Crouzeix's conjecture

    Definition. Given closed convex subsets $C_1$ and $C_2$ of $\mathbb{C}$, define \[ W(C_1,C_2):=\bigcup\Bigl\{W(f(T)): W(T)\subset C_1,~f(C_1)\subset C_2\Bigr\}. \] Here the union is taken over:

• all operators $T$ (on any $H$) such that $W(T)\subset C_1$, and
• all functions $f$ holomorphic near $C_1$ such that $f(C_1)\subset C_2$.

Examples. Let $D$ be a closed disk, let $rD$ be the disk with the same centre and radius scaled by a factor $r$, and let $H$ be a closed half-plane. Then:

• $W(D,D)=(5/4)D$ (Drury, ’08).
• $W(C_1,D)\subset\sqrt2\,D$ (Crouzeix–Palencia, ’17).
• $W(H,C_2)=C_2$ (Kato, ’65).
• $W(D,H)=\mathbb{C}$ (Kato, ’65).

Remark. The Crouzeix conjecture is equivalent to the statement that $W(C_1,D)\subset (5/4)D$ for all $C_1,D$.