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On a family of domains associated with ΞΌ-synthesis in M2(β„‚)

  • Indranil Biswas
  • Sourav Pal
  • Nitin Tomar
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Abstract

In this paper, we show that the domain ℍ serves as a unifying domain for 𝔾2, 𝔼 and β„™ by proving that each of these domains is an analytic retract of ℍ. Further, we prove that the bidisc 𝔻2 and the Euclidean unit ball 𝔹2 are also analytic retracts of ℍ. Then the rational ℍ̅-inner functions, i.e., the rational maps from 𝔻 into ℍ̅ whose boundary values lie in the distinguished boundary of ℍ, are characterized. Two different versions of Schwarz type lemma for ℍ are established of which one provides a necessary and sufficient condition. Finally, capitalizing on the theory of ℍ, we present alternative proofs to a few known results on β„™ from the literature.

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Cite

BibTeXRIS
BibTeX
@misc{biswas2026family,
  title = {{On a family of domains associated with $ΞΌ$-synthesis in $M_2(\mathbb C)$}},
  author = {Indranil Biswas and Sourav Pal and Nitin Tomar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38677},
  eprint = {2610.38677},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38677},
}
RIS
TY  - UNPB
TI  - On a family of domains associated with ΞΌ-synthesis in M_2(β„‚)
AU  - Indranil Biswas
AU  - Sourav Pal
AU  - Nitin Tomar
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38677
UR  - https://arxiv.org/abs/2610.38677
AB  - In this paper, we show that the domain ℍ serves as a unifying domain for 𝔾_2, 𝔼 and β„™ by proving that each of these domains is an analytic retract of ℍ. Further, we prove that the bidisc 𝔻^2 and the Euclidean unit ball 𝔹_2 are also analytic retracts of ℍ. Then the rational ℍ̅-inner functions, i.e., the rational maps from 𝔻 into ℍ̅ whose boundary values lie in the distinguished boundary of ℍ, are characterized. Two different versions of Schwarz type lemma for ℍ are established of which one provides a necessary and sufficient condition. Finally, capitalizing on the theory of ℍ, we present alternative proofs to a few known results on β„™ from the literature.
ER  -