Square roots of complex symmetric operators

Abstract

In this paper we study square roots of complex symmetric operators. In particular, we prove that if T∈L(H) is a square root of a complex symmetric operator, then T∗ has the single-valued extension property if and only if so does T. Moreover, in this case, T has the Bishop's property (β) if and only if T is decomposable. Finally, we show that if T has a nontrivial hyperinvariant subspace, then T∗ has a nontrivial invariant subspace.