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 $L \colon U \to H$ is paired with a finite-dimensional constraint operator $\pi \colon H \to H_0$. 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 $\|Lu - h\| < \varepsilon$ and $\pi(Lu) = \pi h$ is solvable for all $\varepsilon > 0$ 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 $L$, $\Gamma$, or $\pi$ leads to a failure of the equivalence. When $\pi \colon H \to H_0$ 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 $(\alpha(I - \pi_n) + \Gamma)^{-1}$.