The abstract issue of 'Krylov solvability' is extensively discussed for the inverse problem $Af = g$ where $A$ is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and $g$ is a datum in the range of $A$. The question consists of whether the solution $f$ can be approximated in the Hilbert norm by finite linear combinations of $g, Ag, A^2 g, \dots$ , and whether solutions of this sort exist and are unique. After revisiting the known picture when $A$ is bounded, we study the general case of a densely defined and closed $A$. Intrinsic operator-theoretic mechanisms are identified that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques.

中文翻译：

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无界逆线性问题的 Krylov 可解性

'Krylov 可解性' 的抽象问题针对反问题 $Af = g$ 进行了广泛讨论，其中 $A$ 是无限维希尔伯特空间上的（可能是无界的）线性算子，而 $g$ 是范围内的数据$A$。问题包括解决方案 $f$ 是否可以通过 $g, Ag, A^2 g, \dots$ 的有限线性组合在希尔伯特范数中近似，以及此类解决方案是否存在且唯一。在重新审视 $A$ 有界时的已知图片后，我们研究了一个密集定义且闭合的 $A$ 的一般情况。确定了保证或阻止 Krylov 可解性的内在算子理论机制，并且由于无界而产生了新特征。在自伴随情况下检查这种机制，其中 Krylov 可解性也通过基于共轭梯度的技术证明。