Complex symmetric operator, operator norm, triangle inequality, selfadjoint operator, Cartesian decomposition, approximate antilinear eigenvalue problem, antilinear, spectrum
If T is a complex symmetric operator on a separable complex Hilbert space H, then the spectrum σ ( |T| ) of √(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 self-adjoint operators Re T and Im T, restricting the possible location of elements of σ ( |T| ) . A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.
© 2007 American Mathematical Society
Garcia, S.R., Approximate antilinear eigenvalue problems and related inequalities, Proc. Amer. Math. Soc. 136 (2008), 171-179. MR2350402 (2008k:47022)