Approximate antilinear eigenvalue problems and related inequalities

Garcia, Stephan

doi:10.1090/S0002-9939-07-08945-9

Approximate antilinear eigenvalue problems and related inequalities
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Proc. Amer. Math. Soc. 136 (2008), 171-179 Request permission

Abstract:

If $T$ is a complex symmetric operator on a separable complex Hilbert space $\mathcal H$, then the spectrum $\sigma (|T|)$ 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 $\sigma (|T|)$. A sharp inequality for the operator norm is produced, and the extremal operators are shown to be complex symmetric.

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