Certain non-homogeneous matricial domains and Pick-Nevanlinna interpolation problem

Chandel, Vikramjeet Singh

doi:10.48550/arxiv.2009.01834

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2021

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“…The above result is a paraphrasing of [17,Theorem 3]. See [9], [10], [16] for results of the similar type in different settings and [8] for the function-theoretic study of S n (Ω). Result 1.1 naturally motivates the following problem.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 68%

On a spectral version of Cartan's theorem

Bera¹,

Chandel²,

Londhe³

2021

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For a domain Ω in the complex plane, we consider the domain Sn(Ω) consisting of those n × n complex matrices whose spectrum is contained in Ω. Given a holomorphic self-map Ψ of Sn(Ω) such that Ψ(A) = A and the derivative of Ψ at A is identity for some A ∈ Sn(Ω), we investigate when the map Ψ would be spectrum-preserving. We prove that if the matrix A is either diagonalizable or non-derogatory then for most domains Ω, Ψ is spectrum-preserving on Sn(Ω). Further, when A is arbitrary, we prove that Ψ is spectrumpreserving on a certain analytic subset of Sn(Ω).

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