Krein-Šmul'jan Theorem Revisited

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