Some properties of a complex symmetric operator

A square matrix with complex entries which is equal to its transpose is called a complex symmetric matrix. Such matrices are found in many areas of pure and applied mathematics. A conjugation on H is defined as a conjugate linear isometric involution C on a Hilbert Space H. A bounded linear operator T : H → H has a complex symmetric matrix with regard to an orthonormal basis if, for every conjugation C on H, T = CT *C. Such an operator is called a Complex symmetric operator. Several characteristics of Complex Symmetric matrices and associated operators are discussed in this article.