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8. Optimal constant dependence for Ritt-operators

A problem that is a bit less connected to Crouzeix’s conjecture, but may require similar techniques coming from function theory.
    1.     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]).

      Problem 8.1.

      [F. Schwenninger] Is this $C$-dependence optimal? What about finite dimensions? In the examples I know the dependence is linear.
        •     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

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

          Problem 8.2.

          [O. Nevanlinna] What is the best constant $C$ here?

              Cite this as: AimPL: Crouzeix's conjecture, available at http://aimpl.org/crouzeix.