  Long Range Dependence (LRD) in functional sequences is characterized in the
spectral domain under suitable conditions. Particularly, multifractionally
integrated functional autoregressive moving averages processes can be
introduced in this framework. The convergence to zero in the Hilbert-Schmidt
operator norm of the integrated bias of the periodogram operator is proved.
Under a Gaussian scenario, a weak--consistent parametric estimator of the
long--memory operator is then obtained by minimizing, in the norm of bounded
linear operators, a divergence information functional loss.
