Stationary Process Invertibility and the Unilateral Shift Operator

The bilateral shift operator $B$ has been the mainstay of stationary process modeling whereas we argue that the unilateral shift operator $T$ may be better suited to analyze invertibility. While doing so, we partially unify the notion of stationary process invertibility (associated with a sufficent but not necessary ${\mathrm{\ell }}^1$ condition) with the algebraic invertibility of the transfer function $f(T)$. We establish a rigorous operator theoretic foundation for these arguments proving that for $f\in {\mathbb{W}}_{+}$, the Wiener algebra, $f(T)$ is well defined, that $\mathrm{\parallel }f(T)\mathrm{\parallel }=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ and that $f(T)=T_f$, the Toeplitz operator.