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## Studia Mathematica

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## Operator ranges and quasicomplemented subspaces of Banach spaces

### Volume 246 / 2019

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

#### Abstract

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.

#### Authors

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

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