Complex symmetric operators and applications II

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 $T = CJ\vert T\vert$, where $J$ is an auxiliary conjugation commuting with $\vert T\vert = \sqrt{T^*T}$. We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition $T = CJ\vert T\vert$ 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.