Abstract: The Crouzeix conjecture claims that the numerical range of a bounded operator on a complex Hilbert space is a 2-spectral set. Sharpness of the bound can, for example, be seen with the two by two nilpotent [0 1; 0 0]. This summer, for many in the operator theory adjacent community, it was the proof of the scalar Crouzeix conjecture (two proofs actually, both by ChatGPT) that acted the part of the canary. The two proofs exhibit different methods, despite a shared dependence on positivity conditions. On one side function theory is used to prove an equivalent statement, the Ransford-Schwenninger condition, on the other classic dilation theory is shown to suffice. Neither method extends directly to the completely bounded case, which as of last week is a casualty of the OpenAI SlopDrop (I credit Burnstein, Iosevich, Krause, and Yavicoli with coining the term). 

 

Faculty host: Alan Chang