Summary: We consider the two-dimensional differential operator \[ A^{(t,x)}u(t,x) = -a_{11} (t,x)u_{tt} -a_{22}(t,x)u_{xx} +\sigma u \] defined on functions on the half-plane \(\mathbb{R}^+ \times \mathbb{R}\) with the boundary condition \(u(0,x) = 0, x \in \mathbb{R}\) where \(a_{ii}(t,x), i = 1,2\) are continuously differentiable and satisfy the uniform ellipticity condition \[a^2_{11}(t,x) + a^2_{22}(t,x)\geq \delta > 0, \sigma > 0. \] The structure of fractional spaces \(E_{\alpha,1} \big(L_1 (\mathbb{R}^+ \times \mathbb{R}), A^{(t,x)}\big)\) generated by the operator \(A^{(t,x)}\) is investigated. The positivity of \(A^{(t,x)}\) in \(L_1 \big(W^{2\alpha}_1 (\mathbb{R}^+ \times \mathbb{R})\big)\) spaces is established. In applications, theorems on well-posedness in \(L_1 \big(W^{2\alpha}_1 (\mathbb{R}^+ \times \mathbb{R})\big)\) spaces of elliptic problems are obtained.