On Some Applications of a Special Integrodifferential Operators - Suna Saltan; Yasemin Özel (Journal of Function Spaces and Applications)
Let 𝐶(𝑛)(𝔻×𝔻) be a Banach space of complex-valued functions 𝑓(𝑥,𝑦) that are continuous on 𝔻×𝔻, where 𝔻={𝑧∈ℂ∶|𝑧|<1} is the unit disc in the complex plane ℂ, and have 𝑛th partial derivatives in 𝔻×𝔻 which can be extended to functions continuous on 𝔻×𝔻, and let 𝐶𝐴(𝑛)=𝐶𝐴(𝑛)(𝔻×𝔻) denote the subspace of functions in 𝐶(𝑛)(𝔻×𝔻) which are analytic in 𝔻×𝔻 (i.e., 𝐶𝐴(𝑛)=𝐶(𝑛)(𝔻×𝔻)∩ℋ𝑜𝑙(𝔻×𝔻)). The double integration operator is defined in 𝐶𝐴(𝑛) by the formula ∫𝑊𝑓(𝑧,𝑤)=𝑧0∫𝑤0𝑓(𝑢,𝑣)𝑑𝑣𝑑𝑢. 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.