Summary: We consider a difference-operator approximation \(A_h^x\) of the differential operator \[A^x u(x) = -a_{1 1}(x) u_{x_1 x_1}(x) - a_{2 2}(x){u}_{x_2 x_2}(x) + \sigma u(x),\quad x=(x_1, x_2),\] defined in the region \(\mathbb{R}^+ \times \mathbb{R}\) with the boundary condition \[u(0, x_2) = 0, \quad x_2 \in \mathbb{R}.\] Here, the coefficients \(a_{ii}(x)\), \(i = 1, 2\), are continuously differentiable, satisfy the condition of uniform ellipticity \(a_{11}^2(x) + a_{22}^2(x) \ge \delta > 0\), and \(\sigma > 0\). We study the structure of the fractional spaces generated by the analyzed difference operator. The theorems on well-posedness of difference elliptic problems in a Hölder space are obtained as applications.