Abstract
The Fréchet sequence spaces \( ces(p+) \) are very different to the Fréchet sequence spaces \( \ell _{p+}, 1 \le p < \infty ,\) that generate them, (Albanese et al. in J Math Anal Appl 458:1314–1323, 2018). The aim of this paper is to investigate various properties (eg. continuity, compactness, mean ergodicity) of certain linear operators acting in and between the spaces \( ces (p+),\) such as the Cesàro operator, inclusion operators and multiplier operators. Determination of the spectra of such classical operators is an important feature. It turns out that both the space of multiplier operators \( {{\mathcal {M}}}(ces (p+)) \) and its subspace \( {{\mathcal {M}}}_c (ces (p+)) \) consisting of the compact multiplier operators are independent of p. Moreover, \( {{\mathcal {M}}}_c (ces (p+)) \) can be topologized so that it is the strong dual of the Fréchet–Schwartz space \( ces ( 1+)\) and \( ( {{\mathcal {M}}}_c (ces (p+))'_\beta \simeq ces (1+) \) is a proper subspace of the Köthe echelon Fréchet space \( {{\mathcal {M}}}(ces (p+)) = \lambda ^\infty (A) , 1 \le p < \infty , \) for a suitable matrix A.
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Acknowledgements
The research of the first two authors was partially supported by the projects MTM2016-76647-P and GV Prometeo/2017/102 (Spain). The authors are thankful to the referees for their careful reading of the manuscript and their suggestions which improved the presentation of the article.
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Albanese, A.A., Bonet, J. & Ricker, W.J. Operators on the Fréchet sequence spaces \({\varvec{ces(p+)}}\), \(1\le p<\infty \). RACSAM 113, 1533–1556 (2019). https://doi.org/10.1007/s13398-018-0564-2
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DOI: https://doi.org/10.1007/s13398-018-0564-2
Keywords
- Fréchet space
- Sequence space \( ces(p+)\)
- Spectrum
- Multiplier operator
- Cesàro operator
- Mean ergodic operator.