Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Optimal Weyl inequality in Banach spaces


Author: Aicke Hinrichs
Journal: Proc. Amer. Math. Soc. 134 (2006), 731-735
MSC (2000): Primary 47B10, 43A25
Posted: July 18, 2005
MathSciNet review: 2180891
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A well-known multiplicative Weyl inequality states that the sequence of eigenvalues $(\lambda_k(T))$ and the sequence of approximation numbers $(a_k(T))$ of any compact operator $T$ in a Banach space satisfy

\begin{displaymath}\prod_{k=1}^n \vert\lambda_k(T)\vert \le n^{n/2} \prod_{k=1}^n a_k(T)\end{displaymath}

for all $n$. We prove here that the constant $n^{n/2}$ is optimal, which solves a longstanding problem.

References

  • [CH04] B. Carl, A. Hinrichs, Optimal Weyl type inequalities for operators in Banach spaces. To appear in Positivity.
  • [Koe84] H. König, Some inequalities for the eigenvalues of compact operators. In Proc. Oberwolfach Conference General Inequalities 1983, Birkhäuser, Basel, 1984, 213-219.MR 0821799 (87g:47032)
  • [Koe86] H. König, Eigenvalue distribution of compact operators. Operator Theory: Advances and Applications 16, Birkhäuser, Basel, 1986. MR 0889455 (88j:47021)
  • [Koe01] H. König, Eigenvalues of operators and applications. In Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, 941-974. MR 1863710 (2003f:47035)
  • [Pie66] A. Pietsch, Absolut $p$-summierende Abbildungen in normierten Rumen. Studia Math. 28 (1966/1967), 333-353. MR 0216328 (35:7162)
  • [Pie74] A. Pietsch, $s$-numbers of operators in Banach spaces. Studia Math. 51 (1974), 201-223. MR 0361883 (50:14325)
  • [Pie80a] A. Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces. Math. Ann. 247 (1980), 149-168. MR 0568205 (82i:47073a)
  • [Pie80b] A. Pietsch, Operator ideals. North Holland, Amsterdam, New York, Oxford, 1980.MR 0582655 (81j:47001)
  • [Pie87] A. Pietsch, Eigenvalues and $s$-numbers. Cambridge Studies in Advanced Mathematics 13, Cambridge University Press, 1987. MR 0890520 (88j:47022b)
  • [Wey49] H. Weyl, Inequalities between two kinds of eigenvalues of a linear transformation. Proc. Nat. Acad. Sci. U. S. A. 35 (1949), 408-411. MR 0030693 (11:37d)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B10, 43A25

Retrieve articles in all journals with MSC (2000): 47B10, 43A25

Additional Information

Aicke Hinrichs
Affiliation: Mathematisches Institut, FSU Jena, Ernst-Abbe-Platz 1-3, D-07743 Jena, Germany
Email: hinrichs@minet.uni-jena.de

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08019-6
PII: S 0002-9939(05)08019-6
Keywords: Weyl inequality, eigenvalue estimates, approximation numbers, $s$-numbers.
Received by editor(s): October 6, 2004
Posted: July 18, 2005
Additional Notes: The research of the author was supported by the DFG Emmy-Noether grant Hi 584/2-3.
Dedicated: Dedicated to Professor Albrecht Pietsch on the occasion of his \textup{70}th birthday
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia