PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1994 | 30 | 1 | 337-361
Tytuł artykułu

Additive combinations of special operators

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
30
Numer
1
Strony
337-361
Opis fizyczny
Daty
wydano
1994
Twórcy
autor
  • Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 30050, Republic of China
Bibliografia
  • [1] J. Anderson, Commutators of compact operators, J. Reine Angew. Math. 291 (1977), 128-132.
  • [2] I. D. Berg, An extension of the Weyl-von Neumann theory to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365-371.
  • [3] I. D. Berg and B. Sims, Denseness of operators which attain their numerical radius, J. Austral. Math. Soc. 36A (1984), 130-133.
  • [4] R. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (1981), 513-517.
  • [5] A. Brown and C. Pearcy, Structure of commutators of operators, Ann. of Math. 82 (1965), 112-127.
  • [6] M.-D. Choi and P. Y. Wu, Convex combinations of projections, Linear Algebra Appl. 136 (1990), 25-42.
  • [7] M.-D. Choi and P. Y. Wu, Sums of projections, to appear.
  • [8] Ch. Davis, Separation of two linear subspaces, Acta Sci. Math. (Szeged) 19 (1958), 172-187.
  • [9] P. Fan and C. K. Fong, Which operators are the self-commutators of compact operators?, Proc. Amer. Math. Soc. 80 (1980), 58-60.
  • [10] P. Fan and C. K. Fong, Operators similar to zero diagonal operators, Proc. Roy. Irish Acad. 87A (1988), 147-153.
  • [11] P. A. Fillmore, Sums of operators with square zero, Acta Sci. Math. (Szeged) 28 (1967), 285-288.
  • [12] P. A. Fillmore, On similarity and the diagonal of a matrix, Amer. Math. Monthly 76 (1969), 167-169.
  • [13] P. A. Fillmore, On sums of projections, J. Funct. Anal. 4 (1969), 146-152.
  • [14] P. A. Fillmore and D. M. Topping, Sums of irreducible operators, Proc. Amer. Math. Soc. 20 (1969), 131-133.
  • [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.
  • [16] C. K. Fong and G. J. Murphy, Averages of projections, J. Operator Theory 13 (1985), 219-225.
  • [17] C. K. Fong and G. J. Murphy, Ideals and Lie ideals of operators, Acta Sci. Math. (Szeged) 51 (1987), 441-456.
  • [18] C. K. Fong and A. R. Sourour, Sums and products of quasi-nilpotent operators, Proc. Royal Soc. Edinburgh 99A (1984), 193-200.
  • [19] C. K. Fong and P. Y. Wu, Diagonal operators: dilation, sum and product, Acta Sci. Math. (Szeged) 57 (1993), 125-138.
  • [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.
  • [21] P. R. Halmos, Commutators of operators, Amer. J. Math. 74 (1952), 237-240.
  • [22] P. R. Halmos, Commutators of operators, II, ibid. 76 (1954), 191-198.
  • [23] P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381-389.
  • [24] P. R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer, New York, 1982.
  • [25] R. E. Hartwig and M. S. Putcha, When is a matrix a difference of two idempotents?, Linear Multilinear Algebra 26 (1990), 267-277.
  • [26] R. E. Hartwig and M. S. Putcha, When is a matrix a sum of idempotents?, ibid. 26 (1990), 279-286.
  • [27] R. B. Holmes, Best approximation by normal operators, J. Approx. Theory 12 (1974), 412-417.
  • [28] S. Izumino, Inequalities on operators with index zero, Math. Japon. 5 (1979), 565-572.
  • [29] R. V. Kadison and G. K. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249-266.
  • [30] N. J. Kalton, Trace-class operators and commutators, J. Funct. Anal. 86 (1989), 41-74.
  • [31] J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139-148.
  • [32] K. Matsumoto, Self-adjoint operators as a real span of 5 projections, Math. Japon. 29 (1984), 291-294.
  • [33] Y. Nakamura, Every Hermitian matrix is a linear combination of four projections, Linear Algebra Appl. 61 (1984), 133-139.
  • [34] K. Nishio, The structure of a real linear combination of two projections, ibid. 66 (1985), 169-176.
  • [35] C. L. Olsen, Unitary approximation, J. Funct. Anal. 85 (1989), 392-419.
  • [36] C. L. Olsen and G. K. Pedersen, Convex combinations of unitary operators in von Neumann algebras, ibid. 66 (1986), 365-380.
  • [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.
  • [38] C. Pearcy and D. M. Topping, Sums of small numbers of idempotents, Michigan Math. J. 14 (1967), 453-465.
  • [39] H. Radjavi, Structure of A*A-AA*, J. Math. Mech. 16 (1966), 19-26.
  • [40] H. Radjavi, Every operator is the sum of two irreducible ones, Proc. Amer. Math. Soc. 21 (1969), 251-252.
  • [41] F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar, New York, 1955.
  • [42] M. Rοrdam, Advances in the theory of unitary rank and regular approximation, Ann. of Math. 128 (1988), 153-172.
  • [43] J. G. Stampfli, Sums of projections, Duke Math. J. 31 (1964), 455-461.
  • [44] B.-S. Tam, A simple proof of the Goldberg-Straus theorem on numerical radii, Glasgow Math. J. 28 (1986), 139-141.
  • [45] D. M. Topping, On linear combinations of special operators, J. Algebra 10 (1968), 516-521.
  • [46] J.-H. Wang, Decomposition of operators into quadratic types, Ph.D. dissertation, National Chiao Tung Univ., Hsinchu, Taiwan, 1991.
  • [47] J.-H. Wang, The length problem for idempotent sum, Linear Algebra Appl., to appear.
  • [48] J.-H. Wang, Linear combination of idempotents, to appear.
  • [49] J.-H. Wang and P. Y. Wu, Sums of square-zero operators, Studia Math. 99 (1991), 115-127.
  • [50] J.-H. Wang and P. Y. Wu, Difference and similarity models of two idempotent operators, Linear Algebra Appl., to appear.
  • [51] P. Y. Wu, Sums of idempotent matrices, ibid. 142 (1990), 43-54.
  • [52] P. Y. Wu, Sums and products of cyclic operators, Proc. Amer. Math. Soc., to appear.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv30z1p337bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.