Operator generalization of one Rubel's result (Q1760032)

\textit{L. A. Rubel} [Funkc. Ekvacioj, Ser. Int. 10, 225--227 (1967; Zbl 0184.17301)] described all derivation pairs \((L,M)\) of linear and continuous functionals on the space \(\mathrm{Hol}(G)\) of all functions analytic on a domain \(G\), endowed with the topology of compact convergence. This means that \(L(fg) = L(f)M(g) + L(g)M(f)\) for every \(f\) and \(g\). In this note, the author gives a description of all pairs of linear continuous operators \(A\) and \(B\) on \(\mathrm{Hol}(G)\) satisfying \(A(fg)(z) = (Af)(z)(Bg)(z) + (Ag)(z)(Bf)(z)\) for all \(z \in G\) and all functions \(f\) and \(g\).