Skip to main content

Some Generalities

  • Chapter
  • First Online:
Loewner's Theorem on Monotone Matrix Functions

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 354))

  • 1315 Accesses

Abstract

In this chapter, we will discuss several issues that will help us prove the easy half of Loewner’s theorem in the next chapter and then, we will extend Theorem 1.6 to infinite dimensions.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Bendat and S. Sherman, Monotone and convex operator functions, Trans. A.M.S. 79 (1955), 58–71.

    Google Scholar 

  2. D. T. Hoa and O. E. Tikhonov, On the theory of operator-monotone and operator-convex functions, Izv. Vyssh. Uchebn. Zaved. Mat. (2010), 9–14; translation: Russian Math. (Iz. VUZ) 54 (2010), 7–11.

    Google Scholar 

  3. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer, Berlin-New York, 1976.

    Google Scholar 

  4. H. Osaka, S. Silvestrov and J. Tomiyama, Monotone operator functions on C –algebras, Internat. J. Math. 16 (2005), 181–196.

    Google Scholar 

  5. M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.

    Google Scholar 

  6. B. Simon, Lower semicontinuity of positive quadratic forms, Proc. Roy. Soc. Edin. 29 (1977), 267–273.

    Google Scholar 

  7. B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28 (1978), 377–385.

    Google Scholar 

  8. B. Simon, Trace Ideals and Their Applications, London Mathematical Society Lecture Note Series, 35, Cambridge Univ. Press, Cambridge-New York, 1979; second edition, American Mathematical Society, Providence, RI, 2005.

    Google Scholar 

  9. B. Simon A Comprehensive Course in Analysis, Part 1: Real Analysis, American Mathematical Society, Providence, RI, 2015.

    Google Scholar 

  10. B. Simon A Comprehensive Course in Analysis, Part 4: Operator Theory, American Mathematical Society, Providence, RI, 2015.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Simon, B. (2019). Some Generalities. In: Loewner's Theorem on Monotone Matrix Functions. Grundlehren der mathematischen Wissenschaften, vol 354. Springer, Cham. https://doi.org/10.1007/978-3-030-22422-6_2

Download citation

Publish with us

Policies and ethics