JOURNAL OF FUNCTION SPACES AND APPLICATIONS, 2012 (Journal Indexed in SCI)
Let C((n))((D x D) over bar) be a Banach space of complex-valued functions f(x, y) that are continuous on (D x D) over bar, where D = {z epsilon C : vertical bar z vertical bar < 1} is the unit disc in the complex plane C, and have nth partial derivatives in D x D which can be extended to functions continuous on <(D x D)over bar>, and let C(A)((n)) = C(A)((n)) (D x D) denote the subspace of functions in C((n))((D x D) over bar) which are analytic in D x D (i.e., C(A)((n)) = C((n))((D x D) over bar)boolean AND Hol(D x D)). The double integration operator is defined in C(A)((n)) by the formula W f (z, w) = integral(z)(0)integral(w)(0) f(u, v)dv du. By using the method of Duhamel product for the functions in two variables, we describe the commutant of the restricted operator W vertical bar E(zw), where E(zw) = {f is an element of C(A)((n)) : f(z, w) = f(z, w)} is an invariant subspace of W, and study its properties. We also study invertibility of the elements in C(A)((n)) with respect to the Duhamel product.