Increasable doubly substochastic matrices with application to infinite linear equations

It has been proved that every doubly substochastic matrix can be “increased” into a doubly stochastic matrix in the sense that there is a doubly stochastic matrix with ([Marshall AW, Olkin I, Arnold BC. Inequalities; theory of majorization and its applications. 2nd ed. New York (NY): Springer; 2011], Theorem C.1). In this paper, we show this assertion does not necessarily hold for all infinite doubly substochastic matrices. Then we characterize all such matrices which are called increasable doubly substochastic matrices. We also provide an application of this result to the existence of a non-negative solution for some infinite linear equations.