A one parameter family of Volterra-type operators

For every α ∈ (0, +∞) and p, q ∈ (1, +∞) let T_α be the operator L^p[0, 1] → L^q [0, 1] defined via the equality (T_α f )(x) := \int_0^{x^α} f (y) dy. We study the norms of Tα for every p, q. In the case p = q we further study its spectrum, point spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case p = q = 2 we determine the point spectrum and eigenfunctions for T_α^* T_α, where T_α^* is the adjoint operator.