Higher-rank numerical ranges and dilations

Hwa Long Gau; Chi Kwong Li; Pei Yuan Wu

Higher-rank numerical ranges and dilations

Hwa Long Gau, Chi Kwong Li, Pei Yuan Wu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

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 (I_n - A*A). This extends and refines previous results of Choi and Li on constrained unitary dilations, and a result of Mirman on S_n- matrices.

Original language	English
Pages (from-to)	181-189
Number of pages	9
Journal	Journal of Operator Theory
Volume	63
Issue number	1
State	Published - Dec 2010

Keywords

Higher-rank numerical range
Unitary dilation

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title = "Higher-rank numerical ranges and dilations",

abstract = "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.",

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T1 - Higher-rank numerical ranges and dilations

AU - Gau, Hwa Long

AU - Li, Chi Kwong

AU - Wu, Pei Yuan

PY - 2010/12

Y1 - 2010/12

N2 - 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.

AB - 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.

KW - Higher-rank numerical range

KW - Unitary dilation

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M3 - 期刊論文

AN - SCOPUS:77955034603

SN - 0379-4024

VL - 63

SP - 181

EP - 189

JO - Journal of Operator Theory

JF - Journal of Operator Theory

IS - 1

ER -

Higher-rank numerical ranges and dilations

Abstract

Keywords

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