Abstract
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.
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