Approximations of the Korovkin type in Banach lattices

Let E, G denote two Banach lattices, and let (T_n) be a sequence of continuous linear operators E → G . We prove that if (T_n) satisfies the difference condition |T_n - T_m| x = |T_n x - T_m x| for all x ∈ E^+ , and if the sequence (T_n x_0) converges for some x_0 ∈ E , then (T_n) converges pointwise on the principal ideal A_x_0 generated by x_0 . This result allows us to strengthen essentially an approximate-spectral theorem of the Freudenthal type obtained recently by A. W. Wickstead.