Composition Operators Isolated in the Uniform Operator Topology

Berkson, Earl

doi:10.2307/2044200

Cited by 29 publications

(39 citation statements)

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“…The inequality above yields for each n which upon iteration shows that g(β n ) < g(a n^) < g{β n -λ ) 2 …”

Section: )mentioning

confidence: 98%

“…\\C φ -C ψ \\>\j measls The norm here is the usual one for bounded linear operators on H 2 , and the measure is normalized Lebesgue measure on the unit circle. Berkson , endowed with the operator norm metric.…”

Section: Berkson's Isolation Theorem ([2]) If φ Has Radial Limits Ofmentioning

confidence: 99%

“…Recall also the crucial assumption on the degree of contact that Ω has with the positive real axis: (2) Γ\ogh(x)dx = -oo.…”

Section: Choosing the Proper Test Functionsmentioning

confidence: 99%

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Isolation amongst the composition operators

Shapiro¹,

Sundberg²

1990

Pacific J. Math.

138

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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...

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“…The inequality above yields for each n which upon iteration shows that g(β n ) < g(a n^) < g{β n -λ ) 2 …”

Section: )mentioning

confidence: 98%

Section: Berkson's Isolation Theorem ([2]) If φ Has Radial Limits Ofmentioning

confidence: 99%

See 1 more Smart Citation

Isolation amongst the composition operators

Shapiro¹,

Sundberg²

1990

Pacific J. Math.

138

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show abstract

“…Finally, we need a variant of a result of Berkson [4] and Shapiro and Sundberg [19], which states that if ψ 1 , · · · , ψ n are distinct analytic self-maps of D and J(ψ i ) = {e iθ : |ψ i (e iθ )| = 1}, then for any complex constants c 1 , · · · , c n ,…”

Section: Necessary Conditionsmentioning

confidence: 99%

Composition operators within singly generated composition C*-algebras

2010

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Abstract. Let ϕ be a linear-fractional self-map of the open unit disk D, not an automorphism, such that ϕ(ζ) = η for two distinct points ζ, η in the unit circle ∂D. We consider the question of which composition operators lie in C * (Cϕ, K), the unital C * -algebra generated by the composition operator Cϕ and the ideal K of compact operators, acting on the Hardy space H 2 . This necessitates a companion study of the unital C * -algebra generated by the composition operators induced by all parabolic non-automorphisms with common fixed point on the unit circle.

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“…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 , . .…”

Section: ϕ(Z)−ψ(z) 1−ϕ(z)ψ(z)mentioning

confidence: 99%