Invariant subspaces of the integration operators on Hörmander algebras and Korenblum type spaces

[EN] We describe the proper closed invariant subspaces of the integration operator when it acts continuously on countable intersections and countable unions of weighted Banach spaces of holomorphic functions on the unit disc or the complex plane. Applications are given to the case of Korenblum type spaces and Hormander algebras of entire functions.This research was partially supported by the Projects MTM2016-76647-P and GV Prometeo/2017/102.Bonet Solves, JA.; Galbis, A. (2020). Invariant subspaces of the integration operators on Hörmander algebras and Korenblum type spaces. Integral Equations and Operator Theory. 92(4):1-13. https://doi.org/10.1007/s00020-020-02593-6113924Abanin, A.V.: Effective and sampling sets for Hörmander spaces. Complex Anal. Oper. Theory 12(6), 1401–1419 (2018)Abanin, A.V., Tien, P.T.: Differentiation and integration operators on weighted Banach spaces of holomorphic functions. Math. Nachr. 290(8–9), 1144–1162 (2017)Abanin, A.V., Tien, P.T.: Invariant subspaces for classical operators on weighted spaces of holomorphic functions. Integr. Equ. Oper. Theory 89(3), 409–438 (2017)Abanin, A.V., Tien, P.T.: Compactness of classical operators on weighted Banach spaces of holomorphic functions. Collect. Math. 69(1), 1–15 (2018)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator on Korenblum type spaces of analytic functions. Collect. Math. 69, 263–281 (2018)Aleman, A., Korenblum, B.: Volterra invariant subspaces of $$H^p$$. Bull. Sci. Math. 132, 510–528 (2008)Beltrán, M.J., Bonet, J., Fernández, C.: Classical operators on weighted Banach spaces of entire functions. Proc. Am. Math. Soc. 141(12), 4293–4303 (2013)Beltrán, M.J., Bonet, J., Fernández, C.: Classical operators on the Hörmander algebras. Discrete Contin. Dyn. Syst. 35(2), 637–652 (2015)Berenstein, C.A., Gay, R.: Complex Analysis and Special Topics in Harmonic Analysis. Springer, New York (1995)Berenstein, C.A., Li, B.Q., Vidras, A.: Geometric characterization of interpolating varieties for the (FN)-space $$A^0_p$$ of entire functions. Can. J. Math. 47, 28–43 (1995)Bierstedt, K.D., Bonet, J.: Some aspects of the modern theory of Fréchet spaces. RACSAM Rev. R. Acad. Cien. Serie A Mat. 97, 159–188 (2003)Bierstedt, K.D., Bonet, J.: Weighted (LB)-spaces of holomorphic functions: $$VH(G)=V_0H(G)$$ and completeness of $$V_0H(G)$$. J. Math. Anal. Appl. 323, 747–767 (2006)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., Meise, R., Summers, W.H.: A projective description of weighted inductive limits. Trans. Am. Math. Soc. 272(1), 107–160 (1982)Bonet, J., Fernández, C.: The range of the restriction map for a multiplicity variety in Hörmander algebras of entire functions. Mediterr. J. Math. 11(2), 643–652 (2014)Bonet, J., Lusky, W., Taskinen, J.: Monomial basis in Korenblum type spaces of analytic functions. Proc. Am. Math. Soc. 146(12), 5269–5278 (2018)Constantin, O.: A Volterra-type integration operator on Fock spaces. Proc. Am. Math. Soc. 140, 4247–4257 (2012)Constantin, O., Peláez, J.A.: Integral operators, embedding theorems and a Littlewood–Paley formula on weighted Fock spaces. J. Geom. Anal. 26, 1109–1154 (2016)Douglas, R.G., Shapiro, H.S., Shields, A.L.: Cyclic vectors and invariant subspaces for the backward shift operator. Ann. Inst. Fourier 20, 37–76 (1970)Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184(3), 233–247 (2008)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)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–45 (2006)Meise, R.: Sequence space representations for $$({\rm DFN})$$-algebras of entire functions modulo closed ideals. J. Reine Angew. Math. 363, 59–95 (1985)Meise, R., Taylor, B.A.: Sequence space representations for (FN)-algebras of entire functions modulo closed ideals. Studia Math. 85, 203–227 (1987)Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford University Press, New York (1997)Trèves, F.: Locally Convex Spaces and Linear Partial Differential Equations. Die Grundlehren der mathematischen Wissenschaften 146. Springer, New York (1967)Vogt, D.: Non-natural topologies on spaces of holomorphic functions. Ann. Polon. Math. 108(3), 215–217 (2013)Wolf, E.: Weighted Fréchet spaces of holomorphic functions. Studia Math. 174(3), 255–275 (2006)Zhu, K.: Operator Theory on Function Spaces. Mathematical Surveys and Monographs, vol. 138. American Mathematical Society, Providence (2007)Zhu, K.: Analysis on Fock Spaces. Graduate Texts in Mathematics, vol. 263. Springer, New York (2012