A range-dependent refinement of the Crouzeix--Palencia spectral constant

Crouzeix and Palencia proved that if Ω is a bounded convex domain containing theclosure of the numerical range of an operator A, then Ω is a(1+√2)-spectral set for A. Their proof estimates a certain product $fg$, whereg is the Cauchy transform of f̄, by fg≤ 1; this uses only thefirst half of their Lemma 2.1, namely g≤f. We observe that the secondhalf of that lemma---the statement that g(∂Ω) lies in the convex hull off(∂Ω)̄---has not been exploited, and that it controls theargument of $fg$, not merely its modulus. Replacing the modulus bound by a bound on(fg) yields the inequality \|f(A)\|²≤ 2CΩ+η(E), whereE=f(∂Ω)̄ and η(E)=max\-(u w):u,w∈ E\. Twoconsequences follow. First, for every f with f=1 we obtain the unconditionalestimate \|f(A)\|≤((1+√2)²-d²/2)1/2, whered=(E,-E); the bound is strictly smaller than 1+√2 whenever $d>0$ anddegenerates to the Crouzeix--Palencia constant exactly when $d=0$. Secondly, for functionswhose values lie in a sector of half-angle β with vertex at the origin, aself-improving version of the argument gives the constant 1+√2\,sinβ forβ∈[π/4,π/2]. This is a continuous interpolation between two results ofCrouzeix and Palencia: it reproduces their Remark 3.5 (the constant $2$ for a right-angledsector) at β=π/4 and their main theorem at β=π/2, and appears to be new inbetween. All estimates hold for arbitrary bounded convex Ω, arbitrary complexHilbert spaces and arbitrary dimension. We also explain why this circle of ideas cannotlower the universal constant Q.