Mean Ergodic Weighted Composition Operator 𝝀𝑪𝝋 on Bloch Space

Subject Areas : Statistics

Fakhreddin falahat ¹ , Zahra Kamali ²

1 - Department of Mathematics, Islamic Azad University, Shiraz Branch, Shiraz, Iran
2 - faculty membrer of Islamic Azad University, Shiraz branch

Received: 2020-11-29 Accepted : 2021-09-09 Published : 2022-03-21

Keywords: عملگر ترکیبی وزن‌دار, عملگر ارگودیک میانگین, نقطه دنجوی-ولف, فضای بلوچ,

Abstract :

Investigating the mean ergodicity of composition operators on various Banach Spaces has always been of interest to mathematicians and many authors studied this topics intensively, in many different spaces, such as, the space of all holomorphic functions on unit disk, Hardy space and Bloch space. In this paper, for a self map of the unit disk, φ and λ∈ℂ, we consider weighted composition operator, (λ𝐶φ)𝑓=λ𝑓𝑜φ , for every 𝑓 in Bloch space and Little Bloch space and inquiry the conditions under which the weighted composition operator 𝜆𝐶𝜑, is mean ergodic or uniformly mean ergodic on the Bloch and Little Bloch Space. In fact, we will show, if |λ|>1,𝜆𝐶𝜑, cannot be power bounded, mean ergodic or uniformly mean ergodic, in contrast, if |λ|<1, 𝜆𝐶𝜑, is always power bounded, mean ergodic or uniformly mean ergodic. In the case, |λ|=1, we will see that it depends directly to the Denjoy-Wolff point of 𝜑.

References:

 

[1] J. M. Anderson, Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math., 279 (1974), 12-37.

 

[2] W. Arendt, I. Chalendar, M. Kumar, S. Srivastava, Asymptotic behavior of the powers of composition operators on Banach spaces of holomorphic functions, Indiana Univ. Math. J., 67 (4) (2018) 1571–1595.

 

[3] W. Arendt, I Chalendar and M Kumar, Powers of composition operators: asymptotic behaviour on Bergman, Dirich-

let and Bloch spaces, J. Australian Math. So., 108(3) (2020) 289-320.

 

[4] M.J. Beltr n Meneu, M.C. G mez-Collado, E. Jord  and D. Jornet, Mean ergodic composition operators on Banach spaces of holomorphic functions, J. Funct. Anal., 270 (12) (2016) 4369–4385.

 

[5] M.J. Beltr n Meneu, M.C. G mez -Collado, E. Jord , and D. Jornet, Mean ergodicity of weighted composition operators on spaces of holomorphic functions, J. Math. Anal. Appl., 444 (2) (2016) 1640–1651.

 

[6] J. Bonet and P. Dom nski, A note on mean ergodic composition operators on spaces of holomorphic functions, Math., RACSAM 105 (2) (2011) 389-396.

 

[7] J. Bonet and W. Ricker, Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions, Arch. Math., 92 (2009) 428–437

 

[8] J. Cima, The basic properties of Bloch functions, Int. J. Math. Math. Sci., 3 (1979) 369-413.

 

[9] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.

 

[10] V. P. Fonf, M. Lin and P. Wojtaszczyk, Ergodic characterizations of reflexivity in Banach spaces, J. Funct. Anal., 187 (2001) 146-162.

 

[11] E. Jord  and A. Rodr guez-Arenas, Ergodic properties on Banach spaces of analytic functions, J. Math. Ana. App., 486(1) (2020) 1-22.

 

[12] U. Krengel, Ergodic Theorems, De Gruyter, Berlin, 1985.

 

[13] M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc., 43(2) (1974) 337–340.

 

[14] E.R. Lorch, Means of iterated transformations in reflexive vector spaces, Bull. Amer. Math. Soc., 45 (12) (1939) 945-947.

 

[15] H.P. Lotz, Uniform convergence of operators on  and similar spaces, Math. Z., 190 (1985) 207–220.

 

[16] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc., 347 (7) (1995) 2679—2687. 

[17] V. Neumann J, Proof of the quasi-ergodic hypothesis, Proc. Natl. Acad. Sci., USA, 18(1) (1932) 70-82. 

 

[18] K. Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press, Cambridge, 1983.  

 

[19] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer, Berlin, 1993.

 

[20] E. Wolf, Power bounded composition operators, Comput. Methods Funct. Theory, 12 (1) (2012) 105–117.

 

[21] C. Xiong, Norm of composition operators on the Bloch space, Bull. Austral. Math. Soc., 70 (2004), 293-299.

 

[22] K. Yosida, Mean ergodic theorem in Banach spaces, Proc. Imp. Acad., 14 (8) (1938) 292–294.

 

[23] K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff’s process and mean ergodic theorem