## Abstract

For an n-by-n 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 3-by-3 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 infinite-dimensional Hilbert space.

Original language | English |
---|---|

Article number | OaM-13-72 |

Pages (from-to) | 1035-1054 |

Number of pages | 20 |

Journal | Operators and Matrices |

Volume | 13 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2019 |

## Keywords

- Idempotent matrix
- Irreducible matrix
- Numerical radius
- Operator norm