Remarks on conjugation and antilinear operators and their numerical range (Research on structure of operators by order and related topics)

Abstract

In this paper, we investigate the numerical ranges of conjugations and antilinear operators on a Hilbert space, which will be shown to be annuli in general. This result proves that Toeplitz-Hausdorff Theorem, which says the convexity on the numerical ranges of linear operators, does not hold for the ones of antilinear operators. Moreover, we extend these results to a Banach space.