223 research outputs found

    On the isomorphic classification of weighted spaces of holomorphic functions

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    Holomorphic Bloch spaces on the unit ball in CnC^n

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    summary:This work is an introduction to anisotropic spaces of holomorphic functions, which have ω\omega-weight and are generalizations of Bloch spaces on a unit ball. We describe the holomorphic Bloch space in terms of the corresponding Lω∞L_\omega ^\infty space. We establish a description of (Ap(ω))∗(A^p(\omega ))^* via the Bloch classes for all 0<p≀10<p\leq 1

    A proof of uniqueness of the Gurarii space

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    We present a short and elementary proof of isometric uniqueness of the Gurarii space.Comment: 6 pages, some improvements incorporate

    A note on completeness of weighted normed spaces of analytic functions

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    [EN] Given a non-negative weight v, not necessarily bounded or strictly positive, defined on a domain G in the complex plane, we consider the weighted space H-v(infinity) (G)of all holomorphic functions on G such that the product v vertical bar f vertical bar is bounded in G and study the question of when such a space is complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight v, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.The first author was partially supported by MTM2013-43540-P and MTM2016-76647-P by MINECO/FEDER-EU and GVA Prometeo II/2013/013. The second author was partially supported by the MINECO/FEDER-EU Grant MTM2015-65792-P. Both authors were partially supported by Thematic Research Network MTM2015-69323-REDT, MINECO, Spain.Bonet Solves, JA.; Vukotic, D. (2017). A note on completeness of weighted normed spaces of analytic functions. Results in Mathematics. 72(1-2):263-279. https://doi.org/10.1007/s00025-017-0696-2S263279721-2Arcozzi, 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)Berenstein, C.A., Gay, R.: Complex Variables, An Introduction. Springer, New York (1991)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)Björn, A.: Removable singularities for weighted Bergman spaces. Czechoslov. Math. J. 56, 179–227 (2006)Bonet, J., DomaƄski, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., Vogt, D.: Weighted spaces of holomorphic functions and sequence spaces. Note Mat. 17, 87–97 (1997)Conway, J.B.: Functions of One Complex Variable, Second Edition, Graduate Texts in Mathematics, vol. 11. Springer, New York (1978)Gaier, D.: Lectures on Complex Approximation. BirkhĂ€user, Boston (1987)Grosse-Erdmann, K.-G.: A weak criterion for vector-valued holomorphic functions. Math. Proc. Camb. Philos. Soc. 136, 399–411 (2004)Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland, Amsterdam (1979)HorvĂĄth, J.: Topological Vector Spaces and Distributions. Addison-Wesley, Reading (1966)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175, 19–45 (2006)Nakazi, T.: Weighted Bloch spaces which are Banach spaces. Rend. Circ. Mat. Palermo 62, 427–440 (2013)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971

    Superposition operators between weighted Banach spaces of analytic functions of controlled growth

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    The final publication is available at Springer via: http://dx.doi.org/10.1007/s00605-012-0441-6[EN] We characterize the entire functions which transform a weighted Banach space of holomorphic functions on the disc of type H∞ into another such space by superposition. We also show that all the superposition operators induced by such entire functions map bounded sets into bounded sets and are continuous. Superposition operators that map bounded sets into relatively compact sets are also considered. © 2012 Springer-Verlag Wien.The research of Bonet was partially supported by MICINN and FEDER Project MTM2010-15200, by GV project Prometeo/2008/101, and by ACOMP/2012/090. The research of Vukotic was partially supported by MICINN grant MTM2009-14694-C02-01, Spain and by the European ESF Network HCAA ("Harmonic and Complex Analysis and Its Applications").Bonet Solves, JA.; Vukotić, D. (2013). Superposition operators between weighted Banach spaces of analytic functions of controlled growth. Monatshefte fĂŒr Mathematik. 170(3-4):311-323. https://doi.org/10.1007/s00605-012-0441-6S3113231703-4Álvarez, V., MĂĄrquez, M.A., Vukotić, D.: Superposition operators between the Bloch space and Bergman spaces. Ark. Mat. 42, 205–216 (2004)Appell, J., Zabrejko, P.P.: Nonlinear Superposition Operators, Cambridge Tracts in Mathematics 95. Cambridge University Press, London (1990)Appell, J., Zabrejko, P.P.: Remarks on the superposition operator problem in various function spaces. Complex Var. Elliptic Equ. 55(8–10), 727–737 (2010)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on bounded domains. Michigan 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)Bonet, J., DomaƄski, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., DomaƄski, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. (Ser. A) 64, 101–118 (1998)Boyd, C., Rueda, P.: Holomorphic superposition operators between Banach function spaces. Preprint (2011)Boyd, C., Rueda, P.: Superposition operators between weighted spaces of analytic functions. Preprint (2011)Buckley, S.M., FernĂĄndez, J.L., Vukotić, D.: Superposition operators on Dirichlet type spaces. In: Papers on Analysis: A Volume dedicaed to Olli Martio on the occasion of his 60th birthday. Rep. Univ. JyvĂ€skyla Dept. Math. Stat, vol. 83, pp. 41–61. Univ. JyvĂ€skyla, JyvĂ€skyla (2001)Buckley, S.M., Vukotić, D.: Univalent interpolation in Besov spaces and superposition into Bergman spaces. Potential Anal. 29(1), 1–16 (2008)CĂĄmera, G.A.: Nonlinear superposition on spaces of analytic functions. In: Harmonic Analysis and Operator Theory (CarĂĄcas, 1994), Contemp. Math, vol. 189, pp. 103–116. Am. Math. Soc, Providence (1995)CĂĄmera, G.A., GimĂ©nez, J.: The nonlinear superposition operators acting on Bergman spaces. Compositio Math. 93, 23–35 (1994)Castillo, R.E., Ramos FernĂĄndez, J.C., Salazar, M.: Bounded superposition operators between Bloch-Orlicz and α\alpha -Bloch spaces. Appl. Math. Comp. 218, 3441–3450 (2011)Dineen, S.: Complex Analysis in Locally Convex Spaces, vol. 57. North-Holland Math. Studies, Amsterdam (1981)Girela, D., MĂĄrquez, M.A.: Superposition operators between QpQ_p spaces and Hardy spaces. J. Math. Anal. Appl. 364, 463–472 (2010)Grosse-Erdmann, K.-G.: A weak criterion for vector-valued holomorphic functions. Math. Proc. Camb. Publ. Soc. 136, 399–41 (2004)Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)Langenbruch, M.: Continuation of Gevrey regularity for solutions of partial differential operators. In: Functional Analysis (Trier, 1994), pp. 249–280. de Gruyter, Berlin (1996)Levin, B.Ya.: Lectures on Entire Functions. Translations of Mathematical Monographs, vol. 150, Amer. Math. Soc., Providence (1996).Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–45 (2006)Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Springer, Berlin (1992)Ramos FernĂĄndez, J.C.: Bounded superposition operators between weighted Banach spaces of analytic functions. Preprint, Available from http://arxiv.org/abs/1203.5857Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971)Vukotić, D.: Integrability, growth of conformal maps, and superposition operators, Technical Report 10. Aristotle University of Thessaloniki, Department of Mathematics (2004)Xiong, C.: Superposition operators between QpQ_p spaces and Bloch-type spaces. Complex Var. Theory Appl. 50, 935–938 (2005)Xu, W.: Superposition operators on Bloch-type spaces. Comput. Methods Funct. Theory 7, 501–507 (2007)Zhu, K.: Operator Theory in Function Spaces, 2nd edn. Am. Math. Soc., Providence (2007

    Weighted Banach spaces of harmonic functions

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    “The final publication is available at Springer via http://dx.doi.org/10.1007/s13398-012-0109-z."We study Banach spaces of harmonic functions on open sets of or endowed with weighted supremum norms. We investigate the harmonic associated weight defined naturally as the analogue of the holomorphic associated weight introduced by Bierstedt, Bonet, and Taskinen and we compare them. We study composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions characterizing the continuity, the compactness and the essential norm of composition operators among these spaces in terms of associated weights.The research of the first author was partially supported by MEC and FEDER Project MTM2010-15200 and by GV project ACOMP/2012/090.Jorda Mora, E.; Zarco GarcĂ­a, AM. (2014). Weighted Banach spaces of harmonic functions. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 108(2):405-418. https://doi.org/10.1007/s13398-012-0109-zS4054181082Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, Berlin (2001)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. J. 40(2), 271–297 (1993)Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127(2), 137–168 (1998)Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 54(1), 70–79 (1993)Bonet, J., DomaƄski, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., DomaƄski, P., Lindström, M.: Weakly compact composition operators on weighted vector-valued Banach spaces of analytic mappings. Ann. Acad. Sci. Fenn. Math. Ser. A I 26, 233–248 (2001)Bonet, J., DomaƄski, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 64, 101–118 (1998)Bonet, J., Friz, M., JordĂĄ, E.: Composition operators between weighted inductive limits of spaces of holomorphic functions. Publ. Math. Debr. Ser. A 67, 333–348 (2005)Boyd, C., Rueda, P.: The v-boundary of weighted spaces of holomorphic functions. Ann. Acad. Sci. Fenn. Math. 30, 337–352 (2005)Boyd, C., Rueda, P.: Complete weights and v-peak points of spaces of weighted holomorphic functions. Isr. J. Math. 155, 57–80 (2006)Boyd, C., Rueda, P.: Isometries of weighted spaces of harmonic functions. Potential Anal. 29(1), 37–48 (2008)Carando, D., Sevilla-Peris, P.: Spectra of weighted algebras of holomorphic functions. Math. Z. 263, 887–902 (2009)Contreras, M.D., HernĂĄndez-DĂ­az, G.: Weighted composition operators in weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 69(1), 41–60 (2000)GarcĂ­a, D., Maestre, M., Rueda, P.: Weighted spaces of holomorphic functions on Banach spaces. Stud. Math. 138(1), 1–24 (2000)GarcĂ­a, D., Maestre, M., Sevilla-Peris, P.: Composition operators between weighted spaces of holomorphic functions on Banach spaces. Ann. Acad. Sci. Fenn. Math. 29, 81–98 (2004)Gunning, R., Rossi, H.: Analytic Functions of Several Complex Variables. AMS Chelsea Publishing, Providence (2009)Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs (1962)Krantz, S.G.: Function Theory of Several Complex Variables. AMS, Providence (2001)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175(1), 19–45 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford University Press, Oxford (1997)Montes-RodrĂ­guez, A.: Weight composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(2), 872–884 (2000)Ng, K.F.: On a theorem of Diximier. Math. Scand. 29, 279–280 (1972)Rudin, W.: Real and Complex Analysis. MacGraw-Hill, NY (1970)Rudin, W.: Functional analysis. In: International series in pure and applied mathematics, 2nd edn. McGraw-Hill, Inc., New York (1991)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of harmonic functions. J. Reine Angew. Math. 299(300), 256–279 (1978)Shields, A.L., Williams, D.L.: Bounded projections and the growth of harmonic conjugates in the unit disc. Mich. Math. J. 29, 3–25 (1982)Zheng, L.: The essential norms and spectra of composition operators on H∞H^\infty . Pac. J. Math. 203(2), 503–510 (2002

    Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

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    The final publication is available at Springer via http://dx.doi.org/10.1007/s00013-012-0458-zWe investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H∞. Applications for composition operators on weighted Bloch spaces are given. © 2012 Springer Basel.1. The authors are thankful to the referee for pointing to us the references [15] and [16] and their relevance in the present research. 2. The research of Bonet was partially supported by MICINN and FEDER Project MTM2010-15200 and by GV project Prometeo/2008/101 and project ACOMP/2012/090.Bonet Solves, JA.; Lindström, M.; Wolf, E. (2012). Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions. Archiv der Mathematik. 99(6):537-546. https://doi.org/10.1007/s00013-012-0458-zS537546996Bierstedt K.D., Bonet J., Galbis A.: Weighted spaces of holomorphic functions on bounded domains. Michigan 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)J. Bonet, P. DomaƄski, and M. Lindström, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Canad, Math. Bull. 42, no. 2, (1999), 139–148Bonet J. et al.: Composition operators between weighted Banach spaces of analytic functions. J. Austral. Math. Soc. Ser. A 64, 101–118 (1998)Bonet J., Lindström M, Wolf E.: Isometric weighted composition operators on weighted Banach spaces of type H ∞. Proc. Amer. Math. Soc. 136, 4267–4273 (2008)Bonet J, Wolf E.: A note on weighted spaces of holomorphic functions. Archiv Math. 81, 650–654 (2003)Contreras M.D, HernĂĄndez-DĂ­az A.G.: Weighted composition operators in weighted banach spaces of analytic functions. J. Austral. Math. Soc. Ser. A 69, 41–60 (2000)Cowen C., MacCluer B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)J. Diestel, Geometry of Banach Spaces. Selected Topics, Lecture Notes in Math. vol. 485, Springer, Berlin, 1975.Hammond C.: On the norm of a composition operator with linear fractional symbol. Acta Sci. Math. (Szeged) 69, 813–829 (2003)Hosokawa T., Izuchi K., Zheng D.: Isolated points and essential components of composition operators on H ∞. Proc. Amer. Math. Soc. 130, 1765–1773 (2001)Hosokava T., Ohno S.: Topological strusctures of the sets of composition operatorson the Bloch spaces. J. Math. anal. Appl. 303, 499–508 (2005)Lusky W.: On the isomorphy classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175, 19–45 (2006)MartĂ­n M.: Norm-attaining composition operators on the Bloch spaces. J. Math. Anal. Appl. 369, 15–21 (2010)A. Montes-RodrĂ­guez, The Pick-Schwarz lemma and composition operators on Bloch spaces, International Workshop on Operator Theory (Cefalu, 1997), Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 167–170.Montes-RodrĂ­guez A.: The essential norm of a composition operator on Bloch spaces. Pacific J. Math. 188, 339–351 (1999)Montes-RodrĂ­guez A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. London Math. Soc. 61, 872–884 (2000)J.H. Shapiro, Composition Operators and Classical Function Theory, Springer, 1993.K. Zhu, Operator Theory in Function Spaces, Second Edition. Amer. Math. Soc., 2007

    Weighted composition operators on Korenblum type spaces of analytic functions

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    [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 FĂ­sicas y Naturales Serie A MatemĂĄticas. 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 CesĂ ro operator in the FrĂ©chet spaces ℓp+\ell ^{p+} and Lp−L^{p-}. Glasgow Math. J. 59, 273–287 (2017)Albanese, A.A., Bonet, J., Ricker, W.J.: The CesĂ ro 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 FrĂ©chet sequence spaces ces(p+),1≀p≀∞ces(p+), 1\le p\le \infty . Rev. R. Acad. Cienc. Exactas FĂ­s. 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≀p≀∞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., DomaƄski, 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., HernĂĄndez-DĂ­az, 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∞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 uCφuC_{\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-RodrĂ­guez, A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(3), 872–884 (2000)QueffĂ©lec, H., QueffĂ©lec, 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
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