On complex symmetric operator matrices

An operator T∈L(H)T∈L(H) is said to be complex symmetric if there exists a conjugation JJ on HH such that T=JT∗JT=JT∗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=(AB0JA∗J) where JJ is a conjugation on HH. We show that if AA is complex symmetric, then TT is decomposable if and only if AA is. Furthermore, we provide some conditions so that aa-Weyl’s theorem holds for the operator matrix TT.