Onl p-complemented copies in Orlicz spaces II

Domination problems for strictly singular operators and other related classes

“…In the context of Banach lattices, a useful variant of strict singularity is the following [28]: given a Banach lattice E and a Banach space Y , an operator T : E → Y is called disjointly strictly singular if it is not invertible on any subspace of E generated by a disjoint sequence.…”

Strictly Singular and Regular Integral Operators

“…It involves the use of a lattice version of the class of strictly singular operators, introduced by Hernández and Rodríguez-Salinas in [11]. Precisely, a bounded operator T from a Banach lattice E into a Banach space Y is said to be disjointly strictly singular if there is no disjoint sequence of non-null vectors (x n ) n in E such that the restriction of T to the subspace [x n ] spanned by the sequence (x n ) n is an isomorphism.…”

Eigenvalue estimates for operators on symmetric Banach sequence spaces

“…Now we show that the convexity and concavity assumptions in the preceding two theorems are in fact essential. To see this, fix 1 < p < ∞ and take any symmetric Banach sequence space E strictly contained in p , but such that the inclusion map E → p is not strictly singular, i.e., it is an isomorphism on some infinitedimensional subspace of E (for Orlicz sequence spaces E of this type we refer to [8] and [9]; see also [14, 4.c.3]). Recall the well-known fact that any separable p -space is complementably minimal (i.e., each of its infinite-dimensional subspaces contains a subspace which is isomorphic to p and complemented in p ).…”