  The numerical solution of dynamical systems with memory requires the
efficient evaluation of Volterra integral operators in an evolutionary manner.
After appropriate discretisation, the basic problem can be represented as a
matrix-vector product with a lower diagonal but densely populated matrix. For
typical applications, like fractional diffusion or large scale dynamical
systems with delay, the memory cost for storing the matrix approximations and
complete history of the data then would become prohibitive for an accurate
numerical approximation. For Volterra-integral operators of convolution type,
the \emph{fast and oblivious convolution quadrature} method of Sch\"adle,
Lopez-Fernandez, and Lubich allows compute the discretized valuation with $N$
time steps in $O(N \log N)$ complexity and only requiring $O(\log N)$ active
memory to store a compressed version of the complete history of the data. We
will show that this algorithm can be interpreted as an $\mathcal{H}$-matrix
approximation of the underlying integral operator and, consequently, a further
improvement can be achieved, in principle, by resorting to
$\mathcal{H}^2$-matrix compression techniques. We formulate a variant of the
$\mathcal{H}^2$-matrix vector product for discretized Volterra integral
operators that can be performed in an evolutionary and oblivious manner and
requires only $O(N)$ operations and $O(\log N)$ active memory. In addition to
the acceleration, more general asymptotically smooth kernels can be treated and
the algorithm does not require a-priori knowledge of the number of time steps.
The efficiency of the proposed method is demonstrated by application to some
typical test problems.
