Generalized adjoints and applications to composition operators
DOI:
https://doi.org/10.7146/math.scand.a-119684Abstract
We generalize the classical notion of adjoint of a linear operator and the Aron-Schottenloher notion of adjoint of a homogeneous polynomial. The general (nonlinear) notion is shown to enjoy several properties enjoyed by the classical (linear) ones, nevertheless new interesting phenomena arise in the nonlinear theory. The proofs are not always simple adaptations of the linear cases, actually nonlinear arguments are often required. Applications of the generalized adjoints to Lindström-Schlüchtermann type theorems for composition operators are provided.
References
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Aron, R. M. and Schottenloher, M., Compact holomorphic mappings on Banach spaces and the approximation property, J. Functional Analysis 21 (1976), no. 1, 7–30. https://doi.org/10.1016/0022-1236(76)90026-4
Berge, C., Principles of combinatorics, Translated from the French. Mathematics in Science and Engineering, Vol. 72, Academic Press, New York-London, 1971.
Botelho, G., Çalışkan, E., and Moraes, G., The polynomial dual of an operator ideal, Monatsh. Math. 173 (2014), no. 2, 161–174. https://doi.org/10.1007/s00605-013-0569-z
Botelho, G., Pellegrino, D., and Rueda, P., On composition ideals of multilinear mappings and homogeneous polynomials, Publ. Res. Inst. Math. Sci. 43 (2007), no. 4, 1139–1155.
Botelho, G. and Polac, L., A polynomial Hutton theorem with applications, J. Math. Anal. Appl. 415 (2014), no. 1, 294–301. https://doi.org/10.1016/j.jmaa.2014.01.056
Botelho, G. and Torres, E. R., Hyper-ideals of multilinear operators, Linear Algebra Appl. 482 (2015), 1–20. https://doi.org/10.1016/j.laa.2015.05.012
Botelho, G. and Torres, E. R., Two-sided polynomial ideals on Banach spaces, J. Math. Anal. Appl. 462 (2018), no. 1, 900–914. https://doi.org/10.1016/j.jmaa.2017.12.054
Çalışkan, E. and Rueda, P., Compactness and $s$-numbers for polynomials, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 1, 93–107. https://doi.org/10.4171/RLM/795
Defant, A. and Floret, K., Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993.
Dineen, S., Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. https://doi.org/10.1007/978-1-4471-0869-6
Floret, K., Natural norms on symmetric tensor products of normed spaces, in “Proceedings of the Second International Workshop on Functional Analysis (Trier, 1997)”, vol. 17, 1997, pp. 153–188.
Floret, K. and Garc\'ıa, D., On ideals of polynomials and multilinear mappings between Banach spaces, Arch. Math. (Basel) 81 (2003), no. 3, 300–308. https://doi.org/10.1007/s00013-003-0439-3
Lassalle, S. and Turco, P., Polynomials and holomorphic functions on $\mathcal A$-compact sets in Banach spaces, J. Math. Anal. Appl. 463 (2018), no. 2, 1092–1108. https://doi.org/10.1016/j.jmaa.2018.03.070
Lindström, M. and Schlüchtermann, G., Composition of operator ideals, Math. Scand. 84 (1999), no. 2, 284–296. https://doi.org/10.7146/math.scand.a-13880
Mujica, J., Complex analysis in Banach spaces: Holomorphic functions and domains of holomorphy in finite and infinite dimensions, Notas de Matemática, 107, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986.
Mujica, J., Linearization of bounded holomorphic mappings on Banach spaces, Trans. Amer. Math. Soc. 324 (1991), no. 2, 867–887. https://doi.org/10.2307/2001745
Pietsch, A., Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980.
Ryan, R. A., Applications of topological tensor products to infinite dimensional holomorphy, Ph.D. thesis, Trinity College Dublin, 1980.
Ryan, R. A., Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2002. https://doi.org/10.1007/978-1-4471-3903-4
Turco, P., $\mathcal A$-compact mappings, Rev. R. Acad. Cienc. Exactas F\'ıs. Nat. Ser. A Mat. RACSAM 110 (2016), no. 2, 863–880. https://doi.org/10.1007/s13398-015-0269-8
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2020-05-06
How to Cite
Botelho, G., & Torres, L. A. (2020). Generalized adjoints and applications to composition operators. MATHEMATICA SCANDINAVICA, 126(2), 367–386. https://doi.org/10.7146/math.scand.a-119684
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