instituto de matemáticas universidad de sevilla
Antonio de Castro Brzezicki
Some recent results around the Volterra and Cesàro operators.
We shall first focus on the Volterra operator  $V(f)(x)=\displaystyle \int_0^x f(t)\; dt$ viewed in the extreme case $V:L^1(0,1)\longrightarrow C([0,1])$. We shall prove that although it is not compact, it satisfies some weak form of compactness (finite strict singularity) and compute its Bernstein numbers, its approximation numbers and its essential norm. We shall also consider the Cesáro operator  $\displaystyle\Gamma(f)(x)=\frac{1}{x}\int_0^x f(t)\; dt$ (not bounded on $L^1$), with the same kind of questions.