Skip to main content
SpringerLink
Log in

Essentially Normal Operators

  • Chapter
A Glimpse at Hilbert Space Operators

Part of the book series: Operator Theory Advances and Applications ((OT,volume 207))

Abstract

This is a survey of essentially normal operators and related developments. There is an overview of Weyl—von Neumann theorems about expressing normal operators as diagonal plus compact operators. Then we consider the Brown—Douglas—Fillmore theorem classifying essentially normal operators. Finally we discuss almost commuting matrices, and how they were used to obtain two other proofs of the BDF theorem.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Anderson, A G*-algebra A for which Ext(A) is not a group, Ann. Math. 107 (1978), 455–458.

    Article  Google Scholar 

  2. W. Arveson, A note on essentially normal operators, Proc. Royal Irish Acad. 74 (1974), 143–146.

    MATH  MathSciNet  Google Scholar 

  3. W. Arveson, Notes in extensions of C*-algebras, Duke Math. J. 44 (1977), 329–355.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Atiyah, K-theory, W.A. Benjamin Inc., New York, 1967.

    Google Scholar 

  5. H. Bercovici and D. Voiculescu, The analogue of Kuroda’s theorem for n-tuples, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988), 57–60, Oper. Theory Adv. Appl. 41, Birkhäuser, Basel, 1989.

    Google Scholar 

  6. I.D. Berg, An extension of the Weyl-von Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365–371.

    Article  MATH  MathSciNet  Google Scholar 

  7. I.D. Berg and K. Davidson, Almost commuting matrices and a quantitative version of the Brown-Douglas-Fillmore theorem, Acta Math. 166 (1991), 121–161.

    Article  MATH  MathSciNet  Google Scholar 

  8. Bratteli, O., Inductive limits of finite-dimensional C*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234.

    MATH  MathSciNet  Google Scholar 

  9. L. Brown, R. Douglas and P. Fillmore, Unitary equivalence modulo the compact operators and extensions of C*-algebras, Proc. conference on Operator theory, Halifax, NS, Lect. Notes Math. 3445, Springer Verlag, Berlin, 1973.

    Google Scholar 

  10. L. Brown, R. Douglas and P. Fillmore, Extensions of C*-algebras and K-homology, Ann. Math. 105 (1977), 265–324.

    Article  MathSciNet  Google Scholar 

  11. M.D. Choi, Almost commuting matrices need not be nearly commuting, Proc. Amer. Math. Soc. 102 (1988), 529–533.

    Article  MATH  MathSciNet  Google Scholar 

  12. M.D. Choi and E. Effros, The completely positive lifting problem for C*-algebras, Ann. Math. 104 (1976), 585–609.

    Article  MathSciNet  Google Scholar 

  13. K. Davidson, Almost commuting Hermitian matrices, Math. Scand. 56 (1985), 222–240.

    MATH  MathSciNet  Google Scholar 

  14. K. Davidson, Normal operators are diagonal plus Hilbert-Schmidt, J. Operator Theory 20 (1988) 241–250.

    MATH  MathSciNet  Google Scholar 

  15. K. Davidson, C*-Algebras by Example, Fields Institute Monograph Series 6, American Mathematical Society, Providence, RI, 1996.

    Google Scholar 

  16. R. Douglas, C*-algebra extensions and K-homology, Annals Math. Studies 95, Princeton University Press, Princeton, N.J., 1980.

    Google Scholar 

  17. E. Effros, D. Handelman and C. Shen, Dimension groups and their affine transformations, Amer. J. Math. 102 (1980), 385–402.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. Elliott, On the classification of inductive limits of sequences of semi-simple finite-dimensional algebras, J. Algebra 38 (1976), 29–44.

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Elliott, On totally ordered groups and Ko, Proc. Ring Theory conf., Waterloo, D. Handelman and J. Lawrence (eds.), Lect. Notes Math. 734 (1978), 1–49, Springer-Verlag, New York, 1978.

    Google Scholar 

  20. R. Exel and T. Loring, Almost commuting unitary matrices, Proc. Amer. Math. Soc. 106 (1989), 913–915.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Exel and T. Loring, Invariants of almost commuting unitaries, J. Funct. Anal. 95 (1991), 364–376.

    Article  MATH  MathSciNet  Google Scholar 

  22. P. Friis and M. Rordam, Almost commuting self-adjoint matrices—a short proof of Huaxin’s theorem, J. reine angew. Math. 479 (1996), 121–131.

    Article  MATH  MathSciNet  Google Scholar 

  23. P. Friis and M. Rordam, Approximation with normal operators with finite spectrum, and an elementary proof of the Brown—Douglas—Fillmore theorem, Pacific J. Math. 199 (2001), 347–366.

    Article  MATH  MathSciNet  Google Scholar 

  24. U. Haagerup and S. Thorbjörnsen, A new application of random matrices: Ext\( (C_{red}^* (\mathbb{F}_2 )) \) is not a group, Ann. Math. 162 (2005), 711–775.

    Article  MATH  Google Scholar 

  25. D. Hadwin, An operator-valued spectrum, Indiana Univ. Math. J. 26 (1977), 329–340.

    Article  MATH  MathSciNet  Google Scholar 

  26. P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933.

    Article  MATH  MathSciNet  Google Scholar 

  27. P.R. Halmos, What does the spectral theorem say?, Amer. Math. Monthly 70 (1963), 241–247.

    Article  MATH  MathSciNet  Google Scholar 

  28. P.R. Halmos, Some unsolved problems of unknown depth about operators on Hilbert space, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), 67–76.

    Google Scholar 

  29. M. Hastings, Making almost commuting matrices commute, Coram. Math. Phys. 291 (2009), no. 2, 321–345.

    Article  MATH  MathSciNet  Google Scholar 

  30. G. Kasparov, The operator K-functor and extensions of C*-algebras (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 44, (1980), 571–636.

    MATH  MathSciNet  Google Scholar 

  31. T. Kato, Perturbation of continuous spectra by trace class operators, Proc. Japan Acad. 33 (1957), 260–264.

    Article  MATH  MathSciNet  Google Scholar 

  32. S. Kuroda, On a theorem of Weyl-von Neumann, Proc. Japan Acad. 34 (1958), 11–15.

    Article  MATH  MathSciNet  Google Scholar 

  33. H. Lin, Almost commuting Hermitian matrices and applications, Fields Institute Comm. 13 (1997), 193–233.

    Google Scholar 

  34. T. Loring, K-theory and asymptotically commuting matrices, Canad. J. Math. 40 (1988), 197–216.

    Article  MATH  MathSciNet  Google Scholar 

  35. W. Luxemburg and R. Taylor, Almost commuting matrices are near commuting matrices, Indag. Math. 32 (1970), 96–98.

    MathSciNet  Google Scholar 

  36. C. Pearcy and A. Shields, Almost commuting matrices, J. Funct. Anal. 33 (1979), 332–338.

    Article  MATH  MathSciNet  Google Scholar 

  37. M. Rosenblum, Perturbation of the continuous spectrum and unitary equivalence, Pacific J. Math. 7 (1957), 997–1010.

    MATH  MathSciNet  Google Scholar 

  38. P. Rosenthal, Research Problems: Are Almost Commuting Matrices Near Commuting Matrices?, Amer. Math. Monthly 76 (1969), 925–926.

    Article  MathSciNet  Google Scholar 

  39. W. Sikonia, The von Neumann converse of Weyl’s theorem, Indiana Univ. Math. J. 21 (1971/1972), 121–124.

    Article  MATH  MathSciNet  Google Scholar 

  40. D. Voiculescu, A non-commutative Weyl—von Neumann Theorem, Rev. Roum. Pures Appl. 21 (1976), 97–113.

    MATH  MathSciNet  Google Scholar 

  41. D. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, J. Operator Theory 2 (1979), 3–37.

    MATH  MathSciNet  Google Scholar 

  42. D. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, II, J. Operator Theory 5 (1981), 77–100.

    MATH  MathSciNet  Google Scholar 

  43. D. Vioculescu, Remarks on the singular extension in the C*-algebra of the Heisenberg group, J. Operator Theory 5 (1981), 147–170.

    MathSciNet  Google Scholar 

  44. D. Vioculescu, Asymptotically commuting finite rank unitaries without commuting approximants, Acta Sci. Math. (Szeged) 45 (1983), 429–431.

    MathSciNet  Google Scholar 

  45. J. von Neumann, Charakterisierung des Spektrums eines Integraloperators, Actualités Sci. Indust. 229, Hermann, Paris, 1935.

    Google Scholar 

  46. H. Weyl, Über beschränkte quadratische Formen deren Differenz vollstetig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pure Math. Dept., University of Waterloo, Waterloo, ON, N2L-3G1, Canada

    Kenneth R. Davidson

Authors
  1. Kenneth R. Davidson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, San Francisco State University, San Francisco, CA, 94132-3163, USA

    Sheldon Axler

  2. Department of Mathematics, University of Toronto, Toronto, ON, MS5 2E4, Canada

    Peter Rosenthal

  3. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA

    Donald Sarason

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Basel AG

About this chapter

Cite this chapter

Davidson, K.R. (2010). Essentially Normal Operators. In: Axler, S., Rosenthal, P., Sarason, D. (eds) A Glimpse at Hilbert Space Operators. Operator Theory Advances and Applications, vol 207. Springer, Basel. https://doi.org/10.1007/978-3-0346-0347-8_13

Download citation

Publish with us

Policies and ethics

Search

Navigation