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PreprintCC BY-NC-ND 4.0arXiv

Berezin Ranges of Composition Operators on H2

  • Paul Bourdon
  • Trieu Le
  • Derek Thompson
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Abstract

We explore properties of the Berezin range B(Cϕ) of a composition operator Cϕ on the Hardy space H2, including constraints on its location in the complex plane ℂ, topological properties, shape, symmetry, and convexity. For example, we show that B(Cϕ) ⊆ {z∈ ℂ: ℜ(z) > 0 and |z| ≤ 1 + |ϕ(0)|}, that B(Cϕ) is closed if and only if Cϕ is the identity operator, and that B(Cϕ) containa all its boundary points except for $0$ if Cϕ is compact. Addressing questions raised by Cowen and Felder, we show that Cϕ can be a (nondegenerate) disk and characterize when B(Cϕ) is convex if ϕ is linear.

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Cite

BibTeXRIS
BibTeX
@misc{bourdon2026berezin,
  title = {{Berezin Ranges of Composition Operators on $H^2$}},
  author = {Paul Bourdon and Trieu Le and Derek Thompson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17634},
  eprint = {2610.17634},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17634},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Berezin Ranges of Composition Operators on H^2
AU  - Paul Bourdon
AU  - Trieu Le
AU  - Derek Thompson
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.17634
UR  - https://arxiv.org/abs/2610.17634
AB  - We explore properties of the Berezin range B(C_ϕ) of a composition operator C_ϕ on the Hardy space H^2, including constraints on its location in the complex plane ℂ, topological properties, shape, symmetry, and convexity. For example, we show that B(C_ϕ) ⊆ {z∈ ℂ: ℜ(z) > 0 and |z| ≤ 1 + |ϕ(0)|}, that B(C_ϕ) is closed if and only if C_ϕ is the identity operator, and that B(C_ϕ) containa all its boundary points except for $0$ if C_ϕ is compact. Addressing questions raised by Cowen and Felder, we show that C_ϕ can be a (nondegenerate) disk and characterize when B(C_ϕ) is convex if ϕ is linear.
ER  -