Strongly Kreiss bounded operators in UMD Banach spaces

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