Spectra of some weighted composition operators on H2

Carl C. Cowen; Eungil Ko; Derek Thompson; Feng Tian

doi:10.14232/actasm-014-542-y

Spectra of some weighted composition operators on H²

Carl C. Cowen, Eungil Ko, Derek Thompson, Feng Tian

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

We completely characterize the spectrum of a weighted composition operator W Ψ,ℓ on H²(D) when ℓ has Denjoy-Wolff point a with 0 < |ℓ'′(α)| < 1, the iterates, ℓ_n, converge uniformly to a, and is in H∞ (the space of bounded analytic functions on D) and continuous at a. We also give bounds and some computations when |α| = 1 and ℓ'′(α) = 1 and, in addition, show that these symbols include all linear fractional ℓ that are hyperbolic and parabolic nonautomorphisms. Finally, we use these results to eliminate possible weights Ψ so that W Ψ,ℓ is seminormal.

Original language	English
Pages (from-to)	221-234
Number of pages	14
Journal	Acta Scientiarum Mathematicarum
Volume	82
Issue number	1-2
DOIs	https://doi.org/10.14232/actasm-014-542-y
State	Published - 2016

Keywords

Hyponormal operator
Spectrum of an operator
Weighted composition operator

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10.14232/actasm-014-542-y

Cite this

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N2 - We completely characterize the spectrum of a weighted composition operator W Ψ,ℓ on H2(D) when ℓ has Denjoy-Wolff point a with 0 < |ℓ'′(α)| < 1, the iterates, ℓn, converge uniformly to a, and is in H∞ (the space of bounded analytic functions on D) and continuous at a. We also give bounds and some computations when |α| = 1 and ℓ'′(α) = 1 and, in addition, show that these symbols include all linear fractional ℓ that are hyperbolic and parabolic nonautomorphisms. Finally, we use these results to eliminate possible weights Ψ so that W Ψ,ℓ is seminormal.

AB - We completely characterize the spectrum of a weighted composition operator W Ψ,ℓ on H2(D) when ℓ has Denjoy-Wolff point a with 0 < |ℓ'′(α)| < 1, the iterates, ℓn, converge uniformly to a, and is in H∞ (the space of bounded analytic functions on D) and continuous at a. We also give bounds and some computations when |α| = 1 and ℓ'′(α) = 1 and, in addition, show that these symbols include all linear fractional ℓ that are hyperbolic and parabolic nonautomorphisms. Finally, we use these results to eliminate possible weights Ψ so that W Ψ,ℓ is seminormal.

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