Numerical ranges of composition operators

When is zero in the numerical range of a composition operator?

“…We have already seen that that the desired "interior zero-containment" happens whenever ϕ is not univalent (Theorem 3.6). The next result, part of which also appears in [26,Section 3], shows that interior zero-containment also happens whenever the derivative of ϕ at the fixed point is non-positive.…”

“…We remark that Matache, who studies numerical ranges of composition operators in [26], presents some results (Theorems 3.1, 3.4, and 3.5) whose conclusions imply 0 ∈ int W (C ϕ ); in each of these results ϕ is not univalent. Acknowledgment.…”

“…(c) the convex hull of a sequence that spirals monotonically towards the origin if r is not real (hence a polygon containing the origin in its interior, see [26,Proposition 2.2]). Case (a) shows that the origin need not always belong to the numerical range of a composition operator not the identity, and (b) shows that the origin can belong to W (C ϕ ) but not its interior.…”

“…Corner points of numerical ranges of composition operators are discussed in [26], which contains, e.g., the following result: if C ϕ is compact with ϕ a nonconstant function satisfying ϕ(0) = 0 and ϕ (0) = 0, then 0 is an interior point of W (C ϕ ) and 1 is the only possible corner point of W (C ϕ ) (which equals W (C ϕ ) because 0 ∈ W (C ϕ ) and C ϕ is compact). Here, we obtain a complete characterization of corner points W (C ϕ ) when C ϕ is compact.…”

“…In the proof of [26,Theorem 3.6], Matache shows that if ϕ induces a compact operator and fixes a point p ∈ U\{0} such that p doesn't belong to the iterate sequence (ϕ [n] (0)), then the set {ϕ (p) n : n ≥ 1} belongs to the interior of W (C ϕ ). The following Theorem shows that the restrictions of compactness and p ∈ {ϕ…”