In the framework of inverse linear problems on infinite-dimensional Hilbert space, we prove the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint, non-negative operator is unbounded and with minimal, technically unavoidable assumptions on the initial guess of the iterative algorithm. The convergence is proved to always hold in the Hilbert space norm (error convergence), as well as at other levels of regularity (energy norm, residual, etc.) depending on the regularity of the iterates. We also discuss, both analytically and through a selection of numerical tests, the main features and differences of our convergence result as compared to the case, already available in the literature, where the operator is bounded.

翻译：在无限维度希尔伯特空间的反线性问题的框架内,我们证明,在最一般的情况下,自我联合、非消极操作者不受约束,对迭代算法的最初猜测有最低、技术上不可避免的假设,在最普遍的情况下,同梯度交错与反向问题的确切解决办法相趋同;在Hilbert空间规范(高度趋同)中,以及根据迭代国的规律性(能源规范、剩余物等)中,这种趋同始终是相同的。 我们还通过分析和选择数字测试,讨论我们趋同结果的主要特点和与文献中已有的案例(即经营者受约束的情况)相比的差异。