ON A QUESTION OF HASKELL P. ROSENTHAL CONCERNING A CHARACTERIZATION OF ${c}_{0}$ AND ${\ell}_{p}$

Abstract

The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of c 0 or p . An analogous problem concerning spreading models is also addressed.

Disjointly Homogeneous Banach Lattices and Applications

“…Also, recall that a normalized basis (e n ) in a Banach space X is said to be a Rosenthal basis if every normalized block-sequence of (e n ) contains a subsequence equivalent to (e n ). It is an open question whether such a basis is necessarily equivalent to the unit basis of ℓ p or c 0 , see [13] for further details and partial results in this direction. In particular, it was observed in [13, p. 397] that a Rosenthal basis (x n ), always satisfies that every subsequence (x n i ) is equivalent to (x n ).…”

Banach spaces and Ramsey Theory: some open problems

“…Problem 3. (Rosenthal basis problem -H. P. Rosenthal [26]) Is it true that every Rosenthal basis (x n ) is equivalent to the standard unit vector basis of c 0 or ℓ p for some 1 p < +∞?…”

“…It is known (see [26]) that the problem has an affirmative answer provided that there exists a constant C 1 such that every normalized block sequence (v n ) of (x n ) has a subsequence which is C-equivalent to (x n ).…”

Infinite asymptotic games

“…We recall that a normalised basic sequence (e i ) is called a Rosenthal basic sequence if any normalised block has a subsequence equivalent to (e i ) (see Ferenczi, Pelczar, and Rosendal [4] for more on such bases). It is still an open question whether Rosenthal sequences are equivalent to the standard unit vector bases in c 0 or ℓ p , though the answer is positive in case there is some uniformity or the subsequence can be chosen continuously.…”

“…Modulo the non-trivial fact that ℓ p , p = 2, and c 0 are not homogeneous, a positive answer would of course provide another solution to the homogeneous space problem. 4. Subspaces of spaces with F.D.D.…”