Matric pencils with the numerical range equal to the whole complex plane

The main purpose of this article is to show that the numerical range of a linear pencil $\lambda A + B$ is equal to $\mathbb{C}$ if and only if $0$ belongs to the joint numerical range of $A$ and $B$. We also prove that if the numerical range of a linear pencil $\lambda A + B$ is equal to $\mathbb{C}$ and $A + A^*, B + B^* \geq 0$, then $A$ and $B$ have a common isotropic vector. Moreover, we improve the classical result which describes Hermitian linear pencils.