2,635 research outputs found
A_{n-1} singularities and nKdV hierarchies
According to a conjecture of E. Witten proved by M. Kontsevich, a certain
generating function for intersection indices on the Deligne -- Mumford moduli
spaces of Riemann surfaces coincides with a certain tau-function of the KdV
hierarchy. The generating function is naturally generalized under the name the
{\em total descendent potential} in the theory of Gromov -- Witten invariants
of symplectic manifolds. The papers arXiv: math.AG/0108100 and arXive:
math.DG/0108160 contain two equivalent constructions, motivated by some results
in Gromov -- Witten theory, which associate a total descendent potential to any
semisimple Frobenius structure. In this paper, we prove that in the case of
K.Saito's Frobenius structure on the miniversal deformation of the
-singularity, the total descendent potential is a tau-function of the
KdV hierarchy. We derive this result from a more general construction for
solutions of the KdV hierarchy from solutions of the KdV hierarchy.Comment: 29 pages, to appear in Moscow Mathematical Journa
Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey
[EN]
In this survey we report about recent work on weighted Banach spaces of analytic functions on the unit disc and on the whole complex plane defined with sup-norms and operators between them. Results about the solid hull and core of these spaces and distance formulas are reviewed. Differentiation and integration operators, Cesaro and Volterra operators, weighted composition and superposition operators and Toeplitz operators on these spaces are analyzed. Boundedness, compactness, the spectrum, hypercyclicity and (uniform) mean ergodicity of these operators are considered.This research was partially supported by the project MCIN PID2020-119457GB-I00/AEI/10.13039/501100011033.
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Bonet Solves, JA. (2022). Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey. Revista de la Real Academia de Ciencias Exactas FÃsicas y Naturales Serie A Matemáticas. 116(4):1-40. https://doi.org/10.1007/s13398-022-01323-41401164Abakumov, E., Doubtsov, E.: Reverse estimates in growth spaces. Math. Z. 271(1–2), 399–413 (2012)Abakumov, E., Doubtsov, E.: Moduli of holomorphic functions and logarithmically convex radial weights. Bull. Lond. Math. Soc. 47(3), 519–532 (2015)Abakumov, E., Doubtsov, E.: Approximation by proper holomorphic maps and tropical power series. Constr. Approx. 47(2), 321–338 (2018)Abakumov, E., Doubtsov, E.: Volterra type operators on growth Fock spaces. Arch. Math. (Basel) 108(4), 383–393 (2017)Abakumov, E., Doubtsov, E.: Univalent symbols of Volterra operators on growth spaces. Anal. Math. Phys. 9(3), 911–917 (2019)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. Integral 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 Cesà ro operator in growth Banach spaces of analytic functions. Integral Equ. Oper. Theory 86(1), 97–112 (2016)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesà ro operator on Korenblum type spaces of analytic functions. Collect. Math. 69(2), 263–281 (2018)Aleman, A., Constantin, O.: Spectra of integration operators on weighted Bergman spaces. J. Anal. Math. 109, 199–231 (2009)Aleman, A., Montes-RodrÃguez, A., Sarafoleanu, A.: The eigenfunctions of the Hilbert matrix. Constr. Approx. 36(3), 353–374 (2012)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 Cesà ro operators. J. Funct. Anal. 258, 67–98 (2010)Aleman, A., Siskakis, A.G.: An integral operator on . 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)Anderson, A., Jovovic, M., Smith, W.: Some integral operators acting on . Integral Equ. Oper. Theory 80, 275–291 (2014)Anderson, J.M., Shields, A.L.: Coefficient multipliers of Bloch functions. Trans. Am. Math. Soc. 224, 255–265 (1976)Arcozzi, N., Björn, A.: Dominating sets for analytic and harmonic functions and completeness of weighted Bergman spaces. Math. Proc. R. Ir. Acad. 102A, 175–192 (2002)Arendt, W., Célariès, B., Chalendar, I.: In Koenigs’ footsteps: diagonalization of composition operators. J. Funct. Anal. 278(2), 108313 (2002)Arendt, W., Chalendar, I., Kumar, M., Srivastava, S.: Asymptotic behaviour of the powers of composition operators on Banach spaces of holomorphic functions. Indiana Univ. Math. J. 67(4), 1571–1595 (2018)Arendt, W., Chalendar, I., Kumar, M., Srivastava, S.: Powers of composition operators: asymptotic behaviour on Bergman, Dirichlet and Bloch spaces. J. Aust. Math. Soc. 108(3), 289–320 (2020)Aron, R., Lindström, M.: Spectra of weighted composition operators on weighted Banach spaces of analytic functions. Isr. J. Math. 141, 263–276 (2004)Atzmon, A., Brive, B.: Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions, Bergman spaces and related topics in complex analysis, pp. 27–39, Contemp. Math., vol. 404. Amer. Math. Soc., Providence (2006)Basallote, M., Contreras, M.D., Hernández-Mancera, C., MartÃn, M.J., Paúl, P.J.: Volterra operators and semigroups in weighted Banach spaces of analytic functions. Collect. Math. 65(2), 233–249 (2014)Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)Beltrán-Meneu, M.J.: Dynamics of differentiation and integration operators on weighted spaces of entire functions. Studia Math. 221(1), 35–60 (2014)Beltrán-Meneu, M.J.: Dynamics of weighted composition operators on weighted Banach spaces of entire functions. J. Math. Anal. Appl. 492(1), 124422 (2020)Beltrán-Meneu, M.J., Jordá, E.: Dynamics of weighted composition operators on spaces of entire functions of exponential and infraexponential type. Mediterr. J. Math. 18(5), Paper No. 212 (2021)Beltrán-Meneu, 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-Meneu, M.J., Bonet, J., Fernández, C.: Classical operators on the Hörmander algebras. Discrete Contin. Dyn. Syst. 35(2), 637–652 (2015)Beltrán-Meneu, M.J., Gómez-Collado, M.C., Jordá, E., Jornet, D.: Mean ergodic composition operators on Banach spaces of holomorphic functions. J. Funct. Anal. 270(12), 4369–4385 (2016)Beltrán-Meneu, M.J., Gómez-Collado, M.C., Jordá, E., Jornet, D.: Mean ergodicity of weighted composition operators on spaces of holomorphic functions. J. Math. Anal. Appl. 444(2), 1640–1651 (2016)Bennet, G., Stegenga, D.A., Timoney, R.M.: Coefficients of Bloch and Lipschitz functions. Ill. J. Math. 25, 520–531 (1981)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. Studia Math. 127, 137–168 (1998)Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. 54, 70–79 (1993)Blasco, O.: Operators on Fock-type and weighted spaces of entire functions. Funct. Approx. Comment. Math. 59(2), 175–189 (2018)Blasco, O.: Boundedness of Volterra operators on spaces of entire functions. Ann. Acad. Sci. Fenn. Math. 43(1), 89–107 (2018)Bonet, J.: Weighted spaces of holomorphic functions and operators between them. Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), pp. 117–138, Colecc. Abierta, vol. 64. Univ. Sevilla Secr. Publ., Seville (2003)Bonet, J.: A note about the spectrum of composition operators induced by a rotation. Rev. R. Acad. Cienc. Exactas FÃs. Nat. Ser. A Mat. RACSAM 114(2), 63 (2020)Bonet, J.: The spectrum of Volterra operators on Korenblum type spaces of analytic functions. Integral Equ. Oper. Theory 91(5), Paper No. 46 (2019)Bonet, J.: The spectrum of Volterra operators on weighted spaces of entire functions. Q. J. Math. 66(3), 799–807 (2015)Bonet, J.: Dynamics of the differentiation operator on weighted spaces of entire functions. Math. Z. 261(3), 649–657 (2009)Bonet, J.: Hausdorff operators on weighted Banach spaces of type . Complex Anal. Oper. Theory 16(1), Paper No. 12 (2022)Bonet, J., Bonilla, A.: Chaos of the differentiation operator on weighted Banach spaces of entire functions. Complex Anal. Oper. Theory 7, 33–42 (2013)Bonet, J., DomaÅ„ski, P.: A note on mean ergodic composition operators on spaces of holomorphic functions. Rev. R. Acad. Cienc. Exactas FÃs. Nat. Ser. A Mat. RACSAM 105(2), 389–396 (2011)Bonet, J., DomaÅ„ski, P.: Power bounded composition operators on spaces of analytic functions. Collect. Math. 62(1), 69–83 (2011)Bonet, J., DomaÅ„ski, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42, 139–148 (1999)Bonet, J., DomaÅ„ski, P., Lindström, M.: Pointwise multiplication operators on weighted Banach spaces of analytic functions. Studia Math. 137, 177–194 (1999)Bonet, J., DomaÅ„ski, P., Lindström, M.: Weakly compact composition operators on analytic vector-valued function spaces. Ann. Acad. Sci. Fenn. 26, 233–248 (2001)Bonet, J., DomaÅ„ski, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. 64, 101–118 (1998)Bonet, J., Galbis, A.: Invariant subspaces of the integration operators on Hörmander algebras and Korenblum type spaces. Integral Equ. Oper. Theory 92(4), Paper No. 34 (2020)Bonet, J., Galindo, P., Lindström, M.: Spectra and essential spectral radii of composition operators on weighted Banach spaces of analytic functions. J. Math. Anal. Appl. 340(2), 884–891 (2008)Bonet, J., Gómez-Collado, M.C., Jordá, E., Jornet, D.: Nuclear weighted composition operators on weighted Banach spaces of analytic functions. Proc. Am. Math. Soc. 149(1), 311–321 (2021)Bonet, J., Lindström, M., Wolf, E.: Isometric weighted composition operators on weighted Banach spaces of type . Proc. Am. Math. Soc. 136(12), 4267–4273 (2008)Bonet, J., Lusky, W., Taskinen, J.: Distance formulas on weighted Banach spaces of analytic functions. Complex Anal. Oper. Theory 13(3), 893–900 (2019)Bonet, J., Lusky, W., Taskinen, J.: Solid hulls and cores of weighted -spaces. Rev. Mat. Complut. 31(3), 781–804 (2019)Bonet, J., Lusky, W., Taskinen, J.: Solid cores and solid hulls of weighted Bergman spaces. Banach J. Math. Anal. 13(2), 468–485 (2019)Bonet, J., Lusky, W., Taskinen, J.: On boundedness and compactness of Toeplitz operators in weighted -spaces. J. Funct. Anal. 278(10), 108456 (2020)Bonet, J., Lusky, W., Taskinen, J.: On the boundedness of Toeplitz operators with radial symbols over weighted sup-norm spaces of holomorphic functions. J. Math. Anal. Appl. 493(1), 124515 (2021)Bonet, J., Lusky, W., Taskinen, J.: Unbounded Bergman projections on weighted spaces with respect to exponential weights. Integral Equ. Oper. Theory 93(6), 61 (2021)Bonet, J., Mangino, E.: Associated weights for spaces of p-integrable entire functions. Quaest. Math. 43(5–6), 747–760 (2020)Bonet, J., Mengestie, T., Worku, M.: Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces. Results Math. 74(4), 197 (2019)Bonet, J., Taskinen, J.: A note about Volterra operators on weighted Banach spaces of entire functions. Math. Nachr. 288(11–12), 1216–1225 (2015)Bonet, J., Taskinen, J.: Solid hulls of weighted Banach spaces of analytic functions on the unit disc with exponential weights. Ann. Acad. Sci. Fenn. 43, 521–530 (2018)Bonet, J., Taskinen, J.: Solid hulls of weighted Banach spaces of entire functions. Rev. Mat. Iberoam. 34(2), 593–608 (2018)Bonet, J., Vukotić, D.: Superposition operators between weighted Banach spaces of analytic functions of controlled growth. Monatsh. Math. 170(3–4), 311–323 (2013)Bonet, J., Vukotić, D.: A note on completeness of weighted normed spaces of analytic functions. Results Math. 72(1–2), 263–279 (2017)Bonet, J., Wolf, E.: A note on weighted Banach spaces of holomorphic functions. Arch. Math. 81, 650–654 (2003)Bourdon, P.S.: Essential angular derivatives and maximum growth of Koenigs eigenfunction. J. Funct. Anal. 160, 561–580 (1998)Bourdon, P.S.: Bellwethers for boundedness of composition operators on weighted Banach spaces of analytic functions. J. Aust. Math. Soc. 86(3), 305–314 (2009)Bourdon, P.S.: Invertible weighted composition operators. Proc. Am. Math. Soc. 142(1), 289–299 (2014)Boyd, C., Rueda, P.: Isometries between spaces of weighted holomorphic functions. Studia Math. 190(3), 203–231 (2009)Boyd, C., Rueda, P.: Isometries of weighted spaces of holomorphic functions on unbounded domains. Proc. R. Soc. Edinb. Sect. A 139(2), 253–271 (2009)Boyd, C., Rueda, P.: Holomorphic superposition operators between Banach function spaces. J. Aust. Math. Soc. 96(2), 186–197 (2014)Boyd, C., Rueda, P.: Superposition operators between weighted spaces of analytic functions. Quaest. Math. 36(3), 411–419 (2013)Boyd, C., Rueda, P.: Surjectivity of isometries between weighted spaces of holomorphic functions. Rev. R. Acad. Cienc. Exactas FÃs. Nat. Ser. A Mat. RACSAM 113(3), 2461–2477 (2019)Bracci, F., Contreras, M.D., DÃaz-Madrigal, S.: Continuous Semigroups of Holomorphic Self-maps of the Unit Disc. Springer, Berlin (2020)Carroll, T., Gilmore, C.: Weighted composition operators on Fock spaces and their dynamics. J. Math. Anal. Appl. 502(1), 125234 (2021)Carswell, B.J., MacCluer, B.D., Schuster, A.: Composition operators on the Fock space. Acta Sci. Math. (Szeged) 69, 871–887 (2003)Colonna, F., MartÃnez-Avendaño, R.A.: Hypercyclicity of composition operators on Banach spaces of analytic functions. Complex Anal. Oper. Theory 12(1), 305–323 (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(2), 1109–1154 (2016)Constantin, O., Peláez, J.A.: Boundedness of the Bergman projection on -spaces with exponential weights. Bull. Sci. Math. 139, 245–268 (2015)Constantin, O., Persson, A.-M.: The spectrum of Volterra-type integration operators on generalized Fock spaces. Bull. Lond. Math. Soc. 47(6), 958–963 (2015)Contreras, M.D., Peláez, J.A., Pommerenke, C., Rättyä, J.: Integral operators mapping into the space of bounded analytic functions. J. Funct. Anal. 271, 2899–2943 (2016)Contreras, M.D., Hernández-DÃaz, A.G.: Weighted composition operators in weighted Banach spaces of analytic functions. J. Aust. Math. Soc. 69, 41–60 (2000)Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)Dai, J.: Norm of the Hilbert matrix operator on the Korenblum space. J. Math. Anal. Appl. 514, 126270 (2022)Diamantopoulos, E., Siskakis, A.G.: Composition operators and the Hilbert matrix. Studia Math. 140(2), 191–198 (2000)DomaÅ„ski, P., Lindström, M.: Sets of interpolation and sampling for weighted Banach spaces of holomorphic functions. Ann. Polon. Math. 79, 233–264 (2002)DomÃnguez, S., Girela, D.: Superposition operators between mixed norm spaces of analytic functions. Mediterr. J. Math. 18(1), 18 (2021)Dostanic, M.: Unboundedness of the Bergman projections on spaces with exponential weights. Proc. Edinb. Math. Soc. 47, 111–117 (2004)Doubtsov, E.: Integration and differentiation in Hardy growth spaces. Complex Anal. Oper. Theory 13(4), 1883–1893 (2019)Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory, 2nd Printing. Wiley Interscience Publ., New York (1964)Duren, P.L.: Theory of Spaces. Academic Press, New York (1970)Eklund, T., Galindo, P., Lindström, M.: Königs eigenfunction for composition operators on Bloch and type spaces. J. Math. Anal. Appl. 445(2), 1300–1309 (2017)Eklund, T., Galindo, P., Lindström, M., Nieminen, I.: Norm, essential norm and weak compactness of weighted composition operators between dual Banach spaces of analytic functions. J. Math. Anal. Appl. 451(1), 1–13 (2017)Eklund, T., Lindström, M., Pirasteh, M.M., Sanatpour, A.H., Wikman, N.: Generalized Volterra operators mapping between Banach spaces of analytic functions. Monatsh. Math. 189(2), 243–261 (2019)Fares, T., Lefèvre, P.: Nuclear composition operators on Bloch spaces. Proc. Am. Math. Soc. 148(6), 2487–2496 (2020)Galindo, P., Lindström, M.: Fredholm composition operators on analytic function spaces. Collect. Math. 63(2), 139–145 (2012)Galindo, P., Lindström, M.: Spectra of some weighted composition operators on dual Banach spaces of analytic functions. Integral Equ. Oper. Theory 90(3), 31 (2018)Galindo, P., Lindström, M., Wikman, N.: Spectra of weighted composition operators on analytic function spaces. Mediterr. J. Math. 17(1), 34 (2020)Gómez-Collado, M.C., Jornet, D.: Fredholm weighted composition operators on weighted Banach spaces of analytic functions of type . J. Funct. Spaces 2015, 982135 (2015)Grosse-Erdmann, K., Peris, A.: Linear Chaos. Springer, London (2011)Guo, K., Izuchi, K.: Composition operators on Fock type space. Acta Sci. Math. (Szeged) 74, 807–828 (2008)Hai, P.V., Khoi, L.H.: Boundedness and compactness of weighted composition operators on Fock spaces . Acta Math. Vietnam. 41(3), 531–537 (2016)Han, S.-A., Zhou, Z.-H.: Mean ergodicity of composition operators on Hardy space. Proc. Indian Acad. Sci. Math. Sci. 129(4), 45 (2019)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. Grad. Texts in Math., vol. 199. Springer, New York (2000)Heuser, H.R.: Functional Analysis. Wiley, New York (1982)Hu, Z.: Extended Cesà ro operators on mixed-norm spaces. Proc. Am. Math. Soc. 131, 2171–2179 (2003)Hu, Z., Lv, X., Schuster, A.: Bergman spaces with exponential weights. J. Funct. Anal. 276, 1402–1429 (2019)Hyvärinen, O., Lindström, M., Nieminen, I., Saukko, E.: Spectra of weighted composition operators with automorphic symbols. J. Funct. Anal. 265(8), 1749–1777 (2013)Hyvärinen, O., Kemppainen, M., Lindström, M., Rautio, A., Saukko, E.: The essential norm of weighted composition operators on weighted Banach spaces of analytic functions. Integral Equ. Oper. Theory 72(2), 151–157 (2012)Jevtić, M., Vukotić, D., Arsenović, M.: Taylor Coefficients and Coefficient Multipliers of Hardy and Bergman-Type Spaces, RSME Springer Series, vol. 2. Springer, Berlin (2016)Jordá, E., RodrÃguez-Arenas, A.: Ergodic properties of composition operators on Banach spaces of analytic functions. J. Math. Anal. Appl. 486(1), 123891 (2020)Karapetyants, A., Liflyand, E.: Defining Hausdorff operators on Euclidean spaces. Math. Methods Appl. Sci. 43(16), 9487–9498 (2020)Krengel, U.: Ergodic Theorems. de Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter Co., Berlin (1985)Liang, Y.-X., Zhou, Z.-H.: Hypercyclic behaviour of multiples of composition operators on weighted Banach spaces of holomorphic functions. Bull. Belg. Math. Soc. Simon Stevin 21(3), 385–401 (2014)Lin, Q.: Volterra type operators between Bloch type spaces and weighted Banach spaces. Integral Equ. Oper. Theory 91(2), 13 (2019)Lindström, M., Miihkinen, S., Wikman, N.: Norm estimates of weighted composition operators pertaining to the Hilbert matrix. Proc. Am. Math. Soc. 147(6), 2425–2435 (2019)Lindström, M., Miihkinen, S., Norrbo, D.: Exact essential norm of generalized Hilbert matrix operators on classical analytic functions spaces. Adv. Math. 408, 108598 (2022)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the Fourier series of unbounded harmonic functions. J. Lond. Math. Soc. 61(2), 568–580 (2000)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175(1), 19–40 (2006)Malman, B.: Spectra of generalized Cesà ro operators acting on growth spaces. Integral Equ. Oper. Theory 90(3), 26 (2018)MartÃn, M.J., Vukotić, D.: Isometries of the Bloch space among the composition operators. Bull. Lond. Math. Soc. 39(1), 151–155 (2007)Mas, A., Vukotić, D.: Invertible and isometric composition operators. Maediterr. J. Math. 19, 173 (2022)Meise, R., Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford University Press, New York (1997)Mengestie, T.: A note on the differential operator on generalized Fock spaces. J. Math. Anal. Appl. 458(2), 937–948 (2018)Mengestie, T.: On the spectrum of Volterra-type integral operators on Fock–Sobolev spaces. Complex Anal. Oper. Theory 11(6), 1451–1461 (2017)Mengestie, T., Ueki, S.-I.: Integral, differential and multiplication operators on generalized Fock spaces. Complex Anal. Oper. Theory 13(3), 935–958 (2019)Miralles, A., Wolf, E.: Hypercyclic composition operators on H0v-spaces. Math. Nachr. 286(1), 34–41 (2013)Mirotin, A.R.: Hausdorff operators on some spaces of holomorphic functions on the unit disc. Complex Anal. Oper. Theory 15, 85 (2021)Montes-RodrÃguez, A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61, 872–884 (2000)Nowak, M., Pavlovic, M.: On the Libera operator. J. Math. Anal. Appl. 370(2), 588–599 (2010)Pavlovic, M.: Definition and Properties of the Libera operators on mixed norm spaces. Sci. World J. 2014, 590656 (2014)Peláez, J.A., Rättyä, J.: Bergman projection induced by radial weight. Adv. Math. 391, 107950 (2021)Perfekt, K.-M.: Duality and distance formulas in spaces defined by means of oscillation. Ark. Mat. 51, 345–361 (2013)Perfekt, K.-M.: On M-ideals and type spaces. Math. Scand. 121(1), 151–160 (2017)Persson, A.-M.: On the spectrum of the Cesà ro operator on spaces of analytic functions. J. Math. Anal Appl. 340, 1180–1203 (2008)Pommerenke, Ch.: Schlichte Funktionen un analytische Functionen von beschrnkter mittlerer Oszilation. Comment. Math. Helv. 52, 591–602 (1977)Ramos Fernández, J.C.: Bounded superposition operators between weighted Banach spaces of analytic functions. Appl. Math. Comput. 219(10), 4942–4949 (2013)Rudin, W.: Real and Complex Analysis. Mc Grawn Hill, New York (1974)Schindl, G.: S
Complexifier Coherent States for Quantum General Relativity
Recently, substantial amount of activity in Quantum General Relativity (QGR)
has focussed on the semiclassical analysis of the theory. In this paper we want
to comment on two such developments: 1) Polymer-like states for Maxwell theory
and linearized gravity constructed by Varadarajan which use much of the Hilbert
space machinery that has proved useful in QGR and 2) coherent states for QGR,
based on the general complexifier method, with built-in semiclassical
properties. We show the following: A) Varadarajan's states {\it are}
complexifier coherent states. This unifies all states constructed so far under
the general complexifier principle. B) Ashtekar and Lewandowski suggested a
non-Abelean generalization of Varadarajan's states to QGR which, however, are
no longer of the complexifier type. We construct a new class of non-Abelean
complexifiers which come close to the one underlying Varadarajan's
construction. C) Non-Abelean complexifiers close to Varadarajan's induce new
types of Hilbert spaces which do not support the operator algebra of QGR. The
analysis suggests that if one sticks to the present kinematical framework of
QGR and if kinematical coherent states are at all useful, then normalizable,
graph dependent states must be used which are produced by the complexifier
method as well. D) Present proposals for states with mildened graph dependence,
obtained by performing a graph average, do not approximate well coordinate
dependent observables. However, graph dependent states, whether averaged or
not, seem to be well suited for the semiclassical analysis of QGR with respect
to coordinate independent operators.Comment: Latex, 54 p., no figure
Lectures on Loop Quantum Gravity
Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has
matured over the past fifteen years to a mathematically rigorous candidate
quantum field theory of the gravitational field. The features that distinguish
it from other quantum gravity theories are 1) background independence and 2)
minimality of structures. Background independence means that this is a
non-perturbative approach in which one does not perturb around a given,
distinguished, classical background metric, rather arbitrary fluctuations are
allowed, thus precisely encoding the quantum version of Einstein's radical
perception that gravity is geometry. Minimality here means that one explores
the logical consequences of bringing together the two fundamental principles of
modern physics, namely general covariance and quantum theory, without adding
any experimentally unverified additional structures. The approach is purposely
conservative in order to systematically derive which basic principles of
physics have to be given up and must be replaced by more fundamental ones. QGR
unifies all presently known interactions in a new sense by quantum mechanically
implementing their common symmetry group, the four-dimensional diffeomorphism
group, which is almost completely broken in perturbative approaches. These
lectures offer a problem -- supported introduction to the subject.Comment: 90 pages, Latex, 18 figures, uses graphicx and pstricks for coloured
text and graphics, based on lectures given at the 271st WE Heraeus Seminar
``Aspects of Quantum Gravity: From Theory to Experimental Search'', Bad
Honnef, Germany, February 25th -- March 1st, to appear in Lecture Notes in
Physic
- …