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Higher rank numerical ranges of rectangular matrices. (English) Zbl 1391.15078
Summary: 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 $$\varepsilon > 0,$$ the notion of Birkhoff-James approximate orthogonality sets for $$\varepsilon$$-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.

##### MSC:
 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 15A18 Eigenvalues, singular values, and eigenvectors 81P68 Quantum computation
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