Resumen de The differences of inclusion map operators between rearrangement invariant spaces on finite and o-finite measures spaces

S. Ya. Novikov

• Let $X$ be a quasi-Banach $\textbf{RIS (QBRIS)}$ on [0,1]. Then the following inclusions are valid: $L_\infty\subset X\subset L_p$, where $p=p(X)?0$. In classical Banach case $p=1$ and for canonical injection operators $I:L_\infty\rightarrow X; I:X\rightarrow L_1$ 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 $\textbf{QBRIS}$ on $[0,\infty]$, then it is $\gamma$-normed for some $0 ?\gamma\leq 1$ and $L_\infty\cap\L_\gamma\subset X\subset L_p+L_\infty$, for some $p=p(X)?0$. On the contrary to the finite measure case, when $I(L_\infty,X)\in SS$ for any $\not= L\infty$, there are many examples of spaces on $[0,\infty)$ such that $I\notin DSS(L_1\cap L_\infty,X)$. Another deep difference is : on [0,1] : $I(X,L_1)\in DSS$ for any Banach $X\not= L_1$; but on $[0,\infty):I(X,L_p+L_\infty)\notin DSS$ for $X$ such that $L_{r,\infty}\subset X$ for some $r?p$.