Estimates for the norm of powers of Volterra's operator through its singular values (Q5932127)

The author proves that any Volterra operator \(T\) on a Hilbert space satisfies \[ \|T^{cm}\|\leq b^{cm}(s_1(T)\cdots s_m(T))^c,\quad m\in\mathbb{N} \] for \(c=2\) and some absolute constant \(b> 0\). Here \(s_j(T)\) denote the singular numbers of \(T\). For some larger \(c\), e.g. \(c=3\), this was proved by Solomjak; for \(c=1\) it is conjectured but not known. The constant \(b\) needs to be \(\geq\pi/2\) if \(c=2\). The proof uses the functional calculus and complex analytic estimates.