A one parameter family of Volterra-type operators

For every $\alpha \in (0,+\mathrm{\infty })$ and $p,q\in (1,+\mathrm{\infty })$ let $T_{\alpha }$ be the operator $L^p[0,1]\to L^q[0,1]$ defined via the equality $(T_{\alpha }f)(x):={\int }_0^{x^{\alpha }}f(y)dy$. We study the norms of $T_{\alpha }$ 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 $T_{\alpha }^{\ast }{T}_{\alpha }$, where $T_{\alpha }^{\ast }$ is the adjoint operator.