Skip to main content
SpringerLink
Log in

Majorisation and the Carpenter’s Theorem

Integral Equations and Operator Theory volume 82, pages 33–49 (2015)Cite this article

Abstract

We discuss Kadison’s Carpenter’s Theorems in the context of their relation to majorisation, and we offer a new proof of his striking characterisation of the set of diagonals of orthogonal projections on Hilbert space.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Antezana, J., Massey, P., Ruiz, M., Stojanoff, D.: The Schur–Horn theorem for operators and frames with prescribed norms and frame operator. Ill. J. Math. 51(2), 537–560 (2007). (electronic)

  2. Argerami, M., Massey, P.: A Schur–Horn theorem in II1 factors. Indiana Univ. Math. J. 56(5), 2051–2059 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Argerami, M., Massey, P.: A contractive version of a Schur–Horn theorem in II1 factors. J. Math. Anal. Appl. 337(1), 231–238 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Argerami, M., Massey, P.: The local form of doubly stochastic maps and joint majorization in II1 factors. Integral Equ. Oper. Theory 61(1), 1–19 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Argerami, M., Massey, P.: Towards the Carpenter’s theorem. Proc. Am. Math. Soc. 137(11), 3679–3687 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Argerami, M., Massey, P.: Schur–Horn theorems in II ∞ factors. Pac. J. Math. 261(2), 283–310 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Arveson, W.: Diagonals of normal operators with finite spectrum. Proc. Natl. Acad. Sci. USA 104(4), 1152–1158 (2007). (electronic)

  8. Arveson, W., Kadison, R.V.: Diagonals of self-adjoint operators. In: Operator Theory, Operator Algebras, and Applications, volume 414 of Contemp. Math., pp. 247–263. American Mathematical Society, Providence (2006)

  9. Avron, J., Seiler, R., Simon, B.: The index of a pair of projections. J. Funct. Anal. 120(1), 220–237 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bercovici, H., Collins, B., Dykema, K., Li, W.S., Timotin, D.: Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor. J. Funct. Anal. 258(5), 1579–1627 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bercovici, H., Li, W.S.: Eigenvalue inequalities in an embeddable factor. Proc. Am. Math. Soc. 134(1), 75–80 (2006). (electronic)

  12. Bercovici, H., Li, W.S., Timotin, D.: The Horn conjecture for sums of compact selfadjoint operators. Am. J. Math. 131(6), 1543–1567 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bhatia, R.: Matrix analysis, volume 169 of Graduate Texts in Mathematics. Springer, New York (1997)

    Google Scholar 

  14. Bownik, M., Jasper, J.: The Schur–Horn theorem for operators with finite spectrum. Trans. Am. Math. Soc. (in press)

  15. Bownik, M., Jasper, J.: Constructive proof of Carpenter’s Theorem. Can. Math. Bull. 57(3), 463–476 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  16. Carlen, E.A., Lieb, E.H.: Short proofs of theorems of Mirsky and Horn on diagonals and eigenvalues of matrices. Electr. J. Linear Algebra 18, 438–441 (2009)

    MATH  MathSciNet  Google Scholar 

  17. Chan, N.N., Li, K.H.: Diagonal elements and eigenvalues of a real symmetric matrix. J. Math. Anal. Appl. 91(2), 562–566 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  18. Collins, B., Dykema, K.: A linearization of Connes’ embedding problem. N. Y. J. Math. 14, 617–641 (2008)

    MATH  MathSciNet  Google Scholar 

  19. Collins, B., Dykema, K.: On a reduction procedure for Horn inequalities in finite von Neumann algebras. Oper. Matrices 3(1), 1–40 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Collins, B., Dykema, K.J.: A nonconvex asymptotic quantum Horn body. N. Y. J. Math. 17, 437–444 (2011)

    MATH  MathSciNet  Google Scholar 

  21. Dhillon, I.S., Heath, R.W., Jr., Sustik, M.A., Tropp, J.A.: Generalized finite algorithms for constructing Hermitian matrices with prescribed diagonal and spectrum. SIAM J. Matrix Anal. Appl. 27(1), 61–71 (2005), (electronic)

  22. Dykema, K.J., Fang, J., Hadwin, D.W., Smith, R.R.: The carpenter and Schur–Horn problems for masas in finite factors. Ill. J. Math. 56(4), 1313–1329 (2012)

    MATH  MathSciNet  Google Scholar 

  23. Effros, E.G.: Why the circle is connected: an introduction to quantized topology. Math. Intell. 11(1), 27–34 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hiai, F.: Majorization and stochastic maps in von Neumann algebras. J. Math. Anal. Appl. 127(1), 18–48 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hiai, F.: Spectral majorization between normal operators in von Neumann algebras. In: Operator Algebras and Operator Theory (Craiova, 1989), volume 271 of Pitman Res. Notes Math. Ser., pp. 78–115. Longman Sci. Tech., Harlow (1992)

  26. Hiai, F., Nakamura, Y.: Majorizations for generalized s-numbers in semifinite von Neumann algebras. Math. Z. 195(1), 17–27 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  27. Horn, A.: Doubly stochastic matrices and the diagonal of a rotation matrix. Am. J. Math. 76, 620–630 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  28. Jasper, J.: The Schur–Horn theorem for operators with three point spectrum. J. Funct. Anal. 265(8), 1494–1521 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kadison, R.V.: The Pythagorean theorem. I. The finite case. Proc. Natl. Acad. Sci. USA 99(7), 4178–4184 (2002). (electronic)

  30. Kadison, R.V.: The Pythagorean theorem. II. The infinite discrete case. Proc. Natl. Acad. Sci. USA 99(8), 5217–5222 (2002). (electronic)

  31. Kaftal, V., Weiss, G.: A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem. In: Hot Topics in Operator Theory, volume 9 of Theta Ser. Adv. Math., pp. 101–135. Theta, Bucharest (2008)

  32. Kaftal, V., Weiss, G.: An infinite dimensional Schur–Horn theorem and majorization theory. J. Funct. Anal. 259(12), 3115–3162 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kamei, E.: Majorization in finite factors. Math. Japon. 28(4), 495–499 (1983)

    MATH  MathSciNet  Google Scholar 

  34. Klyachko, A.A.: Stable bundles, representation theory and Hermitian operators. Selecta Math. (N.S.) 4(3), 419–445 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  35. Knutson, A., Tao, T.: The honeycomb model of GL n (C) tensor products. I. Proof of the saturation conjecture. J. Am. Math. Soc. 12(4), 1055–1090 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  36. Knutson, A., Tao, T., Woodward C.: The honeycomb model of \({{\rm GL}_n(\mathbb{C})}\) tensor products. II. Puzzles determine facets of the Littlewood–Richardson cone. J. Am. Math. Soc. 17(1), 19–48 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Leite, R.S., Richa, T.R.W., Tomei, C.: Geometric proofs of some theorems of Schur–Horn type. Linear Algebra Appl. 286(1–3), 149–173 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  38. Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: theory of majorization and its applications. Springer Series in Statistics, 2nd edn. Springer, New York (2011)

    Book  Google Scholar 

  39. Mirsky, L.: Matrices with prescribed characteristic roots and diagonal elements. J. Lond. Math. Soc. 33, 14–21 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  40. Neumann, A.: An infinite-dimensional version of the Schur–Horn convexity theorem. J. Funct. Anal. 161(2), 418–451 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ravichandran, M.: The Schur–Horn Theorem in von Neumann algebras. preprint. arXiv:1209.0909

  42. Schur, I.: Über eine Klasse von Mittelbildungen mit Anwendung auf die Determinantentheorie. S.-Ber. Berliner math. Ges. 2, 9–20 (1923)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2, Canada

    Martín Argerami

Authors
  1. Martín Argerami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martín Argerami.

Additional information

This work was supported in part by the NSERC Discovery Grant program.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Argerami, M. Majorisation and the Carpenter’s Theorem. Integr. Equ. Oper. Theory 82, 33–49 (2015). https://doi.org/10.1007/s00020-014-2180-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-014-2180-7

Mathematics Subject Classification

Keywords

search

Navigation