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Transactions of the American Mathematical Society

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Summing inclusion maps between symmetric sequence spaces

Authors: Andreas Defant, Mieczyslaw Mastylo and Carsten Michels
Journal: Trans. Amer. Math. Soc. 354 (2002), 4473-4492
MSC (2000): Primary 47B10; Secondary 46M35, 47B06
Published electronically: July 2, 2002
MathSciNet review: 1926884
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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 $(\Vert x_n \Vert _{\ell_2})$ 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 $(\Vert x_n \Vert _{\ell_2})$ 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.

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Additional Information

Andreas Defant
Affiliation: Fachbereich Mathematik, Carl von Ossietzky University of Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany

Mieczyslaw Mastylo
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

Carsten Michels
Affiliation: Fachbereich Mathematik, Carl von Ossietzky University of Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany

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
Article copyright: © Copyright 2002 American Mathematical Society

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