Somewhat quasireflexive Banach spaces

Abstract

The question of "What kind of subspaces must a nonreflexive Banach space X have?" has received a lot of attention. Pelczynski [23] (in 1962) has given the most general answer to date: X contains a basic sequence which is not shrinking (and hence spanning a nonreflexive space). For special cases more is known. Johnson and Rosenthal [8] have shown that X and X* contain reflexive subspaces if X** is separable. (This was extended to the case when X**/X is separable by Clark [2].) In another direction, Davis and Johnson [5] have shown that if X**/X is infinite dimensional then X contains a basic sequence that spans a nonquasireflexive subspace. Perhaps the main reason for this interest are the following two long open questions:(1) Does each Banach space contain an unconditional basic sequence? (2) Does each Banach space contain a subspace isomorphic to c0,/1 or to a reflexive space?Indeed, James [6] has shown that a positive answer to (1) implies a positive answer to (2). And clearly these results are partial answers to (2). On the other hand, consider the collection of spaces to which the special cases apply. James [7], Lindenstrauss [9], Davis, Figiel, Johnson and Pelczynski [4] and the author [1] show how to construct an X so that X**/X is a pregiven Z (with restrictions on Z). All these constructions depend on reflexivity or quasireflexivity in a strong way and the constructed X has lots of quasireflexive subspaces. This paper attempts to unite these results. It is shown that if X**/X is separable then each element of X**/X is "reachable" by an order one quasireflexive subspace ZcX, so that Z has a shrinking basis (Theorem 8). If X** is separable, both X and X* have subspaces and quotients which are order one quasireflexive with bases (Theorem 9). And if X* is separable then X has a nonreflexive quotient

Frequently Hypercyclic Bilateral Shifts

“…It is easily seen that X is a Banach sequence space that contains X as a closed subspace. The space appears, for example, in Singer [13, p. 39] or in Bellenot [5], but does not seem to have been given a name or a universally accepted notation. We have, for example, that c 0 (Z) = ℓ ∞ (Z).…”

More quasireflexive subspaces

“…In [3] (or see [2]), Bourgain and Delbaen construct a collection of somewhat reflexive .Coo-spaces. It was suggested to the author that these spaces might yield a negative answer to question (1). This note shows that these t^-spaces contain quasi-reflexive spaces, and hence (1) is still open.…”

“…It was suggested to the author that these spaces might yield a negative answer to question (1). This note shows that these t^-spaces contain quasi-reflexive spaces, and hence (1) is still open.…”

“…We mention a short footnote to [1]. In [1] it was shown that most positive answers to question (2) also have a positive answer to question (1).…”

“…The author in [1] has shown that most collections of Banach spaces which are known to contain a reflexive subspace also contain a quasi-reflexive subspace. (Of course, the words "infinite dimensional" are needed to make this correct.…”