On Some Applications of a Special Integrodifferential Operators
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 subspa...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
|
| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2012/894527 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 0972-6802 1758-4965 |