To solve operator equations numerically, matrix representations are employing bases or more recently frames. For finding the numerical solution of operator equations a decomposition in subspaces is needed in many applications. To combine those two approaches, it is necessary to extend the known methods of matrix representation to the utilization of fusion frames. In this paper, we investigate this representation of operators on a Hilbert space $\mathcal{H}$ with Bessel fusion sequences, fusion frames and fusion Riesz bases. Fusion frames can be considered as a frame-like family of subspaces. Taking the particular property of the duality of fusion frames into account, this allows us to define a matrix representation in a canonical as well as an alternate way, the later being more efficient and well behaved in respect to inversion. We will give the basic definitions and show some structural results, like that the functions assigning the alternate representation to an operator is an algebra homomorphism. We give formulas for pseudo-inverses and the inverses (if existing) of such matrix representations. We apply this idea to Schatten p-class operators. Consequently, we show that tensor products of fusion frames are fusion frames in the space of Hilbert-Schmidt operators. We will show how this can be used for the solution of operator equations and link our approach to the additive Schwarz algorithm. Consequently, we propose some methods for solving an operator equation by iterative methods on the subspaces. Moreover, we implement the alternate Schwarz algorithms employing our perspective and provide small proof-of-concept numerical experiments. Finally, we show the application of this concept to overlapped convolution and the non-standard wavelet representation of operators.

为了以数值方式求解算子方程，矩阵表示采用基或更新的框架。为了找到算子方程的数值解，在许多应用中需要进行子空间分解。为了结合这两种方法，有必要将已知的矩阵表示方法扩展到融合帧的利用。在本文中，我们研究了希尔伯特空间上算子的这种表示$\mathcal{H}$具有 Bessel融合序列、融合框架和融合 Riesz 基。融合框架可以被认为是一个类似框架的子空间族。考虑到融合框架对偶性的特殊属性，这使我们能够以规范和替代方式定义矩阵表示，后者在反演方面更加高效且表现良好。我们将给出基本定义并展示一些结构结果，例如将替代表示分配给运算符的函数是代数同态。我们给出了此类矩阵表示的伪逆和逆（如果存在）的公式。我们将这个想法应用于 Schatten p类运算符。因此，我们证明融合框架的张量积是希尔伯特-施密特算子空间中的融合框架。我们将展示如何将其用于算子方程的求解，并将我们的方法与加法 Schwarz 算法联系起来。因此，我们提出了一些通过子空间上的迭代方法求解算子方程的方法。此外，我们利用我们的观点实现了替代 Schwarz 算法，并提供了小型概念验证数值实验。最后，我们展示了这个概念在重叠卷积和算子的非标准小波表示中的应用。