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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
