions, J. London Math. Soc., 61 (2000), 568-580. doi: 10.1112/S0024610799008443.  Google Scholar

[21]

F. Martínez-Giménez and A. Peris, Chaos for backward shift operators, Int. J. Bifurcation and Chaos, 12 (2002), 1703-1715. doi: 10.1142/S0218127402005418.  Google Scholar

[22]

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.  Google Scholar

[23]

R. Meise, Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals, J. Reine Angew. Math., 363 (1985), 59-95. doi: 10.1515/crll.1985.363.59.  Google Scholar

[24]

R. Meise and B. A. Taylor, Sequence space representations for (FN)-algebras of entire functions modulo closed ideals, Studia Math., 85 (1987), 203-227.  Google Scholar

[25]

R. Meise and D. Vogt, Introduction to Functional Analysis, The Clarendon Press, Oxford University Press, New York, 1997.  Google Scholar

[26]

H. Neus, Über die Regularitätsbegriffe induktiver lokalkonvexer Sequenzen, Manuscripta Math., 25 (1978), 135-145. doi: 10.1007/BF01168605.  Google Scholar

[27]

E. Wolf, Weighted Fréchet spaces of holomorphic functions, Stud. Math., 174 (2006), 255-275. doi: 10.4064/sm174-3-3.  Google Scholar

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