Representation of Locally Convex Algebras

Sin resume

Mean ergodic multiplication operators on weighted spaces of continuous functions

[EN] Multiplication operators on weighted Banach spaces and locally convex spaces of continuous functions have been thoroughly studied. In this note, we characterize when continuous multiplication operators on a weighted Banach space and on a weighted inductive limit of Banach spaces of continuous functions are power bounded, mean ergodic or uniformly mean ergodic. The behaviour of the operator on weighted inductive limits depends on the properties of the defining sequence of weights and it differs from the Banach space case.The research of Bonet was partially supported by Project Prometeo/2017/102 of the Generalitat Valenciana. The authors authors were also partially supported by MINECO Project MTM2016-76647-P. Rodriguez also thanks the support of the Grant PAID-01-16 of the Universitat Politecnica de Valencia.Bonet Solves, JA.; Jorda Mora, E.; Rodríguez-Arenas, A. (2018). Mean ergodic multiplication operators on weighted spaces of continuous functions. Mediterranean Journal of Mathematics. 15(3):1:108-11:108. https://doi.org/10.1007/s00009-018-1150-8S1:10811:108153Bierstedt, K.D.: An introduction to locally convex inductive limits, Functional analysis and its applications (Nice, 1986), 35–133, ICPAM Lecture Notes. World Sci. Publishing, Singapore (1988)Bierstedt, K.D.: A survey of some results and open problems in weighted inductive limits and projective description for spaces of holomorphic functions. Bull. Soc. Roy. Sci. Liège 70(4–6), 167–182 (2001)Bierstedt, K.D., Bonet, J.: Some recent results on VC(X). In: Advances in the theory of Fréchet spaces (Istanbul, 1988), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 287, pp. 181–194. Kluwer Acad. Publ., Dordrecht (1989)Bierstedt, K.D., Bonet, J.: Completeness of the (LB)-spaces VC(X). Arch. Math. (Basel) 56(3), 281–285 (1991)Bierstedt, K.D., Bonet, J.: Some aspects of the modern theory of Fréchet spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat 97(2), 159–188 (2003)Bierstedt, K.D., Meise, R., Summers, W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272(1), 107–160 (1982)Bierstedt, K.D., Meise, R., Summers, W.H.: Köthe sets and Köthe sequence spaces. In: Functional analysis, holomorphy and approximation theory, Rio de Janeiro, pp. 27–91 (1980)Bonet, J., Ricker, W.J.: Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions. Arch. Math. 92, 428–437 (2009)Klilou, M., Oubbi, L.: Multiplication operators on generalized weighted spaces of continuous functions. Mediterr. J. Math. 13(5), 3265–3280 (2016)Krengel, U.: Ergodic Theorems. de Gruyter, Berlin (1985)Lin, M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 2 (1974)Lotz, H.P.: Uniform convergence of operators on $$L^\infty$$ L ∞ and similar spaces. Math. Z. 190, 207–220 (1985)Manhas, J.S.: Compact multiplication operators on weighted spaces of vector-valued continuous functions. Rocky Mt. J. Math. 34(3), 1047–1057 (2004)Manhas, J.S.: Compact and weakly compact multiplication operators on weighted spaces of vector-valued continuous functions. Acta Sci. Math. (Szeged) 70(1–2), 361–372 (2004)Manhas, J.S., Singh, R.K.: Compact and weakly compact weighted composition operators on weighted spaces of continuous functions. Integral Equ. Oper. Theory 29(1), 63–69 (1997)Meise, R., Vogt, D.: Introduction to Functional Analysis. The Clarendon Press, Oxford University Press, New York (1997)Oubbi, L.: Multiplication operators on weighted spaces of continuous functions. Port. Math. (N.S.) 59(1), 111–124 (2002)Oubbi, L.: Weighted composition operators on non-locally convex weighted spaces. Rocky Mt. J. Math. 35(6), 2065–2087 (2005)Singh, R.K., Manhas, J.S.: Multiplication operators on weighted spaces of vector-valued continuous functions. J. Austral. Math. Soc. Ser. A 50(1), 98–107 (1991)Singh, R.K., Manhas, J.S.: Composition operators on function spaces. North-Holland Publishing Co., Amsterdam (1993)Singh, R.K., Manhas, J.S.: Operators and dynamical systems on weighted function spaces. Math. Nachr. 169, 279–285 (1994)Wilanski, A.: Topology for Analysis. Ginn, Waltham (1970)Yosida, K.: Functional Analysis. Springer, Berlin (1980

Spectra of weighted algebras of holomorphic functions

We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach-Stone type question is addressed.Comment: 25 pages Corrected typo

The Arzéla-Ascoli theorem for non-locally convex weighted spaces

We deal with Arzéla-Ascoli type theorems in non-locally convex weighted spaces CVo(X, E) and CVp(X,E

Algèbres à bases bornées

In this paper, we introduce a new class of locally-convex algebras containing the one of unital uniformly $A$-convex algebras. It is the collection of those separated locally-convex algebras which have an algebraic basis whose idempotent hull is bounded. Such algebras are said to be “with $m$- bounded bases”. Under different notions of completeness, we endow these algebras with a complete algebra norm which has the same bounded sets of different kinds. Different questions on the factorization and commutativity are considered