A characterization of numerical ranges for antilinear operators

Abstract This article aims to study the problem of determining the numerical ranges of antilinear operators on complex Hilbert spaces. First, we provide a concrete description of the numerical range $W(R)$ for every bounded antilinear operator R on a complex Hilbert space $\mathcal {H}$ , solving the preceding problem. Second, given a bounded linear operator T on $\mathcal {H}$ , we determine the possible value of the numerical radius $w(CT)$ of $CT$ when C ranges over the collection of all conjugations on $\mathcal {H}$ .