In this paper, we consider a linear operator of the Korteweg-de Vries type Lu = ∂u ∂y + R 2 (y) ∂ 3 u ∂x 3 + R 1 (y) ∂u ∂x + R 0 (y)u initially defined on C ∞ 0,π (Ω), where Ω = {(x, y) : −π ≤ x ≤ π, −∞ < y < ∞}. C ∞ 0,π (Ω) is a set of infinitely differentiable compactly supported function with respect to a variable y and satisfying the conditions: u (i) x (−π, y) = u (i) x (π, y), i = 0, 1, 2. With respect to the coefficients of the operator L , we assume that these are continuous functions in R(−∞, +∞) and strongly growing functions at infinity. In this paper, we proved that there exists a bounded inverse operator and found a condition that ensures the compactness of the resolvent under some restrictions on the coefficients in addition to the above conditions. Also, two-sided estimates of singular numbers (s-numbers) are obtained and an example is given of how these estimates allow finding estimates of the eigenvalues of the considered operator.