This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the parallel-in-time integration leading to a class of algorithms called ParaDiag. In order to avoid the use of iterative refinement needed by ParaDiag and related

space-time approaches for attaining good accuracy, we develop and analyze two novel approaches for the numerical solution of such equations . Our first approach is based on the observation that the modification of these equations performed by ParaDiag in order to solve them in parallel has low rank. Building upon previous work on low-rank updates of matrix equations, this allows us to make use of  tensorized Krylov subspace methods to account for the modification. Our second approach is based on interpolating the solution of the matrix equation from the solutions of several modifications.