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

DOI:
https://doi.org/10.1090/S0002-9947-02-03056-8

Published electronically:
July 2, 2002

MathSciNet review:
1926884

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Abstract | References | Similar Articles | Additional Information

Abstract: In 1973/74 Bennett and (independently) Carl proved that for the identity map id: is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a -concave symmetric Banach sequence space the identity map is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in . 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 on with values in a -concave symmetric Banach sequence space is a multiplier from into . Furthermore, we prove an asymptotic formula for the -th approximation number of the identity map , where denotes the linear span of the first standard unit vectors in , 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

Email:
defant@mathematik.uni-oldenburg.de

**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

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

DOI:
https://doi.org/10.1090/S0002-9947-02-03056-8

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