A one parameter family of Volterra-type operators

For every \(\alpha \in (0,+\infty )\) $\alpha \in (0,+\infty )$ and \(p,q \in (1,+\infty )\) $p,q\in (1,+\infty )$ let \(T_\alpha \) $T_{\alpha }$ be the operator \(L^p[0,1]\rightarrow L^q[0,1]\) $L^p[0,1]\to L^q[0,1]$ defined via the equality \((T_\alpha f)(x):= \int _0^{x^\alpha } f(y) \,\textrm{d}y\) $(T_{\alpha }f)(x):={\int }_0^{x^{\alpha }}f(y)\text{d}y$ . We study the norms of \(T^{\phantom {*}}_\alpha \) $T_{\alpha }^{\phantom{\ast }}$ for every p, q. In the case \(p=q\) $p=q$ we further study its spectrum, point spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case \(p=q=2\) $p=q=2$ we determine the point spectrum and eigenfunctions for \(T^*_\alpha T^{\phantom {*}}_\alpha \) $T_{\alpha }^{\ast }T_{\alpha }^{\phantom{\ast }}$ , where \(T^*_\alpha \) $T_{\alpha }^{\ast }$ is the adjoint operator.