Higher-rank numerical ranges and dilations
Hwa-Long Gau, Chi-Kwong Li, Pei Yuan Wu · 2010
ABSTRACT. For any n-by-n complex matrix A and any k, 1 ≤ k ≤ n, let Λ k(A) = {λ ∈ C: X ∗ AX = λI k for some n-by-k X satisfying X ∗ X = I k} be its rank-k numerical range. It is shown that if A is an n-by-n contraction, then Λ k(A) = ∩{Λ k(U) : U is an (n + d A)-by-(n + d A) unitary dilation of A}, where d A = rank (In − A ∗ A). This extends and refines previous results of Choi and Li on constrained unitary dilations, and a result of Mirman on Snmatrices. KEYWORDS: Higher-rank numerical range, unitary dilation. MSC (2000): 15A21, 15A24, 15A60, 15A90, 81P68. 1.