Classical operators on the Hörmander algebras

Previous Article
Chaos for the Hyperbolic Bioheat Equation
DCDS Home
This Issue
Next Article
Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model

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

Abstract
Related Papers

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 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.

[1]	Núria Fagella, Christian Henriksen. Deformation of entire functions with Baker domains. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 379-394. doi: 10.3934/dcds.2006.15.379
[2]	Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure and Applied Analysis, 2009, 8 (1) : 5-36. doi: 10.3934/cpaa.2009.8.5
[3]	Zhihua Liu, Pierre Magal. Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2271-2292. doi: 10.3934/dcdsb.2019227
[4]	Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395
[5]	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
[6]	Andrea Bonfiglioli, Ermanno Lanconelli. Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1587-1614. doi: 10.3934/cpaa.2012.11.1587
[7]	Harun Karsli, Purshottam Narain Agrawal. Rate of convergence of Stancu type modified $ q $-Gamma operators for functions with derivatives of bounded variation. Mathematical Foundations of Computing, 2022 doi: 10.3934/mfc.2022002
[8]	Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157
[9]	Karim Boulabiar, Gerard Buskes and Gleb Sirotkin. A strongly diagonal power of algebraic order bounded disjointness preserving operators. Electronic Research Announcements, 2003, 9: 94-98.
[10]	JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure and Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042
[11]	Chuan-Fu Yang, Natalia Pavlovna Bondarenko, Xiao-Chuan Xu. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems and Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068
[12]	Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217
[13]	Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335
[14]	Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751
[15]	Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707
[16]	Valerii Los, Vladimir A. Mikhailets, Aleksandr A. Murach. An isomorphism theorem for parabolic problems in Hörmander spaces and its applications. Communications on Pure and Applied Analysis, 2017, 16 (1) : 69-98. doi: 10.3934/cpaa.2017003
[17]	Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259
[18]	Lei Lei, Wenli Ren, Cuiling Fan. The differential spectrum of a class of power functions over finite fields. Advances in Mathematics of Communications, 2021, 15 (3) : 525-537. doi: 10.3934/amc.2020080
[19]	Yunho Kim, Luminita A. Vese. Image recovery using functions of bounded variation and Sobolev spaces of negative differentiability. Inverse Problems and Imaging, 2009, 3 (1) : 43-68. doi: 10.3934/ipi.2009.3.43
[20]	Ana-Maria Acu, Laura Hodis, Ioan Rasa. Multivariate weighted kantorovich operators. Mathematical Foundations of Computing, 2020, 3 (2) : 117-124. doi: 10.3934/mfc.2020009

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
María José Beltrán José Bonet Carmen Fernández
on Google Scholar
María José Beltrán José Bonet Carmen Fernández

[Back to Top]