The higher rank numerical range of matrix polynomials

The notion of the higher rank numerical range ${\mathrm{\Lambda }}_k(L(\lambda ))$ for matrix polynomials $L(\lambda )=A_m{\lambda }^m+\mathrm{.}\mathrm{.}\mathrm{.}+A_1\lambda +A_0$ is introduced here and some fundamental geometrical properties are investigated. Further, the sharp points of ${\mathrm{\Lambda }}_k(L(\lambda ))$ are defined and their relation to the numerical range $w(L(\lambda ))$ is presented. A connection of ${\mathrm{\Lambda }}_k(L(\lambda ))$ with the vector-valued higher rank numerical range ${\mathrm{\Lambda }}_k(A_0,\mathrm{.}\mathrm{.}\mathrm{.},A_m)$ is also discussed.