Mean ergodicity and spectrum of the Cesàro operator on weighted c0 spaces

[EN] A detailed investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator C acting on the weighted Banach sequence space c0(w) for a bounded, strictly positive weight w. New features arise in the weighted setting (e.g. existence of eigenvalues, compactness, mean ergodicity) which are not present in the classical setting of c0.The research of the first two authors was partially supported by the Projects MTM2013-43540-P, GVA Prometeo II/2013/013 and ACOMP/2015/186 (Spain).Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2016). Mean ergodicity and spectrum of the Cesàro operator on weighted c0 spaces. Positivity. 20:761-803. https://doi.org/10.1007/s11117-015-0385-xS76180320Akhmedov, A.M., Başar, F.: On the fine spectrum of the Cesàro operator in $$c_0$$ c 0 . Math. J. Ibaraki Univ. 36, 25–32 (2004)Akhmedov, A.M., Başar, F.: The fine spectrum of the Cesàro operator $$C_1$$ C 1 over the sequence space $$bv_p, (1 \le p < \infty )$$ b v p , ( 1 ≤ p < ∞ ) . Math. J. Okayama Univ. 50, 135–147 (2008)Albanese, A.A., Bonet, J., Ricker, W.J.: Convergence of arithmetic means of operators in Fréchet spaces. J. Math. Anal. Appl. 401, 160–173 (2013)Albanese, A.A., Bonet, J., Ricker, W.J.: Spectrum and compactness of the Cesàro operator on weighted $$\ell _p$$ ℓ p spaces. J. Aust. Math. Soc. 99, 287–314 (2015)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in the Fréchet spaces $$\ell ^{p+}$$ ℓ p + and $$L ^{p-}$$ L p - . Glasg. Math. J (to appear)Ansari, S.I., Bourdon, P.S.: Some properties of cyclic operators. Acta Sci. Math. Szeged 63, 195–207 (1997)Brown, A., Halmos, P.R., Shields, A.L.: Cesàro operators. Acta Sci. Math. Szeged 26, 125–137 (1965)Curbera, G.P., Ricker, W.J.: Spectrum of the Cesàro operator in $$\ell ^p$$ ℓ p . Arch. Math. 100, 267–271 (2013)Curbera, G.P., Ricker, W.J.: Solid extensions of the Cesàro operator on $$\ell ^p$$ ℓ p and $$c_0$$ c 0 . Integr. Equ. Oper. Theory 80, 61–77 (2014)Curbera, G.P., Ricker, W.J.: The Cesàro operator and unconditional Taylor series in Hardy spaces. Integr. Equ. Oper. Theory 83, 179–195 (2015)Diestel, J.: Sequences and Series in Banach Spaces. Springer, New York (1984)Dowson, H.R.: Spectral Theory of Linear Operators. Academic Press, London (1978)Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory, 2nd Printing. Wiley Interscience Publ, New York (1964)Emilion, R.: Mean-bounded operators and mean ergodic theorems. J. Funct. Anal. 61, 1–14 (1985)Goldberg, S.: Unbounded Linear Operators: Theory and Applications. Dover Publ, New York (1985)Hille, E.: Remarks on ergodic theorems. Trans. Am. Math. Soc. 57, 246–269 (1945)Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)Krengel, U.: Ergodic Theorems. de Gruyter, Berlin (1985)Leibowitz, G.: Spectra of discrete Cesàro operators. Tamkang J. Math. 3, 123–132 (1972)Lin, M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 337–340 (1974)Megginson, R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)Mureşan, M.: A Concrete Approach to Classical Analysis. Springer, Berlin (2008)Okutoyi, J.I.: On the spectrum of $$C_1$$ C 1 as an operator on $$bv_0$$ b v 0 . J. Aust. Math. Soc. Ser. A 48, 79–86 (1990)Radjavi, H., Tam, P.-W., Tan, K.-K.: Mean ergodicity for compact operators. Studia Math. 158, 207–217 (2003)Reade, J.B.: On the spectrum of the Cesàro operator. Bull. Lond. Math. Soc. 17, 263–267 (1985)Rhoades, B.E., Yildirim, M.: The spectra and fine spectra of factorable matrices on $$c_0$$ c 0 . Math. Commun. 16, 265–270 (2011)Taylor, A.E.: Introduction to Functional Analysis. Wiley, New York (1958