In this paper, the notions of rank$-k$ numerical range and $k-$spectrum of rectangular complex matrices are introduced. Some algebraic and geometrical properties are investigated. Moreover, for $\epsilon \gt 0,$ the notion of Birkhoff-James approximate orthogonality sets for $\epsilon$-higher rank numerical ranges of rectangular matrices is also introduced and studied. The proposed definitions yield a natural generalization of standard higher rank numerical ranges.
"Higher rank numerical ranges of rectangular matrices." Ann. Funct. Anal. 6 (2) 133 - 142, 2015. https://doi.org/10.15352/afa/06-2-12