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Question: Let $V$ be a complex vector space and $T$ be an invertible linear operator on $V$. Show that there is a polynomial $p(x)\in\Bbb{C}[x]$ such that $T^{-1}=p(T)$.

We have if $V$ finite dimensional then using Caley-Hamilton theorem we can show it. But how to approach for infinite case? Please help.

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    $\begingroup$ It is not true in infinite dimensions. For example $(I-T)^{-1}=I+T+T^2+\cdots$, ie $A^{-1}=I+(I-A)+(I-A)^2+\cdots$ which cannot necessarily be reduced to a polynomial. $\endgroup$ Oct 21, 2020 at 7:19
  • $\begingroup$ Ok, let me remaind that it has allowed the polynomial in $\Bbb{C}[x]$. $\endgroup$ Oct 21, 2020 at 8:36

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Consider $V=Vect(e_i,i\in\mathbb{Z})$ and $T$ the shift operator, $T(e_i)=e_{i+1}$.

. Write $P(X)=a_0+a_1X+...+a_nX^n$, $P(T)(e_0)=a_0+a_1e_1+...+a_ne_n=T^{-1}(e_0)=e_{-1}$ is impossible.

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