[Math] Does every linear operator have a minimal polynomial

Question

I know that a linear operator $T$ defined on a finite-dimensional vector space has a minimal polynomial since, by Caley-Hamilton, $g(T)=0$, where $g$ is the characteristic polynomial. Is there a linear operator defined on an infinite-dimensional vector space that has no minimal polynomial?

Answer 1

I would say most linear operators don't have a minimal polynomial. Take for example the space $\mathbb{R}^{\mathbb{N}}$ of all real sequences and define $$T((u_n)_{n\ge 0}) = (n u_n)_{n\ge 0}$$ Every integer is an eigenvalue of $T$, therefore there is no minimal polynomial.

Another simple example is the derivation operator $D(P)= P^\prime$ on the space $\mathbb{R}[X]$ of all real polynomials. If there was a minimal polynomial, all polynomials would be solutions of the same linear differential equation with constant coefficients.