We give a characterization of conditional expectation operators through a disjointness type property similar to band-preserving operators. We say that the operator T:X→ X on a Banach lattice X is semi-band-preserving if and only if for all f, g ∈ X, f ⊥ Tg implies that Tf ⊥ Tg. We prove that when X is a purely atomic Banach lattice, then an operator T on X is a weighted conditional expectation operator if and only if T is semi-band-preserving.
We give an example of a uniform quotient map from R to R which has non-locally connected level sets.
Suppose that a real nonatomic function space on is equipped with two rearrangement-invariant norms and . We study the question whether or not the fact that is isometric to implies that for all in . We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, respĿorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class...
We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p ≠ 2, does not admit a Schauder basis with constant one.
We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces , where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then...
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