无限维空间里是否也存在最小多项式？

abababa

如题。在有限维空间里一定存在，很好证明。如果是无限维空间，是否还一定存在最小多项式？
最小多项式：线性变换$\sigma$的次数最低的首一零化多项式称为$\sigma$的最小多项式。
零化多项式：对于数域$\mathbb{F}$上的线性空间$V$上的线性变换$\sigma$，若存在$f\in\mathbb{F}[x]$使得$f(\sigma)$是零变换，则称$f$是$\sigma$的零化多项式。

hbghlyj

Does every linear operator have a minimal polynomial?
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.

Czhang271828

没仔细看二楼, 已经写了(看来多数人最先给出的反例都是这个).
原答案:
反例一堆啊, 比如 $\ell:\mathbb N^{\mathbb N}\to \mathbb N^{\mathbb N}, (a_n)_{n\in \mathbb N}\to (na_n)_{n\in \mathbb N}$ 就是. 显然 $\{\mathrm{id},\ell,\ell^2,\ldots\}$ 是线性无关向量(证明思路: 反证法, 构造范德蒙德行列式, 矛盾).

hbghlyj

hbghlyj 发表于 2023-5-12 15:23
If there was a minimal polynomial, all polynomials would be solutions of the same linear differential equation with constant coefficients.

为什么linear differential equation?

abababa

hbghlyj 发表于 2023-5-12 22:23
Does every linear operator have a minimal polynomial?
I would say most linear operators don't have a ...

嗯，微分方程这个我到是想过，但没构造出来，线性空间里的元素是多项式，最小多项式也是多项式，给我弄乱了。