Summing inclusion maps between symmetric sequence spaces
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- by Andreas Defant, Mieczyslaw Mastyło and Carsten Michels PDF
- Trans. Amer. Math. Soc. 354 (2002), 4473-4492 Request permission
Abstract:
In 1973/74 Bennett and (independently) Carl proved that for $1 \le u \le 2$ the identity map id: $\ell _u \hookrightarrow \ell _2$ is absolutely $(u,1)$-summing, i. e., for every unconditionally summable sequence $(x_n)$ in $\ell _u$ the scalar sequence $(\|x_n \|_{\ell _2})$ is contained in $\ell _u$, which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a $2$-concave symmetric Banach sequence space $E$ the identity map $\text {id}: E \hookrightarrow \ell _2$ is absolutely $(E,1)$-summing, i. e., for every unconditionally summable sequence $(x_n)$ in $E$ the scalar sequence $(\|x_n \|_{\ell _2})$ is contained in $E$. Various applications are given, e. g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator $T$ on $\ell _2$ with values in a $2$-concave symmetric Banach sequence space $E$ is a multiplier from $\ell _2$ into $E$. Furthermore, we prove an asymptotic formula for the $k$-th approximation number of the identity map $\text {id}: \ell _2^n \hookrightarrow E_n$, where $E_n$ denotes the linear span of the first $n$ standard unit vectors in $E$, and apply it to Lorentz and Orlicz sequence spaces.References
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Additional Information
- Andreas Defant
- Affiliation: Fachbereich Mathematik, Carl von Ossietzky University of Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany
- Email: defant@mathematik.uni-oldenburg.de
- Mieczyslaw Mastyło
- Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University, and Institute of Mathematics (Poznań branch), Polish Academy of Sciences, Matejki 48/49, 60-769 Poznań, Poland
- Email: mastylo@amu.edu.pl
- Carsten Michels
- Affiliation: Fachbereich Mathematik, Carl von Ossietzky University of Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany
- Email: michels@mathematik.uni-oldenburg.de
- Received by editor(s): June 20, 2000
- Published electronically: July 2, 2002
- Additional Notes: The second named author is supported by KBN Grant 2 P03A 042 18
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 4473-4492
- MSC (2000): Primary 47B10; Secondary 46M35, 47B06
- DOI: https://doi.org/10.1090/S0002-9947-02-03056-8
- MathSciNet review: 1926884