On Some Applications of a Special Integrodifferential Operators

Abstract EN

Let C(n)(?×?¯) be a Banach space of complex-valued functions f(x,y) that are continuous on ?×?¯, where ?={z∈ℂ:|z|<1} is the unit disc in the complex plane ℂ, and have nth partial derivatives in ?×? which can be extended to functions continuous on ?×?¯, and let CA(n)=CA(n)(?×?) denote the subspace of functions in C(n)(?×?¯) which are analytic in ?×? (i.e., CA(n)=C(n)(?×?¯)∩ℋol(?×?)).

The double integration operator is defined in CA(n) by the formula Wf(z,w)=∫0z∫0wf(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∣Ezw, where Ezw={f∈CA(n):f(z,w)=f(zw)} is an invariant subspace of W, and study its properties.

We also study invertibility of the elements in CA(n) with respect to the Duhamel product.