Strongly Kreiss Bounded Operators in UMD Banach Spaces

In this paper we give growth estimates for $\mathrm{\parallel }{T}^n\mathrm{\parallel }$ for $n\to \mathrm{\infty }$ in the case $T$ is a strongly Kreiss bounded operator on a UMD Banach space $X$. In several special cases we provide explicit growth rates. This includes known cases such as Hilbert and $L^p$-spaces, but also intermediate UMD spaces such as non-commutative $L^p$-spaces and variable Lebesgue spaces.