Crouzeix's conjecture, compressions of shifts, and classes of nilpotent matrices
Kelly Bickel, Georgia Corbett, Annie Glenning, Changkun Guan, Martin Vollmayr-Lee · Concrete Operators · 2024
Abstract This article studies the level set Crouzeix (LSC) conjecture, which is a weak version of Crouzeix’s conjecture that applies to finite compressions of the shift. Among other results, this article establishes the LSC conjecture for several classes of 3 × 3 3\times 3 , 4 × 4 4\times 4 , and 5 × 5 5\times 5 matrices associated with compressions of the shift via a geometric analysis of their numerical ranges. This study also establishes Crouzeix’s conjecture for several classes of nilpotent matrices whose studies are motivated by related compressions of shifts.