**The Crouzeix conjecture and deformations of the numerical range.**

Crouzeix observed in [1] that for any operator A in a Hilbert space and any polynomial p

∥p(A)∥ ≤ C sup_{W(A)} |p|

where W(A) is the numerical range of A and the constant C>0 is universal, i.e. does not depend neither on the operator nor on the space.

He also proved in the same paper that 2 ≤ C ≤ 11.08 and conjectured that C=2.

We will review recent developments on proving the conjecture (C ≤1+√ 2 in [2]) and show some deformations of the numerical range that lead to new constants.

The talk is based on joint work with P. Pagacz and P. Pietrzycki.

- Crouzeix, Michel. “Numerical range and functional calculus in Hilbert space.” Journal of Functional Analysis 244.2 (2007): 668-690.
- Crouzeix, Michel, and César Palencia. “The Numerical Range is a (1+\sqrt 2)-Spectral Set.” SIAM Journal on Matrix Analysis and Applications 38.2 (2017): 649-655.