# Popov M. M.

### Theorems on decomposition of operators in *L*_{1} and their generalization to vector lattices

Maslyuchenko O. V., Mykhailyuk V. V., Popov M. M.

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 26-35

The Rosenthal theorem on the decomposition for operators in *L*_{1} is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in *L*_{1} is a projective component, which yields the known fact that a sum of narrow operators in *L*_{1} is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in *L*_{1}, in particular the Liu decomposition.

### On the Lyapunov convexity theorem with appications to sign-embeddings

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1192–1200

It is proved (Theorem 1) that for a Banach space $X$ the following assertions are equivalent:

(1) the range of every $X$- valued $σ$- additive nonatomic measure of finite variation possesses a convex closure;

(2) $L_1$ does not signembed in $X$.