Aspects of unconditionality of bases in spaces of compact operators

James R. Holub

Annales Polonici Mathematici (1998)

  • Volume: 68, Issue: 1, page 27-30
  • ISSN: 0066-2216

Abstract

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E. Tutaj has introduced classes of Schauder bases termed "unconditional-like" (UL) and "unconditional-like*" (UL*) whose intersection is the class of unconditional bases. In view of this association with unconditional bases, it is interesting to note that there exist Banach spaces which have no unconditional basis and yet have a basis of one of these two types (e.g., the space 𝓞[0,1]). In the same spirit, we show in this paper that the space of all compact operators on a reflexive Banach space with an unconditional basis has a basis of type UL*, even though it is well-known that this space has no unconditional basis.

How to cite

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James R. Holub. "Aspects of unconditionality of bases in spaces of compact operators." Annales Polonici Mathematici 68.1 (1998): 27-30. <http://eudml.org/doc/270166>.

@article{JamesR1998,
abstract = {E. Tutaj has introduced classes of Schauder bases termed "unconditional-like" (UL) and "unconditional-like*" (UL*) whose intersection is the class of unconditional bases. In view of this association with unconditional bases, it is interesting to note that there exist Banach spaces which have no unconditional basis and yet have a basis of one of these two types (e.g., the space 𝓞[0,1]). In the same spirit, we show in this paper that the space of all compact operators on a reflexive Banach space with an unconditional basis has a basis of type UL*, even though it is well-known that this space has no unconditional basis.},
author = {James R. Holub},
journal = {Annales Polonici Mathematici},
keywords = {unconditional basis; unconditional-like basis; tensor product basis; compact operator space; unconditional-like; Schauder bases},
language = {eng},
number = {1},
pages = {27-30},
title = {Aspects of unconditionality of bases in spaces of compact operators},
url = {http://eudml.org/doc/270166},
volume = {68},
year = {1998},
}

TY - JOUR
AU - James R. Holub
TI - Aspects of unconditionality of bases in spaces of compact operators
JO - Annales Polonici Mathematici
PY - 1998
VL - 68
IS - 1
SP - 27
EP - 30
AB - E. Tutaj has introduced classes of Schauder bases termed "unconditional-like" (UL) and "unconditional-like*" (UL*) whose intersection is the class of unconditional bases. In view of this association with unconditional bases, it is interesting to note that there exist Banach spaces which have no unconditional basis and yet have a basis of one of these two types (e.g., the space 𝓞[0,1]). In the same spirit, we show in this paper that the space of all compact operators on a reflexive Banach space with an unconditional basis has a basis of type UL*, even though it is well-known that this space has no unconditional basis.
LA - eng
KW - unconditional basis; unconditional-like basis; tensor product basis; compact operator space; unconditional-like; Schauder bases
UR - http://eudml.org/doc/270166
ER -

References

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  1. [1] B. R. Gelbaum and J. Gil de Lamadrid, Bases of tensor products of Banach spaces, Pacific J. Math. 11 (1961), 1281-1286. Zbl0106.08604
  2. [2] S. Karlin, Bases in Banach spaces, Duke Math. J. 15 (1948), 971-985. Zbl0032.03102
  3. [3] J. Lindenstrauss, On a certain subspace of l¹, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 539-542. Zbl0133.06604
  4. [4] A. Pełczyński and S. Kwapień, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43-63. Zbl0189.43505
  5. [5] I. Singer, Bases in Banach Spaces I, Grundlehren Math. Wiss. 154, Springer, New York, 1970. 
  6. [6] E. Tutaj, On Schauder bases which are unconditional-like, Bull. Polish Acad. Sci. Math. 32 (1985), 137-146. Zbl0614.46007
  7. [7] E. Tutaj, Some observations concerning the classes of unconditional-like basic sequences, Bull. Polish Acad. Sci. Math. 35 (1987), 35-42. Zbl0645.46013

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