Strictly singular non-compact operators between $L_p$ spaces

Abstract

We study the structure of strictly singular non-compact operators between $L\_p$ spaces. Answering a question raised in earlier work on interpolation properties of strictly singular operators, it is shown that there exist operators $T$, for which the set of points $(1/p,1/q)\in(0,1)\times (0,1)$ such that $T\colon L\_p\rightarrow L\_q$ is strictly singular but not compact contains a line segment in the triangle ${(1/p,1/q):1\<p\<q<\infty}$ of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular (i.e., difference of positive) operators defined on subspaces of $L\_p$ is also explored.