Abstract:Let $T$ be a power-bounded operator on a Banach space $X$. We treat the sequence of polynomials $(p_{n;a})_{n\ge 0}$ such that the entire group generated by the fractional power operator $-(I-T)^a$ is given by $$ e^{-t(I-T)^a}=e^{-t}\sum_{n=0}^\infty p_{n;a}(t)T^n, \qquad t\in \CC, \quad \Re a>0. $$
We provide a self-contained introduction to the polynomial family $(p_{n;a})_{n\ge 0}$, for $a\in \CC$, whose coefficients are determined by means of a suitable recurrence relation. The sequence $(n!p_{n;a})_{n\ge 0}$ forms a family of Sheffer polynomials. For $\Re a>0$ and $t\in \CC$, the sequence $(p_{n;a}(t))_{n\ge 0}$ belongs to the Lebesgue sequence space $\ell^1$ of absolutely summable sequences. Moreover, these polynomials are closely related to the Lévy density functions $(f_{t,\alpha})_{t>0}$ defined for $0<\alpha<1$.
Finally, we discuss several particular cases corresponding to specific values of $a\in \CC$, as well as applications to fractional powers in the Banach algebra $\ell^1$, multiplication operators, and Cesàro means.
From: Pedro Miana [view email]
[v1]
Mon, 15 Jun 2026 16:07:04 UTC (21 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。