Higher-rank numerical ranges and dilations

Hwa Long Gau, Chi Kwong Li, Pei Yuan Wu

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


For any n-by-n complex matrix A and any k, 1 ≤ k ≤ n, let Λk(A) = {λ ∈ C: X*AX = λIk for some n-by-k X satisfying X*X = Ik} be its rank-k numerical range. It is shown that if A is an n-by-n contraction, then Λk(A) = {Λk(U): U is an (n + dA)-by-(n + dA) unitary dilation of A}, where dA = rank (In - A*A). This extends and refines previous results of Choi and Li on constrained unitary dilations, and a result of Mirman on Sn- matrices.

Original languageEnglish
Pages (from-to)181-189
Number of pages9
JournalJournal of Operator Theory
Issue number1
StatePublished - Dec 2010


  • Higher-rank numerical range
  • Unitary dilation


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