6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Michał Wojtylak

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 supW(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+√ 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.

  1. Crouzeix, Michel. “Numerical range and functional calculus in Hilbert space.” Journal of Functional Analysis 244.2 (2007): 668-690.
  2. 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.
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