Results on the generalized numerical ranges in max algebra

Let $n$ and $k$ be two positive integers with $k\le n$ and $C$ an $n\times n$ matrix with nonnegative entries. In this paper, the rank-$k$ numerical range in the max algebra setting is introduced and studied. The related notions of the max joint $k$-numerical range and the max joint $C$-numerical range of an entry-wise nonnegative matrix and an $m$-tuple of nonnegative matrices are also introduced. Some interesting algebraic properties of these concepts are investigated.