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Composition operators in the Dirichlet series setting

Hervé Queffélec (2007)

Banach Center Publications

In this work, we begin with a survey of composition operators on the Hardy space H² and on the Wiener algebra A⁺ of absolutely convergent Taylor series, with special emphasis on their compactness, or invertibility, or isometric character. The main results are due respectively to J. Shapiro and D.~Newman. In a second part, we present more recent results, due to Gordon and Hedenmalm on the one hand, and to Bayart, the author et al. on the other hand, concerning the analogues of H² and A⁺ in the setting...

Composition operators: $N_{α}$ to the Bloch space to $Q_{β}$

Jie Xiao (2000)

Studia Mathematica

Let $N_{α}$ ,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_{β}$ are Möbius invariant, but $N_{α}$ is not. We characterize, in function-theoretic terms, when the composition operator $C_{ϕ} f = f ◦ ϕ$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_{ϕ} : N_{α} \to B$ , $B \to Q_{β}$ , $N_{α} \to Q_{β}$ which is bounded resp. compact.

Composition operators on Banach spaces of formal power series

B. Yousefi, S. Jahedi (2003)

Bollettino dell'Unione Matematica Italiana

Let ${\{β (n)\}}_{n = 0}^{\infty}$ be a sequence of positive numbers and $1 \leq p < \infty$ . We consider the space $H^{p} (β)$ of all power series $f (z) = \sum_{n = 0}^{\infty} \overset{⌃}{f} (n) z^{n}$ such that $\sum_{n = 0}^{\infty} {|\overset{⌃}{f} (n)|}^{p} β {(n)}^{p} < \infty$ . Suppose that $\frac{1}{p} + \frac{1}{q} = 1$ and $\sum_{n = 1}^{\infty} \frac{n^{q j}}{β {(n)}^{q}} = \infty$ for some nonnegative integer $j$ . We show that if $C_{φ}$ is compact on $H^{p} (β)$ , then the non-tangential limit of $φ^{(j + 1)}$ has modulus greater than one at each boundary point of the open unit disc. Also we show that if $C_{φ}$ is Fredholm on $H_{p} (β)$ , then $φ$ must be an automorphism of the open unit disc.

Composition operators on Musielak-Orlicz spaces of Bochner type

Kuldip Raj, Sunil K. Sharma (2012)

Mathematica Bohemica

The invertible, closed range, compact, Fredholm and isometric composition operators on Musielak-Orlicz spaces of Bochner type are characterized in the paper.

Composition operators on the Dirichlet space and related problems.

Chacón, Gerardo A., Chacón, Gerardo R., Giménez, José (2006)

Boletín de la Asociación Matemática Venezolana

Composition operators on vector-valued Hardy spaces.

S. D. Sharma, U. Bhanu (1999)

Extracta Mathematicae

Composition operators on W 1 X are necessarily induced by quasiconformal mappings

Luděk Kleprlík (2014)

Open Mathematics

Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to L q(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W 1 X to W 1 X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.

In this paper, we discuss the hypercyclicity, supercyclicity and cyclicity of the adjoint of a weighted composition operator on a Hilbert space of analytic functions.

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