The factorization of a linear conjugate symmetric involution in Hilbert space

Abstract:

Let $X$ be a closed linear transformation whose domain is dense in the complex separable Hilbert space $H$ and whose adjoint is denoted by ${X^ * }$. The operator $X$ is said to be conjugate symmetric if $\Gamma (X) \subset \Gamma (Q{X^ * }Q)$, where $\Gamma (X)$ represents the graph of $X$ in $H \otimes H$ and $Q$ is a conjugation on $H$. The main theorem in this note states that a conjugate symmetric linear involution $X$ satisfies the operator equation $X = Q{X^ * }Q$.