Crouzeix's conjecture

    Let $T$ be an operator on a Banach space $X$ and assume that the spectrum is contained in $\bar{\mathbb{D}}$ as well as $C=\sup_{|z|>1}\|(z-1)(zI-T)^{-1}\|<\infty$. Then, using contour integrals, it can be shown that $$\sup_{n\in\mathbb{N}}\|T^{n}\|\leq K C \log(1+C)$$ for some absolute constant $K$, see Bakaev’03 ("Constant-size control in stability estimates under some resolvent estimates" in "Numerical methods and programming", 2003 (4)), and [MR3474352]).

    If we have additionally the spectrum in the open unit disc

$$|| (zI-T)^{-1} || \le M$$ for $|z| \ge 1$.

Then it is known that with $C=4$ the following holds

$$\sum_{j=1}^\infty || T^j || \le C M (M-1)$$