# For a invertible linear operator $T$ on a complex vector space, $T^{-1}=p(T)$ for some polynomial $p(x)$.

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.

• 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. Oct 21, 2020 at 7:19
• Ok, let me remaind that it has allowed the polynomial in $\Bbb{C}[x]$. Oct 21, 2020 at 8:36

## 1 Answer

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.