Complex symmetric weighted composition operators on Bergman spaces and Lebesgue spaces

Hai, Pham Viet; Severiano, Osmar R.

doi:10.48550/arxiv.2105.14979

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Complex Symmetric Composition Operators1

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2021

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“…Using the same rule as J, but in other spaces, [19,20,21,23] also described weighted composition operators and composition operators that are J-symmetric.…”

Section: Complex Symmetric Composition Operatorsmentioning

confidence: 99%

Composition operators on Hardy-Smirnov spaces

V.¹,

V.²,

Pellegrino³

et al. 2021

Preprint

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We investigate composition operators CΦ on the Hardy-Smirnov space H 2 (Ω) induced by analytic self-maps Φ of an open simply connected proper subset Ω of the complex plane. When the Riemann map τ : U → Ω used to define the norm of H 2 (Ω) is a linear fractional transformation, we characterize the composition operators whose adjoints are composition operators. As applications of this fact, we provide a new proof for the adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez and we give a new approach to describe all Hermitian and unitary composition operators on H 2 (Ω). Additionally, if the coefficients of τ are real, we exhibit concrete examples of conjugations and describe the Hermitian and unitary composition operators which are complex symmetric with respect to specific conjugations on H 2 (Ω). We finish this paper showing that if Ω is unbounded and Φ is a non-automorphic self-map of Ω with a fixed point, then CΦ is never complex symmetric on H 2 (Ω).

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“…Using the same rule as J, but in other spaces, [19,20,21,23] also described weighted composition operators and composition operators that are J-symmetric.…”

Section: Complex Symmetric Composition Operatorsmentioning

confidence: 99%