Weighted slant Toep-Hank Operators

A $\it{weighted~slant~Toep}$-$\it{Hank}$ operator $L_{\phi}^{\beta}$ with symbol $\phi\in L^{\infty}(\beta)$ is an operator on $L^2(\beta)$ whose representing matrix consists of all even (odd) columns from a weighted slant Hankel (slant weighted Toeplitz) matrix, $\beta=\{\beta_n\}_{n\in \mathbb{Z}}$ be a sequence of positive numbers with $\beta_0=1$. A matrix characterization for an operator to be $\it{weighted~slant~Toep}$-$\it{Hank}$ operator is also obtained.