Numerical ranges of Foguel operators

We study properties of the numerical ranges of Foguel operators F-T = [S* T 0 S], where Sis the simple unilateral shift and Tis some operator, both acting on l(2). Among other things, we show that (1) if Tis nonzero compact, then the numerical radius w(F-T) is strictly less than 1 +(parallel to T parallel to/2), (2) if T is a diagonal unitary operator, then root 5/2 < w(F-T) <= 3/2, and (3) if Tis a scalar operator aI, then the numerical range W(F-T) is open and is not a circular disc unless a = 0. (C) 2020 Elsevier Inc. All rights reserved.