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## 1. Investigation of the Crouzeix-Palencia proof

This section deals with the careful analysis of the proof in order to think of possible adaption in order to improve the constant.
1. #### Problem 1.1.

[E. Wegert] Better estimate for $fg$ than $\|fg\|\leq \|f\|\|g\|$. Modifications of the Cauchy integral, especially concerning its behavior on constant functions.
The motivation is that the current proof does not deliver $C_{\mathbb{D}}=2$ (even not for $f(0)=0$).
1. Remark. see A. Greenbaum’s talk
• #### Problem 1.2.

[T. Ransford] Let $\theta$ be a continuous homomorphism from a uniform algebra $A$ into a $C^{*}$-algebra $B$ and $\alpha:A\rightarrow A$ be antilinear and such that
• [*] $\|\alpha f\|\leq \|f\|$ for all $f\in A$,
• [*] $\|\theta f +(\theta\alpha f)^{*}\|\leq 2\|f\|$ for all $f\in A$, and
• [*] $\alpha 1=1$.
Does this imply that $\|\theta\|=2$?
Clearly, the proof of Crouzeix-Palencia is a particular instance of the above situation, however with a weaker constant in the implication. However, their proof does not use $\alpha 1=1$! If one drops this assumption, then the implication is not true in general. There exist a counterexample (where $\alpha$ is not the Cauchy transform).
• #### Problem 1.3.

[J. Burke] Does it help to replace $\lambda^2$ in the proof by $p(\lambda)$ for polynomial other than $\lambda^2$?
• #### Problem 1.4.

[T. Ransford] Let $f$ be such that $|f(z)|=1$ for $z\in\partial\Omega$. Then $$C(\bar{f},z)=\frac{1}{f(z)}-\sum_{\xi_{j}=\text{zero of }f} \frac{R(\frac{1}{f},\xi_{j})}{z-\xi_{j}}.$$ How to prove that $C(\bar{f},z)|\leq 1$? (what about Blaschke products?)
• #### Problem 1.5.

[T. Ransford] Let $f$ be such that $|f(z)|=1$ for $z\in\partial\Omega$. Then $$C(\bar{f},z)=\frac{1}{f(z)}-\sum_{\xi_{j}=\text{zero of }f} \frac{R(\frac{1}{f},\xi_{j})}{z-\xi_{j}}.$$ How to prove that $C(\bar{f},z)|\leq 1$ directly?
1. Remark. B. Beckermann: For $z\in\partial \Omega$, $C(\bar{f},z)=$ Kovari et al. Green’s formula “vanished” in the recent developments, Faber operators
• #### Problem 1.6.

[A. Greenbaum] Can anything be said about $\delta,\hat{\delta}$, in particular $$\delta_{\phi(A)}=-\int_{\partial \mathbb{D}}\lambda_{min}(\mu(\sigma(s),\phi(A))ds?$$
• #### Problem 1.7.

[A. Greenbaum] In what situations (other than in Caldwell-Greenbaum-Li) does the estimate $$\|f(A)+g(A)^{*}\|\geq \|f(A)\|$$ hold? Here, $f=B\circ \phi$ is such that $B$ is a Blaschke product, $\phi$ is a conformal mapping from $\Omega$ to $\mathbb{D}$ and such that $\|f(A)\|$ is maximized over all all $\|f\|_{\Omega}=1$.
In this case, the conjecture holds.
• #### Problem 1.8.

[A. Greenbaum] What if $\Omega$ consists of a union of disjoint regions? It still holds that $$\|f(A)+g(A)^{*}\|\leq 2 +\delta,$$ but how does is this related to $\|f(A)\|$? What is the optimal $f$, like in the case of the union of two disks?

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