Infinite distohastic square operators in $l_1$

The article is devoted to bistochastic quadratic operators. The concept of a bistochastic quadratic operator is introduced and the property of such operators is studied. A necessary and sufficient condition for bistochasticity is given. An analogue of Birkhoff's theorem is studied for the class of bistochastic quadratic operators. A sufficient condition for the extremity of bistochastic quadratic operators is obtained, and a necessary condition is also obtained for small dimensions. The concept of bistochastic quadratic operators is generalized, and the concept of a bistochastic operator of arbitrary order is introduced.