Does $\langle f(x),x\rangle=\langle g(x),x\rangle$ imply that $f=g$?

Question

Let $V$ be a a Euclidean or unitary vector space and $f,g$ be linear endomorphisms from $V$ such that $\langle f(x),x\rangle=\langle g(x),x\rangle,\,\forall x\in V$ holds. Does $f=g$ hold? Does it hold if $V$ is finite-dimensional?

I thought that this looked like a good application of the polarisation identities since the terms in there look similar to the above terms but I haven't been able to show the validity of the claim above with this or anything else so far (nor have I been able to disprove it).

@AnginaSeng It means Euclidean or unitary, I've edited the question. — , Jul 04 '20 at 13:37
Over $\Bbb R$, it's false. Over $\Bbb C$ (with a positive definitive Hermitian product) it's true. — Angina Seng, Jul 04 '20 at 13:39
@AnginaSeng Thanks! Could you maybe give me a hint on why it's true in the case of $\mathbb{C}$ with a positive definite Hermitian product? — , Jul 04 '20 at 14:13
This is a duplicate, cf https://math.stackexchange.com/questions/2033703 — Hanno, Jul 04 '20 at 14:26

José Carlos Santos · Answer 1 · 2020-07-04T13:32:41.730

3

No. Take $f,g\colon\Bbb R^2\longrightarrow\Bbb R^2$ defined by $f(x,y)=(-y,x)$ and $g(x,y)=(y,-x)$. For each $v\in\Bbb R^2$, you have$$\bigl\langle f(v),v\bigr\rangle=\bigl\langle g(v),v\bigr\rangle=0.$$However, $f\ne g$. In fact, unless $v=(0,0)$, $f(v)\ne g(v)$.

edited Jul 04 '20 at 13:32

answered Jul 04 '20 at 13:31

José Carlos Santos

440,053

3

Or, more easily, take $g(x,y)=(0,0)$. – Angina Seng Jul 04 '20 at 13:32
@AnginaSeng Sure! Nice remark. – José Carlos Santos Jul 04 '20 at 13:33

score 0 · Answer 2 · answered Jul 04 '20 at 14:36

0

As José Carlos Santos pointed it out, it's not true in a real Hilbert-space. But on a complex Hilbert-space it's true, thanks to the following polarization identity: $$(Ax|y)=\frac{1}{4}\sum_{k=0}^3 i^k(A(x+i^ky)|x+i^ky)$$

answered Jul 04 '20 at 14:36

Botond

12,134

Does $\langle f(x),x\rangle=\langle g(x),x\rangle$ imply that $f=g$?

2 Answers2