On the invertibility of elementary operators

Preprint English OPEN
Boudi, Nadia ; Bračič, Janko (2013)
  • Subject: Mathematics - Operator Algebras | Mathematics - Functional Analysis
    arxiv: Mathematics::General Topology | Mathematics::Functional Analysis

Let $\mathscr{X}$ be a complex Banach space and $\mathcal{L}(\mathscr{X})$ be the algebra of all bounded linear operators on $\mathscr{X}$. For a given elementary operator $\Phi$ of length $2$ on $\mathcal{L}(\mathscr{X})$, we determine necessary and sufficient conditions for the existence of a solution of the equation ${\rm X} \Phi=0$ in the algebra of all elementary operators on $\mathcal{L}(\mathscr{X})$. Our approach allows us to characterize some invertible elementary operators of length $2$ whose inverses are elementary operators.
  • References (10)

    Write Ψ = P2 i=1 MAi,Bi, where Ai ∈ L(Ψ) and Bi ∈ R(Ψ). Then there exist E1, . . . , En ∈ U and F1, . . . , Fn ∈ V such that

    [1] M. D. Atkinson, N. M. Stephens, Spaces of matrices of bounded rank, Quart. J. Math. Oxford (2) 29 (1978), 221-223.

    [2] P.-H. Lee, J.-S. Lin, R.-J. Wang, T.-L Wong, Commuting traces of multiadditive mappings, J. Alg. 193 (1997), 709-723.

    [3] M. Breˇsar and P. Sˇemrl, On locally linearly dependent operators and derivations, Trans. Amer. Math. Soc. 351 (1999), 1257-1275.

    [4] C. K. Fong, A. R. Sourour, On the operator identity P AkXBk ≡ 0, Can. J. Math., Vol. XXXI, No. 4, (1979), 845-857.

    [5] W.S. Martindale 3rd, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc. 142 (1969), 437-455.

    [6] V. Mu¨ller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Second Edition, Operator Theory: Advances and Applications, Vol. 139, Birkh¨auser (2007).

    [7] M. Rosenblum, On the operator equation BX − XA = Q, Duke Math. J. 23 (1956), 263-269.

    [8] J. Sylvester, On the dimension of spaces of linear transformations satisfying rank conditions, Lin. Alg. Appl. 78 (1986), 1-10.

    [9] Westwick, Spaces of matrices of fixed rank, Linear and Multilinear Algebra 20 (1987), 171-174.

  • Metrics
    No metrics available
Share - Bookmark