# Aspects of unconditionality of bases in spaces of compact operators

Annales Polonici Mathematici (1998)

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

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topJames 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

top- [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] S. Karlin, Bases in Banach spaces, Duke Math. J. 15 (1948), 971-985. Zbl0032.03102
- [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] A. Pełczyński and S. Kwapień, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43-63. Zbl0189.43505
- [5] I. Singer, Bases in Banach Spaces I, Grundlehren Math. Wiss. 154, Springer, New York, 1970.
- [6] E. Tutaj, On Schauder bases which are unconditional-like, Bull. Polish Acad. Sci. Math. 32 (1985), 137-146. Zbl0614.46007
- [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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