Mapping theorems for numerical range · Question 11.1

Question context

{\bf Definition.} Given closed convex subsets $C_1$ and $C_2$ of $\mathbb{C}$, define

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 $f(C_1)\subset C_2$.

{\bf 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).

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

Dated source record, inspected 2026-09-10. Current status has not been independently reassessed. Source-reported results have not been independently verified.

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