Approximated Solution Based on the Frame Bounds in Hilbert C∗-Modules

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Given \(\mathcal{H}\) is a finitely or countably generated Hilbert \(\mathcal{A}\)-module over a unital \(C^*\)-algebra \(\mathcal{A}\), and given a frame in \(\mathcal{H}\). We introduce several iterative methods for solving the operator equation:\[Lu=f\]where \(L\) is a bounded, invertible, and symmetric operator on \(\mathcal{H}\). We present algorithms that utilize frame bounds, the Chebyshev method, and the conjugate gradient method to provide approximate solutions to the problem. Additionally, we analyze the convergence and optimality of these methods.

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