WHEN IS ZERO IN THE NUMERICAL RANGE OF A COMPOSITION OPERATOR?

PAUL S. BOURDON AND JOEL H. SHAPIRO

Abstract. We work on the Hardy space H^{2} of the open unit disc 𝕌, and consider the numerical ranges of composition operators C_{φ} induced by holomorphic self-maps φ of 𝕌. For maps φ that fix a point of 𝕌 we determine precisely when 0 belongs to the numerical range W of C_{φ}, and in the process discover the following dichotomy: either 0 ∈ W or the real part of C_{φ} admits a decomposition that reveals it to be strictly positive-definite. In this latter case we characterize those operators that are sectorial. For compact composition operators our work has the following consequences: it yields a complete description of the corner points of the closure of W, and it establishes when W is closed. In the course of our investigation we uncover surprising connections between composition operators, Chebyshev polynomials, and Pascal matrices.

1. Introduction

The study of composition operators turns the lens of operator theory upon the classical theory of holomorphic self-maps of the open unit disc. Let 𝕌 denote this disc, and H(𝕌) the space of all functions holomorphic thereon. The idea is to associate with each holomorphic φ : 𝕌 → 𝕌 the composition operator C_{φ} defined on H(𝕌) by the equation

C_{φ}f = f ∘ φ (f ∈ H(𝕌)).

The operator so defined is linear on H(𝕌), and continuous in the natural (compact-open) topology of that space; the goal is to connect the function-theoretic properties of φ with the operator-theoretic behavior of C_{φ}.

The subject’s true appeal originates with J.E. Littlewood’s remarkable Subordination Theorem [24, 1925], which guarantees that each composition operator restricts to a bounded linear transformation on the Hardy space H^{2} of 𝕌. Littlewood’s Theorem provides a striking way of connecting the function theory of the unit disc with the theory of bounded linear operators on Hilbert space. The past three decades have witnessed enthusiastic exploration of this connection, as partially recorded in the monographs [8] and [32], and the recent conference proceedings [17].

In this paper we study how the numerical range of a composition operator on H^{2} is influenced by the properties of its inducing map. The numerical range of a bounded linear operator T on a Hilbert space H is the set of complex numbers

W (T) = {⟨T f, f⟩ : f ∈ H, ∥f∥ = 1}.

Some important properties of the numerical range are:

Date: October 17, 2000.

This research was supported in part by the National Science Foundation.

(a) it lies in the disc {|w| ≤ ∥T ∥} (obvious);

(b) it contains every eigenvalue of T (also obvious);

(c) its closure contains the spectrum of T (elementary, see e.g., [15, Theorem 1.2-1, p. 6] or [18, Problem 214, p. 115])

(d) it is not similarity invariant (elementary, but a major theme in this paper);

(e) it is convex (very mysterious!). Note in particular that according to property (b) the numerical range of any composition operator contains the point 1, since 1 is an eigenvalue (composition operators fix constant functions). Also note that by properties (c) and (e), the convex hull of the spectrum of T is contained in the closure of W (T) for any Hilbert space operator T.

Convexity was first proved for the boundary of the numerical range by Toeplitz in 1918 [34], and the result was completed the following year by Hausdorff [19]. This Toeplitz-Hausdorff Theorem has fascinated mathematicians ever since. Many proofs have been given, perhaps the simplest being that of Dekker [12] (see [18, Problem 210]), and the most conceptual that of Davis [9]. However much remains to be understood; for example, the numerical range is easily seen to be compact if H is finite dimensional, but for H merely separable the numerical range was only relatively recently shown (by Agler) to be a Borel set [1, 1982].

In this paper we focus on the following:

Zero-Inclusion Question. For which φ does W (C_{φ}) contain the origin?

We are drawn to this question for three reasons:

• The numerical range of any composition operator, except for the identity, contains the origin in its closure (Theorem 3.1 below).

• The numerical range of a compact operator is closed if and only if it contains the origin [11].

• Resolution of the Zero-Inclusion Question for W (C_{φ}) leads to a description of the corner points of the closure of W (C_{φ}) when C_{φ} is compact (Theorem 7.5 below).

When C_{φ} is compact, φ necessarily fixes a point in 𝕌 (see, e.g., [32, p. 84]). For those symbols φ that fix a point p ∈ 𝕌 we answer the Zero-Inclusion Question completely (Theorem 5.3): 0 does not belong to W (C_{φ}) if and only if there is an r ∈ (0, 1] such that

|p| ≤ \sqrt{r} and φ(z) = α_{p}(rα_{p}(z)) (z ∈ 𝕌),

where α_{p} is the self-inverse automorphism of 𝕌 given by α_{p}(z) = (p − z)/(1 − p̄z). In addition, we show that 0 ∈ W (C_{φ}) implies 0 ∈ int W (C_{φ}) as long as φ doesn’t have the form φ(z) = tz for some t ∈ [−1, 0].

Our work on the Zero-Inclusion Question occupies Sections 3, 4, and 5 of this paper. This work is completed in Section 5, where we consider composition operators whose symbols have the form

(1) φ = α_{p} ∘ (rα_{p}), for some p ∈ 𝕌 and r ∈ (0, 1).

We emphasize that for fixed r each of these “conformal dilations” φ is conformally conjugate to the ordinary dilation z → rz; hence, the corresponding composition operators C_{φ} are similar to the simple composition operator f(z) → f(rz) whose numerical range does not include zero (it is easily seen to be the interval (0, 1]). Despite the similarity of these operators, there is, as we indicated in the preceding paragraph, a zero-inclusion dichotomy: 0 ∉ W (C_{φ}) if |p| ≤ \sqrt{r} and 0 ∈ int W (C_{φ}) if |p| > \sqrt{r}. This points out the crucial role played in our work by the fact that the numerical range is not similarity-invariant. Both this lack of invariance and the zero-inclusion dichotomy are illustrated in the following sequence of figures, the second and third of which are computer-generated approximations. In all three figures φ = α_{p} ∘ (rα_{p}) with r = 1/4: in the first figure p = 0, i.e. φ is a dilation; in the second one p = 3/8, so that p < \sqrt{r}; and in the third p = 3/4, illustrating the behavior for p > \sqrt{r}.

Numerical ranges of three similar composition operators

Observe that in Figure 2, W (C_{φ}) appears to lie in the open right half-plane. Our work on the zero-inclusion dichotomy in Section 5 shows that Figure 2 representative: we prove that that the real part of C_{φ} is strictly positive whenever φ has the form (1) and |p| ≤ \sqrt{r}.

In Section 6 we continue our analysis of W (C_{φ}) for φ conformally conjugate to a positive dilation. We prove that when φ has the form (1) and |p| < r, then not only does W (C_{φ}) lie in the open right half-plane, it lies in a sector {z : | arg(z)| < θ} for some positive acute angle θ (Figure 2 is, once again, representative). However, we show that when |p| = \sqrt{r}, then W (C_{φ}) does not lie in such a sector. We thus obtain a characterization of the sectorial composition operators (among those whose symbols fix a point in 𝕌). Behind our analysis of the zero-inclusion and sectorial issues at the point p = \sqrt{r} lie interesting connections between composition operators, Pascal’s matrix, and Chebyshev polynomials.

In Section 7 we show how our work on the Zero-Inclusion Question leads to a complete characterization of the corner points of the closure of W (C_{φ}) when C_{φ} is compact. (An element p of a set E ⊂ ℂ is a corner point of E provided E is contained in a half-cone with vertex p and angular opening less than π.)

In the final section of this paper we discuss the Zero-Inclusion Question for mappings φ that do not fix a point of 𝕌. Our work in this section and in several others is simplified by the following result (Corollary 3.6), which requires no fixed-point hypothesis:

Suppose φ is a nonconstant holomorphic self-map of 𝕌. If φ is not one-to-one, then 0 ∈ int W (C_{φ}).

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. We wish to thank Wayne Dymacek of Washington and Lee University for providing us with helpful references concerning Pascal matrices and Chebyshev polynomials.

2. Preliminaries

We gather here, often without proof, some well known results about Hardy spaces, numerical ranges, and conformal automorphisms that will be needed for the sequel.

2.1. The space H^{2}. For functions f holomorphic on 𝕌 we use f̂(n) to denote n-th coefficient of f in its Taylor expansion about the origin. The Hardy space H^{2} is the collection of functions f holomorphic on 𝕌 with

∥f∥^{2} := ∑_{n=0}^{∞} |f̂(n)|^{2} < ∞.

The function ∥·∥ so defined is a norm that turns H^{2} into a Hilbert space with inner product

⟨f, g⟩ = ∑_{n=0}^{∞} f̂(n)\overline{ĝ(n)}.

By the Riesz-Fisher theorem the trigonometric series ∑_{0}^{∞} f̂(n)e^{inθ} associated with an H^{2} function f is the Fourier series of some function f^{∗} ∈ L^{2} = L^{2}(m, ∂𝕌), where m denotes arclength measure on ∂𝕌, normalized to have total mass one. Clearly the map f → f^{∗} takes H^{2} isometrically onto the closed subspace of L^{2} consisting of functions whose Fourier coefficients of negative index vanish. The boundary function f^{∗} is, in fact, the natural extension of f to ∂𝕌; it is just the radial limit function

f^{∗}(ζ) = lim_{r→1−} f(rζ),

where the limit exists for m almost every ζ ∈ ∂𝕌. From now on we simply write f(ζ) instead of f^{∗}(ζ) for each ζ ∈ ∂𝕌 at which this radial limit exists, letting the context determine the meaning of the symbol f.

With these observations the norm and inner product in H^{2} can be computed on the boundary of the unit disc as:

∥f∥^{2} = ∫_{∂𝕌} |f|^{2} dm (f ∈ H^{2})

and

⟨f, g⟩ = ∫_{∂𝕌} f\overline{g} dm (f, g ∈ H^{2}).

2.2. Reproducing kernels. Associated with each point a ∈ 𝕌 is its reproducing kernel

K_{a}(z) := \frac{1}{1 − \overline{a}z} = ∑_{n=0}^{∞} \overline{a}^{n}z^{n} (z ∈ 𝕌).

Clearly K_{a} ∈ H^{2} for every a ∈ 𝕌, and the series definition of the inner product for H^{2} yields immediately the reproducing property

⟨f, K_{a}⟩ = f(a) (f ∈ H^{2}, a ∈ 𝕌). (2)

The importance for composition operators of the collection of reproducing kernels stems in part from the adjoint property

C_{φ}^{∗}K_{a} = K_{φ(a)} (a ∈ 𝕌), (3)

which follows quickly from the reproducing property.

2.3. Compression and the numerical range. We have already defined the numerical range W (T) of an operator T on a Hilbert space H and noted its salient features. W (T) is just the image of the unit sphere of H under the quadratic form Q_{T} : H → ℂ associated with T:

Q_{T} (f) = ⟨T f, f⟩ (f ∈ H).

If M is a closed linear subspace of H and P the orthogonal projection of H onto M, then the restriction T_{M} of the operator P T to M is called the compression of T to M. It is easy to check that the quadratic form of T_{M} is just the restriction of Q_{T} to M, and so W (T_{M}) ⊂ W (T). It is an easy step from this to conclude that the convexity of the numerical range for any Hilbert space operator follows from the two dimensional case.

2.4. The two dimensional case. There are only a few possibilities for the numerical range of a linear operator on a two dimensional Hilbert space. One of these is trivial; if the operator is a scalar multiple of the identity then its numerical range is a single point (that scalar). So assume T is not a scalar multiple of the identity on H. Then there are only three possibilities:

(a) T has just one eigenvalue. Then W (T) is a closed disc centered at that eigenvalue.

(b) T is normal. Then T has two distinct eigenvalues (else it is a scalar multiple of the identity), and W (T) is the closed line segment joining these eigenvalues.

(c) T is not normal and has two distinct eigenvalues. Then W (T) is a closed elliptical disc with foci at the eigenvalues. Case (a) can be viewed as the limiting case of this, where the foci coincide.

The last case will be particularly important for us. If a non-normal T has distinct eigenvalues λ_{1} and λ_{2} then the dimensions of the elliptical disc W (T) are determined by the angle between the eigenvectors. More precisely, let f_{j} be an eigenvector for λ_{j} with ∥f_{j}∥ = 1 (j = 1, 2), and set γ = |⟨f_{1}, f_{2}⟩|. Then the major axis of W (T) has length

|λ_{1} − λ_{2}|/\sqrt{1 − γ^{2}},

and the minor axis has length γ times that of the major axis [18, page 113]. In particular, if T has matrix

[ \begin{matrix}λ_{1} & a \\ 0 & λ_{2} \end{matrix} ]

with respect to an orthonormal basis of ℂ^{2}, then the length of the major axis of W (T) is \sqrt{|a|^{2} + |λ_{1} − λ_{2}|^{2}}, while that of the minor axis is just |a| (see [15, Chapter 1] or [21, Chapter 1] for more details).

In studying composition operators, the case where the inducing function is a constant— whereupon the operator itself has rank one—is usually of little interest. The same is true here, but disposing of this simple case takes a little more work than one might expect. We do this right now (cf. [26, Theorem 2.4]).

2.5. Proposition. Suppose T is a bounded linear operator on a Hilbert space of dimension ≥ 2, and suppose the range of T is one dimensional, so that there exist nonzero vectors g and h with ∥h∥ = 1 such that

T f = ⟨f, g⟩h (f ∈ H).

(a) If g = ch for some c ∈ ℂ\\{0\} then W (T) is the closed line segment from 0 to \overline{c}.

(b) If g ⊥ h then W (T) is the closed disc centered at the origin, of radius ∥g∥/2

(c) Otherwise W (T) is a closed elliptical disc with foci at 0 and ⟨h, g⟩.

Proof. Since H has dimension > 1 it contains a nonzero vector orthogonal to g, and T therefore annihilates this vector, hence 0 ∈ W (T).

Let M be the linear span of g and h, and P the orthogonal projection of H onto M. Then an easy calculation shows that

⟨T f, f⟩ = ⟨P T P f, f⟩ (f ∈ H),

so that W(T)=W(PTP). Now PTP is the orthogonal direct sum of the compression T_{M} of T to M, operating on M, and the zero-operator acting on M^{⊥}. Thus W (P T P) is the convex hull of 0 and W (T_{M}), which equals W (T_{M}) because 0 ∈ W (T). Thus W (T) = W (T_{M}), so we may without loss of generality assume that H = M and T = T_{M}, which we do for the rest of the proof.

If g = ch for some nonzero scalar c we have

⟨T f, f⟩ = \overline{c}⟨f, h⟩⟨h, f⟩ = \overline{c}|⟨f, h⟩|^{2}

from which it follows easily that W (T) is the closed line segment from the origin to \overline{c}. If g is not a scalar multiple of h then set e_{1} = h and

e_{2} = a(g − ⟨g, h⟩h), where a^{−1} = ∥g − ⟨g, h⟩h∥.

Then the pair (e_{1}, e_{2}) is an orthonormal basis for H relative to which T has matrix

A = [ \begin{matrix}⟨h, g⟩ & b \\ 0 & 0 \end{matrix} ] , where b = a(∥g∥^{2} − |⟨g, h⟩|^{2}).

The point here is that b ≠ 0 because g is not a scalar multiple of h. Thus if g ⊥ h then

A = [ \begin{matrix}0 & ∥g∥ \\ 0 & 0 \end{matrix} ] , which has numerical range equal to the closed disc with center at the origin

and radius ∥g∥/2; otherwise W (A) is a closed elliptical disc with foci at 0 and ⟨h, g⟩.

2.6. Corollary. Suppose φ ≡ a ∈ 𝕌. If a = 0 then W (C_{φ}) is the closed interval [0, 1], otherwise it is a closed elliptical disc with foci at 0 and 1.

Proof. We have for every f ∈ H^{2}:

C_{φ}f = f(a) = ⟨f, K_{a}⟩ · 1,

where K_{a} is the reproducing kernel of Section 2.2. The result follows upon applying the previous Proposition with g = K_{a} and h ≡ 1.

2.7. Symmetry of W (C_{φ}). The following observation will prove useful in Sections 6 and 7:

If φ is real on the real axis (or, equivalently, has real Taylor coefficients at the origin), then W (C_{φ}) is symmetric with respect to the real axis.

If φ has real Taylor coefficients at 0, then the matrix of C_{φ} with respect to the basis {z^{n}} of H^{2} will have real entries. In this case, if w = ⟨C_{φ}f, f⟩ for some f in the unit sphere of H^{2}, then w̄ = ⟨C_{φ}f̃, f̃⟩, where f̃(z) = \overline{f(\overline{z})} is also in the unit sphere. Thus W (C_{φ}) will be symmetric relative to the real axis.

2.8. Conformal automorphisms and dilations. A conformal automorphism is a univalent holomorphic mapping of 𝕌 onto itself. Each such map is linear fractional, and can be represented as a product ω · α_{p}, where

α_{p}(z) := \frac{p − z}{1 − \overline{p}z} (z ∈ 𝕌) (4)

for some fixed p ∈ 𝕌 and ω ∈ ∂𝕌 [30, Theorem 12.6, page 255]. The special automorphism α_{p} is particularly convenient to use because it is self-inverse and exchanges the origin with the point p (see [32, Chapter 0], for example).

A dilation (more precisely, an r-dilation) is a map of the form δ_{r}(z) = rz, where r ∈ 𝕌. We call r the dilation parameter of δ_{r}. If r > 0 we call δ_{r} a positive dilation. Clearly the composition operator C_{δ_{r}} on H^{2} is compact and normal, having for each non-negative integer n the monomial z^{n} as an eigenvector with eigenvalue r^{n}. Easy arguments show that W (C_{δ_{r}}) is:

(a) the semi-open interval (0, 1] if r > 0,

(b) the closed interval [r, 1] if −1 < r ≤ 0.

(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. For non-dilations fixing the origin, however, the situation is different, as we will see in Theorem 3.8.

A conformal dilation (more precisely, a conformal r-dilation) is a map that is conformally conjugate to an r-dilation, i.e., a map φ = α^{−1} ∘ δ_{r} ∘ α, where r ∈ 𝕌 and α is a conformal automorphism of 𝕌. Each such map fixes the point p = α^{−1}(0) ∈ U. Now as we have previously noted, α = ρ_{ω} ∘ α_{p} for some ω ∈ ∂𝕌, where α_{p} is the special (self-inverse) automorphism defined by (4), and ρ_{ω} is the rotation defined by ρ_{ω}(z) ≡ ωz. From this it follows easily that φ = α_{p} ∘ δ_{r} ∘ α_{p}.

Finally, it is worth noting that for φ as in the last paragraph and η ∈ ∂𝕌, the map ψ := ρ_{η} ∘ φ ∘ (ρ_{η})^{−1} is a conformal r-dilation that fixes the point ηp. Now C_{ψ} = V^{−1}C_{φ}V, where V is the unitary composition operator induced on H^{2} by ρ_{η}, hence given any conformal dilation we may rotationally conjugate its fixed point p to any other point of the circle |z| = |p| at the cost of a unitary equivalence at the composition operator level. Since unitary equivalence does not change the numerical range we may, if it is convenient, assume with no loss of generality that the fixed point lies on the unit interval.

We will see in Section 4 that the conformal dilations that do not fix the origin, although simple in form, play a pivotal role in the study of zero-inclusion for the numerical range.

2.9. Zero-Inclusion for Automorphic Composition Operators. Numerical ranges of composition operators induced by conformal automorphisms are the focus of the authors’ paper [4], where it is shown such automorphic composition operators usually, but not always, have numerical ranges whose closures are discs centered at the origin. Conformal automorphisms of 𝕌, which as we have already noted are always linear-fractional maps, are classified as follows.

Hyperbolic: exactly two fixed points, both on ∂𝕌.

Parabolic: exactly one fixed point on ∂𝕌, no other fixed point on the Riemann sphere. Elliptic: Conformally conjugate to a rotation about the origin; two fixed points in the

sphere, one in 𝕌, one outside its closure. By Theorem 3.1 of [4], if φ is an automorphism of 𝕌 that is either of hyperbolic or parabolic type, then the closure of W (C_{φ}) is a disc centered at the origin, from which it follows that 0 ∈ int W (C_{φ}). The situation for elliptic automorphisms is more interesting.

An elliptic automorphism φ of 𝕌 must have the form

φ = α_{p} ∘ ρ_{ω} ∘ α_{p} (5)

for some p ∈ 𝕌 and some ω ∈ ∂𝕌. Suppose that φ is given by (5). If ω is not a root of unity, then by Theorem 4.1 of [4], the closure of W (C_{φ}) is a disk centered at the origin so that, just as for the other automorphic composition operators, W (C_{φ}) contains 0 in its interior. To answer the Zero-Inclusion Question when ω is a root of unity, we observe that for every nonnegative integer n, the map α_{p}^{n} will be an eigenvector for C_{φ} with eigenvalue ω^{n}. It follows, by convexity, that if ω is neither 1 nor −1, then 0 ∈ int W (C_{φ}). If ω = 1, then, of course, φ is the identity mapping and W (C_{φ}) = {1}. If ω = −1, then by Corollary 4.4 of [4], W (C_{φ}) is a ellipse with foci −1 and 1 that is degenerate if and only if p = 0 (in which case φ(z) = −z). Summarizing, we see that if φ is an elliptic automorphism, then 0 ∈ W (C_{φ}) unless φ(z) = z, and 0 ∈ int W (C_{φ}) except when φ(z) = z or φ(z) = −z. Finally, note that when φ(z) = −z, then W (C_{φ}) = [−1, 1], which is consistent with our description of numerical ranges of composition operators induced by dilations.

2.10. The Denjoy-Wolff Theorem. Every holomorphic self-map φ of the unit disc 𝕌 that is not conformally conjugate to a rotation has a unique attractive fixed point p in the closure of 𝕌 which is called the Denjoy-Wolff point of φ. If p ∈ ∂𝕌, then p is fixed in the sense that lim_{r→1^{−}} φ(rp) = p. The Denjoy-Wolff point p of φ is globally attractive: for each z ∈ 𝕌, the limit of the iterate sequence (φ^{[n]}(z)) is p. Hence, in particular, a self-map of 𝕌 that is not the identity may fix at most one point of 𝕌, a fact we will use without comment in the sequel. For a proof and further discussion of the Denjoy-Wolff theorem, the reader may consult, e.g., [32, Chapter 5].

3. Zero containment: Part I

Here we begin our investigation of when 0 ∈ W (C_{φ}). We commented earlier that for each 0 < r < 1 the positive-dilation map φ(z) ≡ rz gives rise to a normal composition operator whose numerical range is the half-open real interval (0, 1], so 0 is not always in W (C_{φ}). On the other hand, if φ is not univalent then C_{φ}^{∗} annihilates a difference of reproducing kernels, in which case 0 ∈ W (C_{φ}).

In this section we prove that for any composition operator not the identity, the origin lies in the closure of the numerical range. We prove that the origin lies in the interior of W (C_{φ}) whenever C_{φ} fails to have dense range (which happens, for example, whenever φ is not univalent). Then we focus on holomorphic self-maps φ of 𝕌 that fix the origin, proving that 0 ∈ int W (C_{φ}) unless φ has the form φ(z) = tz for some t ∈ [−1, 1]. This result, together with our observations in Sections 2.8 and 2.9 concerning numerical ranges for composition operators induced by dilations and rotations, settles the zero inclusion issue for composition operators with symbols that fix the origin. We wait until Section 4 to take up the fascinating case of maps that fix points of 𝕌 other than the origin.

We employ the notation \overline{W} (T) for the closure of the numerical range of an operator T. Note that a composition operator is the identity map on H^{2} if and only if its inducing function is the identity map on 𝕌.

3.1. Theorem. If φ is any holomorphic self-map of 𝕌 that is not the identity map then 0 ∈ \overline{W} (C_{φ}).

Proof. For a ∈ 𝕌 let f_{a} = K_{a}/∥K_{a}∥, where K_{a} is the reproducing kernel discussed in Section 2.2, and let

w_{a} = ⟨C_{φ}f_{a}, f_{a}⟩ = \frac{⟨C_{φ}K_{a}, K_{a}⟩}{∥K_{a}∥^{2}},

a point of W (C_{φ}). Since

∥K_{a}∥^{2} = ⟨K_{a}, K_{a}⟩ = K_{a}(a) = \frac{1}{1 − |a|^{2}} ,

we have for each point a ∈ 𝕌:

w_{a} = (1 − |a|^{2})⟨K_{a}, C_{φ}^{∗}K_{a}⟩ = (1 − |a|^{2})⟨K_{a}, K_{φ(a)}⟩

= (1 − |a|^{2})K_{a}(φ(a)) = \frac{1 − |a|^{2}}{1 − \overline{a}φ(a)}.

Now because φ is not the identity map on 𝕌 its radial limit function is not almost-everywhere equal to the identity map on ∂𝕌, so there exists ζ ∈ 𝕌 for which the radial limit φ(ζ) exists and is not equal to ζ. Thus, by the result of the last calculation, if a → ζ then the points w_{a} of W (C_{φ}) converge to zero, hence 0 ∈ \overline{W} (C_{φ}), as desired.

Next we generalize the observation that 0 ∈ W (C_{φ}) whenever φ is not univalent. The crucial idea is the notion of normal eigenvalue.

3.2. Normal eigenvalues. Suppose T is a bounded linear operator on Hilbert space, and λ is an eigenvalue of T. We say λ normal for T if

ker(T − λI) = ker(T^{∗} − \overline{λ}I).

This terminology reflects the elementary fact that the eigenvalues of a normal operator are normal eigenvalues.

By Theorem 2 of [20], every eigenvalue of an arbitrary Hilbert space operator T that lies on the boundary of W (T) must be a normal eigenvalue. We supply a proof different from the one in [20], which is based on the following.

3.3. Proposition. Suppose that T is a bounded linear operator on a Hilbert space H and that f is a unit vector in H such that ⟨T f, f⟩ belongs to the boundary of W (T). Then T^{∗}f is a linear combination of f and T f.

Proof. Suppose, in order to prove the contrapositive, that T^{∗}f is not a linear combination of f and T f. Then there is a unit vector g ∈ H that is orthogonal to both f and T f yet ⟨g, T^{∗}f⟩ ≠ 0. Let M denote the two-dimensional subspace of H with orthonormal basis {f, g} and let λ = ⟨T f, f⟩. The compression of T to M, which we denote T_{M}, has matrix

[ \begin{matrix}λ & ⟨T g, f⟩ \\ 0 & ⟨T g, g⟩ \end{matrix} ]

relative to {f, g}. Because ⟨T g, f⟩ = ⟨g, T^{∗}f⟩ is nonzero, the numerical range of T_{M} is a nondegenerate ellipse with foci λ and ⟨T g, g⟩. In particular, λ lies in the interior of W (T_{M}) ⊆ W (T).

3.4. Corollary. Suppose that T is a bounded operator on a Hilbert space H; then every eigenvalue of T that lies on the boundary of W (T) is a normal eigenvalue.

Proof. Suppose f is a unit vector in H and λ ∈ ∂W (T) satisfies T f = λf. By Proposition 3.3, T^{∗}f = αf for some scalar α. We have α = ⟨T^{∗}f, f⟩ = ⟨f, T f⟩ = λ̄, as desired. Because λ̄ belongs to ∂W (T^{∗}) whenever λ belongs to ∂W (T), the same argument shows ker(T^{∗} − \overline{λ}I) ⊆ ker(T − λI).

The preceding corollary shows that 0 belongs to the interior of W (T) whenever T is injective and fails to have dense range. We present the argument in the context of composition operators.

3.5. Corollary. Suppose φ is a nonconstant holomorphic self-map of 𝕌. If the polynomials in φ are not dense in H^{2}, then 0 ∈ int W (C_{φ}).

Proof. Because φ is nonconstant, C_{φ} is one-to-one. The hypothesis on φ means that the range of C_{φ} is not dense, hence its orthogonal complement, the kernel of C_{φ}^{∗}, is not zero. Thus zero is an eigenvalue of C_{φ}^{∗}, and therefore belongs to W (C_{φ}^{∗}). But zero is not a normal eigenvalue, so by Theorem 3.4 it cannot be a boundary point of W (C_{φ}^{∗}). Thus, because the numerical range of the adjoint operator is the complex conjugate of the original numerical range, 0 lies in the interior of W (C_{φ}).

3.6. Corollary. Suppose φ is a nonconstant holomorphic self-map of 𝕌. If φ is not oneto-one then 0 ∈ int W (C_{φ}).

Proof. As we noted earlier, if φ is not one-to-one then C_{φ}^{∗} annihilates a nontrivial difference of reproducing kernels, so the kernel of C_{φ}^{∗} is not {0}. Thus the closure of the range of C_{φ}, which is (ker C_{φ}^{∗})^{⊥}, is not all of H^{2}, i.e., ran C_{φ} is not dense. Thus, by Corollary 3.5, 0 ∈ int W (C_{φ}).

3.7. Remark. The work of Section 2.6 disposes of the case φ ≡ constant; if the constant is nonzero then zero is in the interior of W (C_{φ}), otherwise W (C_{φ}) is the closed unit interval.

We now consider the question of zero-inclusion for composition operators induced by maps φ that fix the origin. If φ is a dilation or rotation, then work of Sections 2.8 and 2.9 yields complete information about zero inclusion; in particular, 0 ∈ int W (C_{φ}) unless φ belongs to the family {z ↦ tz : t ∈ [−1, 1]}. We now prove that this family provides the only exception to the interior zero-inclusion rule among all holomorphic self-maps of 𝕌 that fix the origin.

3.8. Theorem. Suppose that φ is a holomorphic self-map of 𝕌 fixing the origin and that φ does not have the form φ(z) = tz for some t ∈ [−1, 1]. Then 0 ∈ int W (C_{φ}).

Proof. φ not constant, for otherwise it would be identically equal to zero (since φ(0) = 0), and hence would have the forbidden form φ(z) = tz with t = 0. Let λ = φ′(0). We may assume that λ ≠ 0 for otherwise φ is not univalent (and nonconstant) and the desired conclusion follows from Corollary 3.6. We may also assume that φ is neither a rotation or dilation (by the work of Sections 2.8 and 2.9, as discussed above). Thus φ’s MacLaurin expansion can be written in the form

φ(z) = λz[1 + bz^{m}(1 + h(z))],

where m is a positive integer, b ≠ 0, and h is holomorphic in 𝕌 with h(0) = 0. The binomial theorem then shows that for each positive integer n,

φ(z)^{n} = λ^{n}z^{n} + nbλ^{n}z^{n+m} + terms of higher order in z.

Now the matrix of C_{φ} with respect to the orthonormal basis {z^{k}}_{0}^{∞} has as its n-th column the sequence of power series coefficients of φ^{n}, and any compression of C_{φ} to the subspace spanned by a subset of these monomials has as its matrix the corresponding principal submatrix. We are going to focus on the linear span M_{n} of {z^{n}, z^{m+n}}; because m > 0 this subspace is two dimensional. Let T_{n} be the compression of C_{φ} to M_{n}, so matrix of T_{n} with respect to the orthonormal basis {z^{n}, z^{n+m}} is

B_{n} := [ \begin{matrix}φ̂^{n}(n) & 0 \\ φ̂^{n}(n + m) & { \atop φ^{n+m}}(n + m) \end{matrix} ]

= [ \begin{matrix}λ^{n} & 0 \\ nbλ^{n} & λ^{n+m} \end{matrix} ]
= λ^{n}A_{n},

where

A_{n} = [ \begin{matrix}1 & 0 \\ nb & λ^{m} \end{matrix} ]

It is enough to show that 0 ∈ int W (A_{n}) for large enough n. By the discussion of Section 2.4 we know that W (A_{n}) is a closed elliptical disc with foci at the points 1 and λ^{m} and minor axis of length n|b|. For n large enough the length of the minor semi-axis of W (A_{n}) will exceed the modulus of its center, thus placing the origin in the interior; this happens whenever n > |1 + λ^{m}|/|b|.

The preceding Theorem completes our solution to the Zero-Inclusion Question for composition operators induced by mappings φ that fix the origin: 0 does not belong to W (C_{φ}) if and only if φ(z) = tz for some t ∈ (0, 1], and 0 ∈ W (C_{φ}) implies 0 ∈ int W (C_{φ}) unless φ has the form φ(z) = tz for some t ∈ [−1, 0]. In the next section, we consider the much more subtle situation that arises when φ fixes a non-zero point of 𝕌.

4. Zero Containment: Part II

Our goal in this section is to show that if φ fixes a nonzero point in 𝕌 and is neither the identity map nor a positive conformal dilation in the sense of Section 2.8, then 0 ∈ int W (C_{φ}). For the moment, however, we assume only that φ is a holomorphic self-map of 𝕌 that fixes a point p ∈ 𝕌\{0}, and that φ is not an automorphism (the zero-containment story for automorphisms has already been told in Section 2.9). As in the proof of Theorem 3.8 we seek to compute appropriate two by two principal submatrices of the matrix of C_{φ}, but instead of computing this matrix relative to the monomial basis (z^{n})_{0}^{∞} of H^{2} we use instead Guyker’s basis. This is the sequence of functions (b_{n})_{0}^{∞} defined for z ∈ 𝕌 by:

b_{n}(z) := \frac{\sqrt{1 − |p|^{2}}}{1 − \overline{p}z} (\frac{z − p}{1 − \overline{p}z} )^{n} = \frac{K_{p}}{∥K_{p}∥}(−α_{p}(z))^{n} (6)

where K_{p} is the reproducing kernel for p (as defined in Section 2.2) and α_{p} is the special automorphism defined by (4).

In [16] Guyker established that (b_{n})_{0}^{∞} is an orthonormal basis for H^{2} relative to which the matrix of C_{φ} is lower triangular, with the derivative-power φ′(p)^{n} as the n-th element of the main diagonal (n = 0, 1, 2, …). Guyker used his basis to study the reducing subspaces of C_{φ}. Actually Guyker’s basis is a special case of an orthonormal basis used by F. Malmquist in 1928 to study interpolation problems in H^{2} ([25], see also [35, §10.7, pp. 304–309]). We are indebted to Professor Harold S. Shapiro for pointing out these references.

Here we focus on the compression of C_{φ} to the two dimensional subspace

M_{n,k} := span {b_{n}, b_{n+k}} (n ≥ 0, k ≥ 1).

Relative to the orthonormal basis (b_{n}, b_{n+k}) this compression has matrix

A_{n,k} := [ \begin{matrix}φ′(p)^{n} & 0 \\ ⟨C_{φ}b_{n}, b_{n+k}⟩ & φ′(p)^{n+k} \end{matrix} ] ,

so, upon noting that the Schwarz-Pick Lemma insures that |φ′(p)| < 1 and proceeding as in the proof of Theorem 3.8, we can establish that 0 ∈ int W (C_{φ}) by showing that for some fixed positive integer k:

lim_{n→∞} \frac{⟨C_{φ}b_{n}, b_{n+k}⟩}{|φ′(p)^{n}|} = ∞. (7)

Without further ado, let us compute the crucial inner product entry.

4.1. Lemma. Suppose φ is a holomorphic self-map of 𝕌 that fixes a point p ∈ 𝕌. Let (b_{n}) be the Guyker basis for the point p. Then for each non-negative integer n and each positive integer k,

⟨C_{φ}b_{n}, b_{n+k}⟩ = \frac{1 − |p|^{2}}{k!} H_{n,k}^{(k)}(p) (8)

where

H_{n,k}(z) = \frac{(1 − \overline{p}z)^{n+k}}{(1 − \overline{p}φ(z))^{n+1}} (\frac{φ(z) − p}{z − p})^{n} .

Proof. We begin by writing the H^{2} inner product in question as a complex contour integral

⟨C_{φ}b_{n}, b_{n+k}⟩ = \frac{1}{2πi} ∫_{∂𝕌} b_{n}(φ(z))\overline{b_{n+k}(z)} \frac{dz}{z} ,

where ∂𝕌 is positively oriented. We compute the integrand by noticing that |α_{p}(z)| = 1 when |z| = 1, so that the complex conjugate of α_{p}(z) is its reciprocal. This observation, along with some algebraic manipulation yields, for |z| = 1,

b_{n}(φ(z))\overline{b_{n+k}(z)} = (1 − |p|^{2}) H_{n,k}(z)\frac{z}{(z − p)^{k+1}} ,

hence

⟨C_{φ}b_{n}, b_{n+k}⟩ = \frac{1 − |p|^{2}}{2πi} ∫_{∂𝕌} H_{n,k}(z) \frac{dz}{(z − p)^{k+1}} = \frac{(1 − |p|^{2})}{k!}H_{n,k}^{(k)}(p),

as desired.

When k = 1, the preceding Lemma yields the following.

4.2. Corollary. Under the hypotheses of Lemma 4.1 we have for each non-negative integer n,

⟨C_{φ}b_{n}, b_{n+1}⟩ = φ′(p)^{n−1} [n(1 − |p|^{2})\frac{φ′′(p)}{2} − (n + 1) \overline{p} φ′(p)(1 − φ′(p))] .

This last corollary allows some preliminary observations that focus attention on the heart of our problem. 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.

4.3. Proposition. Suppose φ is a holomorphic self-map of 𝕌 that is univalent but not the identity, and fixes a non-zero point p ∈ 𝕌. If φ′(p) is not strictly positive then 0 ∈ int W (C_{φ}).

Proof. Univalence forces φ′(p) ≠ 0, which we assume from now on. The form of the matrix of C_{φ} with respect to (b_{n})_{0}^{∞} shows that the complex conjugate of each number φ′(p)^{n} is an eigenvalue of C_{φ}^{∗} (see, e.g., [32, Section 6.2] for the details), and therefore lies in the numerical range of that operator. Thus φ′(p)^{n} ∈ W (C_{φ}) for each n.

Suppose that φ′(p) is non-real. Then as n → ∞ the sequence (φ′(p)^{n}) spirals in towards the origin, circling it infinitely often, thus (by the Toeplitz-Hausdorff Theorem) placing the origin in the interior of W (C_{φ}).

The only remaining possibility is φ′(p) < 0, for which the argument just given at least locates the origin in W (C_{φ}). To see that it is in the interior one need only note that by the case n = 0 of Corollary 4.2,

⟨C_{φ}b_{0}, b_{1}⟩ = \overline{p}(φ′(p) − 1) ≠ 0.

Thus the numerical range of the matrix

[ \begin{matrix}1 & 0 \\ ⟨C_{φ}b_{0}, b_{1}⟩ & φ′(p) \end{matrix} ]

is a nondegenerate ellipse with foci at 1 and φ′(p). Since φ′(p) < 0, this ellipse has the origin in its interior, and because the matrix represents a compression of C_{φ} (to the subspace spanned by b_{0} and b_{1}), the origin lies in the interior of W (C_{φ}) as well.

According to the discussion above, if φ fixes a point p ∈ U \ {0}, then the containment of 0 in the interior of W (C_{φ}) is in doubt only when φ is univalent and φ′(p) > 0; for example, when φ is a positive conformal dilation (terminology as in Section 2.8). The next result— the principal one of this section—reduces the zero-containment issue to precisely the class of positive conformal dilations.

4.4. Theorem. Suppose φ is a holomorphic self-map of 𝕌 that fixes a non-zero point p ∈ 𝕌 and is neither the identity map on 𝕌 nor a positive conformal dilation. Then 0 ∈ int W (C_{φ}).

Proof. By Corollary 3.6 we may assume that φ is univalent, and by the work of Section 2.9, we may assume that φ is not an elliptic automorphism. By Proposition 4.3 we may require further that φ not be any sort of conformal dilation. Note that for integers n ≥ 0 and k > 0 the function H_{n,k} that is the major player in Lemma 4.1 can be written as

H_{n,k} = L^{n+1}M^{k−1}Q^{n}

where L, M, and Q are holomorphic functions on 𝕌 defined by:

L(z) = \frac{1 − \overline{p}z}{1 − \overline{p}φ(z)} , M(z) = 1 − \overline{p}z, and Q(z) = \frac{φ(z) − p}{z − p} (9)

(Q being holomorphic at p, with Q(p) = φ′(p), because φ(p) = p).

Clearly L is non-vanishing on 𝕌, and the univalence of φ insures that the same is true of Q. Note also that

L(p) = 1, M(p) = 1 − |p|^{2}, and Q(p) = φ′(p),

so that

H_{n,k}(p) = (1 − |p|^{2})^{k−1}φ′(p)^{n} = const. φ′(p)^{n},

where “const.” does not depend on n.

Recall that our goal is to establish (7) for some fixed k > 0, so by Lemma 4.1 it is enough to show that

lim_{n→∞} \frac{H_{n,k}^{(k)}(p)}{H_{n,k}(p)} = ∞. (10)

To motivate what is to follow we first consider the case k = 1, and for convenience write H_{n,1} simply as H_{n}. Thus H_{n} = L^{n+1}Q^{n}, whereupon logarithmic differentiation yields

\frac{H′_{n}}{H_{n}} = (n + 1)\frac{L′}{L} + n\frac{Q′}{Q} = n [\frac{L′}{L} + \frac{Q′}{Q} ] + \frac{L′}{L} ,

which we rewrite as:

(11) \frac{H′_{n}}{H_{n}} = nF + B, where F := \frac{L′}{L} + \frac{Q′}{Q} and B := \frac{L′}{L} .

Note that F and B are holomorphic on 𝕌 because neither L nor Q vanishes anywhere on 𝕌. Moreover (11) shows that if F (p) ≠ 0 then (10) holds for k = 1, hence

F (p) ≠ 0 ⟹ 0 ∈ int W (C_{φ}).

Although we will not need this in the sequel, it is interesting to note that

F (p) ≠ 0 ⟺ (1 − |p|^{2})\frac{φ′′(p)}{2} ≠ \overline{p}φ′(p)(1 − φ′(p)),

so the reader who is tiring of the zero-containment issue can rest assured that for at least all the maps φ satisfying the condition above, 0 ∈ int W (C_{φ}).

For those who wish to continue, the crucial point is that (relaxing for the moment our requirement that φ not be a conformal dilation)

F ≡ 0 on 𝕌 ⟺ φ is a conformal dilation with fixed point at p.

This follows from the fact that F is the logarithmic derivative of LQ = (α_{p}∘φ)/α_{p} (from (9)), hence F ≡ 0 if and only if α_{p} ∘ φ is a constant multiple of α_{p}, i.e., if and only if φ is a conformal dilation.

Since we have already done the case F (p) ≠ 0 let’s suppose that F (p) = 0. Because we are demanding that φ not be a conformal dilation, F is not identically zero on 𝕌, and since F is holomorphic on 𝕌 this guarantees an integer k ≥ 2 such that

F (p) = F′(p) = ⋯ = F^{(k−2)}(p) = 0, but F^{(k−1)}(p) ≠ 0.

It is for this integer k that we are going to prove (10).

From (11) we see that for each integer j ≥ 0:

(\frac{H′_{n}}{H_{n}} )^{(j)} = n F^{(j)} + B^{(j)}, (12)

so that the numerical sequence ((H′_{n}/H_{n})^{(j)}(p) : n ≥ 0) is bounded for 0 ≤ j ≤ k − 2, and it → ∞ for j = k − 1. Using

H_{n}^{(k)} = (H_{n} · \frac{H′_{n}}{H_{n}} )^{(k−1)}

and the product rule, we obtain

\frac{H_{n}^{(k)}(p)}{H_{n}(p)} = (\frac{H′_{n}}{H_{n}} )^{(k−1)}(p) + ∑_{j=0}^{k−2} \binom{k − 1}{j}\frac{H_{n}^{(k−1−j)}(p)}{H_{n}(p)} (\frac{H′_{n}}{H_{n}} )^{(j)}(p). (13)

Because (H′_{n}/H_{n})^{(k−1)}(p) → ∞ as n → ∞ the same will be true of H_{n}^{(k)}(p)/H_{n}(p) provided the sum on the right of (13) remains bounded as n → ∞. We already have observed that the terms in the sum of the form (H′_{n}/H_{n})^{(j)}(p) are bounded independent of n (given the range of the index j). The other terms in the sum take the form H_{n}^{(J)}(p)/H_{n}(p), where J varies from 1 to k − 1. To see that the sequence (H_{n}^{(J)}(p)/H_{n}(p)) is bounded (in n) for this range of index J, apply an inductive argument. The boundedness of (H_{n}^{(1)}(p)/H_{n}(p)) has already been noted (j = 0 in (12)). That the boundedness of (H_{n}^{(v)}(p)/H_{n}(p)) for v = 1, …, J − 1 implies that of (H_{n}^{(J)}(p)/H_{n}(p)), when 1 < J ≤ k − 1, follows from (13): simply replace k with J. We have established

lim_{n→∞} \frac{H_{n}^{(k)}(p)}{H_{n}(p)} = ∞, (14)

which is almost the desired result. To finish the job note that H_{n,k} = H_{n}M_{k}, where

M_{k}(z) = M(z)^{k−1} = (1 − \overline{p}z)^{k−1} (z ∈ 𝕌).

Thus

H_{n,k}^{(k)} = H_{n}^{(k)}M_{k} + ∑_{j=0}^{k−1} \binom{k}{j}H_{n}^{(j)}M_{k}^{(k−j)},

so that

\frac{H_{n,k}^{(k)}(p)}{H_{n,k}(p)} = \frac{H_{n}^{(k)}(p)}{H_{n}(p)} + ∑_{j=0}^{k−1} c_{j} \frac{H_{n}^{(j)}(p)}{H_{n}(p)} , (15)

where on the right-hand side, each c_{j} is a non-zero constant that does not depend on n. We have previously shown that for each 0 ≤ j ≤ k − 1 the sequence

(\frac{H_{n}^{(j)}(p)}{H_{n}(p)} : n ≥ 0)

is bounded, and that it → ∞ for j = k. Thus the left-hand side of (15) → ∞ as n → ∞, as we wished to show.

5. Zero Containment: Part III

The work of Sections 3 and 4 has established that if φ is a holomorphic self-map of 𝕌 that fixes a point p ∈ 𝕌 and neither the identity nor a positive conformal dilation, then 0 ∈ W (C_{φ}). What happens when φ is a positive conformal dilation? We have already observed (Section 2.8) that 0 ∉ W (C_{φ}) whenever φ is a positive conformal dilation that fixes the origin (i.e. an “ordinary” positive dilation). However the next result shows that zero-containment reappears whenever the fixed point is located sufficiently far from the origin.

5.1. Theorem. Suppose φ is a positive conformal dilation that fixes the point p ∈ 𝕌 and has dilation parameter r (0 < r < 1). Then the following three conditions are equivalent:

(a) 0 ∈ W (C_{φ}).

(b) 0 ∈ int W (C_{φ}).

(c) |p| > \sqrt{r}. Moreover if |p| ≤ \sqrt{r} then Re C_{φ} := \frac{1}{2}(C_{φ} + C_{φ}^{∗}) is strictly positive-definite on H^{2}.

Proof. By the work of Section 2.8 we know that φ(z) = α_{p}(rα_{p}(z)) for z ∈ 𝕌, where r ∈ (0, 1) is the dilation parameter of φ.

(c) → (b): Suppose |p| > r. We know that C_{φ}(α_{p}^{n}) = r^{n}α_{p}^{n} for each non-negative integer n. For each positive integer k let M_{k} be the two dimensional C_{φ}-invariant subspace spanned by 1 and α_{p}^{k}, and let A_{k} be the restriction of C_{φ} to M_{k}. By the discussion of Section 2.4 we know that W (A_{k}) is a nondegenerate ellipse with foci at 1 and r^{k}, with major axis of length

ℓ = \frac{1 − r^{k}}{\sqrt{1 − γ_{k}^{2}}} where γ_{k} := |⟨α_{p}^{k}, 1⟩| = |α_{p}^{k}(0)| = |p|^{k}

(the nondegeneracy comes from the fact that the vectors 1 and α_{p}^{k} are not orthogonal in H^{2}, as the above calculation shows). The center c of the ellipse is the midpoint of the foci, i.e., c = (1 + r^{k})/2 on the positive real axis. Now 0 ∈ int W (A_{k}) if and only if c < ℓ/2, i.e., if and only if

1 + r^{k} < \frac{1 − r^{k}}{\sqrt{1 − |p|^{2k}}} .

Some algebraic manipulation shows this last condition to be equivalent to

|p| > (\frac{2}{1 + r^{k}} )^{1/k} \sqrt{r} ,

which, because we are assuming |p| > \sqrt{r}, is satisfied for all sufficiently large k. For any such k

0 ∈ int W (A_{k}) ⊂ int W (C_{φ}),

as we wished to show.

(b) → (a): Trivial.

(a) → (c): Suppose |p| ≤ \sqrt{r}. We will prove the strict positivity of Re C_{φ}, which implies that 0 ∉ W (C_{φ}).

By the discussion of Section 2.8 we may, without loss of generality, assume that 0 < p < 1. Then a little calculation proceeding from the formula φ(z) = α_{p}(rα_{p}(z)) shows that

φ(z) = \frac{az + 1}{d − z}, (16)

where

a = \frac{r − p^{2}}{p(1 − r)} and d = \frac{1 − p^{2}r}{p(1 − r)} (17)

(the assumption that p is real simply avoids some complex conjugates that would otherwise have appeared in these last two formulas).

Clearly d > 0; furthermore the hypothesis p ≤ \sqrt{r} implies that a ≥ 0, with equality if and only if p = \sqrt{r}. In this interesting transitional case, about which we will say more later, φ(z) ≡ (d − z)^{−1} with d = (1 + r)/\sqrt{r}.

Our hypothesis relating p and r also implies that d ≥ (1 + r)/\sqrt{r} > 2, from which it follows that φ takes strictly positive values on the closed unit interval. In particular

(a + 1)/(d − 1) = φ(1) < 1 hence d − a > 2. (18)

Because of this last inequality the auxiliary mapping

ν(z) := \frac{δz + 1}{d} where δ := \sqrt{ad + 1} (19)

maps 𝕌 into itself; indeed for z ∈ 𝕌:

|ν(z)| < ν(1) = \frac{δ + 1}{d} < 1.

Let M_{z} denote the operator of “multiplication by z” on H^{2} (i.e., the “forward shift”), set T = C_{ν}M_{z}^{∗}, and let P_{0} denote the orthogonal projection of H^{2} onto the closed subspace of constant functions:

P_{0}f = ⟨f, 1⟩ = f(0) (f ∈ H^{2}).

With this notation we can state what is really the main result of this section:

C_{φ} + C_{φ}^{∗} = P_{0} + C_{ν}^{∗}C_{ν} + \frac{a}{d} T^{∗}T (20)

This decomposition, whose existence was suggested by matrix arguments (see the discussion following the proof of Theorem 6.2 below), exhibits C_{φ} + C_{φ}^{∗} = 2 Re C_{φ} as the sum of three positive operators, hence Re C_{φ} is positive. To see that it is strictly positive we need only observe that C_{ν} is one-to-one (as is any composition operator induced by a non-constant map), hence if f ∈ H^{2} is not the zero-function then

⟨C_{ν}^{∗}C_{ν}f, f⟩ = ⟨C_{ν}f, C_{ν}f⟩ = ∥C_{ν}f∥^{2} > 0,

i.e., C_{ν}^{∗}C_{ν} is strictly positive, and therefore the same is true of Re C_{φ}.

It remains to prove (20). The idea will be to check that both sides of this equation yield the same result when applied to the reproducing kernels K_{λ} (λ ∈ 𝕌) introduced in Section 2.2. Because of the reproducing property (2), any function in H^{2} orthogonal to K_{λ} vanishes at λ, hence if λ runs through a subset of 𝕌 with a limit point in 𝕌 then the linear span of the corresponding K_{λ}’s is dense in H^{2}. Thus it is enough to check our equation on any such family of reproducing kernels; for convenience we choose to restrict λ to the real interval (−1, 1).

First Subclaim: C_{ν}^{∗}C_{ν}K_{λ} = \frac{d}{d−λ}K_{φ(λ)} (−1 < λ < 1).

Proof. A straightforward calculation shows that for z ∈ 𝕌:

C_{ν}K_{λ}(z) = \frac{d}{d − λ} \frac{1}{1 − βz} where β = \frac{λδ}{d − λ} .

Clearly β > 0, and since C_{ν}K_{λ} ∈ H^{2} we must also have β < 1, hence the last equation can be rephrased:

C_{ν}^{∗}C_{ν}K_{λ} = \frac{d}{d − λ} K_{ν(β)}.

Another straightforward calculation shows that

ν(β) = \frac{λa + 1}{d − λ} = φ(λ),

which completes the proof of this Subclaim.

Second Subclaim: T^{∗}T K_{λ}(z) = \frac{λdz}{d−λ} K_{φ(λ)} for each z ∈ 𝕌.

Proof. Once checks easily that M_{z}^{∗}K_{λ} = \overline{λ}K_{λ} for each z ∈ 𝕌, so if 1− < λ < 1 then T K_{λ} = λC_{ν}K_{λ}. Thus for z ∈ 𝕌:

T^{∗}T K_{λ}(z) = M_{z}C_{ν}^{∗}(λC_{ν}K_{λ})(z) = zλC_{ν}^{∗}C_{ν}K_{λ}(z),

which, along with the First Subclaim, yields the desired result.

To complete the proof of Theorem 5.1, fix λ ∈ (−1, 1), and note that by the reproducing property (2),

(C_{φ} + C_{φ}^{∗})K_{λ} = K_{λ} ∘ φ + K_{φ(λ)}

which, after some routine calculation, results for each z ∈ 𝕌 in:

(C_{φ} + C_{φ}^{∗})K_{λ}(z) = 1 + \frac{d + λaz}{(d − λ) − (λa + 1)z}

= 1 + \frac{d + λaz}{d − λ} \frac{1}{1 − \frac{aλ+1}{d−λ}z}

= 1 + [\frac{d}{d − λ} + \frac{λaz}{d − λ}] K_{φ(λ)}(z)

= P_{0}K_{λ}(z) + C_{ν}^{∗}C_{ν}K_{λ}(z) + \frac{a}{d} T^{∗}T K_{λ}(z),

where the last line follows from the two Subclaims above.

5.2. Remarks. (a) In the argument above we borrowed the idea of using reproducing kernels as test functions from Cowen’s derivation of the formula for the adjoint of a composition operator induced on H^{2} by a linear fractional transformation [7].

(b) In the final section of this paper we use the argument above to prove strict positivity for Re C_{φ} for a class of composition operators induced by linear fractional maps that fix no point of 𝕌.

Theorem 5.1 completes our solution to the Zero-Inclusion Question for composition operators induced by self-maps of 𝕌 that fix a point in 𝕌. We summarize our work in the following.

5.3. Theorem. Suppose that φ is a holomorphic self-map of 𝕌 fixing a point p ∈ 𝕌. Then 0 ∉ W (C_{φ}) if and only if there is an r ∈ (0, 1] such that

|p| ≤ \sqrt{r} and φ(z) = α_{p}(rα_{p}(z)).

Moreover, 0 ∈ W (C_{φ}) implies 0 ∈ int W (C_{φ}) as long as φ does not have the form φ(z) = tz for some t ∈ [−1, 0].

As we have mentioned, if C_{φ} is compact, then φ fixes a point in 𝕌. Because the numerical range of a compact operator is closed if and only if it contains zero, the preceding theorem also provides a characterization of when the numerical range of a compact composition operator is closed. (The compact composition operators on H^{2} are characterized in [31], see also [32, Chapter 10].)

6. Sectorial Composition Operators

We call a Hilbert space operator T sectorial provided its numerical range is contained in a sector of the form {z : | arg(z)| < θ} for some positive acute angle θ (cf. [14, page 37]). The work of the preceding sections shows that if φ, not the identity, fixes a point p ∈ 𝕌 then C_{φ} can be sectorial only if there is an r ∈ (0, 1) such that φ = α_{p} ∘ (rα_{p}) where |p| ≤ \sqrt{r}. In this section, we prove that, for such a map φ, C_{φ} is sectorial when |p| < \sqrt{r}, but not sectorial when |p| = \sqrt{r}.

We begin our work with the case |p| < \sqrt{r}. Until further notice, we assume that r ∈ (0, 1) and

φ(z) = α_{p}(rα_{p}(z))

where 0 < p < \sqrt{r} (recall there is no loss of generality in assuming p > 0). Hence, as we saw in Section 5,

(16)  φ(z) = \frac{az + 1}{d − z},

where

(17)  a = \frac{r − p^{2}}{p(1 − r)} and d = \frac{1 − p^{2}r}{p(1 − r)}.

Note well that our assumption p < \sqrt{r} implies a > 0 and d > 2. Also note a < d. Finally, recall that the auxiliary mapping

(19)  ν(z) := \frac{δz + 1}{d} where δ := \sqrt{ad + 1},

is a self-map of 𝕌. Our proof that C_{φ} is sectorial depends on representation (20) for C_{φ}+C_{φ}^{∗} and a corresponding representation for C_{φ} − C_{φ}^{∗}.

6.1. Lemma. Suppose that φ is given by (16) and (17) and that p < \sqrt{r}. Then

C_{φ} − C_{φ}^{∗} = \frac{1}{d} (C_{ν}^{∗}T − T^{∗}C_{ν}) ,

where T = C_{ν}M_{z}^{∗} and ν is defined by (19).

Proof. The Lemma may be verified—as was the decomposition (20)—by using the H^{2} reproducing kernels as test functions. What follows is a more informative proof. Let ψ(z) = δz/(d − z). Using [7, Theorem 2] we have

C_{ν}^{∗} = M_{d/(d−z)}C_{ψ},

where M_{d/(d−z)} is the operator of multiplication by d/(d−z) on H^{2}. The preceding formula and the easily verified relationship C_{ψ}C_{ν} = C_{φ} yield

C_{φ} = M_{(d−z)/d}C_{ν}^{∗}C_{ν}

so that C_{φ} = C_{ν}^{∗}C_{ν} − (1/d)T^{∗}C_{ν}. Hence

C_{φ} − C_{φ}^{∗} = \frac{1}{d} (C_{ν}^{∗}T − T^{∗}C_{ν}) ,

as desired.

6.2. Theorem. Suppose that φ is given by (16) and (17) and that p < \sqrt{r}. Then the numerical range of C_{φ} lies in the sector {z : | arg z| ≤ π/2 − arctan(a/2)}.

Proof. Let f be an arbitrary unit vector in H^{2} and let θ > 0 be such that tan(θ) ≤ a/2. Observe

(21) 2Re (e^{iθ}⟨C_{φ}f, f⟩) = cos(θ)⟨(C_{φ} + C_{φ}^{∗})f, f⟩ + i sin(θ)⟨(C_{φ} − C_{φ}^{∗})f, f⟩.

Applying the decomposition (20) of C_{φ} + C_{φ}^{∗}, we see that the first summand on the right of the preceding equation is greater than or equal to

cos(θ) (∥C_{ν}f∥^{2} + (a/d)∥T f∥^{2}) . (22)

Using Lemma 6.1, we obtain the following estimate on the size of the second summand on the right of (21)

| sin(θ)⟨(C_{φ} − C_{φ}^{∗})f, f⟩| ≤ sin(θ)\frac{1}{d} (|⟨T f, C_{ν}f⟩| + |⟨C_{ν}f, T f⟩|) .

By the Cauchy-Schwarz Inequality,

| sin(θ)⟨(C_{φ} − C_{φ}^{∗})f, f⟩| ≤ sin(θ)\frac{2}{d}∥T f∥∥C_{ν}f∥ (23)

Suppose that ∥C_{ν}f∥ ≥ ∥T f∥. Then starting with (21) and applying estimates (22) and (23), we have

2Re (e^{iθ}⟨C_{φ}f, f⟩) ≥ (a/d) cos(θ)∥T f∥^{2} + cos(θ)∥C_{ν}f∥^{2} − (2/d) sin(θ)∥C_{ν}f∥^{2}
= (a/d) cos(θ)∥T f∥^{2} + ∥C_{ν}f∥^{2} cos(θ)[1 − (2/d) tan(θ)]
≥ (a/d) cos(θ)∥T f∥^{2} + ∥C_{ν}f∥^{2} cos(θ) [1 − (2/a) tan(θ)]
≥ 0,

where we have used a < d to obtain the next-to-last inequality.

Now suppose that ∥T f∥ ≥ ∥C_{ν}f∥. Again, starting with (21) and applying estimates (22) and (23), we have

2Re (e^{iθ}⟨C_{φ}f, f⟩) ≥ cos(θ)∥C_{ν}f∥^{2} + (a/d) cos(θ)∥T f∥^{2} − (2/d) sin(θ)∥T f∥^{2}
= cos(θ)∥C_{ν}f∥^{2} + (a/d)∥T f∥^{2} cos(θ)[1 − (2/a) tan(θ)]
≥ 0.

We have proved that if we rotate W (C_{φ}) by an angle θ satisfying 0 < θ ≤ arctan(a/2), the resulting region is still in the closed right half-plane. Since W (C_{φ}) is symmetric with respect to the real axis (see Section 2.7), it follows that W (C_{φ}) lies in the sector

{z : | arg z| ≤ π/2 − arctan(a/2)},

as desired.

Our goal now becomes that of proving W (C_{φ}) is not sectorial when φ has the form α_{p} ∘ (rα_{p}) and p = \sqrt{r}. We employ matrix methods that contributed to our development of Theorem 5.1. The major element in our proof of Theorem 5.1, the decomposition (20) of C_{φ} + C_{φ}^{∗}, evolved from a matrix analysis of the transitional case p = \sqrt{r}. For this case a = 0 in (17) so the expression (16) for φ becomes

φ(z) = \frac{1}{d − z} with d = \frac{1 + r}{\sqrt{r}} > 2, (24)

and the decomposition (20) simplifies to

C_{φ} + C_{φ}^{∗} = P_{0} + C_{ν}C_{ν}^{∗} where ν(z) = \frac{1 + z}{d} . (25)

The discussion below is also valid when d = 2.

For an operator T on H^{2}, let us denote its matrix with respect to the orthonormal basis {z^{j}}_{0}^{∞} by [T]. Note that for φ as above, [C_{φ}] has as its j-th column the MacLaurin coefficient sequence of φ^{j}. Let w = 1/d and apply the binomial theorem to obtain that the (k, j) element of this matrix is:

(26) a_{k,j} := \{ \begin{matrix}\binom{k+j−1}{k}w^{j+k} & if & j ≥ 1, \\ 1 & if & (k, j) = (0, 0), and \\ 0 & if & j = 0 and k > 0. \end{matrix}

Thus for j > 0 the (k, j) entry of [C_{φ} + C_{φ}^{∗}] is

a_{k,j} + a_{j,k} = [\binom{k + j − 1}{k} + \binom{j + k − 1}{j}] w^{k+j}

= \binom{k + j}{k}w^{k+j} = \binom{j + k}{j}w^{k+j},

a formula which also gives the correct answer for j = 0, k ≠ 0. The only anomaly is the (0, 0) entry, which is 2, instead of the value 1 predicted by the formula. From this we obtain the matrix equation

[C_{φ} + C_{φ}^{∗}] = [P_{0}] + T (w) (27)

where P_{0} is, as before, the orthogonal projection of H^{2} onto the space of constant functions ([P_{0}] has (0, 0) entry 1 and all other entries zero), and T (w) is the infinite matrix whose m-th cross-diagonal is w^{m} times the m-th row of Pascal’s Triangle (m = 0, 1, 2, …).

The matrix T (1) (which does not correspond to any composition operator) is called Tartaglia’s table or Pascal’s matrix, and figures in a surprising variety of applications (see [2] for example).

The operator decompositions (25) and (27) yield the matrix factorization (at least for w ≤ 1/2)

T (α) = [C_{ν}]^{∗}[C_{ν}], (28)

with the superscript “^{∗}” now denoting matrix transpose. Now [C_{ν}], because of the simple form of ν given in (25), is upper triangular with w^{j} times the j-th row of Pascal’s triangle as its j-th column. Thus (28) furnishes an LU-decomposition of T (w).

The analogous decomposition of the n × n upper left-hand submatrix of T (w) is known (see [5, Theorem 3]), and in fact follows quickly (for w ≤ 1/2) from the discussion above and the fact that C_{ν} leaves invariant the subspace spanned by the monomials {z^{j} : 0 ≤ j ≤ n}. It was, in fact, this result that first led us to the auxiliary mapping ν and the decomposition (25).

These matrix methods permit us to complete our characterization of sectorial composition operators. For the remainder of the section, we assume

φ(z) = \frac{1}{d − z} ,

where d ≥ 2.

6.3. Theorem. The operator C_{φ} is not sectorial.

We will show that for each positive integer n, there is a point in W (C_{φ}) whose argument is π/2 − π/(n + 1). Our proof will ultimately depend on a functional equation satisfied by Chebyshev polynomials of the second kind.

Let n be an arbitrary positive integer, let t ∈ R, let M(t) be the matrix of (e^{it}C_{φ} + e^{−it}C_{φ}^{∗}) : H^{2} → H^{2} with respect to the basis {1, z, z^{2}, … } of H^{2}, and let A_{n}(t) be the n×n principal submatrix of M(t). For example,

A_{3}(t) = [ \begin{matrix}2 cos(t) & w cos(t) + iw sin(t) & w^{2} cos(t) + iw^{2} sin(t) \\ w cos(t) − iw sin(t) & 2w^{2} cos(t) & 3w^{3} cos(t) + iw^{3} sin(t) \\ w^{2} cos(t) − iw^{2} sin(t) & 3w^{3} cos(t) − iw^{3} sin(t) & 6w^{4} cos(t) \end{matrix} ] ,

where w = 1/d.

To prove Theorem 6.3, we show that when t = π/(n + 1), then det A_{n}(t) = 0. Suppose det A_{n}(t) = 0; then the numerical range W (A_{n}(t)) contains 0, which means 0 also lies in W (e^{it}C_{φ} + e^{−it}C_{φ}^{∗}). In fact, there is a polynomial p of degree n and H^{2} norm 1 such that

⟨e^{it}(C_{φ} + e^{−it}C_{φ}^{∗})p, p⟩ = 0 (29)

(just choose the coefficients of p to form a unit vector in the kernel of A_{n}(t)). Hence Re (e^{it}⟨C_{φ}p, p⟩) = 0, which implies that ⟨C_{φ}p, p⟩ lies on the ray {z : arg z = π/2 − t} since Re ⟨C_{φ}p, p⟩ > 0 by Theorem 5.1. Thus, our goal is to prove det A_{n}(t) vanishes for t = π/(n + 1).

Let us describe the entries in A_{n}(t), considering the top row of A_{n}(t) and the left-most column of A_{n}(t) to be its 0-th row and 0-th column respectively. Using (26), we see that for 0 ≤ k ≤ n − 1 and 0 < j ≤ n − 1,

[A_{n}(t)]_{k,j} = cos(t)w^{k+j}\binom{k + j}{j} + i sin(t)w^{k+j} \frac{j − k}{j} \binom{k + j − 1}{k}.

Also,

[A_{n}(t)]_{0,0} = 2 cos(t), and [A_{n}(t)]_{k,0} = w^{k} cos(t) − iw^{k} sin(t) for k ≥ 1.

Note that for 1 ≤ k, j ≤ n − 1, each entry in row k of A_{n}(t) contains a factor of w^{k} while each entry in column j contains a factor of the form w^{j}. It follows that

det A_{n}(t) = w^{n(n−1)} det B_{n}(t),

where B_{n}(t) is free of factors of w:

[B_{n}(t)]_{k,j} = cos(t)\binom{k + j}{j} + i sin(t)\frac{j − k}{j} \binom{k + j − 1}{k}

for 0 ≤ k ≤ n − 1 and 0 < j ≤ n − 1. Also the 0-th column of B_{n}(t) consists of 2 cos(t) followed by n − 1 entries, each of which is cos(t) − i sin(t).

Our goal now becomes that of proving det B_{n}(t) = 0 when t = π/(n + 1).

6.4. Lemma. For n ≥ 1,

det B_{n}(t) = U_{n}(cos(t)),

where U_{n} is the n-th Chebyshev polynomial of the second kind.

The polynomial U_{n} is given by

U_{n}(x) = ∑_{k=0}^{[n/2]}(−1)^{k}\binom{n − k}{k}(2x)^{n−2k}

and is defined recursively by

U_{0} = 1, U_{1}(x) = 2x, and U_{n}(x) = 2xU_{n−1}(x) − U_{n−2}(x) for n ≥ 2.

Moreover, we have the following functional equation

U_{n}(cos(t)) = \frac{sin((n + 1)t)}{sin(t)},

[29, p. 7, pp. 39–40]. The preceding equation shows that U_{n} vanishes at cos(t) whenever sin((n + 1)t) = 0 while sin(t) ≠ 0. Thus, in particular, cos(π/(n + 1)) is always a zero of U_{n} (for n ≥ 1). By the Lemma above det B_{n}(π/(n + 1)) = 0. Thus, given the discussion in the paragraphs preceding the statement of Lemma 6.4, we see that Theorem 6.3 follows from the Lemma.

Because we have been unable to find Lemma 6.4 in the literature, we present a detailed proof.

Proof of Lemma 6.4. Note det B_{1}(t) = 2 cos(t) = U(cos(t)) and det B_{2}(t) = 4 cos^{2}(t) − 1 = U_{2}(cos(t)). Thus the Lemma holds when n = 1 and n = 2. To see that it holds for all n we show

(30) det B_{n}(t) = 2 cos(t) det B_{n−1}(t) − det B_{n−2}(t) for n ≥ 3.

We describe a sequence of elementary row and column operations on B_{n}(t) that will produce a matrix L_{n}(t) whose determinant (which is the same as that of B_{n}(t)) is easily seen to equal the sum on the right of (30). This will complete the proof of the lemma. The following observations will be used to produce the matrix L_{n}(t).

Observation 1. For 1 ≤ k ≤ n − 2,

row (k + 1) of B_{n}(t) − (row k of B_{n}(t))

equals the row matrix whose first entry is 0 and whose remaining entries constitute the 0-th through (n − 2)-nd entries of row (k + 1) of B_{n}(t).

In other words, if row k of B_{n}(t) is subtracted from row k + 1, the result is the “forward shift” of row (k + 1) with the last entry dropped. In symbols this difference equals

[0 R_{k+1}] ,

where R_{k+1} represents row (k + 1) of B_{n}(t) with its final entry truncated.

Observation 2. For 1 ≤ k, j ≤ n − 1,

(−1)[B_{n}(t)]_{k,j−1} + [B_{n}(t)]_{k,j} = [B_{n}(t)]_{k−1,j}

Apply Observation 1 repeatedly, first replace row (n − 1) of B_{n}(t) with the difference

row (n − 1) of B_{n}(t) − (row (n − 2) of B_{n}(t));

then replace row (n − 2) with the difference of rows (n − 2) and (n − 3), and continue this process until row 2 has been replaced with the difference row 2 − row 1. (Recall that the top row of B_{n}(t) is row 0.) What results is the matrix

[ \begin{matrix}2 cos(t) & cos(t) + i sin(t) & cos(t) + i sin(t) & ⋯ & cos(t) + i sin(t) \\ cos(t) − i sin(t) & 2 cos(t) & 3 cos(t) + i sin(t) & ⋯ & n cos(t) + i(n − 2) sin(t) \\ 0 & R_{2} \\ 0 & R_{3} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & R_{n−1} \end{matrix} ] ,

which we label E_{n}(t). Now apply Observation 2 repeatedly to E_{n}(t), first replace its (n − 1) column of with the difference

column (n − 1) of E_{n}(t) − (column (n − 2) of E_{n}(t));

then replace column (n−2) with the difference of columns (n−2) and (n−3), and continue this process until column 2 has been replaced with the difference column 2 − column 1. What results is the matrix

[ \begin{matrix}2 cos(t) & cos(t) + i sin(t) & 0 & ⋯ & 0 \\ cos(t) − i sin(t) & 2 cos(t) & cos(t) + i sin(t) & ⋯ & cos(t) + i sin(t) \\ 0 & R_{1} \\ 0 & R_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & R_{n−2} \end{matrix} ] ,

which we label L_{n}(t). Calculate det L_{n}(t) by expanding along the first column of L_{n}(t):

(31) det L_{n}(t) = 2 cos(t) det B_{n−1}(t) − e^{−it} det [ \begin{matrix}cos(t) + i sin(t) & 0 & ⋯ & 0 \\ R_{1} \\ R_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ R_{n−2} \end{matrix} ] .

Now apply Observations 1 and 2, exactly as they were applied to B_{n}(t), to reduce the matrix on the right of the preceding display to

[ \begin{matrix}cos(t) + i sin(t) & 0 & ⋯ & 0 \\ 0 & R̃_{0} \\ 0 & R̃_{1} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & R̃_{n−3} \end{matrix} ] ,

where for k = 0, 1, …, n−3, R̃_{k} is the k-th row of B_{n}(t) with its final two entries truncated. This reduced matrix permits us to complete our calculation (31):

det L_{n}(t) = 2 cos(t) det B_{n−1}(t) − e^{−it}e^{it} det [ R̃_{0} ; R̃_{1} ; ⋮ ; R̃_{n−3} ]

= 2 cos(t) det B_{n−1}(t) − det B_{n−2}(t),

as desired.

We have proved Theorem 6.3 by showing that when φ(z) = 1/(d − z) for d ≥ 2, W (C_{φ}) contains points in the first quadrant arbitrary close to the imaginary axis. Recall from Section 2.7, that since φ is real on the real axis, W (C_{φ}) is symmetric with respect to the real axis; hence, Theorem 6.3 shows that 0 is not a corner point of \overline{W} (C_{φ}). We discuss corner points in detail in the next section.

7. Corner Points

An element p of a set E ⊂ ℂ is a corner point of E provided E is contained in a half-cone with vertex p and angular opening less than π. It is well known that for any Hilbert space operator T, a corner point of W (T) must be an eigenvalue of T ([10]). A corner point of \overline{W} (T) must be an approximate eigenvalue; in particular, any corner point of \overline{W} (T) must belong to the spectrum of T ([20]).

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 \overline{W} (C_{φ}) because 0 ∈ W (C_{φ}) and C_{φ} is compact). Here, we obtain a complete characterization of corner points \overline{W} (C_{φ}) when C_{φ} is compact.

We begin with the simpler task of characterizing the corner points not of \overline{W} (C_{φ}) but of W (C_{φ}), under the assumption that φ fix a point p in 𝕌. These corner points must be in the point spectrum of C_{φ}, which is known to be a subset of {φ′(p)^{n} : n = 0, 1, 2, … } ([6, Theorem 4.1]).

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^{[n]}(0) : n ≥ 1} may be removed. We do not consider composition operators induced by dilations and rotations since the numerical ranges for such operators are easily described (Sections 2.8 and 2.9).

7.1. Theorem. Suppose φ is a self-map of 𝕌 that is neither a dilation nor a rotation and φ(p) = p for some p ∈ 𝕌. Then {φ′(p)^{n} : n ≥ 1} is contained in the interior of W (C_{φ}).

Proof. If φ′(p) = 0, the Theorem follows immediately from Corollary 3.6 because φ is not univalent. Thus we assume φ′(p) ≠ 0.

In case p = 0, the proof of Theorem 3.8 shows that there is a nonzero number b and a positive integer m such the compression of C_{φ} to the two-dimensional subspace of H^{2} with basis {z^{n}, z^{n+m}} has matrix

[ \begin{matrix}φ′(0)^{n} & 0 \\ nbφ′(0)^{n} & φ′(0)^{n+m} \end{matrix} ] .

Because the entry in the first column and second row is always nonzero, the numerical range of C_{φ} contains a nondegenerate ellipse with foci λ^{n} and λ^{n+m} for each n ≥ 1. Thus {φ′(0)^{n} : n ≥ 1} ⊂ int W (C_{φ}), as desired.

In case p ≠ 0, the work of Section 4 shows that for each n ≥ 0 , there is an an orthonormal set {b_{n}, b_{n+1}} in H^{2} generating a subspace M of H^{2} such that the compression of C_{φ} to M has matrix

[ \begin{matrix}φ′(p)^{n} & 0 \\ ⟨C_{φ}b_{n}, b_{n+1}⟩ & φ′(p)^{n+1} \end{matrix} ] .

By Corollary 4.2, the (2, 1) entry of this matrix can be 0 for at most one positive integer n_{0} (we have already observed, in the proof of Proposition 4.3 that it is not zero if n = 0). Thus, for all nonnegative integers n ≠ n_{0}, we see that W (C_{φ}) contains a nondegenerate ellipse with foci φ′(p)^{n} and φ′(p)^{n+1}. Hence for each positive integer n, including n_{0}, φ′(p)^{n} is a focal point of a nondegenerate ellipse contained in W (C_{φ}), so that the entire sequence {φ′(p)^{n} : n ≥ 1} lies in the interior of W (C_{φ}), as desired.

7.2. Corollary. Suppose φ is a self-map of 𝕌 that is neither a dilation nor a rotation and φ(p) = p for some p ∈ 𝕌. Then 1 is the only possible corner point of W (C_{φ}).

Proof. All eigenvalues of C_{φ} except for 1 lie in the interior of W (C_{φ}) by Theorem 7.1.

Observe that 1 is a corner point of W (C_{φ}) if and only if it is a corner point of \overline{W} (C_{φ}). Thus we complete our characterization of the corner points of W (C_{φ}) by answering the following.

When is 1 a corner point of \overline{W} (C_{φ})? If φ(0) ≠ 0, then 1 is not a normal eigenvalue of C_{φ}; hence by Theorem 3.4, the point 1

belongs to ∈ int W (C_{φ}), and therefore cannot be a corner point of \overline{W} (C_{φ}). (This observation is a generalization of Theorem 3.3 in [26].) Thus we assume that φ(0) = 0. Then the subspace H_{0}^{2} is invariant for C_{φ} and has as its orthogonal complement the space of constant functions. It is known that φ is an inner function (i.e. φ has radial limits of modulus 1 a.e. on ∂𝕌) if and only if the restriction of C_{φ} to H_{0}^{2} has norm 1 [33, Theorem 5.1]. Thus if φ is not inner then \overline{W} (C_{φ}|H_{0}^{2}) lies in the closed disc with center at the origin and radius ∥C_{φ}|_{H_{0}^{2}}∥ < 1, and therefore \overline{W} (C_{φ}) itself lies in the convex hull of this disc and the singleton \{1\}. In particular 1 is a corner point of \overline{W} (C_{φ}).

Conversely suppose φ is inner, fixes the origin, and is not a rotation. Then C_{φ} is a nonunitary isometry taking H^{2} properly into itself, and the spectrum of any such operator is well known to be the closed unit disc \overline{𝕌}. (Proof: Because T is an isometry σ(T) ⊂ \overline{𝕌}. Also, T^{∗}T = I, so for each λ ∈ 𝕌 there is the factorization T − λI = (I − λT^{∗})T where ∥λT^{∗}∥ = |λ|∥T^{∗}∥ = |λ| < 1. Thus I − λT^{∗} is invertible, but T is not, so neither is T − λI. Thus λ ∈ σ(T), which shows that 𝕌 ⊂ σ(T).)

Now \overline{W} (C_{φ}) must contain the spectrum, but must also lie in the \overline{𝕌}, so it coincides with \overline{𝕌}. In particular 1 is not a corner point of \overline{W} (C_{φ}) in this case. To summarize:

7.3. Theorem. If φ(0) ≠ 0, then 1 ∈ int W (C_{φ}). If φ(0) = 0 and φ is not a rotation, then 1 is a corner point of \overline{W} (C_{φ}) if and only if φ is not inner.

We are now in a position to characterize the corner points of \overline{W} (C_{φ}) for compact C_{φ}. Note well that when C_{φ} is compact, φ cannot be inner: if φ is inner, then the H^{2} norm of φ^{n} is 1 for each n ≥ 1 so that C_{φ} does not take the sequence (z^{n}) in H^{2}, which converges weakly to 0, to a sequence that converges in norm to 0. In particular, the symbol of a compact composition operator cannot be a rotation mapping. Since the spectrum of a compact operator consists of its eigenvalues together with 0 and since only spectral points can be corner points, to obtain the full story on corner points of a compact operator we need only check the origin and the eigenvalues of the operator. Thus for a compact composition operator C_{φ}, the set of points that must be checked is {φ′(p)^{n} : n ≥ 0} ∪ {0} where p ∈ 𝕌 is the point fixed by φ. (Recall that the symbol of compact composition operator must fix a point in 𝕌.) We have seen (Theorem 7.1) that when φ fixes p ∈ 𝕌 and is neither a dilation nor a rotation, then points {φ′(p)^{n} : n ≥ 1} belong to the interior of W (C_{φ}), which yields:

7.4. Corollary. The numbers 1 and 0 are the only possible corner points of \overline{W} (C_{φ}) when C_{φ} is compact and φ is not a dilation.

7.5. Theorem. Let φ be a self-map of 𝕌 that induces a compact composition operator on H^{2}.

(a) Suppose that φ(z) = βz, where |β| < 1. If 0 ≤ β < 1, then 0 and 1 are the corner points of \overline{W} (C_{φ}); otherwise, the set of corner points of \overline{W} (C_{φ}) has the form \{β^{n} : n = 0, 1, … k − 1} where k is the least positive integer such that β^{k} lies in the convex hull of {β^{n} : n = 0, 1, … k − 1}.

(b) If φ(0) = 0 and φ is not a dilation, then 1 is the only corner point of \overline{W} (C_{φ}).

(c) If φ fixes a point in 𝕌 \ \{0\} and is not a positive conformal dilation, then \overline{W} (C_{φ}) has no corner points.

(d) If φ is the positive conformal dilation given by φ(z) = α_{p}(rα_{p}(z)), where 0 < r, |p| < 1, then \overline{W} (C_{φ}) has no corner points when |p| ≥ \sqrt{r} and has 0 as its only corner point when |p| < r.

Proof. The proof of (a) follows immediately from the discussion of Section 2.8.

Suppose the hypothesis of (b) holds. Then Theorem 5.3 shows that 0 belongs to the interior of W (C_{φ}); hence, by Corollary 7.4, 1 is the only possible corner point of \overline{W} (C_{φ}). That 1 is a corner point in this situation follows from Theorem 7.3 because the symbol of a compact composition operator cannot be inner.

Suppose that hypothesis of (c) holds. Then, reasoning just as in the preceding paragraph, we see that 1 is the only possible corner point. However, 1 is not a corner point in this situation: in fact 1 must lie in the interior of W (C_{φ}) because 1 is not an normal eigenvalue (φ(0) ≠ 0 since φ fixes a nonzero point in 𝕌).

That (d) holds follows quickly from the results of Section 6 and the observation that 1 is, again, not a normal eigenvalue.

We remark that Corollary 7.4 and Theorem 7.5 are actually valid for Riesz composition operators. A Riesz operator is spectrally indistinguishable from a compact operator. For a discussion of Riesz composition operators, the reader may consult, e.g., [3].

8. No Fixed Point in 𝕌

We have answered the Zero-Inclusion Question for composition operators whose symbols fix a point in 𝕌. What about those whose symbols don’t fix a point in 𝕌? If φ fails to fix a point in 𝕌, then, by the Denjoy-Wolff Theorem, φ has an attractive fixed point p ∈ ∂𝕌. Moreover, φ′(p) = lim_{r→1^{−}} φ′(rp) exists and 0 < φ′(p) ≤ 1 (see, [32, Chapter 5], e.g.). The mapping φ is said to be of hyperbolic type, when φ′(p) < 1; if φ′(p) = 1 and the iterate sequence (φ^{[n]}(0)) is separated in the hyperbolic metric on 𝕌, then φ is of parabolic automorphism type; finally, if φ′(p) = 1 and the iterate sequence (φ^{[n]}(0)) is not separated in the hyperbolic metric on 𝕌, then φ is of parabolic nonautomorphism type.

When φ is of hyperbolic type or parabolic automorphism type, then the spectrum of the corresponding composition operator must contain at least the unit circle ([6, Theorems 4.5 and Theorem 4.10]), which, by convexity, puts 0 in the interior of W (C_{φ}). However, as we describe below, when φ is of parabolic nonautomorphism type, the work of Section 5 shows that zero need not belong to W (C_{φ}).

Write σ(z) = (1 + z)/(1 − z), a conformal mapping taking 𝕌 onto the open right halfplane RHP, the unit circle onto the imaginary axis, and sending 1 to ∞ and −1 to 0. Fix t, initially in the closed right half-plane, and consider the parabolic map Φ(w) = w + t, which takes RHP into itself. If t is pure imaginary then our translation map is a conformal automorphism of RHP, but otherwise it is not. Thus φ_{t}(z) := σ^{−1}(σ(z) + t) is a parabolic automorphism of 𝕌 precisely when t is pure imaginary, and a parabolic non-automorphic self-map of 𝕌 when Re t > 0.

In case t is not real the spectrum of C_{φ_{t}} is known to be an origin-centered disc in the automorphic case [27] and a curve spiraling into the origin in the non-automorphic case [6, Section 6]. In either case, these spectral considerations place the origin in the interior of W (C_{φ_{t}}). However if t is real (so t > 0) then the spectrum of C_{φ_{t}} is just the closed unit interval ([6, Section 6] again), so for these operators there is a chance that the origin may not lie in the numerical range. This is exactly what happens!

Upon writing out the formula for φ_{t} one obtains (16),

φ(z) = \frac{az + 1}{d − z},

but now with

a = \frac{2}{t} − 1 and d = \frac{2}{t} + 1.

For 0 < t ≤ 2 both a and d are non-negative, and the map ν defined by (19) becomes simply

ν(z) = \frac{2z + t}{2 + t} (z ∈ 𝕌),

which obviously takes 𝕌 into itself. Now the argument that led to the decomposition (20) works once again, with no changes whatsoever, in this new situation, and shows that Re C_{φ_{t}} is strictly positive definite whenever 0 < t ≤ 2. For t > 2 the origin lies in the interior of W (C_{φ_{t}}), as an analysis exactly like that of Theorem 5.1 (c) → (b) shows (with eigenvectors e^{−sσ}, where s is positive and large).

Because any linear fractional mapping φ of parabolic nonautomorphism type is, by rotational conjugation, equivalent to one of the mappings φ_{t} discussed above, we have answered the Zero-Inclusion Question for all composition operators except those with univalent, non linear-fractional symbols that are of parabolic nonautomorphism type.

Suppose φ is of parabolic nonautomorphism type and is not linear fractional.

Must 0 ∈ W (C_{φ})?

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Washington and Lee University, Lexington, VA 24450, USA

  E-mail address: pbourdon@wlu.edu

Michigan State University, East Lansing, MI 48824-1027, USA

  E-mail address: shapiro@math.msu.edu