IJPAM: Volume 21, No. 4 (2005)


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

Abstract.Let $T$ be a (linear, not necessarily bounded) operator on a Banach space $X$, 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 $X$ is Hilbert space, and $T$ 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 $n$ and $\vert z\vert\neq 1$. These resolvent estimates imply Cauchy-type estimates for $f^{(n)}(T)$.

Received: May 9, 2005

AMS Subject Classification: 47A60, 47A63

Key Words and Phrases: operational calculus, Banach space, Hilbert space, Cauchy-type estimates

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 21
Issue: 4