Isolation amongst the composition operators

Abstract

Earl Berkson has shown that certain highly non-compact composition operators on the Hardy space H 2 are, in the operator norm topology, isolated from all the other composition operators. On the other hand, it is easy to see that no compact composition operator is so isolated. Here we explore the intermediate territory, with the following results: (i) Only the extreme points of the H°° unit hall can induce isolated composition operators. In particular, those holomorphic self-maps of the unit disc whose images make at most finite order of contact with the unit circle induce composition operators that are not isolated. However, (ii) extreme points do not tell the whole story about isolation: some of them induce compact, hence non-isolated, composition operators. Nevertheless, (iii) all sufficiently regular univalent extreme points induce isolated composition operators.Introduction. It is a familiar fact of elementary function theory that the composition of holomorphic functions is again holomorphic. More precisely, if φ is a holomorphic function taking a plane domain into itself, and if / is holomorphic on that domain, then so is the composition / o φ. Less familiar is the fact that if the domain is the unit disc U, and / belongs to the Hardy space H 2 of U, then so does / o φ . This is Littlewood's Subordination Principle ([13], [17], [25]), which in modern language states that the composition operator C φ defined on functions holomorphic in U by:restricts to a bounded linear operator on H 2 . The remarkable aspect of Littlewood's Principle is that nothing extra is required of the holomorphic map φ : it need not be univalent, or even boundedly valent, nor is it required to have any regularity at the boundary.Littlewood's Principle raises the possibility of explaining the behavior of the operator C φ in terms of the function theoretic properties of the inducing map φ , and so provides a new point of contact between function theory and functional analysis. Ground in this area was broken about twenty years ago by Eric Nordgren [20], who determined the spectra of composition operators induced by disc automorphisms; , endowed with the operator norm metric. It says that the composition operator C φ is isolated in Comp(// 2 ) whenever \φ\ = 1 on a subset of the unit circle having positive measure. For example this result locates the identity operator, as well as any composition operator induced by an inner function, at least 1/Λ/2 units distant from every other composition operator.At the other extreme, an elementary argument (Proposition 2.2) shows that the compact composition operators are dramatically nonisolated: they all lie in the same path component of Comp(/f 2 ). The problem of characterizing the compact composition operators is a subtle one that has only recently been answered [25]. Early on, Schwartz observed that holomorphic self-maps φ of U which have radial limits of modulus one on a set of positive measure induce noncompact composition operators on H 2 ([24], see also [26]). He also observed that there are...

Hypercyclic operators that commute with the Bergman backward shift

“…Particularly striking is the association with extreme points of the H ∞ unit ball, which we recall are characterized for all bounded analytic functions ϕ with ϕ ∞ = 1 by failure of the logarithmic integrability condition (16). In [26] it is proved that if C ϕ is isolated from other composition operators on H 2 , then ϕ must be an extreme point (but not conversely). We do not know if the analogous result holds for our present problem: Corollary 3.3 can be regarded as providing evidence in favor of an affirmative answer to this question: For the class of mappings considered there, "exponential contact of order 1" can be thought of as a sort of dividing line between extreme points and non-extreme points.…”

“…A similar question arises for composition operators on the Hardy and Bergman spaces, both when one tries to characterize which of these operators are non-compact (see [10, §3.2], [24], [25]), and when one tries to characterize which ones are isolated from the other composition operators in the operator-norm topology (see [10, §9.3] and [26]). Our results on the commutant hypercyclicity problem resemble most closely those obtained in [26] for the isolation problem, although why there should be such a connection remains mysterious. Particularly striking is the association with extreme points of the H ∞ unit ball, which we recall are characterized for all bounded analytic functions ϕ with ϕ ∞ = 1 by failure of the logarithmic integrability condition (16).…”

Linear relations in the Calkin algebra for composition operators

“…A variation on ideas of Berkson [2], Shapiro and Sundberg [26], and MacCluer [13] (see Exercise 9.3.2 in [8]) states that if ϕ 1 , . .…”

“…In [26], Shapiro and Sundberg posed the following interesting question: What is the relationship between the conditions (a) C ϕ − C ψ is compact; (b) C ϕ and C ψ lie in the same component of the topological space of composition operators on D β ? The second author and Toews [17] proposed the scheme discussed in Section 2.6 and used it to give examples of composition operators satisfying (b) but not (a).…”

“…The problem of compact difference was raised in an explicit form by Shapiro and Sundberg [26] and MacCluer [13]; these authors found several criteria, some necessary and some sufficient. More recently, MacCluer, Ohno, and Zhao [14] have shown that compactness of C ϕ − C ψ acting on H ∞ , the space of bounded analytic functions on D, is characterized in terms of the quantity ρ(z) =…”