Abstract: Linear elasticity theory constitutes one of the most fundamental and extensively applied theoretical frameworks in continuum mechanics, serving as the cornerstone for analyzing deformation, stress distribution, and mechanical response in a vast range of engineering materials and structures. Despite its long-standing maturity, the efficient and accurate numerical solution of two-dimensional linear elastic partial differential equations (PDEs) remains a persistent challenge, especially for problems involving highly heterogeneous materials with complex microstructural configurations and for structures subjected to arbitrary, non-trivial boundary conditions. To address this pressing need, this paper proposes a novel computational paradigm termed the Green's Function Fourier Operator (GFFO) method, which uniquely integrates both the physical structure of the underlying differential operators and the profound mathematical essence of Green's functions directly into the architecture of neural networks. The proposed approach begins by expanding the input functions into Fourier series, thereby mapping the original physical-space representation into the frequency domain, where the Green's function kernel together with its associated partial derivatives are explicitly parameterized as a set of learnable complex-valued multipliers. For self-adjoint differential equations, which constitute a broad and important class of problems in elasticity, Hermitian symmetry is strictly enforced within the frequency-domain kernel, ensuring that the learned operator faithfully preserves the intrinsic spectral properties of the true physical system. Furthermore, to properly handle non-homogeneous boundary conditions under which the translation invariance of Green's functions is inevitably lost, the method introduces an adaptive approximation strategy based on enhancing sparsity learning of the frequency-domain kernel matrix. Additionally, by exploiting frequency-domain truncation, the learnable integral kernel is effectively compressed into a finite-dimensional parameter set that is completely independent of the underlying computational grid, substantially reducing both the number of trainable parameters and the associated training cost. To accommodate solution problems of varying complexity, three distinct operator-learning architectures are carefully designed and implemented, namely GFFO-linear, GFFO-lift, and GFFO-ResMLP. The effectiveness of the proposed method is comprehensively validated through two representative numerical examples: a two-dimensional composite Representative Volume Element (RVE) characterized by heterogeneous microstructures, and an elastic plate subjected to mixed boundary conditions. Extensive experimental results consistently demonstrate that GFFO outperforms existing comparison methods across three critical dimensions—approximation accuracy, computational efficiency, and generalization capability—while simultaneously retaining clear physical interpretability owing to its explicit Green's function structure. This work therefore provides a novel, efficient, and physically grounded pathway for the rapid solution and parametric analysis of linear elasticity problems in computational mechanics and engineering applications.