序列空间上广义Volterra算子的拟幂零性
Quasi-nilpotency of generalized Volterra operators on sequence spaces
AI总结：
该研究引入下三角矩阵Schur乘子方法，证明加权ℓᵖ空间上解析符号乘子对应的广义Volterra算子有界且拟幂零，改进了已有结果并构造了相关例子与新的Schur乘子类。
摘要：
我们研究了泰勒系数属于加权ℓᵖ空间（1<p<+∞）的幂级数空间上广义Volterra算子的拟幂零性。我们的主要结果是：当解析符号g是某加权ℓᵖ空间的乘子时，对应的广义Volterra算子T_g在该空间上有界且拟幂零，即其谱为{0}。这在序列空间的情形下改进了A. Limani和B. Malman此前的结果。结合已知的ℓᵖ空间乘子结果，我们还给出了ℓᵖ空间上有界拟幂零广义Volterra算子的非平凡例子。我们通过引入所谓的下三角矩阵Schur乘子来研究该问题，并构造了一族与可和核相关的、作用于ℓᵖ（1<p<∞）上下三角矩阵的Schur乘子。为展示结果的效力，我们还发现了一类新的ℓ²上Hankel算子的Schur乘子，推广了E. Ricard的结果。
英文摘要：
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.
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