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.

我们研究了广义 Volterra 算子在加权\(\ell ^p\)空间\(1<p<+\infty \) 中具有泰勒系数的幂级数空间的拟幂零性。我们的主要结果是，当解析符号g是加权\(\ell ^p\)空间的乘数时，相应的广义 Volterra 算子\(T_g\)有界于相同的空间和拟幂零，即它的频谱是\(\{0\}.\)这改进了 A. Limani 和 B. Malman 在序列空间情况下的先前结果。还结合关于\(\ell ^p\)空间乘数的已知结果，我们给出了有界拟幂零广义 Volterra 算子的非平凡例子\(\ell ^p\)。我们通过为下三角矩阵引入所谓的 Schur 乘子来解决这个问题，并且我们为与可和核相关的\(\ell ^p, 1<p<\infty \)上的下三角矩阵构建了一系列 Schur 乘子。为了证明我们的结果的威力，我们还在\(\ell ^2 \)上为 Hankel 算子找到了一类新的 Schur 乘子，扩展了 E. Ricard 的结果。