On the Structure of Real Operators

An operator T on a complex separable Hilbert space \({\mathcal {H}}\) is called a real operator if T can be represented as a real matrix relative to some orthonormal basis of \({\mathcal {H}}\). In this paper, we provide descriptions of concrete real operators, such as real normal operators, real partial isometries, and real Toeplitz operators, among others. Furthermore, we present several structure theorems of real operators, including the polar decomposition, the Riesz decomposition and the block structure.