Research Article
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Year 2024, , 98 - 113, 15.09.2024
https://doi.org/10.33205/cma.1518651

Abstract

References

  • D. Achour, P. Rueda, A. Sánchez-Pérez and R. Yahi: Lipschitz operator ideals and the approximation property, J. Math. Anal. Appl., 436 (1) (2016), 217–236.
  • J. M. Anderson: Bloch functions: the basic theory. Operators and function theory, (Lancaster, 1984), 1–17, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 153, Reidel, Dordrecht (1985).
  • J. Arazy, S. D. Fisher and J. Peetre: Möbius invariant function spaces, J. Reine Angew. Math., 363 (1985), 110–145.
  • R. Aron, G. Botelho, D. Pellegrino and P. Rueda: Holomorphic mappings associated to composition ideals of polynomials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 21 (3) (2010), 261–274.
  • A. Belaada, K. Saadi and A. Tiaiba: On the composition ideals of Schatten class type mappings, J. Math., (2016), Article ID: 3492934.
  • G. Botelho, E. Çali¸skan and G. Moraes: The polynomial dual of an operator ideal, Monatsh Math., 173 (2014), 161–174.
  • G. Botelho, D. Pellegrino and P. Rueda: On composition ideals of multilinear mappings and homogeneous polynomials, Publ. Res. Inst. Math. Sci., 43 (4) (2007), 1139–1155.
  • M. G. Cabrera-Padilla, A. Jiménez-Vargas and D. Ruiz-Casternado: On composition ideals and dual ideals of bounded holomorphic mappings, Results Math., 78 (3) (2023), Paper No. 103, 21 pp.
  • M. G. Cabrera-Padilla, A. Jiménez-Vargas and D. Ruiz-Casternado: p-Summing Bloch mappings on the complex unit disc, Banach J. Math. Anal., 18 (9) (2024), https://doi.org/10.1007/s43037-023-00318-6.
  • J. Diestel, H. Jarchow and A. Tonge: Absolutely Summing Operators, Cambridge Studies in Adv. Math, vol. 43, Cambridge Univ. Press, Cambridge (1995).
  • K. Floret, D. García: On ideals of polynomials and multilinear mappings between Banach spaces, Arch. Math. (Basel), 81 (3) (2003), 300–308.
  • M. González, J. M. Gutiérrez: Surjective factorization of holomorphic mappings, Comment. Math. Univ. Carolin., 41 (3) (2000), 469–476.
  • A. Jiménez-Vargas, D. Ruiz-Casternado: Compact Bloch mappings on the complex unit disc, http://arxiv.org/abs/2308.02461.
  • A. Persson, A. Pietsch: p-nuklear and p-integrale Abbildungen in Banach raümen, Studia. Math., 33 (1969), 19–62.
  • A. Pietsch: Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, (1980). Translated from German by the author.
  • A. Pietsch: Ideals of multilinear functionals (designs of a theory), in: Proc. Second Int. Conf. on Operator Algebras, Ideals and Their Applications in Theoretical Physics, Teubner-Texte Math. 67, Leipzig, (1983), 185–199.
  • T. Quang: Banach-valued Bloch-type functions on the unit ball of a Hilbert space and weak spaces of Bloch-type, Constr. Math. Anal., 6 (1) (2023), 6–21. https://doi.org/10.33205/cma.1243686
  • K. Saadi: On the composition ideals of Lipschitz mappings, Banach. J. Math. Anal., 11 (4) (2017), 825–840.

New ideals of Bloch mappings which are I-factorizable and Möbius-invariant

Year 2024, , 98 - 113, 15.09.2024
https://doi.org/10.33205/cma.1518651

Abstract

In this paper, we introduce an unified method for generating ideals of Möbius-invariant Banach-valued Bloch mappings on the complex open unit disc $\D$, through the composition with the members of a Banach operator ideal $\I$. Using linearisation of derivatives of Banach-valued normalized Bloch mappings on $\D$, this composition method yields the so-called ideals of $\I$-factorizable normalized Bloch mappings $\I\circ\hat{\B}$, where $\hat{\B}$ denotes the class of normalized Bloch mappings on $\D$. We present new examples of them as ideals of separable (Rosenthal, Asplund) normalized Bloch mappings and $p$-integral (strictly $p$-integral, $p$-nuclear) normalized Bloch mappings for any $p\in[1,\infty)$. Moreover, the Bloch dual ideal $\I^{\hat{\B}\text{-}\d}$ of an operator ideal $\I$ is introduced and shown that it coincides with the composition ideal $\I^\d\circ\hat{\B}$.

Ethical Statement

This is an original paper and we cite all the necessary references.

Supporting Institution

This research has been supported in part by grant PID2021-122126NB-C31 funded by MCIN/AEI/ 10.13039/501100011033 and by ``ERDF A way of making Europe'', and by Junta de Andalucía grant FQM194.

References

  • D. Achour, P. Rueda, A. Sánchez-Pérez and R. Yahi: Lipschitz operator ideals and the approximation property, J. Math. Anal. Appl., 436 (1) (2016), 217–236.
  • J. M. Anderson: Bloch functions: the basic theory. Operators and function theory, (Lancaster, 1984), 1–17, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 153, Reidel, Dordrecht (1985).
  • J. Arazy, S. D. Fisher and J. Peetre: Möbius invariant function spaces, J. Reine Angew. Math., 363 (1985), 110–145.
  • R. Aron, G. Botelho, D. Pellegrino and P. Rueda: Holomorphic mappings associated to composition ideals of polynomials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 21 (3) (2010), 261–274.
  • A. Belaada, K. Saadi and A. Tiaiba: On the composition ideals of Schatten class type mappings, J. Math., (2016), Article ID: 3492934.
  • G. Botelho, E. Çali¸skan and G. Moraes: The polynomial dual of an operator ideal, Monatsh Math., 173 (2014), 161–174.
  • G. Botelho, D. Pellegrino and P. Rueda: On composition ideals of multilinear mappings and homogeneous polynomials, Publ. Res. Inst. Math. Sci., 43 (4) (2007), 1139–1155.
  • M. G. Cabrera-Padilla, A. Jiménez-Vargas and D. Ruiz-Casternado: On composition ideals and dual ideals of bounded holomorphic mappings, Results Math., 78 (3) (2023), Paper No. 103, 21 pp.
  • M. G. Cabrera-Padilla, A. Jiménez-Vargas and D. Ruiz-Casternado: p-Summing Bloch mappings on the complex unit disc, Banach J. Math. Anal., 18 (9) (2024), https://doi.org/10.1007/s43037-023-00318-6.
  • J. Diestel, H. Jarchow and A. Tonge: Absolutely Summing Operators, Cambridge Studies in Adv. Math, vol. 43, Cambridge Univ. Press, Cambridge (1995).
  • K. Floret, D. García: On ideals of polynomials and multilinear mappings between Banach spaces, Arch. Math. (Basel), 81 (3) (2003), 300–308.
  • M. González, J. M. Gutiérrez: Surjective factorization of holomorphic mappings, Comment. Math. Univ. Carolin., 41 (3) (2000), 469–476.
  • A. Jiménez-Vargas, D. Ruiz-Casternado: Compact Bloch mappings on the complex unit disc, http://arxiv.org/abs/2308.02461.
  • A. Persson, A. Pietsch: p-nuklear and p-integrale Abbildungen in Banach raümen, Studia. Math., 33 (1969), 19–62.
  • A. Pietsch: Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, (1980). Translated from German by the author.
  • A. Pietsch: Ideals of multilinear functionals (designs of a theory), in: Proc. Second Int. Conf. on Operator Algebras, Ideals and Their Applications in Theoretical Physics, Teubner-Texte Math. 67, Leipzig, (1983), 185–199.
  • T. Quang: Banach-valued Bloch-type functions on the unit ball of a Hilbert space and weak spaces of Bloch-type, Constr. Math. Anal., 6 (1) (2023), 6–21. https://doi.org/10.33205/cma.1243686
  • K. Saadi: On the composition ideals of Lipschitz mappings, Banach. J. Math. Anal., 11 (4) (2017), 825–840.
There are 18 citations in total.

Details

Primary Language English
Subjects Operator Algebras and Functional Analysis
Journal Section Articles
Authors

Antonio Jiménez Vargas 0000-0002-0572-1697

David Ruiz Casternado 0000-0002-3222-8996

Early Pub Date August 9, 2024
Publication Date September 15, 2024
Submission Date July 18, 2024
Acceptance Date August 6, 2024
Published in Issue Year 2024

Cite

APA Jiménez Vargas, A., & Ruiz Casternado, D. (2024). New ideals of Bloch mappings which are I-factorizable and Möbius-invariant. Constructive Mathematical Analysis, 7(3), 98-113. https://doi.org/10.33205/cma.1518651
AMA Jiménez Vargas A, Ruiz Casternado D. New ideals of Bloch mappings which are I-factorizable and Möbius-invariant. CMA. September 2024;7(3):98-113. doi:10.33205/cma.1518651
Chicago Jiménez Vargas, Antonio, and David Ruiz Casternado. “New Ideals of Bloch Mappings Which Are I-Factorizable and Möbius-Invariant”. Constructive Mathematical Analysis 7, no. 3 (September 2024): 98-113. https://doi.org/10.33205/cma.1518651.
EndNote Jiménez Vargas A, Ruiz Casternado D (September 1, 2024) New ideals of Bloch mappings which are I-factorizable and Möbius-invariant. Constructive Mathematical Analysis 7 3 98–113.
IEEE A. Jiménez Vargas and D. Ruiz Casternado, “New ideals of Bloch mappings which are I-factorizable and Möbius-invariant”, CMA, vol. 7, no. 3, pp. 98–113, 2024, doi: 10.33205/cma.1518651.
ISNAD Jiménez Vargas, Antonio - Ruiz Casternado, David. “New Ideals of Bloch Mappings Which Are I-Factorizable and Möbius-Invariant”. Constructive Mathematical Analysis 7/3 (September 2024), 98-113. https://doi.org/10.33205/cma.1518651.
JAMA Jiménez Vargas A, Ruiz Casternado D. New ideals of Bloch mappings which are I-factorizable and Möbius-invariant. CMA. 2024;7:98–113.
MLA Jiménez Vargas, Antonio and David Ruiz Casternado. “New Ideals of Bloch Mappings Which Are I-Factorizable and Möbius-Invariant”. Constructive Mathematical Analysis, vol. 7, no. 3, 2024, pp. 98-113, doi:10.33205/cma.1518651.
Vancouver Jiménez Vargas A, Ruiz Casternado D. New ideals of Bloch mappings which are I-factorizable and Möbius-invariant. CMA. 2024;7(3):98-113.