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Subspaces of ℓ₂(X) and Rad(X) without local unconditional structure

Ryszard A. Komorowski;  Nicole Tomczak-Jaegermann — 2002

Studia Mathematica

It is shown that if a Banach space X is not isomorphic to a Hilbert space then the spaces ℓ₂(X) and Rad(X) contain a subspace Z without local unconditional structure, and therefore without an unconditional basis. Moreover, if X is of cotype r < ∞, then a subspace Z of ℓ₂(X) can be constructed without local unconditional structure but with 2-dimensional unconditional decomposition, hence also with basis.

Selecting basic sequences in φ-stable Banach spaces

Tadeusz Figiel;  Ryszard Frankiewicz;  Ryszard A. Komorowski;  Czesław Ryll-Nardzewski — 2003

Studia Mathematica

In this paper we make use of a new concept of φ-stability for Banach spaces, where φ is a function. If a Banach space X and the function φ satisfy some natural conditions, then X is saturated with subspaces that are φ-stable (cf. Lemma 2.1 and Corollary 7.8). In a φ-stable Banach space one can easily construct basic sequences which have a property P(φ) defined in terms of φ (cf. Theorem 4.5). This leads us, for appropriate functions φ, to new results on the existence of unconditional...

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