Abstract
Let 𝔻 = {
∈ C : |
| < 1} be the unit disk and Hol(𝔻 × 𝔻) be the space of all holomorphic functions on the bi-disc 𝔻 × 𝔻. We consider the double convolution operator 𝒦
on the subspace Hol
(𝔻 × 𝔻) := {
∈ Hol(𝔻 × 𝔻) :
) =
) for some
∈ Hol(𝔻)} defined by
We study extended eigenvalues of 𝒦
. We characterize extended eigen vectors of 𝒦
in terms of Duhamel operators. Moreover, we describe cyclic vectors of operator 𝒦
by applying the Duhamel product method.