Order continuous extensions of positive compact operators on Banach lattices

Let $E$ and $F$ be Banach lattices. Let $G$ be a vector sublattice of $E$ and $T: G\rightarrow F$ be an order continuous positive compact (resp. weakly compact) operators. We show that if $G$ is an ideal or an order dense sublattice of $E$, then $T$ has a norm preserving compact (resp. weakly compact) positive extension to $E$ which is likewise order continuous on $E$. In particular, we prove that every compact positive orthomorphism on an order dense sublattice of $E$ extends uniquely to a compact positive orthomorphism on $E$.Comment: 7 page

On Transverse-Momentum Dependent Light-Cone Wave Functions of Light Mesons

Transverse-momentum dependent (TMD) light-cone wave functions of a light meson are important ingredients in the TMD QCD factorization of exclusive processes. This factorization allows one conveniently resum Sudakov logarithms appearing in collinear factorization. The TMD light-cone wave functions are not simply related to the standard light-cone wave functions in collinear factorization by integrating them over the transverse momentum. We explore relations between TMD light-cone wave functions and those in the collinear factorization. Two factorized relations can be found. One is helpful for constructing models for TMD light-cone wave functions, and the other can be used for resummation. These relations will be useful to establish a link between two types of factorization.Comment: add more discussions and reference