## 摘要

For any operator T on ℓ^{2}, its associated Foguel operator F_{T} is [Formula presented] on ℓ^{2}⊕ℓ^{2}, where S is the (simple) unilateral shift. It is easily seen that the numerical radius w(F_{T}) of F_{T} satisfies 1≤w(F_{T})≤1+(1/2)‖T‖. In this paper, we study when such upper and lower bounds of w(F_{T}) are attained. For the upper bound, we show that w(F_{T})=1+(1/2)‖T‖ if and only if w(S+T^{⁎}S^{⁎}T)=1+‖T‖^{2}. When T is a diagonal operator with nonnegative diagonals, we obtain, among other results, that w(F_{T})=1+(1/2)‖T‖ if and only if w(ST)=‖T‖. As for the lower bound, it is shown that any diagonal T with w(F_{T})=1 is compact. Examples of various T's are given to illustrate such attainments of w(F_{T}).

原文 | ???core.languages.en_GB??? |
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文章編號 | 127484 |

期刊 | Journal of Mathematical Analysis and Applications |

卷 | 528 |

發行號 | 1 |

DOIs | |

出版狀態 | 已出版 - 1 12月 2023 |