The Volterra operator is not supercyclic
If \((B,\|.\|_B)\) is a Banach space with norm \(\|.\|_B\), then according to statements of the introduction of this paper, a bounded linear operator \(T\) on \(B\) is said to be supercyclic if there is a vector \(f\) in \(B\) such that \[ \{\lambda_n T^n(f),\,\lambda_n\in \mathbb{C},\;n= 0,1,2,3,\dots\} \] constitutes a dense subspace of \(B\). The vector \(f\) is then said to be supercyclic. If \(1\leq p<\infty\), let \(L^p([0,1])\) denote the space \[ \Biggl\{[f]:\| f\|_p= \Biggl(\int^1_0|f(t)|^p\,dt\Biggr)^{1/p}< \infty\Biggr\}, \] where \([f]\) is the equivalence class of functions which differ from \(f\) on sets of measure \(0\). The Volterra operator \(V\) is defined by \[ V(f)(x)= \int^x_0 f(t)\,dt, \] so that \(|V(f)(x)|\leq x^{1/p'}\| f\|_{p'}\leq\| f\|_{p'}\) (\(0< x< 1\); \(f\in L^p([0,1])\)), \((1/p')+ (1/p)= 1\). The main result of the present paper indicates that the Volterra operator is not supercyclic in \(L^p([0,1])\) for any \(p\), \(1\leq p<\infty\).
scientific article; zbMATH DE number 2138207