Impossible relation


Prove that we cannot find any two linear continuous maps f,g : X \to X such that (f\circ g -g\circ f)x=x,\ \forall x \in X.
Solution: Prove by induction that f \circ g^{n+1}-g^{n+1}\circ f =(n+1)g^n. Taking norms, we get (n+1)\|g^n \| \leq 2 \|f\| \|g\| \|g^n\|. If \|g^n\| \neq 0 then n+1 \leq 2 \|f \| \|g\|,\ \forall n, which is a contradiction. On the other hand, if g^n=0 then g^{n-1}=0, which implies g=0 and I=0. Contradiction. Therefore, we cannot find linear continuous function with given property.

Advertisements
  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: