Crouzeix's conjecture, compressions of shifts, and classes of nilpotent matrices

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 3×33\times 3, 4×44\times 4, and 5×55\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.

Keywords

• nilpotent matrices
• compressions of shifts
• crouzeix’s conjecture
• numerical ranges
• primary 47a12
• 47a25
• secondary 30j10
• 15a60