Approximate antilinear eigenvalue problems and related inequalities
Stephan Ramon Garcia · Proceedings of the American Mathematical Society · 2007
If T T is a complex symmetric operator on a separable complex Hilbert space H \mathcal H , then the spectrum σ ( | T | ) \sigma (|T|) of T ∗ T \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 : H → H T:\mathcal H\rightarrow \mathcal H ), in terms of the spectra of the selfadjoint operators Re T \operatorname {Re} T and Im T \operatorname {Im} T , restricting the possible location of elements of σ ( | T | ) \sigma (|T|) . A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.