Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Optimal Weyl inequality in Banach spaces
HTML articles powered by AMS MathViewer

by Aicke Hinrichs PDF
Proc. Amer. Math. Soc. 134 (2006), 731-735 Request permission

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 \[ \prod _{k=1}^n |\lambda _k(T)| \le n^{n/2} \prod _{k=1}^n a_k(T)\] for all $n$. We prove here that the constant $n^{n/2}$ is optimal, which solves a longstanding problem.
References
  • B. Carl, A. Hinrichs, Optimal Weyl type inequalities for operators in Banach spaces. To appear in Positivity.
  • 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
  • Hermann König, Eigenvalue distribution of compact operators, Operator Theory: Advances and Applications, vol. 16, Birkhäuser Verlag, Basel, 1986. MR 889455, DOI 10.1007/978-3-0348-6278-3
  • 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, DOI 10.1016/S1874-5849(01)80024-3
  • A. Pietsch, Absolut $p$-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1966/67), 333–353 (German). MR 216328, DOI 10.4064/sm-28-3-333-353
  • Albrecht Pietsch, $s$-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201–223. MR 361883, DOI 10.4064/sm-51-3-201-223
  • Albrecht Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann. 247 (1980), no. 2, 149–168. MR 568205, DOI 10.1007/BF01364141
  • 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
  • A. Pietsch, Eigenvalues and $s$-numbers, Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and its Applications in Physics and Technology], vol. 43, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987. MR 917067
  • 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 30693, DOI 10.1073/pnas.35.7.408
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
  • 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
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 731-735
  • MSC (2000): Primary 47B10, 43A25
  • DOI: https://doi.org/10.1090/S0002-9939-05-08019-6
  • MathSciNet review: 2180891