Compactness of Riemann–Liouville fractional integral operators

Abstract

We obtain results on compactness of two linear Hammerstein integral operators with singularities, and apply the results to give new proof that Riemann–Liouville fractional integral operators of order $\alpha\in (0,1)$ map $L^{p}(0,1)$ to $C[0,1]$ and are compact for each $p\in \bigl(\frac{1}{1-\alpha},\infty\bigr]$. We show that the spectral radii of the Riemann–Liouville fractional operators are zero.

Keywords

• linear hammerstein integral operator
• riemann–liouville fractional integral operator
• compactness
• spectral radius