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Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the function $f(z)= e^{-\psi(z)}\Phi(z)e^{\psi(z)}$ for $z\in \mathbb{D}.$

What can we know about the derivative $f'(0)$ in relation to $\Phi'(0)$ and $\psi'(0)?$ For example, if $\psi$ varies, what can we say about the mapping $\psi'(0)\mapsto f'(0)?$ E.g, surjectivity?

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  • $\begingroup$ Thank you, but I worry about the derivative of $e^{\psi}$ because of non-commutativity between $\psi'$ and $\psi$ (therefore I think we may not have an explicit formula for $(e^{\psi})'$) $\endgroup$
    – Đức Anh
    Commented May 8, 2015 at 21:29
  • $\begingroup$ Yes, you're right of course, this needs to be done more carefully. You could try to write $e^{\psi}\simeq e^{\psi_0+t\psi'}$ and use Campbell-Hausdorff now. $\endgroup$
    – Christian Remling
    Commented May 8, 2015 at 21:41
  • $\begingroup$ Probably the easiest way to figure this out is to write out the second order expansion of each function. $\endgroup$
    – Deane Yang
    Commented May 8, 2015 at 21:55
  • $\begingroup$ Thank you all very much. I have thought too complicatedly. $\endgroup$
    – Đức Anh
    Commented May 8, 2015 at 21:58

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