Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

On Self-commutator Approximants

  • Received : 2007.10.23
  • Accepted : 2008.03.17
  • Published : 2009.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let B(X) denote the algebra of operators on a complex Banach space X, H(X) = {h ${\in}$ B(X) : h is hermitian}, and J(X) = {x ${\in}$ B(X) : x = $x_1$ + $ix_2$, $x_1$ and $x_2$ ${\in}$ H(X)}. Let ${\delta}_a$ ${\in}$ B(B(X)) denote the derivation ${\delta}_a$ = ax - xa. If J(X) is an algebra and ${\delta}_a^{-1}(0){\subseteq}{\delta}_{a^*}^{-1}(0)$ for some $a{\in}J(X)$, then ${\parallel}a{\parallel}{\leq}{\parallel}a-(x^*x-xx^*){\parallel}$ for all $x{\in}J(X){\cap}{\delta}_a^{-1}(0)$. The cases J(X) = B(H), the algebra of operators on a complex Hilbert space, and J(X) = $C_p$, the von Neumann-Schatten p-class, are considered.

Keywords

References

  1. T. A. Abatzoglou, Norm derivatives on spaces of operators, Math. Ann., 239(1979), 129-135. https://doi.org/10.1007/BF01420370
  2. J. Anderson and C. Foias, Properties which normal operators share with normal derivations and related operators, Pac. J. Math., 61(1975), 313-325. https://doi.org/10.2140/pjm.1975.61.313
  3. F. F. Bonsall and J. Duncan, Numerical ranges I, Lond. Math. Soc. Lecture Notes Series, 2(1971).
  4. F. F. Bonsall and J. Duncan, Numerical ranges II, Lond. Math. Soc. Lecture Notes Series, 10(1973).
  5. F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergebnisse der Math. und ihrer Grenzgebiete, Band 80, Springer-Verlag, (1973).
  6. B. P. Duggal, Range-kernel orthogonality of the elementary operator $X\rightarrow\sum_{i=1}^{n}A_iXB_i-X$, Lin. Alg. Appl., 337(2001), 79-86. https://doi.org/10.1016/S0024-3795(01)00339-1
  7. B. P. Duggal, Subspace gaps and range-kernel orthogonality of an elementary operator, Lin. Alg. Appl., 383(2004), 93-106. https://doi.org/10.1016/j.laa.2003.11.006
  8. B. P. Duggal, A perturbed elementary operator and range-kernel orthogonality, Proc. Amer. Math. Soc., 134(2006), 1727-1734. https://doi.org/10.1090/S0002-9939-05-08337-1
  9. N. Dunford, J. T. Schwartz, Linear Operators, Part I, Interscience, New York, (1964).
  10. R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61(1947), 265-292. https://doi.org/10.1090/S0002-9947-1947-0021241-4
  11. Dragoljub Keckic, Orthogonality of the range and the kernel of some elementary operators, Proc. Amer. Math. Soc., 128(2000), 3369-3377. https://doi.org/10.1090/S0002-9939-00-05890-1
  12. Yuan-Chuan Li and Sen-Yen Shaw, An absolute ergodic theorem and some inequalities for operators on Banach spaces, Proc. Amer. Math. Soc., 125(1997), 111-119. https://doi.org/10.1090/S0002-9939-97-03504-1
  13. P. J. Maher, Self-commutator approximants, Proc. Amer. Math. Soc., 134(2006), 157-165. https://doi.org/10.1090/S0002-9939-05-07871-8
  14. Aleksej Turnsek, Orthogonality in $C_p$ classes, Mh. Math., 132(2001), 349-354. https://doi.org/10.1007/s006050170039