If $A(z)=[A_{ij} (z)] $ is an $n\times n$ unitary matrix valued functions. Is there a characterization of such matrix if:
(1) the entries are analytic functions on a set $D$.
and
(2) if the entries are entire functions.
Thanks in advance
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3$\begingroup$ The entries are bounded (by $1$), hence if they are entire functions, they are constant. $\endgroup$– abxCommented Dec 9, 2019 at 17:23
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$\begingroup$ I think that question (1) was reasonable, since it is not necessarily the case that researchers in one area of maths would know the tools used in Alexandre Eremenko's elegant answer $\endgroup$– Yemon ChoiCommented Dec 10, 2019 at 16:38
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$\begingroup$ @abx. Thank you, that make sense. $\endgroup$– GuestCommented Dec 11, 2019 at 19:41
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1 Answer
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Such matrices can exist only if their entries are constant. For entire functions this is proved in he remark of @abx. For analytic functions in any other region this can be proved as follows: the entiries of the $j$-th column of your matrix must satisfy $$|a_{1,j}|^2+\ldots+|a_{m,j}|^2=1.$$ Notice that $|a_{ij}|^2$ are subharmonic, and their sum is harmonic. Therefore they must be all harmonic, but it is easy to see (for example, by using Cauchy-Riemann equations) that for an analytic function $|f|^2$ can be harmonic only if $f$ is constant.
answered Dec 9, 2019 at 17:33