  We study the structure of strictly singular non-compact operators between
$L_p$ spaces. Answering a question raised in [Adv. Math. 316 (2017), 667-690],
it is shown that there exist operators $T$, for which the set of points
$(\frac1p,\frac1q)\in(0,1)\times (0,1)$ such that $T:L_p\rightarrow L_q$ is
strictly singular but not compact contains a line segment in the triangle
$\{(\frac1p,\frac1q):1<p<q<\infty\}$ 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 operators defined on subspaces of $L_p$ is also
explored.
