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Abstract
For an nbyn complex matrix A, we consider conditions on A for which the operator norms A^{k} (resp., numerical radii w(A^{k})), k ≥ 1, of powers of A are constant. Among other results, we show that the existence of a unit vector x in C^{n} satisfying 〈A^{k} x,x〉 = w(A^{k})=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(C^{k}) ≤ w(B) for 1 ≤ k ≤ 4. This is no longer the case for the norm: there is a 3by3 matrix A with A^{k} x = A^{k} =√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 A^{k}  (resp., w(A^{k})) for k = ±1,±2,… is equivalent to A being unitary. This is not true for invertible operators on an infinitedimensional Hilbert space.
Original language  English 

Article number  OaM1372 
Pages (fromto)  10351054 
Number of pages  20 
Journal  Operators and Matrices 
Volume  13 
Issue number  4 
DOIs  
State  Published  2019 
Keywords
 Idempotent matrix
 Irreducible matrix
 Numerical radius
 Operator norm
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Dive into the research topics of 'Constant norms and numerical radii of matrix powers'. Together they form a unique fingerprint.Projects
 1 Finished

A Study on Matrices with Circular Numerical Ranges
Gau, H.L. (PI)
1/08/19 → 31/07/20
Project: Research