Linear extension operators on products of compact spaces

Let X and Y be the Alexandroff compactifications of the locally compact spaces X and Y, respectively. Denote by Σ(X×Y ) the space of all linear extension operators from C((X×Y )⧹(X×Y)) to C((X×Y )). We prove that X and Y are σ-compact spaces if and only if there exists a T∈Σ(X×Y ) with ‖T‖<2 if and only if there exists a Γ∈Σ(X×Y ) with ‖Γ‖=1. Assuming the existence of a T∈Σ(X×Y ) with ‖T‖<3, it is shown that the pseudocompactness of X and Y is equivalent to the fact that ‖Γ‖⩾2 for every Γ∈Σ(X×Y ).