The class of complex symmetric operators is not norm closed

An operator $T\in\mathcal{B(H)}$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\mathcal{H}\longrightarrow\mathcal{H}$ so that $CTC=T^*$. In this paper, a class of complex symmetric operators on finite dimensional Hilbert spaces is constructed. As an application, it is shown that Kakutani's unilateral weighted shift operator is not complex symmetric; however, it is a norm limit of complex symmetric operators. This gives a negative answer to a question of S. Garcia and W. Wogen: that is, whether or not the class of complex symmetric operators is norm closed.