On some questions for the q-integration operator

We use the q-Duhamel product to provide a Banach algebra structure to some closed subspaces of the Wiener disk- algebra \(W_{+}\left( \mathbb {D}\right) \) of analytic functions on the unit disk \(\mathbb {D}\) of the complex plane \(\mathbb {C.}\) We study the q-integration operator on \(W_{+}\left( \mathbb {D}\right) ,\) namely, we characterize invariant subspaces of this operator and describe its extended eigenvalues and extended eigenvectors. Moreover, we prove an addition formula for the spectral multiplicity of the direct sum of q-integration operator on \(W_{+}\left( \mathbb {D}\right) \) and some Banach space operator.