The differences of inclusion map operators between rearrangement invariant spaces on finite and σ-finite measure spaces

Let X be a quasi-Banach RIS (QBRIS) on [0,1]. Then the following inclusions are valid: L∞⊂X⊂Lp, where p=p(X)>0. In classical Banach case p=1 and for canonical injection operators I:L∞→X; I:X→L1 it’s known conditions for such properties as strict singularity (SS), disjoint strict singularity (DSS), (p,q)-absolutely summing, etc. We prove some similar facts in quasi-Banach case. If X is a QBRIS on [0,∞], then it is γ-normed for some 0<γ≤1 and L∞∩Lγ⊂X⊂ Lp+L∞, for some p=p(X)>0. On the contrary to the finite measure case, when I(L∞,X)∈SS for any X =L∞, there are many examples of spaces on [0,∞) such that I ∈DSS(L1∩L∞,X). Another deep difference is: on [0,1] : I(X,L1)∈DSS for any BanachX =L1; but on [0,∞):I(X,Lp+L∞) ∈DSS forX such thatLr,∞⊂X for some r>p. 1. Definitions and basic notations Let us start with some definitions. We shall use the term operator to mean a bounded linear operator; subspaces are assumed infinite and closed. We shall consider rear-rangement invariant spaces (RIS) of functions, both Banach and quasi-Banach. A quasi-Banach RIS is a complete quasinormed vector space (X, ‖ · ‖) of measurable functions on (0, 1) or (0,∞) such that ‖κA ‖ = 1 if measA = 1, and if g is in X, then f is in X and ‖f ‖ ≤ ‖g ‖ if f is a measurable function satisfying f ∗ ≤ g∗, where h∗ denotes the decreasing rearrangement of the function |h | (cf. [1]). A quasinorm is a function which satisfies the axioms for a norm except that the triangle inequality is replaced by ‖x+ y ‖ ≤ K(‖x‖+ ‖y‖) with some K> 1