Majorization theory is considered as a powerful tool in application, especially in quantum physics. Since many quantum problems occur in the infinite dimension, many mathematicians, such as Ryff and Day, generalized the Majorization theory based on doubly stochastic operators to infinite dimensional space. In this paper, we introduce larger class of operators on $L^1(X, \mu)$ which is more suitable for extending the notion of majorization, called semi doubly stochastic operators. We show that on the finite measure space $(X, \mu)$, doubly stochastic operators and semi doubly stochastic operators on $L^1(X, \mu)$ coincide, but in general we provide a counterexample. Also for $\sigma$-finite measure space $(X, \mu)$, the majorization relation can be characterized on $L^1(X, \mu)$ as $g\prec f$ if and only if there is a sequence of semi doubly stochastic operators $(S_n)_{n\in\mathbb{N}}$ such that $S_nf\overset{L^1}{\rightarrow} g$. Moreover, as an application of our result in quantum physics, the convertibility of pure states of an infinite dimensional composite system by local operations and classical communication has been considered.

This is joint work with Dr. Seyed Mahmoud Manjegani, Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran.