  Let $(X_{k})_{k \in \mathbb Z }$ be a linear process with values in a
separable Hilbert space $\mathbb{H}$ given by $X_{k} =\sum_{j=0}^{\infty}
(j+1)^{-N}\varepsilon_{k-j}$ for each $k \in \mathbb Z$, where $N:\mathbb{H}
\to \mathbb{H}$ is a bounded, linear normal operator and $(\varepsilon_{k})_{ k
\in \mathbb Z }$ is a sequence of independent, identically distributed
$\mathbb{H}$-valued random variables with $E\varepsilon_{0}=0$ and $E\|
\varepsilon_{0} \|^2<\infty$. We investigate the central and the functional
central limit theorem for $(X_{k})_{k \in \mathbb Z }$ when the series of
operator norms $\sum_{j=0}^{\infty} \|(j+1)^{-N}\|_{op}$ diverges. Furthermore
we show that the limit process in case of the functional central limit theorem
generates an operator self-similar process.
