Finite-Approximate Solvability of Linear Operator Equations

TL;DR

This paper studies the conditions under which approximate solutions to linear operator equations in Hilbert spaces can be found, focusing on finite-approximate solvability and the role of structural assumptions.

Contribution

It establishes a necessary and sufficient condition for finite-approximate solvability involving the behavior of \\alpha T_\alpha^{-1}h as \alpha approaches zero, and explores the impact of structural assumptions.

Findings

Finite-approximate solvability is characterized by the convergence of \alpha T_\alpha^{-1}h to zero.

Dropping structural assumptions on operators can cause the equivalence to fail.

A Galerkin scheme with approximate operators recovers solvability when the original operator's invertibility is compromised.

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 \(\pi \colon H \to H_0\). 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 \((\alpha(I - \pi_n) + \Gamma)^{-1}\).