Finite-Approximate Solvability of Linear Operator Equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908990910234624 |
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| author | Mahmudov, Nazim I. |
| author_facet | Mahmudov, Nazim I. |
| contents | 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 \((α(I - π_n) + Γ)^{-1}\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_22279 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite-Approximate Solvability of Linear Operator Equations Mahmudov, Nazim I. Dynamical Systems 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 \((α(I - π_n) + Γ)^{-1}\). |
| title | Finite-Approximate Solvability of Linear Operator Equations |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2604.22279 |