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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Eigenvalue estimates for operators on symmetric Banach sequence spaces


Authors: Andreas Defant, Mieczyslaw Mastylo and Carsten Michels
Journal: Proc. Amer. Math. Soc. 132 (2004), 513-521
MSC (2000): Primary 47B06; Secondary 47B10, 47B37
Published electronically: July 14, 2003
MathSciNet review: 2022377
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Abstract: Using abstract interpolation theory, we study eigenvalue distribution problems for operators on complex symmetric Banach sequence spaces. More precisely, extending two well-known results due to König on the asymptotic eigenvalue distribution of operators on $\ell_p$-spaces, we prove an eigenvalue estimate for Riesz operators on $\ell_p$-spaces with $1 < p < 2$, which take values in a $p$-concave symmetric Banach sequence space $E \hookrightarrow \ell_p$, as well as a dual version, and show that each operator $T$ on a $2$-convex symmetric Banach sequence space $F$, which takes values in a $2$-concave symmetric Banach sequence space $E$, is a Riesz operator with a sequence of eigenvalues that forms a multiplier from $F$ into $E$. Examples are presented which among others show that the concavity and convexity assumptions are essential.


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Additional Information

Andreas Defant
Affiliation: FK 5, Inst. F. 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 Insti- tute of Mathematics, Polish Academy of Sciences (Poznań branch), Umultowska 87, Poznań 61-614, Poland
Email: mastylo@amu.edu.pl

Carsten Michels
Affiliation: FK 5, Inst. F. 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/S0002-9939-03-07106-5
PII: S 0002-9939(03)07106-5
Received by editor(s): April 12, 2002
Received by editor(s) in revised form: October 21, 2002
Published electronically: July 14, 2003
Additional Notes: The second-named author was supported by KBN Grant 2 P03A 042 18
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2003 American Mathematical Society