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Let $M_{k}(\mathbb{C})$ be the set of $k \times k$ complex matrices. I am trying to find a sequence of polynomials $P_{n}: M_{k}(\mathbb{C}) \to M_{k}(\mathbb{C})$ (or continuous functions $f_{n}$) such that $P_{n}$ converges to identity matrix $1_{k}$ uniformly for compact subspaces of all matrices with Hilbert–Schimdt norm $\leq 1$, except $0$ matrix, and $P_{n}(0)=0$.
So far I have tried:

  1. $P_{n}(A)=1_{k}-(1_{k}-(AA^*)^2)^n$, where $A^*=\overline{A^{T}}$ (but I am running into trouble when A is singular diagonal matrices with H.S. norm $\leq 1$, then the $P_{n}$ does not converge to $1_{K}$)

  2. To rectify that I have tried the following: $f_{n}(A)=1_{k}-e^{-nAA^*}$, but still the same issue persists (I think).

I will be highly obliged for any suggestion or hint.

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    $\begingroup$ What is an example for $k = 1$? $\endgroup$
    – LSpice
    Commented Oct 12, 2021 at 22:14
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    $\begingroup$ for k=1, taking $P_{n}(x)=1-(1-x^2)^n$ works I think. $\endgroup$
    – user938363
    Commented Oct 12, 2021 at 22:17
  • $\begingroup$ @markvs if AA*=0, then is it not true then A=0? as the sum of diagonal elements of AA* is \sum\limits_{i,j}|a_{i,j}|^2=0, then it implies a_{i,j}=0. I could be wrong. $\endgroup$
    – user938363
    Commented Oct 12, 2021 at 23:33
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    $\begingroup$ What is your definition of a polynomial from $M_k(\mathbf{C})$ to itself? $\endgroup$
    – YCor
    Commented Oct 13, 2021 at 7:09
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    $\begingroup$ @user938363 this is typically not a complex polynomial, but indeed is (component-wise) polynomial in the $2k^2$ real variables. $\endgroup$
    – YCor
    Commented Oct 13, 2021 at 16:05

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