A functional model and tridiagonalisation for symmetric anti-linear operators

We consider the class of bounded symmetric anti-linear operators $B$ with a cyclic vector. We associate with $B$ the spectral data consisting of a probability measure and a function. In terms of the spectral data of $B$, we introduce a functional model operator $\mathcal{B}$ acting on a model space. We prove an anti-linear variant of the spectral theorem demonstrating that $B$ is unitarily equivalent to $\mathcal{B}$. Next, we show that $B$ is also unitarily equivalent to an anti-linear tridiagonal operator and discuss connection with orthogonal polynomials in the anti-linear setting.