Entire groups generated by fractional powers of operators

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 L\'evy 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 Ces\`aro means.