IJPAM: Volume 21, No. 4 (2005)

CAUCHY ESTIMATES FOR
THE OPERATIONAL CALCULUS

S. Kantorovitz
Bar-Ilan University
52900 Ramat-Gan, ISRAEL
e-mail: kantor@math.biu.ac.il

Abstract.Let be a (linear, not necessarily bounded) operator on a Banach space , whose resolvent contains the open unit disc $\Omega$ (or the set $\{z;\vert z\vert>1\}$ ), and whose resolvent operator $R(\cdot)$ satisfies the inequality $\vert\vert(1-\vert z\vert)R(z)\vert\vert\leq M$ for all $z\in\Omega$ (or $\vert z\vert>1$ , respectively). We show that $\vert\vert[(1-\vert z\vert)R(z)]^n\vert\vert<Men$ for all $n\in\mathbb N$ and $z\in\Omega$ ( $\vert z\vert>1$ , respectively). In case is Hilbert space, and is a contraction satisfying $\vert\vert(1-\vert z\vert)R(z)\vert\vert\leq M$ in $\Omega$ , one has $\vert\vert[(1-\vert z\vert)R(z)]^n\vert\vert=O(1)$ for all and $\vert z\vert\neq 1$ . These resolvent estimates imply Cauchy-type estimates for $f^{(n)}(T)$ .