International Journal of Engineering, Mathematical and Physical Sciences ISSN: 2517-9934 Vol:9, No:8, 2015 478 Some Results on the Generalized Higher Rank Numerical Ranges Mohsen Zahraei Abstract—In this paper, the notion of rank-k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for  > 0, the notion of Birkhoff-James approximate orthogonality sets for -higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural generalization of the standard higher rank numerical ranges. Keywords—Rank-k numerical range, isometry, numerical range, rectangular matrix polynomials. I. I NTRODUCTION AND RELATED WORK L ET M n×m be the vector space of all n × m complex matrices. For the case n = m, M n×n is denoted by M n ; namely, the algebra of all n ×n complex matrices. Throughout the paper, k,m and n are considered as positive integers and k ≤ min{m, n}. Moreover, I k denotes the k × k identity matrix. The set of all n × k isometry matrices is denoted by X n,k , i.e., X n,k = {X ∈ M n×k : X ∗ X = I k }. For the case n = k, X n,n is denoted by U n , namely, the group of all n × n unitary matrices. Motivation of our study comes from quantum information science. A quantum channel is a trace preserving completely positive map such as L : M n → M n . By the structure of completely positive linear maps, e.g., see [3], there are matrices E 1 ,...,E r ∈ M n with ∑ r j=1 E j E ∗ j = I n such that L(A)= ∑ r j=1 E ∗ j AE j . The matrices E 1 ,...,E r are interpreted as the error operators of the quantum channel L. Let V be a k−dimensional subspace of C n and P be the orthogonal projection of C n onto V. Then the k−dimensional subspace V is a quantum error correction code for the channel L if and only if there are scalars γ ij ∈ C with i, j ∈{1,...,r} such that PE ∗ i E j P = γ ij P ; for more information, see [7] and its references, and also see [11]. In this connection, the rank−k numerical range of A ∈ M n is defined and denoted by Λ k (A)= {λ ∈ C : P AP = λP, for some rank − k orthogonal projection P on C n }. It is known, see [4], that Λ k (A)= {λ ∈ C : X ∗ AX = λI k , for some X ∈X n,k }. The sets Λ k (A), where k ∈ {1,...,n}, are generally called higher rank numerical ranges of A. Apparently, for k=1, Λ k (A) reduces to the classical numerical range of A; namely, Λ 1 (A)= W (A) := {x ∗ Ax : x ∈ C n ,x ∗ x =1}, M. Zahraei is with the Department of Mathematics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran, e-mail: m.zahraei@iauahvaz.ac.ir, mzahraei56@yahoo.com. which has been studied extensively for many decades; e.g., see [9] and [10]. Stampfli and Williams in [14], and later Bonsall and Duncan in [2], observed that the numerical range of A ∈ M n can be rewritten as: W (A)= {μ ∈ C : ‖A − λI n ‖ 2 ≥|μ − λ|, ∀λ ∈ C}, where ‖.‖ 2 denotes the spectral matrix norm (i.e., the matrix norm subordinate to the Euclidean vector norm). By this idea, Chorianopoulos, Karanasios and Psarrakos [5] recently introduced a definition of the numerical range for rectangular complex matrices. For any A, B ∈ M n×m with B =0, and any vector norm ‖.‖ on M n×m , they defined the numerical range of A with respect to B as the compact and convex set: W ‖.‖ (A; B)= {μ ∈ C : ‖A − λB‖≥|μ − λ|, ∀λ ∈ C}. (1) It is clear that W ‖.‖2 (A; I n ) = W (A) = Λ 1 (A), where A ∈ M n . Hence, W ‖.‖ (. ; .) is a direct generalization of the classical numerical range. It is known that W ‖.‖ (A; B) = ∅ if and only if ‖B‖≥ 1. So, to avoid trivial consideration, we assume that ‖B‖≥ 1. Suppose P (λ)= A l λ l + A l−1 λ l−1 + ··· + A 1 λ + A 0 (2) is a rectangular matrix polynomial, where A i ∈ M n×m (i ∈ {0, 1, 2,...,s}),A s =0, and λ is a complex variable. The study of matrix polynomials has a long history, especially with regard to their applications on higher order linear systems of differential equations; e.g., see [8], [12] and the references therein. Let B ∈ M n×m and ‖·‖ be a vector norm on M n×m such that ‖B‖≥ 1. Moreover, let P (λ) be an n × m matrix polynomial as in (2). Using (9), Chorianopoulos and Psarrakos [6] recently introduced and studied the numerical range of P (λ) with respect to B as: W ‖·‖ [P (λ); B]= {μ ∈ C :0 ∈ W ‖·‖ (P (μ); B)}. (3) For the case n = m, B = I n and ‖·‖ = ‖·‖ 2 , we have the classical numerical range of the square matrix polynomial P (λ); namely, W ‖·‖2 [P (λ); I n ]= W [P (λ)] : = {μ ∈ C : x ∗ P (μ)x =0, for some nonzero x ∈ C n }. Hence, W ‖·‖ [.; .] is a direct generalization of the classical numerical range of square matrix polynomials, which plays an important role in the study of overdamped vibration systems with a finite number of degrees of freedom, and it also is related to the stability theory; e.g., see [13] and its references. Recently, Aretaki and Maroulas [1] introduced the notion of higher rank numerical ranges of square complex