  Thanks to its great potential in reducing both computational cost and memory
requirements, combining sketching and Krylov subspace techniques has attracted
a lot of attention in the recent literature on projection methods for linear
systems, matrix function approximations, and eigenvalue problems. Applying this
appealing strategy in the context of linear matrix equations turns out to be
far more involved than a straightforward generalization. These difficulties
include analyzing well-posedness of the projected problem and deriving possible
error estimates depending on the sketching properties. Further computational
complications include the lack of a natural residual norm estimate and of an
explicit basis for the generated subspace. In this paper we propose a new
sketched-and-truncated polynomial Krylov subspace method for Sylvester
equations that aims to address all these issues. The potential of our novel
approach, in terms of both computational time and storage demand, is
illustrated with numerical experiments. Comparisons with a state-of-the-art
projection scheme based on rational Krylov subspaces are also included.
