Spectral Conditions for Composition Operators on Algebras of Functions - Johnson, J.; Tonev, T. (Communications in Mathematics and Applications, 2012)
We establish general sufficient conditions for maps between function algebras to be composition or weighted composition operators, which extend previous results in [2,4,6,7]. Let X be a locally compact Hausdorff space and A C(X) a dense subalgebra of a function algebra, not necessarily with unit, such that X = A and p(A)=δ A, where A is the Shilov boundary, δ A – the Choquet boundary, and p(A) – the set of p-points of A. If TA → B is a surjective map onto a function algebra BC(Y) such that either (Tf· Tg)(fg) for all f,g A, or, alternatively, (fg)(Tf· Tg) for all f,g A, then there is a homeomorphism ψδ B→δ A and a function on δ B so that (Tf)(y)=(y) f(ψ(y)) for all f A and y δ B. If, instead, (Tf· Tg)(fg) eq for all f,gA, and either (f)(Tf) for all fA, or, alternatively, (Tf)(f) for all fA, then (Tf)(y)=f(ψ(y)) for all f A and y δ B. In particular, if A and B are uniform algebras and TA → B is a surjective map with (Tf· Tg)(fg) eq for all f,g A, that has a limit, say b, at some aA with a^2=1, then (Tf)(y)=b(y) a(ψ(y)) f(ψ(y)) for every fA and yδ B.