Additive combinations of special operators

Pei Wu

Banach Center Publications (1994)

  • Volume: 30, Issue: 1, page 337-361
  • ISSN: 0137-6934

Abstract

top
This is a survey paper on additive combinations of certain special-type operators on a Hilbert space. We consider (finite) linear combinations, sums, convex combinations and/or averages of operators from the classes of diagonal operators, unitary operators, isometries, projections, symmetries, idempotents, square-zero operators, nilpotent operators, quasinilpotent operators, involutions, commutators, self-commutators, norm-attaining operators, numerical-radius-attaining operators, irreducible operators and cyclic operators. In each case, we are mainly concerned with the characterization of such combinations and the minimal number of the special operators required in them. We will omit the proofs of most of the results here but give some indication or brief sketch of the ideas behind and point out the remaining open problems.

How to cite

top

Wu, Pei. "Additive combinations of special operators." Banach Center Publications 30.1 (1994): 337-361. <http://eudml.org/doc/262750>.

@article{Wu1994,
abstract = {This is a survey paper on additive combinations of certain special-type operators on a Hilbert space. We consider (finite) linear combinations, sums, convex combinations and/or averages of operators from the classes of diagonal operators, unitary operators, isometries, projections, symmetries, idempotents, square-zero operators, nilpotent operators, quasinilpotent operators, involutions, commutators, self-commutators, norm-attaining operators, numerical-radius-attaining operators, irreducible operators and cyclic operators. In each case, we are mainly concerned with the characterization of such combinations and the minimal number of the special operators required in them. We will omit the proofs of most of the results here but give some indication or brief sketch of the ideas behind and point out the remaining open problems.},
author = {Wu, Pei},
journal = {Banach Center Publications},
keywords = {additive combinations; sums; convex combinations; averages of operators; diagonal operators; unitary operators; isometries; projections; symmetries; idempotents; square-zero operators; nilpotent operators; quasinilpotent operators; involutions; commutators; self-commutators; norm-attaining operators; numerical-radius-attaining operators; irreducible operators; cyclic operators},
language = {eng},
number = {1},
pages = {337-361},
title = {Additive combinations of special operators},
url = {http://eudml.org/doc/262750},
volume = {30},
year = {1994},
}

TY - JOUR
AU - Wu, Pei
TI - Additive combinations of special operators
JO - Banach Center Publications
PY - 1994
VL - 30
IS - 1
SP - 337
EP - 361
AB - This is a survey paper on additive combinations of certain special-type operators on a Hilbert space. We consider (finite) linear combinations, sums, convex combinations and/or averages of operators from the classes of diagonal operators, unitary operators, isometries, projections, symmetries, idempotents, square-zero operators, nilpotent operators, quasinilpotent operators, involutions, commutators, self-commutators, norm-attaining operators, numerical-radius-attaining operators, irreducible operators and cyclic operators. In each case, we are mainly concerned with the characterization of such combinations and the minimal number of the special operators required in them. We will omit the proofs of most of the results here but give some indication or brief sketch of the ideas behind and point out the remaining open problems.
LA - eng
KW - additive combinations; sums; convex combinations; averages of operators; diagonal operators; unitary operators; isometries; projections; symmetries; idempotents; square-zero operators; nilpotent operators; quasinilpotent operators; involutions; commutators; self-commutators; norm-attaining operators; numerical-radius-attaining operators; irreducible operators; cyclic operators
UR - http://eudml.org/doc/262750
ER -

References

top
  1. [1] J. Anderson, Commutators of compact operators, J. Reine Angew. Math. 291 (1977), 128-132. Zbl0341.47025
  2. [2] I. D. Berg, An extension of the Weyl-von Neumann theory to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365-371. Zbl0212.15903
  3. [3] I. D. Berg and B. Sims, Denseness of operators which attain their numerical radius, J. Austral. Math. Soc. 36A (1984), 130-133. Zbl0561.47004
  4. [4] R. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), 513-517. Zbl0483.47015
  5. [5] A. Brown and C. Pearcy, Structure of commutators of operators, Ann. of Math. 82 (1965), 112-127. Zbl0131.12302
  6. [6] M.-D. Choi and P. Y. Wu, Convex combinations of projections, Linear Algebra Appl. 136 (1990), 25-42. Zbl0721.15007
  7. [7] M.-D. Choi and P. Y. Wu, Sums of projections, to appear. 
  8. [8] Ch. Davis, Separation of two linear subspaces, Acta Sci. Math. (Szeged) 19 (1958), 172-187. Zbl0090.32603
  9. [9] P. Fan and C. K. Fong, Which operators are the self-commutators of compact operators?, Proc. Amer. Math. Soc. 80 (1980), 58-60. Zbl0442.47024
  10. [10] P. Fan and C. K. Fong, Operators similar to zero diagonal operators, Proc. Roy. Irish Acad. 87A (1988), 147-153. Zbl0651.47013
  11. [11] P. A. Fillmore, Sums of operators with square zero, Acta Sci. Math. (Szeged) 28 (1967), 285-288. Zbl0159.19302
  12. [12] P. A. Fillmore, On similarity and the diagonal of a matrix, Amer. Math. Monthly 76 (1969), 167-169. Zbl0175.02403
  13. [13] P. A. Fillmore, On sums of projections, J. Funct. Anal. 4 (1969), 146-152. Zbl0176.43404
  14. [14] P. A. Fillmore and D. M. Topping, Sums of irreducible operators, Proc. Amer. Math. Soc. 20 (1969), 131-133. Zbl0165.47702
  15. [15] C. K. Fong, C. R. Miers and A. R. Sourour, Lie and Jordan ideals of operators on Hilbert space, ibid. 84 (1982), 516-520. Zbl0509.47035
  16. [16] C. K. Fong and G. J. Murphy, Averages of projections, J. Operator Theory 13 (1985), 219-225. Zbl0614.47011
  17. [17] C. K. Fong and G. J. Murphy, Ideals and Lie ideals of operators, Acta Sci. Math. (Szeged) 51 (1987), 441-456. Zbl0659.47040
  18. [18] C. K. Fong and A. R. Sourour, Sums and products of quasi-nilpotent operators, Proc. Royal Soc. Edinburgh 99A (1984), 193-200. Zbl0598.47016
  19. [19] C. K. Fong and P. Y. Wu, Diagonal operators: dilation, sum and product, Acta Sci. Math. (Szeged) 57 (1993), 125-138. Zbl0819.47047
  20. [20] U. Haagerup, On convex combinations of unitary operators in C*-algebras, in: Mappings of Operator Algebras, H. Araki and R. V. Kadison (eds.), Birkhäuser, Boston, 1991, 1-13. Zbl0772.47020
  21. [21] P. R. Halmos, Commutators of operators, Amer. J. Math. 74 (1952), 237-240. Zbl0046.12302
  22. [22] P. R. Halmos, Commutators of operators, II, ibid. 76 (1954), 191-198. Zbl0055.10705
  23. [23] P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381-389. Zbl0187.05503
  24. [24] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982. 
  25. [25] R. E. Hartwig and M. S. Putcha, When is a matrix a difference of two idempotents?, Linear Multilinear Algebra 26 (1990), 267-277. Zbl0696.15010
  26. [26] R. E. Hartwig and M. S. Putcha, When is a matrix a sum of idempotents?, ibid. 26 (1990), 279-286. Zbl0696.15011
  27. [27] R. B. Holmes, Best approximation by normal operators, J. Approx. Theory 12 (1974), 412-417. Zbl0291.41025
  28. [28] S. Izumino, Inequalities on operators with index zero, Math. Japon. 5 (1979), 565-572. Zbl0411.47006
  29. [29] R. V. Kadison and G. K. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249-266. Zbl0573.46034
  30. [30] N. J. Kalton, Trace-class operators and commutators, J. Funct. Anal. 86 (1989), 41-74. Zbl0684.47017
  31. [31] J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139-148. Zbl0127.06704
  32. [32] K. Matsumoto, Self-adjoint operators as a real span of 5 projections, Math. Japon. 29 (1984), 291-294. Zbl0549.47007
  33. [33] Y. Nakamura, Every Hermitian matrix is a linear combination of four projections, Linear Algebra Appl. 61 (1984), 133-139. Zbl0547.15012
  34. [34] K. Nishio, The structure of a real linear combination of two projections, ibid. 66 (1985), 169-176. Zbl0584.47014
  35. [35] C. L. Olsen, Unitary approximation, J. Funct. Anal. 85 (1989), 392-419. Zbl0684.46049
  36. [36] C. L. Olsen and G. K. Pedersen, Convex combinations of unitary operators in von Neumann algebras, ibid. 66 (1986), 365-380. Zbl0597.46061
  37. [37] A. Paszkiewicz, Any self-adjoint operator is a finite linear combination of projections, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 227-245. Zbl0495.47003
  38. [38] C. Pearcy and D. M. Topping, Sums of small numbers of idempotents, Michigan Math. J. 14 (1967), 453-465. Zbl0156.38102
  39. [39] H. Radjavi, Structure of A*A-AA*, J. Math. Mech. 16 (1966), 19-26. 
  40. [40] H. Radjavi, Every operator is the sum of two irreducible ones, Proc. Amer. Math. Soc. 21 (1969), 251-252. Zbl0182.45903
  41. [41] F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar, New York, 1955. Zbl0070.10902
  42. [42] M. Rοrdam, Advances in the theory of unitary rank and regular approximation, Ann. of Math. 128 (1988), 153-172. Zbl0659.46052
  43. [43] J. G. Stampfli, Sums of projections, Duke Math. J. 31 (1964), 455-461. Zbl0127.06702
  44. [44] B.-S. Tam, A simple proof of the Goldberg-Straus theorem on numerical radii, Glasgow Math. J. 28 (1986), 139-141. Zbl0605.15019
  45. [45] D. M. Topping, On linear combinations of special operators, J. Algebra 10 (1968), 516-521. Zbl0185.38801
  46. [46] J.-H. Wang, Decomposition of operators into quadratic types, Ph.D. dissertation, National Chiao Tung Univ., Hsinchu, Taiwan, 1991. 
  47. [47] J.-H. Wang, The length problem for idempotent sum, Linear Algebra Appl., to appear. 
  48. [48] J.-H. Wang, Linear combination of idempotents, to appear. 
  49. [49] J.-H. Wang and P. Y. Wu, Sums of square-zero operators, Studia Math. 99 (1991), 115-127. Zbl0745.47006
  50. [50] J.-H. Wang and P. Y. Wu, Difference and similarity models of two idempotent operators, Linear Algebra Appl., to appear. Zbl0803.47021
  51. [51] P. Y. Wu, Sums of idempotent matrices, ibid. 142 (1990), 43-54. 
  52. [52] P. Y. Wu, Sums and products of cyclic operators, Proc. Amer. Math. Soc., to appear. Zbl0816.47015

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.