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Operator ranges and quasicomplemented subspaces of Banach spaces

Volume 246 / 2019

V. P. Fonf, S. Lajara, S. Troyanski, C. Zanco Studia Mathematica 246 (2019), 203-216 MSC: Primary 47A05, 46B25; Secondary 46C07. DOI: 10.4064/sm180110-31-1 Published online: 7 September 2018


Given a bounded linear operator $T$ from a separable infinite-dimensional Banach space $E$ into a Banach space $Y$, an operator range $R$ in $E$ and a closed subspace $L\subset E$ such that $L\cap R=\{0\}$ and $\mathop {\rm codim}\nolimits (L+R)=\infty $, we provide a condition to ensure the existence of an infinite-dimensional closed subspace $L_1\subset E$, containing $L$ as an infinite-codimensional subspace, such that $L_1\cap R= \{0\}$ and $\mathop {\rm cl}\nolimits T(L_1) = \mathop {\rm cl}\nolimits T(E)$. This condition enables us to build closed subspaces of $E$ with a special behaviour with respect to an operator range in $E$. In particular, we show that if $R$ is an operator range in a Hilbert space, then for every closed subspace $H_0$ in $H$ satisfying $H_0\cap R = \{0\}$ and $\mathop {\rm codim}\nolimits (H_0+R)=\infty $ there exists an orthogonal decomposition $H=V\oplus _{\perp } W$ such that $V$ contains $H_0$ as an infinite-codimensional subspace and ${V \cap R = W \cap R =\{0\}}$. We also obtain generalizations of some classical results on quasicomplemented subspaces of Banach spaces.


  • V. P. FonfDepartment of Mathematics
    Ben-Gurion University of the Negev
    84105 Beer-Sheva, Israel
  • S. LajaraDepartamento de Matemáticas
    Universidad de Castilla-La Mancha
    Escuela de Ingenieros Industriales
    02071 Albacete, Spain
  • S. TroyanskiDepartamento de Matemáticas
    Universidad de Murcia
    Campus de Espinardo
    30100 Murcia, Spain
    Institute of Mathematics and Informatics
    Bulgarian Academy of Sciences
    bl. 8, Acad. G. Bonchev str.
    1113 Sofia, Bulgaria
  • C. ZancoDipartimento di Matematica
    Università degli Studi di Milano
    Via C. Saldini 50
    20133 Milano (MI), Italy

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