Higher rank numerical ranges of rectangular matrix polynomials

In this paper, the notion of rank??k numerical range of rectangular complex ma- trix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for ? > 0; the notion of Birkho-James approximate orthogonality sets for ???higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed denitions yield a natural generalization of the standard higher rank numerical ranges. c