Quasi-nilpotency of generalized Volterra operators on sequence spaces

We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted $\ell^p$ spaces $1<p<+\infty$ . Our main result is that when an analytic symbol $g$ is a multiplier for a weighted $\ell^p$ space, then the corresponding generalized Volterra operator $T_g$ is bounded on the same space and quasi-nilpotent, i.e. its spectrum is $\{0\}.$ This improves a previous result of A. Limani and B. Malman in the case of sequence spaces. Also combined with known results about multipliers of $\ell^p$ spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on $\ell^p$. We approach the problem by introducing what we call Schur multipliers for lower triangular matrices and we construct a family of Schur multipliers for lower triangular matrices on $\ell^p, 1<p<\infty$ related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on $\ell^2$, extending a result of E. Ricard.