# On isomorphisms of some Köthe function F-spaces

Open Mathematics (2011)

• Volume: 9, Issue: 6, page 1267-1275
• ISSN: 2391-5455

top

## Abstract

top
We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (ΩX; ΣX, µX) and (ΩY; ΣY; µY), respectively, with absolute continuous norms are isomorphic and have the property $\underset{\mu \left(A\right)\to 0}{lim}∥\mu {\left(A\right)}^{-1}{1}_{A}∥=0$ (for µ = µX and µ = µY, respectively) then the measure spaces (ΩX; ΣX; µX) and (ΩY; ΣY; µY) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L p(µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.

## How to cite

top

Violetta Kholomenyuk, Volodymyr Mykhaylyuk, and Mikhail Popov. "On isomorphisms of some Köthe function F-spaces." Open Mathematics 9.6 (2011): 1267-1275. <http://eudml.org/doc/269237>.

@article{ViolettaKholomenyuk2011,
abstract = {We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (ΩX; ΣX, µX) and (ΩY; ΣY; µY), respectively, with absolute continuous norms are isomorphic and have the property $\mathop \{\lim \}\limits _\{\mu (A) \rightarrow 0\} \left\Vert \{\mu (A)^\{ - 1\} 1_A \} \right\Vert = 0$ (for µ = µX and µ = µY, respectively) then the measure spaces (ΩX; ΣX; µX) and (ΩY; ΣY; µY) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L p(µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.},
author = {Violetta Kholomenyuk, Volodymyr Mykhaylyuk, Mikhail Popov},
journal = {Open Mathematics},
keywords = {Köthe function F-spaces; Not locally convex spaces; Narrow operator; Separable quotient problem; Köthe function -spaces; non-locally convex spaces; narrow operator; separable quotient problem; Maharam set},
language = {eng},
number = {6},
pages = {1267-1275},
title = {On isomorphisms of some Köthe function F-spaces},
url = {http://eudml.org/doc/269237},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Violetta Kholomenyuk
AU - Volodymyr Mykhaylyuk
AU - Mikhail Popov
TI - On isomorphisms of some Köthe function F-spaces
JO - Open Mathematics
PY - 2011
VL - 9
IS - 6
SP - 1267
EP - 1275
AB - We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (ΩX; ΣX, µX) and (ΩY; ΣY; µY), respectively, with absolute continuous norms are isomorphic and have the property $\mathop {\lim }\limits _{\mu (A) \rightarrow 0} \left\Vert {\mu (A)^{ - 1} 1_A } \right\Vert = 0$ (for µ = µX and µ = µY, respectively) then the measure spaces (ΩX; ΣX; µX) and (ΩY; ΣY; µY) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L p(µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.
LA - eng
KW - Köthe function F-spaces; Not locally convex spaces; Narrow operator; Separable quotient problem; Köthe function -spaces; non-locally convex spaces; narrow operator; separable quotient problem; Maharam set
UR - http://eudml.org/doc/269237
ER -

## References

top
1. [1] Lacey H.E., The Isometric Theory of Classical Banach Spaces, Grundlehren Math. Wiss., 208, Springer, Berlin-Heidelberg-New York, 1974 Zbl0285.46024
2. [2] Lindenstrauss J., Some open problems in Banach space theory, Séminaire Choquet, Initiation à l’Analyse, 1975–76, 15, #18
3. [3] Maharam D., On homogeneous measure algebras, Proc. Natl. Acad. Sci. USA, 1942, 28, 108–111 http://dx.doi.org/10.1073/pnas.28.3.108 Zbl0063.03723
4. [4] Plichko A.M., Popov M.M., Symmetric Function Spaces on Atomless Probability Spaces, Dissertationes Math. (Rozprawy Mat.), 306, Polish Academy of Sciences, Warsaw, 1990
5. [5] Popov M.M., On codimension of subspaces of L p(µ) for p < 1, Funktsional. Anal. i Prilozhen., 1984, 18(2), 94–95 (in Russian) http://dx.doi.org/10.1007/BF01077844
6. [6] Popov M.M., An isomorphic classification of the spaces L p for 0 < p < 1, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen., 1987, 47, 77–85 (in Russian)
7. [7] Rolewicz S., Metric Linear Spaces, 2nd ed., Math. Appl. (East European Ser.), 20, PWN, Warszawa, 1985
8. [8] Śliwa W., The separable quotient problem for symmetric function spaces, Bull. Polish Acad. Sci. Math., 2000, 48(1), 13–27 Zbl0984.46017
9. [9] Śliwa W., The separable quotient problem for (LF)tv-spaces, J. Korean Math. Soc., 2009, 46(6), 1233–1242 http://dx.doi.org/10.4134/JKMS.2009.46.6.1233 Zbl1190.46004

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.