Finite-Approximate Solvability of Linear Operator Equations

We introduce and study the finite-approximate solvability of operator equations $Lu=h$ in a Hilbert space setting, where a bounded operator $L\text{\hspace{0.17em}⁣}:U\to H$ is paired with a finite-dimensional constraint operator $\pi \text{\hspace{0.17em}⁣}:H\to H_0$. The objective is to match exactly the prescribed component $\pi h$ while approximating the remainder. We prove that the problem of finding $u$ such that $\mathrm{\parallel }Lu-h\mathrm{\parallel }<\epsilon$ and $\pi (Lu)=\pi h$ is solvable for all $\epsilon >0$ if and only if $\alpha T_{\alpha }^{-1}h\to 0$ as $\alpha \to 0^{+}$. We further show that dropping any of the structural assumptions on $L$, $\mathrm{\Gamma }$, or $\pi$ leads to a failure of the equivalence. When $\pi \text{\hspace{0.17em}⁣}:H\to H_0$ has an infinite-dimensional range that is compactly embedded in $H$, the operator $T_{\alpha }$ may no longer be invertible. However, a Galerkin scheme ${\pi }_n\to \pi$ recovers approximate solvability through the resolvents $(\alpha (I-{\pi }_n)+\mathrm{\Gamma }{)}^{-1}$.