using frames in gmres-based iteration method for solving operator equations

In this paper, we delve into frame theory to create an in novative iterative method for resolving the operator equation lu = f. in this case, l : h → h, a bounded, invertible, and self-adjoint lin ear operator, operates within a separable hilbert space denoted by h. our methodology, which is based on the gmres projective method, in troduces an alternate search space, which brings another dimension to the problem-solving process. our investigation continues with the assess ment of convergence, where we look at the corresponding convergence rate. this rate is intricately influenced by the frame bounds, shedding light on the effectiveness of our approach. furthermore, we investigate the ideal scenario in which the equation finds an exact solution, providing useful insights into the practical implications of our work.