Complex symmetric operators and applications II

Garcia, Stephan; Putinar, Mihai

doi:10.1090/S0002-9947-07-04213-4

Complex symmetric operators and applications II
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by Stephan Ramon Garcia and Mihai Putinar PDF

Trans. Amer. Math. Soc. 359 (2007), 3913-3931 Request permission

Abstract:

A bounded linear operator $T$ on a complex Hilbert space $\mathcal {H}$ is called complex symmetric if $T = CT^*C$, where $C$ is a conjugation (an isometric, antilinear involution of $\mathcal {H}$). We prove that $T = CJ|T|$, where $J$ is an auxiliary conjugation commuting with $|T| = \sqrt {T^*T}$. We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition $T = CJ|T|$ also extends to the class of unbounded $C$-selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

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