AbstractWe study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted $$\ell ^p$$
ℓ
p
spaces $$1<p<+\infty $$
1
<
p
<
+
∞
. Our main result is that when an analytic symbol g is a multiplier for a weighted $$\ell ^p$$
ℓ
p
space, then the corresponding generalized Volterra operator $$T_g$$
T
g
is bounded on the same space and quasi-nilpotent, i.e. its spectrum is $$\{0\}.$$
{
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 $$\ell ^p$$
ℓ
p
spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on $$\ell ^p$$
ℓ
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 $$\ell ^p, 1<p<\infty $$
ℓ
p
,
1
<
p
<
∞
related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on $$\ell ^2 $$
ℓ
2
, extending a result of E. Ricard.