The Crouzeix conjecture. Deformations.

Seminarium: 
Dyskretna analiza harmoniczna i niekomutatywna probabilistyka
Osoba referująca: 
Michał Wojtylak (Uniwersytet Jagielloński)
Data: 
czwartek, 24. Maj 2018 - 10:15
Sala: 
604
Opis: 
Crouzeix observed in 2007 that for any operator A in a Hilbert space and any polynomial p \[ \| p(A) \| \leq C \sup_W |p| \] where W is the numerical range of A and the constant C is universal, i.e. does not depend neither on the operator nor on the space. He also proved in the same paper that $2 \leq C \leq 11.08$ and conjectured that $C=2$. We will review recent developments on proving the conjecture ($C=1+\sqrt2$) and show some deformations of the numerical range that may lead to new constants.