ESSENTIAL NORMS OF LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON h<sup>∞</sup>

Izuchi, Keiji; Izuchi, Kou Hei

doi:10.4134/jkms.2013.50.1.137

Cited by 2 publications

(4 citation statements)

References 14 publications

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“…We have p n ∞ = 1 and p n → 0 uniformly on any compact subsets of D as n → ∞. Then (see Proposition 2.1 of [9]), (2) holds.…”

Section: Main Results and Proofsmentioning

confidence: 96%

“…Along the line of study on differences, the study on linear combination of composition operators has been a topic of growing interest. Izuchi and Ohno [9] have given a complete characterization for the compactness of linear combination of composition operators on H ∞ . Hosokawa, Nieminen and Ohno [7] characterized the compactness of the linear combination of composition operators on the Bloch space.…”

Section: Introductionmentioning

confidence: 99%

“…For further results on the linear combination of composition operators, we refer to [1,2,3,4,6,7,9,10,11] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Linear combination of composition operators on $H^\infty$ and the Bloch space

Shi,

2018

Preprint

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“…We have p n ∞ = 1 and p n → 0 uniformly on any compact subsets of D as n → ∞. Then (see Proposition 2.1 of [9]), (2) holds.…”

Section: Main Results and Proofsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Linear combination of composition operators on $H^\infty$ and the Bloch space

Shi,

2018

Preprint

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“…Then f * ∂D = f ∞ . For u ∈ L ∞ (∂D), let u(z) be the harmonic extension of u defined by See [6,11] for other subjects on C(H ∞ ). This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Path connected components in weighted composition operators on $h^\infty $ and $H^\infty $ with the operator norm

Izuchi

Izuchi²,

Ohno

2012

Trans. Amer. Math. Soc.

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We consider the component problem on the sets of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk with the operator norms, respectively. Especially, we shall determine path connected components in the sets of noncompact weighted composition operators. u(z) = ∂D u(e iθ)P z (e iθ) dm(e iθ) for z ∈ D, where P z is the Poisson kernel for the point z ∈ D and m is the normalized Lebesgue measure on ∂D. Then u ∈ h ∞. Besides, for f ∈ h ∞ , f * = f on D. Let S(D) be the set of analytic self-maps of D. For each ϕ ∈ S(D), we may define the composition operator C ϕ by C ϕ f = f •ϕ, where f is analytic on D. Composition operators on various analytic function spaces have been studied extensively during the past few decades. See [2, 14] for an overview of these results. Presently, some of the open questions in this field are related to the topological structure of the set of composition operators. For even f ∈ h ∞ , C ϕ f = f • ϕ is also harmonic on D. So the question of properties of C ϕ on h ∞ naturally arises. In [1], Choa and the first and third authors first considered properties of composition operators on h ∞ and, although h ∞ is not algebra, introduced the concept of weighted composition operators M u C ϕ on h ∞ defined by (M u C ϕ f)(z) = ∂D u(e iθ)(f • ϕ) * (e iθ)P z (e iθ) dm(e iθ)

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ESSENTIAL NORMS OF LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON h^∞

Abstract: Abstract. It is studied the linear combinations of composition operators on the Banach space of bounded harmonic functions on the open unit disk. We determine the essential norm of them.

Cited by 2 publications

References 14 publications

Linear combination of composition operators on $H^\infty$ and the Bloch space

Linear combination of composition operators on $H^\infty$ and the Bloch space

Path connected components in weighted composition operators on $h^\infty $ and $H^\infty $ with the operator norm

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ESSENTIAL NORMS OF LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON h∞

Abstract: Abstract. It is studied the linear combinations of composition operators on the Banach space of bounded harmonic functions on the open unit disk. We determine the essential norm of them.

Cited by 2 publications

References 14 publications

Linear combination of composition operators on $H^\infty$ and the Bloch space

Linear combination of composition operators on $H^\infty$ and the Bloch space

Path connected components in weighted composition operators on $h^\infty $ and $H^\infty $ with the operator norm

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ESSENTIAL NORMS OF LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON h^∞