A 𝐶*-algebra approach to complex symmetric operators

Abstract

In this paper, certain connections between complex symmetric operators and anti-automorphisms of singly generated C ∗ C^* -algebras are established. This provides a C ∗ C^* -algebra approach to the norm closure problem for complex symmetric operators. For T ∈ B ( H ) T\in \mathcal {B(H)} satisfying C ∗ ( T ) ∩ K ( H ) = { 0 } C^*(T)\cap \mathcal {K(H)}=\{0\} , we give several characterizations for T T to be a norm limit of complex symmetric operators. As applications, we give concrete characterizations for weighted shifts with nonzero weights to be norm limits of complex symmetric operators. In particular, we prove a conjecture of Garcia and Poore. On the other hand, it is proved that an essentially normal operator is a norm limit of complex symmetric operators if and only if it is complex symmetric. We obtain a canonical decomposition for essentially normal operators which are complex symmetric.