Wide operators on Kothe function spaces

M. M. Popov; D. O. Vatsek

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.

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.

	Wide operators on Kothe function spaces
Author	M. M. Popov, D. O. Vatsek 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	104-112
Volume	42
Issue	1
Year	2014
Journal	Matematychni Studii
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