On isomorphisms of some Köthe function F-spaces

Violetta Kholomenyuk; Volodymyr Mykhaylyuk; Mikhail Popov

Open Mathematics (2011)

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

Abstract

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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 lim μ ( A ) → 0 μ ( A ) - 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

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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

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  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

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