Skip to main content
SpringerLink
Log in

The level sets of the resolvent norm and convexity properties of Banach spaces

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

We construct a bounded linear operator on a separable, reflexive and strictly convex Banach space with the resolvent norm that is constant in a neighbourhood of zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Böttcher A. (1994) Pseudospectra and singular values of large convolution operators, J. Integral Equations Appl. 6: 267–301

    Article  MATH  MathSciNet  Google Scholar 

  • Böttcher A., Grudsky S., Silbermann B. (1997) Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices, New York J. Math. 3: 1–31

    MATH  Google Scholar 

  • Carothers N.L. (2005) A short course on Banach space theory. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Clarkson J. (1936) Uniformly convex spaces. Trans. Am. Math. Soc. 40: 396–414

    Article  MATH  MathSciNet  Google Scholar 

  • Day M.M. (1941) Reflexive Banach spaces not isomorphic to uniformly convex spaces. Bull. Am. Math. Soc. 47: 313–317

    Article  Google Scholar 

  • Dowling P.N. (2003) On convexity properties of ψ-direct sums of Banach spaces. J. Math. Anal. Appl. 288: 540–543

    Article  MATH  MathSciNet  Google Scholar 

  • Dunford N., Schwartz J.T. (1958) Linear Operators I, General theory. Interscience Publishers, New York

    MATH  Google Scholar 

  • Globevnik J. (1975) On complex strict and uniform convexity, Proc. Am. Math. Soc. 47: 175–178

    Article  MATH  MathSciNet  Google Scholar 

  • Globevnik J. (1976) Norm-constant analytic functions and equivalent norms. Ill. J. Math. 20: 503–506

    MATH  MathSciNet  Google Scholar 

  • Lešnjak G. (1988) Complex convexity and finitely additive vector measures. Proc. Am. Math. Soc. 102: 867–873

    Article  MATH  Google Scholar 

  • Shargorodsky E. (2008) On the level sets of the resolvent norm of a linear operator. Bull. Lond. Math. Soc. 40: 493–504

    Article  MATH  MathSciNet  Google Scholar 

  • Thorp E., Whitley R. (1967) The strong maximum modulus theorem for analytic functions into a Banach space. Proc. Am. Math. Soc. 18: 640–646

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, UK

    Eugene Shargorodsky

  2. Department of Pure Mathematics, Queen’s University Belfast, Belfast, BT7 1NN, UK

    Stanislav Shkarin

Authors
  1. Eugene Shargorodsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stanislav Shkarin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eugene Shargorodsky.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shargorodsky, E., Shkarin, S. The level sets of the resolvent norm and convexity properties of Banach spaces. Arch. Math. 93, 59–66 (2009). https://doi.org/10.1007/s00013-009-0001-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-009-0001-z

Mathematics Subject Classification (2000)

Keywords

Search

Navigation