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On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility

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Abstract

What is an adequate extension of an operator ideal \(\mathcal{I }\) to the polynomial and multilinear settings? This question motivated the appearance of the interesting concepts of coherent sequences of polynomial ideals and compatibility of a polynomial ideal with an operator ideal, introduced by D. Carando et al. We propose a different approach by considering pairs \((\mathcal{U }_{k},\mathcal{M }_{k})_{k=1}^{\infty }\), where \((\mathcal{U }_{k})_{k=1}^{\infty }\) is a polynomial ideal and \((\mathcal{M }_{k})_{k=1}^{\infty }\) is a multi-ideal, instead of considering just polynomial ideals. It is our belief that our approach ends a discomfort caused by the previous theory: for real scalars the canonical sequence \((\mathcal{P }_{k})_{k=1}^{\infty }\) of continuous \(k\)-homogeneous polynomials is not coherent according to the definition of Carando et al. We apply these new notions to test the pairs of ideals of nuclear and integral polynomials and multilinear operators, the factorisation method and different classes that generalise the concept of absolutely summing operator.

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Acknowledgments

The authors thank Geraldo Botelho for important suggestions. The authors are very indebted to the referee, whose careful analysis and insightful suggestions helped to improve the paper.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, 58.051-900, Brazil

    Daniel Pellegrino

  2. Instituto de Matemática, Universidade Federal da Bahia, Av. Adhemar de Barros, s/n, Sala 268, Salvador, 40170-110, Brazil

    Joilson Ribeiro

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  1. Daniel Pellegrino
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Correspondence to Daniel Pellegrino.

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Communicated by G. Teschl.

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Pellegrino, D., Ribeiro, J. On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility. Monatsh Math 173, 379–415 (2014). https://doi.org/10.1007/s00605-013-0534-x

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