Wide operators on Kothe function spaces 

Author 
misham.popov@gmail.com, diana1232208@ukr.net
Chernivtsi National University

Abstract 
We study operators defined on Kothe function spaces which are uniformly bounded from
below at some sign functions supported on any fixed measurable set. Precise definition is a kind
of opposite to the definition of narrow operators, so many questions concerning the relationship
between narrow and wide operators naturally arise. The main questions are to describe how
"large" has to be a wide operator, and how "small" has to be an operator which is "nowhere"
wide. Some easy to formulate problems on wide operators turn out to be more involved than
their analogues for narrow operators, and most of the results have restrictive assumptions on
the domain spaces. We pose some open problems.

Keywords 
vector lattice; orthogonally additive operator; disjointness preserving operator

Reference 
1. Albiac F., Kalton N., Topics in Banach space theory, Graduate texts in mathematics, V.233, Springer,
New York, 2006.
2. Carothers N. L., A short course of Banach space theory, Cambridge Univ. Press, 2004. 3. Lindenstrauss J., Tzafriri L., Classical Banach spaces, V.1, Sequence spaces, Springer–Verlag, Berlin– Heidelberg–New York, 1977. 4. Lindenstrauss J., Tzafriri L., Classical Banach spaces, V.2, Function spaces, Springer–Verlag, Berlin– Heidelberg–New York, 1979. 5. Mykhaylyuk V., Popov M., Randrianantoanina B., Schechtman G., Narrow and $\ell_2$strictly singular operators on $L_p$, Isr. J. Math. (to appear) 6. Popov M., Randrianantoanina B., Narrow operators on function spaces and vector lattices, De Gruyter Studies in Mathematics, V.45, De Gruyter, Berlin–Boston, 2013. 
Pages 
104112

Volume 
42

Issue 
1

Year 
2014

Journal 
Matematychni Studii

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