Uniqueness of (...) Bases and Isometries of Banach Spaces.
We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does . We also give some positive results including a simpler proof that has a unique unconditional basis and a proof that has a unique unconditional basis when , and remains bounded.
We prove that if 0 < p < 1 then a normalized unconditional basis of a complemented subspace of must be equivalent to a permutation of a subset of the canonical unit vector basis of . In particular, has unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss, and Tzafriri have previously proved the same result for .
We observe that a separable Banach space is reflexive iff each of its quotients with Schauder basis is reflexive. Similarly if is not reflexive for reflexive and then is is not reflexive for some , having a basis.
It is shown that an orthonormal wavelet basis for associated with a multiresolution is an unconditional basis for , 1 < p < ∞, provided the father wavelet is bounded and decays sufficiently rapidly at infinity.
2000 Mathematics Subject Classification: 46B30, 46B03.It is shown that most of the well known classes of nonseparable Banach spaces related to the weakly compact generating can be characterized by elementary properties of the closure of the coefficient space of Markusevic bases for such spaces. In some cases, such property is then shared by all Markusevic bases in the space.