Approximate antilinear eigenvalue problems and related inequalities

If T is a complex symmetric operator on a separable complex Hilbert space H, then the spectrum sigma(vertical bar T vertical bar) of root 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 : H -> H), in terms of the spectra of the selfadjoint operators ReT and ImT, restricting the possible location of elements of sigma(vertical bar T vertical bar). A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.