## Abstract

We show that if A is a C_{0} contraction with minimal function φ{symbol} such that w (A) = w (S (φ{symbol})), where w (·) denotes the numerical radius of an operator and S (φ{symbol}) is the compression of the shift on H^{2} ⊖ φ{symbol} H^{2}, and B commutes with A, then w (AB) ≤ w (A) {norm of matrix} B {norm of matrix}. This is in contrast to the known fact that if A = S (φ{symbol}) (even on a finite-dimensional space) and B commutes with A, then w (AB) ≤ {norm of matrix} A {norm of matrix} w (B) is not necessarily true. As a consequence, we have w (AB) ≤ w (A) {norm of matrix} B {norm of matrix} for any quadratic operator A and any B commuting with A.

Original language | English |
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Pages (from-to) | 1509-1516 |

Number of pages | 8 |

Journal | Linear Algebra and Its Applications |

Volume | 430 |

Issue number | 5-6 |

DOIs | |

State | Published - 1 Mar 2009 |

## Keywords

- Compression of the shift
- Numerical radius
- Numerical range
- Quadratic operator