Dynamics of Weighted Composition Operators on Spaces of Entire Functions of Exponential and Infraexponential Type

Given an affine symbol $$\varphi $$ φ and a multiplier w,  we focus on the weighted composition operator $$C_{w, \varphi }$$ C w , φ acting on the spaces Exp and $$Exp^0$$ E x p 0 of entire functions of exponential and of infraexponential type, respectively. We characterize the continuity of the operator and, for w the product of a polynomial by an exponential function, we completely characterize power boundedness and (uniform) mean ergodicity. In the case of multiples of composition operators, we also obtain the spectrum and characterize hypercyclicity.

[1]  J. Bonet,et al.  A note on mean ergodic composition operators on spaces of holomorphic functions , 2011 .

[2]  K. Yosida,et al.  Operator-Theoretical Treatment of Markoff's Process and Mean Ergodic Theorem , 1941 .

[3]  E. Jordá,et al.  Ergodic properties of composition operators on Banach spaces of analytic functions , 2020, Journal of Mathematical Analysis and Applications.

[4]  POWERS OF COMPOSITION OPERATORS: ASYMPTOTIC BEHAVIOUR ON BERGMAN, DIRICHLET AND BLOCH SPACES , 2019, Journal of the Australian Mathematical Society.

[5]  J. Bonet,et al.  THE CESÀRO OPERATOR IN THE FRÉCHET SPACES ℓ p+ AND L p− , 2016, Glasgow Mathematical Journal.

[6]  Mean Ergodic Operators , 2008 .

[7]  R. Meise,et al.  Introduction to Functional Analysis , 1997 .

[8]  E. Jordá,et al.  Mean ergodicity of weighted composition operators on spaces of holomorphic functions , 2016 .

[9]  M. Beltrán-Meneu.  Dynamics of weighted composition operators on weighted Banach spaces of entire functions , 2020, Journal of Mathematical Analysis and Applications.

[10]  D. Jornet,et al.  Mean Ergodic Composition Operators on Banach spaces of holomorphic functions , 2016, 1601.03353.

[11]  C. Berenstein.  Complex Analysis and Special Topics in Harmonic Analysis , 1995 .

[12]  Esther Gómez-Orts.  Weighted composition operators on Korenblum type spaces of analytic functions , 2020, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas.

[13]  J. Bonet,et al.  Associated weights for spaces of p-integrable entire functions , 2020, Quaestiones Mathematicae.

[14]  R. Meise.  Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals. , 1985 .

[15]  J. Bès.  Dynamics of Weighted Composition Operators , 2014 .

[16]  Frédéric Bayart,et al.  Dynamics of Linear Operators , 2009 .

[17]  K. Petersen.  Ergodic Theory by Karl E. Petersen , 1983 .

[18]  R. Meise,et al.  A projective description of weighted inductive limits , 1982 .

[19]  J. Bonet,et al.  The Cesàro operator on Korenblum type spaces of analytic functions , 2018 .

[20]  Linear Operators on the (LB)-Sequence Spaces $$\mathbf{ces(p-), 1< p \le \infty } $$ , 2018, Descriptive Topology and Functional Analysis II.

[21]  C. Caramanis.  What is ergodic theory , 1963 .

[22]  J. Bonet,et al.  Operators on the Fréchet sequence spaces $${\varvec{ces(p+)}}$$ces(p+), $$1\le p<\infty $$1≤p<∞ , 2018, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas.

[23]  P. Tien.  The iterates of composition operators on Banach spaces of holomorphic functions , 2020 .

[24]  José Bonet.  Hypercyclic and Chaotic Convolution Operators , 2000 .