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Numerical range of the Foguel–Halmos operator

Volume 263 / 2022

Hwa-Long Gau, Kuo-Zhong Wang, Pei Yuan Wu Studia Mathematica 263 (2022), 267-291 MSC: 47A12, 15A60, 47B37. DOI: 10.4064/sm201020-30-7 Published online: 22 November 2021

Abstract

We study properties of the numerical range of the Foguel–Halmos operator $F_T=\left [\begin {smallmatrix}S^* &T\\ 0 &S\end {smallmatrix}\right ]$ on $\ell ^2\oplus \ell ^2$, where $S$ is the simple unilateral shift and $T=\operatorname{diag} (a_1, a_2, \ldots )$ with $a_n=1$ if $n=3^k$ for some $k\ge 1$ and $a_n=0$ otherwise. Among other things, we show that the numerical range $W(F_T)$ is neither open nor closed, and give lower and upper bounds for the numerical radius $w(F_T)$.

Authors

  • Hwa-Long GauDepartment of Mathematics
    National Central University
    Chungli 32001, Taiwan
    e-mail
  • Kuo-Zhong WangDepartment of Applied Mathematics
    National Yang Ming Chiao Tung University
    Hsinchu 30010, Taiwan
    e-mail
  • Pei Yuan WuDepartment of Applied Mathematics
    National Yang Ming Chiao Tung University
    Hsinchu 30010, Taiwan
    e-mail

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