Spectral constants for algebraic numerical ranges

Abstract

We study spectral-set constants associated with the algebraic numerical range in unital Banach algebras. If $Γ_n$ denotes the universal constant for algebraic elements of degree at most $n$, then \[ Γ_1=1, \qquad 2n-1\leqslantΓ_n<\infty\qquad(n\geqslant2). \] Thus the algebraic numerical range is a spectral set with a constant depending only on the algebraic degree. In degree two we obtain \[ 3\leqslantΓ_2 \leqslant\sqrt{1+(2\mathrm e-1)^2}<4.55. \] For unital $C^*$-algebras we prove a gap theorem: the algebra-level constant is one precisely in the commutative case, whereas every non-commutative algebra has constant at least two. Moreover, the constant of the Jiang--Su algebra is exactly the universal Crouzeix constant. In the opposite direction, we construct a norm-one operator with a contractive polynomial calculus on the unit disc but infinite numerical-range spectral constant, and we show that the canonical left shift on every spreading combinatorial space in a broad class has infinite constant. These results settle the three questions posed by Blazhko, Homza, Schwenninger, de Vries, and Wojtylak.