On the Regularity of Spectral Densities

Let $W(\theta)$ be an operator function representing the spectral density of a multidimensional stationary random sequence. In the case of finite-dimensional random sequences, it is well known that if $W$ satisfies the Szegö condition
$$\int{\log W(\theta)d\theta\geq-cI,}$$
where $c$ is a constant and $I$ the unit operator, then the error of the best linear prediction of a sequence one step ahead will really be nonzero. In the present note, an example is constructed which shows that this assertion is no longer true in the infinite-dimensional case.