Finite-Approximate Solvability of Linear Operator Equations

Nazim I. Mahmudov

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arXiv · 2604.22279

Finite-Approximate Solvability of Linear Operator Equations

Nazim I. Mahmudov

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Abstract

We introduce and study the finite-approximate solvability of operator equations $Lu = h$ in a Hilbert space setting, where a bounded operator $L \colon U \to H$ is paired with a finite-dimensional constraint operator $π\colon H \to H_0$. The objective is to match exactly the prescribed component $πh$ while approximating the remainder. We prove that the problem of finding $u$ such that $\|Lu - h\| < \varepsilon$ and $π(Lu) = πh$ is solvable for all $\varepsilon > 0$ if and only if $αT_α^{-1}h \to 0$ as $α\to 0^+$. We further show that dropping any of the structural assumptions on $L$, $Γ$, or $π$ leads to a failure of the equivalence. When $π\colon H \to H_0$ has an infinite-dimensional range that is compactly embedded in $H$, the operator $T_α$ may no longer be invertible. However, a Galerkin scheme $π_n \to π$ recovers approximate solvability through the resolvents $(α(I - π_n) + Γ)^{-1}$.

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BibTeX RIS

Nazim I. Mahmudov. 2026-04-24. Finite-Approximate Solvability of Linear Operator Equations. https://arxiv.org/abs/2604.22279

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