The Kreĭn-Šmul'jan theorem revisited

Abstract

We present a generalization of the Kreĭn-Šmul'jan theorem involving several operators: Given bounded selfadjoint operators $A,B_1,\dots ,B_m$ acting on a Hilbert space $H$, we provide sufficient conditions to determine whether there are ${\lambda }_1,\dots ,{\lambda }_m\in R$ such that $A+{\sum }_{i=1}^m{\lambda }_i{B}_i$ is a positive semidefinite operator.