Splitting of the direct image of sheaves under the Frobenius

A generalisation and a new proof are given of a recent result of J. F. Thomsen (1996), which says that for $L$ a line bundle on a smooth toric variety $X$ over a field of positive characteristic, the direct image $F_*L$ under the Frobenius morphism splits into a direct sum of line bundles. (The special case of projective space is due to Hartshorne.) Our method is to interpret the result in terms of Grothendieck differential operators $\operatorname{Diff}^{(1)} (L,L)\cong\operatorname{Hom}_{O_{X^{(1)}}}(F_*L,F_*L)$, and $T$-linearized sheaves.