On incomparability of Banach spaces

Manuel González; Antonio Martinón

Banach Center Publications (1994)

  • Volume: 30, Issue: 1, page 161-174
  • ISSN: 0137-6934

Abstract

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Several concepts of incomparability of Banach spaces have been considered in the literature, which allow one to describe some of the properties of the product of two Banach spaces as a juxtaposition of the corresponding properties of the factors. In this paper we study the relations between these concepts of incomparability, survey the main results and applications, and state some open problems.

How to cite

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González, Manuel, and Martinón, Antonio. "On incomparability of Banach spaces." Banach Center Publications 30.1 (1994): 161-174. <http://eudml.org/doc/262688>.

@article{González1994,
abstract = {Several concepts of incomparability of Banach spaces have been considered in the literature, which allow one to describe some of the properties of the product of two Banach spaces as a juxtaposition of the corresponding properties of the factors. In this paper we study the relations between these concepts of incomparability, survey the main results and applications, and state some open problems.},
author = {González, Manuel, Martinón, Antonio},
journal = {Banach Center Publications},
keywords = {incomparability of Banach spaces},
language = {eng},
number = {1},
pages = {161-174},
title = {On incomparability of Banach spaces},
url = {http://eudml.org/doc/262688},
volume = {30},
year = {1994},
}

TY - JOUR
AU - González, Manuel
AU - Martinón, Antonio
TI - On incomparability of Banach spaces
JO - Banach Center Publications
PY - 1994
VL - 30
IS - 1
SP - 161
EP - 174
AB - Several concepts of incomparability of Banach spaces have been considered in the literature, which allow one to describe some of the properties of the product of two Banach spaces as a juxtaposition of the corresponding properties of the factors. In this paper we study the relations between these concepts of incomparability, survey the main results and applications, and state some open problems.
LA - eng
KW - incomparability of Banach spaces
UR - http://eudml.org/doc/262688
ER -

References

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  1. [1] P. Aiena and M. González, Essentially incomparable Banach spaces and Fredholm theory, Proc. Roy. Irish Acad. 93A (1993), 49-59. Zbl0790.46010
  2. [2] T. Alvarez, M. González and V. M. Onieva, A note on three-space Banach space ideals, Arch. Math. (Basel) 46 (1986), 169-170. Zbl0574.47029
  3. [3] T. Alvarez, M. González and V. M. Onieva, Totally incomparable Banach spaces and three-space ideals, Math. Nachr. 131 (1987), 83-88. Zbl0659.46009
  4. [4] T. Alvarez, M. González and V. M. Onieva, Characterizing two classes of operator ideals, in: Contribuciones Matemáticas. Homenaje Prof. Antonio Plans, Univ. Zaragoza, Zaragoza, 1990, 7-21. Zbl1290.47073
  5. [5] J. Bourgain, New Classes of p -Spaces, Lecture Notes in Math. 889, Springer, Berlin, 1981. 
  6. [6] J. Bourgain and F. Delbaen, A class of special -spaces, Acta Math. 145 (1980), 155-176. Zbl0466.46024
  7. [7] P. G. Casazza, N. J. Kalton and L. Tzafriri, Decompositions of Banach lattices into direct sums, Trans. Amer. Math. Soc. 304 (1987), 771-800. Zbl0676.46012
  8. [8] J. C. Díaz, An example of Fréchet space, not Montel, without infinite dimensional normable subspaces, Proc. Amer. Math. Soc. 96 (1986), 721. Zbl0592.46001
  9. [9] J. Diestel and R. H. Lohman, Applications of mapping theorems to Schwartz spaces and projections, Michigan Math. J. 20 (1973), 39-44. Zbl0238.46043
  10. [10] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977. 
  11. [11] A. Douady, Un espace de Banach dont le groupe linéaire n'est pas connexe, Indag. Math. 27 (1965), 787-789. Zbl0178.26403
  12. [12] L. Drewnowski, On minimally subspace-comparable F-spaces, J. Funct. Anal. 26 (1977), 39-44. Zbl0366.46012
  13. [13] I. S. Edelstein and P. Wojtaszczyk, On projections and unconditional bases in direct sums of Banach spaces, Studia Math. 56 (1976), 263-276. Zbl0362.46017
  14. [14] G. Emmanuele, Banach spaces in which Dunford-Pettis sets are relatively compact, Arch. Math. (Basel) 58 (1992), 477-485. Zbl0761.46010
  15. [15] M. González, On essentially incomparable Banach spaces, Math. Z., to appear. 
  16. [16] M. González and A. Martinón, Operational quantities derived from the norm and generalized Fredholm theory, Comment. Math. Univ. Carolin. 32 (1991), 645-657. Zbl0762.47005
  17. [17] M. González and A. Martinón, Fredholm theory and space ideals, Boll. Un. Mat. Ital. 7-B (1993), 473-488. 
  18. [18] M. González and V. M. Onieva, On incomparability of Banach spaces, Math. Z. 112 (1986), 581-585. Zbl0602.46015
  19. [19] M. González and V. M. Onieva, El problema de los tres espacios, in: Homenaje al Prof. Dr. Nácere Hayek Calil, Univ. La Laguna, La Laguna, 1990, 155-162. 
  20. [20] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), 851-874. Zbl0827.46008
  21. [21] V. I. Gurariĭ, On openings and inclinations of subspaces of Banach spaces, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 1 (1965), 194-204 (in Russian). 
  22. [22] S. Heinrich, On the relation between linear n-widths and approximation numbers, J. Approx. Theory 58 (1989), 315-333. Zbl0699.41022
  23. [23] N. J. Kalton, Quotients of F-spaces, Glasgow Math. J. 19 (1978), 103-108. Zbl0398.46002
  24. [24] N. J. Kalton, B. Turett and J. J. Uhl, Jr., Basically scattered vector measures, Indiana Univ. Math. J. 28 (1979), 803-816. Zbl0461.28005
  25. [25] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322. Zbl0090.09003
  26. [26] N. H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19-30. Zbl0129.38901
  27. [27] A. Lavergne, Remark on sums of complemented subspaces, Colloq. Math. 41 (1979), 103-104. Zbl0447.47002
  28. [28] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, Berlin, 1977. Zbl0362.46013
  29. [29] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Function Spaces, Springer, Berlin, 1977. Zbl0362.46013
  30. [30] R. H. Lohman, An example concerning normed subspaces of locally convex spaces, Proc. Amer. Math. Soc. 41 (1973), 245-246. Zbl0251.46004
  31. [31] A. Martinón, Cantidades operacionales en teoría de Fredholm, Tesis Doctoral, Univ. La Laguna, 1989. 
  32. [32] A. Martinón, Operational quantities and classes of operators, in: Aportaciones matemáticas en memoria del Prof. V. M. Onieva, Univ. Cantabria, Santander, 1991, 227-236. Zbl0756.47012
  33. [33] B. S. Mityagin, The homotopy structure of the linear group of a Banach space, Russian Math. Surveys 25 (1970), 59-103. Zbl0232.47046
  34. [34] G. Neubauer, Der Homotopietyp der Automorphismengruppe in den Räumen l p und c₀, Math. Ann. 174 (1967), 33-40. Zbl0158.14402
  35. [35] A. Pełczyński, On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C(S)-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 31-36. Zbl0138.38604
  36. [36] A. Pełczyński, On strictly singular and strictly cosingular operators. II. Strictly singular and strictly cosingular operators in L(ν)-spaces, ibid., 37-41. Zbl0138.38604
  37. [37] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980. 
  38. [38] A. Pietsch, Inessential operators in Banach spaces, Integral Equations Operator Theory 1 (1978), 589-591. Zbl0399.47040
  39. [39] J. Prada, On idempotent operators on Fréchet spaces, Arch. Math. (Basel) 43 (1984), 179-182. Zbl0537.46005
  40. [40] H. P. Rosenthal, On totally incomparable Banach spaces, J. Funct. Anal. 4 (1969), 167-175. Zbl0184.15004
  41. [41] H. P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from L p ( μ ) to L r ( ν ) , ibid., 176-214. Zbl0185.20303
  42. [42] W. Rudin, Spaces of type H + C , Ann. Inst. Fourier (Grenoble) 25 (1) (1975), 99-125. 
  43. [43] R. J. Whitley, Strictly singular operators and their conjugates, Trans. Amer. Math. Soc. 113 (1964), 252-261. Zbl0124.06603
  44. [44] P. Wojtaszczyk, On projections and unconditional bases in direct sums of Banach spaces II, Studia Math. 62 (1978), 193-201. Zbl0389.46011
  45. [45] V. P. Zahariuta, On the isomorphism of cartesian products of locally convex spaces, ibid. 46 (1973), 201-221. 
  46. [46] J. Zemánek, Geometric characteristics of semi-Fredholm operators and their asymptotic behaviour, ibid. 80 (1984), 219-234. Zbl0556.47008

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