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

Classical operators on the Hörmander algebras

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.
Citation: María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete and 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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[27]

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

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.

[2]

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

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[27]

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

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