Strongly Kreiss bounded operators in UMD Banach spaces

In this paper we give growth estimates for \(\Vert T^n\Vert \) for \(n\rightarrow \infty \) in the case T is a strongly Kreiss bounded operator on a \({{\,\textrm{UMD}\,}}\) 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 \({{\,\textrm{UMD}\,}}\) spaces such as non-commutative \(L^p\)-spaces and variable Lebesgue spaces.