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 + σu defined on functions on the half-plane R + × R with the boundary condition u(0, x) = 0, x ∈ 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) ≥ δ > 0, σ > 0. The structure of fractional spaces E α,1 L 1 (R + × R) , A (t,x) generated by the operator A (t,x) is investigated. The positivity of A (t,x) in L 1 W 2α 1 (R + × R) spaces is established. In applications, theorems on well-posedness in L 1 W 2α 1 (R + × R) spaces of elliptic problems are obtained.