The eigenvalues form a discrete spectrum, when the operator is limited to act on functions on the unit interval of the real number line. More broadly, since the Gauss map is the shift operator on Baire space \mathbb{N}^\omega, the GKW operator can also be viewed as an operator on the function space \mathbb{N}^\omega\to\mathbb{C} (considered as a Banach space, with basis functions taken to be the indicator functions on the cylinders of the product topology). In the later case, it has a continuous spectrum, with eigenvalues in the unit disk |\lambda|<1 of the complex plane. That is, given the cylinder C_n[b]= \{(a_1,a_2,\cdots) \in \mathbb{N}^\omega : a_n = b \}, the operator G shifts it to the left: GC_n[b] = C_{n-1}[b]. Taking r_{n,b}(x) to be the indicator function which is 1 on the cylinder (when x\in C_n[b]), and zero otherwise, one has that Gr_{n,b}=r_{n-1,b}. The series
