Numerical range of the Foguel–Halmos operator
Volume 263 / 2022
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 $w(F_T)$.
