Global regularity in ultradifferentiable classes

Se estudia la w-regularidad de soluciones de ciertos operadores que son globalmente hipoelípticos en el toro N-dimensional. Se aplican estos resultados para probar la w-regularidad global de ciertas clases de sublaplacianos. En este sentido, se extiende trabajo previo en el contexto de la clases analíticas y de Gevrey. Se dan varios ejemplos de w-hipoelipticidad local y global.The research of the authors was partially supported by MEC and FEDER Project MTM2010-15200.Albanese, AA.; Jornet Casanova, D. (2014). Global regularity in ultradifferentiable classes. Annali di Matematica Pura ed Applicata. 193(2):369-387. https://doi.org/10.1007/s10231-012-0279-5S3693871932Albanese A.A., Jornet D., Oliaro A.: Quasianalytic wave front sets for solutions of linear partial differential operators. Integr. Equ. Oper. Theory 66, 153–181 (2010)Albanese A.A., Jornet D., Oliaro A.: Wave front sets for ultradistribution solutions of linear pertial differential operators with coefficients in non-quasianalytic classes. Math. Nachr. 285, 411–425 (2012)Albanese A.A., Zanghirati L.: Global hypoellipticity and global solvability in Gevrey classes on the n–dimensional torus. J. Differ. Equ. 199, 256–268 (2004)Albanese A.A., Popivanov P.: Global analytic and Gevrey solvability of sublaplacians under Diophantine conditions. Ann. Mat. Pura e Appl. 185, 395–409 (2006)Albanese A.A., Popivanov P.: Gevrey hypoellipticity and solvability on the multidimensional torus of some classes of linear partial differential operators. Ann. Univ. Ferrara 52, 65–81 (2006)Baouendi M.S., Goulaouic C.: Nonanalytic–hypoellipticity for some degenerate elliptic operators. Bull. Am. Math. Soc. 78, 483–486 (1972)Bergamasco A.P.: Remarks about global analytic hypoellipticity. Trans. Am. Math. Soc. 351, 4113–4126 (1999)Bonet J., Meise R., Melikhov S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007)Braun R.W., Meise R., Taylor B.A.: Ultradifferentiable functions and Fourier analysis. Results Math. 17, 206–237 (1990)Chen W., Chi M.Y.: Hypoelliptic vector fields and almost periodic motions on the torus $${\mathbb{T}^n}$$ . Commun. Part. Differ. Equ. 25, 337–354 (2000)Christ M.: Certain sums of squares of vector fields fail to be analytic hypoelliptic. Commun. Part. Differ. Equ. 16, 1695–1707 (1991)Christ M.: A class of hypoelliptic PDE admitting non-analytic solutions. Contemp. Math. Symp. Complex Anal. 137, 155–168 (1992)Christ M.: Intermediate optimal Gevrey exponents occur. Commun. Part. Differ. Equ. 22, 359–379 (1997)Cordaro P.D., Himonas A.A.: Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus. Math. Res. Lett. 1, 501–510 (1994)Cordaro P.D., Himonas A.A.: Global analytic regularity for sums of squares of vector fields. Trans. Am. Math. Soc. 350, 4993–5001 (1998)Dickinson D., Gramchev T., Yoshino M.: Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena. Proc. Edinb. Math. Soc. 45, 731–759 (2002)Ferńandez C., Galbis A., Jornet D.: Pseudodifferential operators of Beurling type and the wave front set. J. Math. Appl. Math. 340, 1153–1170 (2008)Gramchev T., Popivanov P., Yoshino M.: Some note on Gevrey hypoellipticity and solvability on torus. J. Math. Soc. Jpn. 43, 501–514 (1991)Gramchev T., Popivanov P., Yoshino M.: Some examples of global Gevrey hypoellipticity and solvability. Proc. Jpn. Acad. 69, 395–398 (1993)Gramchev T., Popivanov P., Yoshino M.: Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients. Rend. Sem. Univ. Pol. Torino 51, 145–172 (1993)Greenfield S., Wallach N.: Global hypoellipticity and Liouville numbers. Proc. Am. Math. Soc. 31, 112–114 (1972)Greenfield S.: Hypoelliptic vector fields and continued fractions. Proc. Am. Math. Soc. 31, 115–118 (1972)Hanges N., Himonas A.A.: Singular solutions for sums of squares of vector fields. Commun. Part. Differ. Equ. 16, 1503–1511 (1991)Hanges N., Himonas A.A.: Analytic hypoellipticity for generalized Baouendi–Goulaouic operators. J. Funct. Anal. 125, 309–325 (1994)Helfer B.: Conditions nécessaires d’hypoanalyticité puor des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué. J. Differ. Equ. 44, 460–481 (1982)Himonas A.A.: On degenerate elliptic operators of infinite type. Math. Z. 220, 449–460 (1996)Himonas A.A.: Global analytic and Gevrey hypoellipticity of sublaplacians under diophantine conditions. Proc. Am. Math. Soc. 129, 2001–2007 (2000)Himonas A.A., Petronilho G.: Global hypoellipticity and simultaneous approximability. J. Funct. Anal. 170, 356–365 (2000)Himonas A.A., Petronilho G.: Propagation of regularity and global hypoellipticity. Mich. Math. J. 50, 471–481 (2002)Himonas A.A., Petronilho G.: On Gevrey regularity of globally C ∞ hypoelliptic operators. J. Differ. Equ. 207, 267–284 (2004)Himonas A.A., Petronilho G.: On C ∞ and Gevrey regularity of sublaplacians. Trans. Am. Math. Soc. 358, 4809–4820 (2006)Himonas A.A., Petronilho G., dos Santos L.A.C.: Regularity of a class of subLaplacians on the 3–dimensional torus. J. Funct. Anal. 240, 568–591 (2006)Hörmander L.: Hypoelliptic second order differential equations. Acta Mat. 119, 147–171 (1967)Langenbruch M.: Ultradifferentiable functions on compact intervals. Math. Nachr. 140, 109–126 (1989)Meise R.: Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals. J. Reine Angew. Math. 363, 59–95 (1985)The Lai P., Robert D.: Sur un probléme aus valeurs propres non linéaire. Israel J. Math. 36, 169–186 (1980)Petronilho G.: On Gevrey solvability and regularity. Math. Nachr. 282, 470–481 (2009)Petzsche H.-J.: Die Nuklearität der Ultradistributionsräume und der Satz von Kern I. Manuscripta Math. 24, 133–171 (1978)Tartakoff D.: Global (and local) analyticity for second order operators constructed from rigid vector fields on product of tori. Trans. Am. Math. Soc. 348, 2577–2583 (1996

Dynamics and spectrum of the Cesàro operator on C-infinity(R+)

[EN] The spectrum and point spectrum of the Cesaro averaging operator C acting on the Frechet space C-infinity(R+) of all C-infinity functions on the interval [0, infinity) are determined. We employ an approach via C-0-semigroup theory for linear operators. A spectral mapping theorem for the resolvent of a closed operator acting on a locally convex space is established; it constitutes a useful tool needed to establish the main result. The dynamical behaviour of C is also investigated.The research of the first two authors was partially supported by the projects MTM2013-43540-P, GVA Prometeo II/2013/013 and GVA ACOMP/2015/186 (Spain).Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2016). Dynamics and spectrum of the Cesàro operator on C-infinity(R+). Monatshefte für Mathematik. 181:267-283. https://doi.org/10.1007/s00605-015-0863-zS267283181Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic semigroups of operators. Rev. R. Acad. Cien. Serie A Mat. RACSAM 106, 299–319 (2012)Albanese, A.A., Bonet, J., Ricker, W.J.: Montel resolvents and uniformly mean ergodic semigroups of linear operators. Quaest. Math. 36, 253–290 (2013)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.: Uniform mean ergodicity of $$C_0$$ C 0 -semigroups in a class of in Fréchet spaces. Funct. Approx. Comment. Math. 50, 307–349 (2014)Albanese, A.A., Bonet, J., Ricker, W.J.: On the continuous Cesàro operator in certain function spaces. Positivity 19, 659–679 (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 - . Glasgow Math. J. (accepted)Arendt, W.: Gaussian estimates and interpolation of the spectrum in $$L^p$$ L p . Diff. Int. Equ. 7, 1153–1168 (1994)Bayart, F., Matheron, E.: Dynamics of linear operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Boyd, D.W.: The spectrum of the Cesàro operator. Acta Sci. Math. (Szeged) 29, 31–34 (1968)Grosse-Erdmann, K.G., Manguillot, A.P.: Linear chaos. Universitext, Springer Verlag, London (2011)Hille, E.: Remarks on ergodic theorems. Trans. Am. Math. Soc. 57, 246–269 (1945)Jarchow, H.: Locally convex spaces. Teubner, Stuttgart (1981)Komura, T.: Semigroups of operators in locally convex spaces. J. Funct. Anal. 2, 258–296 (1968)Lin, M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 337–340 (1974)Malgrange, B.: Idéaux de fonctions différentiables et division des distributions. Distributions, Editions École Polytechnique, Palaiseau, pp. 1–21 (2003)Meise, R., Vogt, D.: Introduction to functional analysis. Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press. Oxford University Press, New York (1997)Seeley, R.T.: Extension of $$C^\infty$$ C ∞ functions defined in a half space. Proc. Am. Math. Soc. 15, 625–626 (1964)Siskakis, A.G.: Composition semigroups and the Cesàro operator. J. London Math. Soc. (2) 36, 153–164 (1987)Yosida, K.: Functional analysis. Springer, New York, Berlin, Heidelberg (1980)Valdivia, M.: Topics in locally convex spaces. North-Holland Math. Stud. 67, North-Holland, Amsterdam (1982

On the continuous Cesàro operator in certain function spaces

“The final publication is available at Springer via http://dx.doi.org/10.1007/s11117-014-0321-5"Various properties of the (continuous) Cesàro operator C, acting on Banach and Fréchet spaces of continuous functions and L p-spaces, are investigated. For instance, the spectrum and point spectrum of C are completely determined and a study of certain dynamics of C is undertaken (eg. hyper- and supercyclicity, chaotic behaviour). In addition, the mean (and uniform mean) ergodic nature of C acting in the various spaces is identified.The research of the first two authors was partially supported by the projects MTM2010-15200 and GVA Prometeo II/2013/013 (Spain). The second author gratefully acknowledges the support of the Alexander von Humboldt Foundation.Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2015). On the continuous Cesàro operator in certain function spaces. Positivity. 19:659-679. https://doi.org/10.1007/s11117-014-0321-5S65967919Albanese, A.A.: Primary products of Banach spaces. Arch. Math. 66, 397–405 (1996)Albanese, A.A.: On subspaces of the spaces $$L^p_{\rm loc}$$ L loc p and of their strong duals. Math. Nachr. 197, 5–18 (1999)Albanese, A.A., Moscatelli, V.B.: Complemented subspaces of sums and products of copies of $$L^1 [0,1]$$ L 1 [ 0 , 1 ] . Rev. Mat. Univ. Complut. Madr. 9, 275–287 (1996)Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)Albanese, A.A., Bonet, J., Ricker, W.J.: On mean ergodic operators. In: Curbera, G.P. (eds.) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol. 201, pp. 1–20. Birkhäuser, Basel (2010)Albanese, A.A., Bonet, J., Ricker, W.J.: $$C_0$$ C 0 -semigroups and mean ergodic operators in a class of Fréchet spaces. J. Math. Anal. Appl. 365, 142–157 (2010)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)Bayart, F., Matheron, E.: Dynamics of linear operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bellenot, S.F., Dubinsky, E.: Fréchet spaces with nuclear Köthe quotients. Trans. Am. Math. Soc. 273, 579–594 (1982)Bonet, J., Frerick, L., Peris, A., Wengenroth, J.: Transitive and hypercyclic operators on locally convex spaces. Bull. Lond. Math. Soc. 37, 254–264 (2005)Boyd, D.W.: The spectrum of the Cesàro operator. Acta Sci. Math. (Szeged) 29, 31–34 (1968)Brown, A., Halmos, P.R., Shields, A.L.: Cesàro operators. Acta Sci. Math. (Szeged) 26, 125–137 (1965)Dierolf, S., Zarnadze, D.N.: A note on strictly regular Fréchet spaces. Arch. Math. 42, 549–556 (1984)Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory (2nd Printing). Wiley-Interscience, New York (1964)Galaz Fontes, F., Solís, F.J.: Iterating the Cesàro operators. Proc. Am. Math. Soc. 136, 2147–2153 (2008)Galaz Fontes, F., Ruiz-Aguilar, R.W.: Grados de ciclicidad de los operadores de Cesàro–Hardy. Misc. Mat. 57, 103–117 (2013)González, M., León-Saavedra, F.: Cyclic behaviour of the Cesàro operator on $$L_2(0,+\infty )$$ L 2 ( 0 , + ∞ ) . Proc. Am. Math. Soc. 137, 2049–2055 (2009)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear chaos. In: Universitext. Springer, London (2011)Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. In: Reprint of the 1952 Edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988)Krengel, U.: Ergodic theorems. In: De Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter Co., Berlin (1985)Leibowitz, G.M.: Spectra of finite range Cesàro operators. Acta Sci. Math. (Szeged) 35, 27–28 (1973)Leibowitz, G.M.: The Cesàro operators and their generalizations: examples in infinite-dimensional linear analysis. Am. Math. Mon. 80, 654–661 (1973)León-Saavedra, F., Piqueras-Lerena, A., Seoane-Sepúlveda, J.B.: Orbits of Cesàro type operators. Math. Nachr. 282, 764–773 (2009)Lin, M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 337–340 (1974)Meise, R., Vogt, D.: Introduction to functional analysis. In: Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press; Oxford University Press, New York (1997)Metafune, G., Moscatelli, V.B.: Quojections and prequojections. In: Terzioğlu, T. (ed.) Advances in the Theory of Fréchet spaces. NATO ASI Series, vol. 287, pp. 235–254. Kluwer Academic Publishers, Dordrecht (1989)Moscatelli, V.B.: Fréchet spaces without norms and without bases. Bull. Lond. Math. Soc. 12, 63–66 (1980)Piszczek, K.: Quasi-reflexive Fréchet spaces and mean ergodicity. J. Math. Anal. Appl. 361, 224–233 (2010)Piszczek, K.: Barrelled spaces and mean ergodicity. Rev R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 104, 5–11 (2010)Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980

Weighted composition operators on Korenblum type spaces of analytic functions

[EN] We investigate the continuity, compactness and invertibility of weighted composition operators W-psi,W-phi: f -> psi(f circle phi) when they act on the classical Korenblum space A(-infinity) and other related Frechet or (LB)-spaces of analytic functions on the open unit disc which are defined as intersections or unions of weighted Banach spaces with sup-norms. Some results about the spectrum of these operators are presented in case the self-map phi has a fixed point in the unit disc. A precise description of the spectrum is obtained in this case when the operator acts on the Korenblum space.This research was partially supported by the research project MTM2016-76647-P and the grant BES-2017-081200.Gomez-Orts, E. (2020). Weighted composition operators on Korenblum type spaces of analytic functions. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 114(4):1-15. https://doi.org/10.1007/s13398-020-00924-1S1151144Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics. Amer. Math. Soc., 50 (2002)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in the Fréchet spaces $$\ell ^{p+}$$ and $$L^{p-}$$. Glasgow Math. J. 59, 273–287 (2017)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator on Korenblum type spaces of analytic functions. Collect. Math. 69(2), 263–281 (2018)Albanese, A.A., Bonet, J., Ricker, W.J.: Operators on the Fréchet sequence spaces $$ces(p+), 1\le p\le \infty$$. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(2), 1533–1556 (2019)Albanese, A.A., Bonet, J., Ricker, W.J.: Linear operators on the (LB)-sequence spaces $$ces(p-), 1\le p\le \infty$$. Descriptive topology and functional analysis. II, 43–67, Springer Proc. Math. Stat., 286, Springer, Cham (2019)Arendt, W., Chalendar, I., Kumar, M., Srivastava, S.: Powers of composition operators: asymptotic behaviour on Bergman, Dirichlet and Bloch spaces. J. Austral. Math. Soc. 1–32. https://doi.org/10.1017/S1446788719000235Aron, R., Lindström, M.: Spectra of weighted composition operators on weighted Banach spaces of analytic funcions. Israel J. Math. 141, 263–276 (2004)Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Austral. Math. Soc., Ser. A, 54(1), 70–79 (1993)Bonet, J.: A note about the spectrum of composition operators induced by a rotation. RACSAM 114, 63 (2020). https://doi.org/10.1007/s13398-020-00788-5Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Austral. Math. Soc., Ser. A, 64(1), 101–118 (1998)Bourdon, P.S.: Essential angular derivatives and maximum growth of Königs eigenfunctions. J. Func. Anal. 160, 561–580 (1998)Bourdon, P.S.: Invertible weighted composition operators. Proc. Am. Math. Soc. 142(1), 289–299 (2014)Carleson, L., Gamelin, T.: Complex Dynamics. Springer, Berlin (1991)Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, FL (1995)Contreras, M., Hernández-Díaz, A.G.: Weighted composition operators in weighted Banach spacs of analytic functions. J. Austral. Math. Soc., Ser. A 69, 41–60 (2000)Eklund, T., Galindo, P., Lindström, M.: Königs eigenfunction for composition operators on Bloch and $$H^\infty$$ spaces. J. Math. Anal. Appl. 445, 1300–1309 (2017)Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Grad. Texts in Math. 199. Springer, New York (2000)Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)Kamowitz, H.: Compact operators of the form $$uC_{\varphi }$$. Pac. J. Math. 80(1) (1979)Korenblum, B.: An extension of the Nevanlinna theory. Acta Math. 135, 187–219 (1975)Köthe, G.: Topological Vector Spaces II. Springer, New York Inc (1979)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomophic functions. Stud. Math. 75, 19–45 (2006)Meise, R., Vogt, D.: Introduction to functional analysis. Oxford Grad. Texts in Math. 2, New York, (1997)Montes-Rodríguez, A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(3), 872–884 (2000)Queffélec, H., Queffélec, M.: Diophantine Approximation and Dirichlet series. Hindustain Book Agency, New Delhi (2013)Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Amer. Math. Soc. 162, 287–302 (1971)Zhu, K.: Operator Theory on Function Spaces, Math. Surveys and Monographs, Amer. Math. Soc. 138 (2007

The Cesàro operator on Korenblum type spaces of analytic functions

[EN] The spectrum of the CesA ro operator , which is always continuous (but never compact) when acting on the classical Korenblum space and other related weighted Fr,chet or (LB) spaces of analytic functions on the open unit disc, is completely determined. It turns out that such spaces are always Schwartz but, with the exception of the Korenblum space, never nuclear. Some consequences concerning the mean ergodicity of are deduced.The research of the first two authors was partially supported by the projects MTM2013-43540-P and MTM2016-76647-P. The second author gratefully acknowledges the support of the Alexander von Humboldt Foundation.Albanese, A.; Bonet Solves, JA.; Ricker, WJ. (2018). The Cesàro operator on Korenblum type spaces of analytic functions. Collectanea mathematica. 69(2):263-281. https://doi.org/10.1007/s13348-017-0205-7S263281692Albanese, A.A., Bonet, J., Ricker, W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)Albanese, A.A., Bonet, J., Ricker, W.J.: Montel resolvents and uniformly mean ergodic semigroups of linear operators. Quaest. Math. 36, 253–290 (2013)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in growth Banach spaces of analytic functions. Integral Equ. Oper. Theory 86, 97–112 (2016)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in the Fréchet spaces $$\ell ^{p+}$$ ℓ p + and $$L^{p-}$$ L p - . Glasgow Math. J. 59, 273–287 (2017)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator on power series spaces. Stud. Math. doi: 10.4064/sm8590-2-2017Aleman, A.: A class of integral operators on spaces of analytic functions, In: Proceedings of the Winter School in Operator Theory and Complex Analysis, Univ. Málaga Secr. Publ., Málaga, pp. 3–30 (2007)Aleman, A., Constantin, O.: Spectra of integration operators on weighted Bergman spaces. J. Anal. Math. 109, 199–231 (2009)Aleman, A., Peláez, J.A.: Spectra of integration operators and weighted square functions. Indiana Univ. Math. J. 61, 1–19 (2012)Aleman, A., Persson, A.-M.: Resolvent estimates and decomposable extensions of generalized Cesàro operators. J. Funct. Anal. 258, 67–98 (2010)Aleman, A., Siskakis, A.G.: An integral operator on $$H^p$$ H p . Complex Var. Theory Appl. 28, 149–158 (1995)Aleman, A., Siskakis, A.G.: Integration operators on Bergman spaces. Indiana Univ. Math. J. 46, 337–356 (1997)Barrett, D.E.: Duality between $$A^\infty$$ A ∞ and $$A^{- \infty }$$ A - ∞ on domains with nondegenerate corners, Multivariable operator theory (Seattle, WA, 1993), pp. 77–87, Contemporary Math. Vol. 185, Amer. Math. Soc., Providence (1995)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on bounded domains. Mich. Math. J. 40, 271–297 (1993)Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127, 137–168 (1998)Bierstedt, K.D., Meise, R., Summers, W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272, 107–160 (1982)Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. (Ser. A) 54, 70–79 (1993)Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. (Ser. A) 64, 101–118 (1998)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Domenig, T.: Composition operators on weighted Bergman spaces and Hardy spaces. Atomic Decompositions and Diagonal Operators, Ph.D. Thesis, University of Zürich (1997). [Zbl 0909.47025]Domenig, T.: Composition operators belonging to operator ideals. J. Math. Anal. Appl. 237, 327–349 (1999)Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory. 2nd Printing. Wiley Interscience Publ., New York (1964)Edwards, R.E.: Functional Analysis. Theory and Applications. Holt, Rinehart and Winston, New York, Chicago San Francisco (1965)Grothendieck, A.: Topological Vector Spaces. Gordon and Breach, London (1973)Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Graduate Texts in Mathematics, vol. 199. Springer, New York (2000)Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)Korenblum, B.: An extension of the Nevanlinna theory. Acta Math. 135, 187–219 (1975)Krengel, U.: Ergodic Theorems. de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter Co., Berlin (1985)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175(1), 19–40 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Melikhov, S.N.: (DFS )-spaces of holomorphic functions invariant under differentiation. J. Math. Anal. Appl. 297, 577–586 (2004)Persson, A.-M.: On the spectrum of the Cesàro operator on spaces of analytic functions. J. Math. Anal Appl. 340, 1180–1203 (2008)Pietsch, A.: Nuclear Locally Convex Spaces. Springer, Berlin (1972)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971)Siskakis, A.: Volterra operators on spaces of analytic functions—a survey. In: Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, Univ. Sevilla Serc. Publ., Seville, pp. 51–68 (2006

The classification of inherited hyperconics in Hall planes of even order

AbstractIn this note we complete the classification of inherited hyperconics in Hall planes of even order that was started by O’Keefe and Pascasio by proving that in the cases left open in [C.M. O’Keefe, A.A. Pascasio, Images of conics under derivation, Discrete Math. 151 (1996) 189–199] there are no inherited hyperconics

Mean ergodic semigroups of operators

We present criteria for determining mean ergodicity of C 0-semigroups of linear operators in a sequentially complete, locally convex Hausdorff space X. A characterization of reflexivity of certain spaces X with a basis via mean ergodicity of equicontinuous C 0-semigroups acting in X is also presented. Special results become available in Grothendieck spaces with the Dunford-Pettis property. © 2011 Springer-Verlag.Research partially supported by MICINN and FEDER Project MTM2010-15200 and GV Project Prometeo/2008/101.Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2012). Mean ergodic semigroups of operators. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 106(2):299-319. https://doi.org/10.1007/s13398-011-0054-2S2993191062Albanese A.A., Bonet J., Ricker W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)Albanese A.A., Bonet J., Ricker W.J.: Grothendieck spaces with the Dunford–Pettis property. Positivity 14, 145–164 (2010)Albanese, A.A., Bonet, J., Ricker, W.J.: On mean ergodic operators. In: Curbera, G.P. et al. (eds.) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol. 201, pp. 1–20. Birkhäuser, Basel (2010)Albanese A.A., Bonet J., Ricker W.J.: C 0-semigroups and mean ergodic operators in a class of Fréchet spaces. J. Math. Anal. Appl. 365, 142–157 (2010)Bonet J., Ricker W.J.: Schauder decompositions and the Grothendieck and the Dunford–Pettis properties in Köthe echelon spaces of infinite order. Positivity 11, 77–93 (2007)Bonet J., Ricker W.J.: Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions. Arch. Math. 92, 428–437 (2009)Bonet J., de Pagter B., Ricker W.J.: Mean ergodic operators and reflexive Fréchet lattices. Proc. R. Soc. Edinb. Sect. A 141, 897–920 (2011)Dunford N., Schwartz J.T.: Linear Operators I: General Theory, 2nd edn. Wiley, New York (1964)Eberlein W.F.: Abstract ergodic theorems and weak almost periodic functions. Trans. Am. Math. Soc. 67, 217–240 (1949)Edwards R.E.: Functional Analysis. Reinhart and Winston, New York (1965)Engel K.-J., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (1999)Floret, K.: Weakly compact sets. In: LNM, vol. 801. Springer, Berlin (1980)Fonf V.P., Lin M., Wojtaszczyk P.: Ergodic characterizations of reflexivity in Banach spaces. J. Funct. Anal. 187, 146–162 (2001)Hille, E., Phillips, R.S.: Functional Analysis and Semigroups, 4th edn. American Math. Soc., Providence (1981, revised)Jarchow H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)Kelley J.L.: General Topology, Rev. Edn, D. van Nostrand Co., Princeton–New York (1961)Komatsu H.: Semi-groups of operators in locally convex spaces. J. Math. Soc. Japan 16, 230–262 (1964)Komura T.: Semigroups of operators in locally convex spaces. J. Funct. Anal. 2, 258–296 (1968)Köthe, G.: Topological Vector Spaces I, 2nd edn. Springer, Berlin (1983, revised)Köthe G.: Topological Vector Spaces II. Springer, Berlin (1979)Krengel U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)Lotz, H.P. (1984) Tauberian theorems for operators on L ∞ and similar spaces. In: Bierstedt, K.D., Fuchssteiner, B. (eds.) Functional Analyis: Surveys and Recent Results III. North Holland, Amsterdam, pp. 117–133Lotz H.P.: Uniform convergence of operators on L ∞ and similar spaces. Math. Z. 190, 207–220 (1985)Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Miyadera I.: Semigroups of operators in Fréchet spaces and applications to partial differential operators. Tôhoku Math. J. 11, 162–183 (1959)Mugnolo D.: A semigroup analogue of the Fonf–Lin–Wojtaszczyk characterization of reflexive Banach spaces with a basis. Studia Math. 164, 243–251 (2004)Piszczek K.: Quasi-reflexive Fréchet spaces and mean ergodicity. J. Math. Anal. Appl. 361, 224–233 (2010)Piszczek K.: Barrelled spaces and mean ergodicity. RACSAM 104, 5–11 (2010)Rudin W.: Functional Analysis. McGraw-Hill, New York (1973)Sato R.: On a mean ergodic theorem. Proc. Am. Math. Soc. 83, 563–564 (1981)Schaefer H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)Shaw S.-Y.: Ergodic projections of continuous and discrete semigroups. Proc. Am. Math. Soc. 78, 69–76 (1980)Yosida K.: Functional Analysis. Springer, Berlin (1980

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