On complex symmetric operator matrices

An operator T ∈ L ( H ) is said to be complex symmetric if there exists a conjugation J on H such that T = J T ∗ J . In this paper, we find several kinds of complex symmetric operator matrices and examine decomposability of such complex symmetric operator matrices and their applications. In particular, we consider the operator matrix of the form T = ( A B 0 J A ∗ J ) where J is a conjugation on H . We show that if A is complex symmetric, then T is decomposable if and only if A is. Furthermore, we provide some conditions so that a -Weyl’s theorem holds for the operator matrix T .