Complex symmetric operators and applications II

A bounded linear operator T T on a complex Hilbert space H H is called complex symmetric if T = C T ∗ C T = CT^*C , where C C is a conjugation (an isometric, antilinear involution of H H ). We prove that T = C J | T | T = CJ|T| , where J J is an auxiliary conjugation commuting with | T | = T ∗ T |T| = √T^*T . We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition T = C J | T | T = CJ|T| also extends to the class of unbounded C 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.