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Aspects of unconditionality of bases in spaces of compact operators

100%

Holub J.

Annales Polonici Mathematici

1998

tom 68

nr 1

27-30

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.

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On general Franklin systems

77%

Gevorkyan G., Kamont A.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS 1. Introduction.....................................................................................5 1.1. Notation......................................................................................7 2. Definition and properties of general Franklin systems..................10 2.1. Piecewise linear functions.........................................................10 2.2. Franklin functions.....................................................................11 2.3. Sequences of partitions and Franklin functions........................13 2.3.1. Regularity of sequences of partitions...................................14 2.4. Sequences of partitions and general Haar systems.................16 2.5. Technical lemmas.....................................................................17 3. Franklin series in $L^p$, 1 < p < ∞...............................................21 4. Franklin series in $L^p$, 0 < p ≤ 1, and $H^p$, 1/2 < p ≤ 1..........27 5. The necessity of strong regularity in $H^p$, 1/2 < p ≤ 1...............42 6. Haar and Franklin series with identical coefficients......................46 7. Characterization of the spaces BMO and Lip(α), 0 < α < 1...........51 References.......................................................................................58

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Gevorkyan G.

1 Holub J.

1 Kamont A.

2 1998

Wyniki wyszukiwania

Aspects of unconditionality of bases in spaces of compact operators

On general Franklin systems