TY - JOUR

T1 - Constant norms and numerical radii of matrix powers

AU - Gau, Hwa Long

AU - Wang, Kuo Zhong

AU - Wu, Pei Yuan

N1 - Publisher Copyright:
© Elemen, Zagreb.

PY - 2019

Y1 - 2019

N2 - For an n-by-n complex matrix A, we consider conditions on A for which the operator norms ||Ak|| (resp., numerical radii w(Ak)), k ≥ 1, of powers of A are constant. Among other results, we show that the existence of a unit vector x in Cn satisfying |〈Ak x,x〉| = w(Ak)=w(A) for 1 ≤ k ≤ 4 is equivalent to the unitary similarity of A to a direct sum (Formula Presented), where |λ| = 1, B is idempotent, and C satisfies w(Ck) ≤ w(B) for 1 ≤ k ≤ 4. This is no longer the case for the norm: there is a 3-by-3 matrix A with ||Ak x|| = ||Ak|| =√2 for some unit vector x and for all k ≥ 1, but without any nontrivial direct summand. Nor is it true for constant numerical radii without a common attaining vector. If A is invertible, then the constancy of ||Ak || (resp., w(Ak)) for k = ±1,±2,… is equivalent to A being unitary. This is not true for invertible operators on an infinite-dimensional Hilbert space.

AB - For an n-by-n complex matrix A, we consider conditions on A for which the operator norms ||Ak|| (resp., numerical radii w(Ak)), k ≥ 1, of powers of A are constant. Among other results, we show that the existence of a unit vector x in Cn satisfying |〈Ak x,x〉| = w(Ak)=w(A) for 1 ≤ k ≤ 4 is equivalent to the unitary similarity of A to a direct sum (Formula Presented), where |λ| = 1, B is idempotent, and C satisfies w(Ck) ≤ w(B) for 1 ≤ k ≤ 4. This is no longer the case for the norm: there is a 3-by-3 matrix A with ||Ak x|| = ||Ak|| =√2 for some unit vector x and for all k ≥ 1, but without any nontrivial direct summand. Nor is it true for constant numerical radii without a common attaining vector. If A is invertible, then the constancy of ||Ak || (resp., w(Ak)) for k = ±1,±2,… is equivalent to A being unitary. This is not true for invertible operators on an infinite-dimensional Hilbert space.

KW - Idempotent matrix

KW - Irreducible matrix

KW - Numerical radius

KW - Operator norm

UR - http://www.scopus.com/inward/record.url?scp=85065521795&partnerID=8YFLogxK

U2 - 10.7153/oam-2019-13-72

DO - 10.7153/oam-2019-13-72

M3 - 期刊論文

AN - SCOPUS:85065521795

VL - 13

SP - 1035

EP - 1054

JO - Operators and Matrices

JF - Operators and Matrices

SN - 1846-3886

IS - 4

M1 - OaM-13-72

ER -