Complex symmetric operators with closed numerical ranges

Abstract

A bounded linear operator T:H→H is said to be complex symmetric if there exists a conjugation C on H such that CT⁎C=T. In this paper, we study the numerical ranges of complex symmetric operators. We show that every complex symmetric operator T on H has a small compact perturbation being complex symmetric and having a closed numerical range. We also show that this holds for skew symmetric operators but fails to hold for unitary operators.