Approximation of complex symmetric operators

Abstract

An operator \(T\) on a complex Hilbert space \(\mathcal {H}\) is called a complex symmetric operator if there exists a conjugate-linear, isometric involution \(C:\mathcal {H}\rightarrow \mathcal {H}\) so that \(CTC=T^*\). In this paper, we study the approximation of complex symmetric operators. By virtue of an intensive analysis of compact operators in singly generated \(C^*\)-algebras, we obtain a complete characterization of norm limits of complex symmetric operators and provide a classification of complex symmetric operators up to approximate unitary equivalence. This gives a general solution to the norm closure problem for complex symmetric operators. As an application, we provide a concrete description of partial isometries which are norm limits of complex symmetric operators.