A characteristic of invariant order-hounded sets in the space $L^p(\Omega,\mu)$

Necessary and sufficient conditions are established for the image $U(B)$ of a set $B\subset L^p$ to be order-bounded in $L^p$ under arbitrary linear continuous mapping $U\colon L^p\to L^p$. The proof is based on properties of absolutely $p$-summing operators.