Strictly singular and power-compact operators on Banach lattices

Disjointly Homogeneous Banach Lattices and Applications

“…In [16] several results in similar spirit were given without the restriction of regularity. An important technique that was exploited in these arguments is the well known Kadec-Pe lczyński's dichotomy (see [14], [33]): given a normalized sequence (x n ) in an order continuous Banach lattice E (i) either ( x n L 1 ) is bounded away from zero, (ii) or there exist a subsequence (x n k ) and a disjoint sequence (z k ) in E such that z k − x n k −→ 0 as k → ∞.…”

“…A characterization of DH Orlicz spaces, over finite [16] and infinite [17] measure spaces, is known. In order to state this, let us first recall the definition of certain subsets of the space of continuous functions C[0, 1] associated to the Orlicz function ϕ (see [31]):…”

“…Later on E. M. Semenov and the authors ( [16]) analyzed the general problem of obtaining compactness of the iterations of a strictly singular operator on a Banach lattice, extending the classical result by V. D. Milman ([35]) which states that strictly singular operators in L p (µ), 1 ≤ p ≤ ∞, have compact square. In fact, one of the purposes of this survey is to offer compelling evidence that the class of disjontly homogeneous Banach lattices constitutes a proper setting for treating these questions.…”

Strictly singular multiplication operators on ℒ(X)

“…The fact that on L p spaces the composition of two strictly singular operators yields a compact operator is due to V. D. Milman [18], and has recently been extended to further classes of Banach spaces (see [5]); these include for instance the Lorentz spaces Λ(w, q) and certain Orlicz spaces. It is conceivable that the results in [15] extend to these larger classes of spaces.…”

The support localization property of the strongly embedded subspaces of Banach function spaces

“…Examples and characterizations of strongly embedded subspaces of Lorentz spaces were studied in the classical papers [3] and [6]. More information on the strongly embedded subspaces of Lorentz L p,q spaces can be found in [17] (see also [8]). Consider the generalized Lorentz spaces Λ(W, p) studied in Section 5 of [6].…”