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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
Escuela de Ingenieros Industriales
02071 Albacete, Spain
e-mail
• S. TroyanskiDepartamento de Matemáticas
Campus de Espinardo
30100 Murcia, Spain
and
Institute of Mathematics and Informatics
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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