Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Optimal Weyl inequality in Banach spaces

Author: Aicke Hinrichs
Journal: Proc. Amer. Math. Soc. 134 (2006), 731-735
MSC (2000): Primary 47B10, 43A25
Published electronically: 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 [Enhancements On Off] (What's this?)

  • [CH04] B. Carl, A. Hinrichs, Optimal Weyl type inequalities for operators in Banach spaces. To appear in Positivity.
  • [Koe84] Hermann König, Some inequalities for the eigenvalues of compact operators, General inequalities, 4 (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 71, Birkhäuser, Basel, 1984, pp. 213–219. MR 821799
  • [Koe86] Hermann König, Eigenvalue distribution of compact operators, Operator Theory: Advances and Applications, vol. 16, Birkhäuser Verlag, Basel, 1986. MR 889455
  • [Koe01] Hermann König, Eigenvalues of operators and applications, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 941–974. MR 1863710, 10.1016/S1874-5849(01)80024-3
  • [Pie66] A. Pietsch, Absolut 𝑝-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1966/1967), 333–353 (German). MR 0216328
  • [Pie74] Albrecht Pietsch, 𝑠-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201–223. MR 0361883
  • [Pie80a] Albrecht Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann. 247 (1980), no. 2, 149–168. MR 568205, 10.1007/BF01364141
  • [Pie80b] Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR 582655
  • [Pie87] Albrecht Pietsch, Eigenvalues and 𝑠-numbers, Cambridge Studies in Advanced Mathematics, vol. 13, Cambridge University Press, Cambridge, 1987. MR 890520
  • [Wey49] Hermann Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci. U. S. A. 35 (1949), 408–411. MR 0030693

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

Keywords: Weyl inequality, eigenvalue estimates, approximation numbers, $s$-numbers.
Received by editor(s): October 6, 2004
Published electronically: 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 70th birthday
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.