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This paper studies the level set Crouzeix conjecture, which is a weak versionof Crouzeix's conjecture that applies to finite compressions of the shift.Amongst other results, this paper establishes the level set Crouzeix conjecturefor several classes of $3\times3$, $4\times4$, and $5\times5$ matricesassociated to compressions of the shift via a geometric analysis of theirnumerical ranges. This paper also establishes Crouzeix's conjecture for severalclasses of nilpotent matrices whose studies are motivated by relatedcompressions of shifts.

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