Approximate antilinear eigenvalue problems and related inequalities

If $T$ is a complex symmetric operator on a separable complex Hilbert space $\mathcal H$, then the spectrum $\sigma(\vert T\vert)$ of $\sqrt{T^*T}$ can be characterized in terms of a certain approximate antilinear eigenvalue problem. This approach leads to a general inequality (applicable to any bounded operator $T:\mathcal H\rightarrow\mathcal H$), in terms of the spectra of the selfadjoint operators $\operatorname{Re} T$ and $\operatorname{Im} T$, restricting the possible location of elements of $\sigma(\vert T\vert)$. A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.