Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47B10, 46M35, 47B06

Retrieve articles in all journals with MSC (2000): 47B10, 46M35, 47B06


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: http://dx.doi.org/10.1090/S0002-9947-02-03056-8
PII: S 0002-9947(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