A one parameter family of Volterra-type operators

For every α ∈ (0,+∞) and p,q ∈ (1,+∞) let T_α be the operator Lᵖ[0,1]→ Lq[0,1] defined via the equality (T_α f)(x) := ∫₀x^α f(y) d y. We study the norms of T_α 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.