An automorphism in a Banach space

[Calabi]

Hello I've got a problem : let be a normed vectorial space E, N and A an continue automorphism.
I suppose E is complete. So by the banach theorem
$$A^{−1}$$
is continue.
So now let be f a k lipshitz application with
$$k<\frac{1}{||A^{−1}||}$$.
.
I'd like to show that f + A is an homomorphism.
I don't even know how to start.

Have you got any idea please?

Thank you in advance and have a nice afternoon.

 

[Samy_A]

I don't understand.
If f is a Lipschitz function (from E to E), f + A will be continuous, and f + A will also be linear if f is linear.

What am I missing?

 

[Calabi]

Oh no. Change homomorphisme in automorphism.
And I never say that f is linear.

 

[Samy_A]

Oh no. Change homomorphisme in automorphism.
And I never say that f is linear.

Ok, you changed the question.

Anyway, an automorphism on the Banach space E would be a bounded linear invertible operator. A necessary condition for f + A to be an automorphism is f being linear.

EDIT: if A is an automorphism of a Banach space, and B is a linear operator on that space satisfying $\|A-B\|<\frac{1}{\|A^{−1}\|}$, then B is invertible too.
That would imply that in your case f + A will be an automorphism if f is linear.

 

[WWGD]

Oh no. Change homomorphisme in automorphism.
And I never say that f is linear.

Like Samy_A said, linearity is necessary, unless you consider some structure other than the vector space/linear structure on your space, which does not seem to be what you are doing. EDIT basically, automorphisms of a space are maps (from the space to itself here) that respect/preserve the structure of the space. A Banach space is by definition a linear/vector space, so automorphisms are maps that must preserve this structure.

 

[S.G. Janssens]

Oh no. Change homomorphisme in automorphism.
And I never say that f is linear.

I think it may be that the OP is trying to merely prove that $A + f$ is invertible, not that it is an automorphism. For this, it is indeed not needed that $f$ is linear.

 

[Samy_A]

I think it may be that the OP is trying to merely prove that $A + f$ is invertible, not that it is an automorphism. For this, it is indeed not needed that $f$ is linear.

I tried to mimic the proof for the basic linear case: if $T$ is a linear operator satisfying $\|T\|<1$, then $I-T$ is invertible.
So the hypothesis is: If $f$ is a Lipschitz function satisfying $\forall x,y: \ \|f(x)-f(y)\|<=k\|x-y\|$, with $k<1$, then $I-f$ is invertible.
In the linear case one constructs the inverse with the series $\sum_{n=0}^\infty T^n(x)$, that obviously converges for linear T.
Does the similar series for the Lipschitz function, $\sum_{n=0}^\infty f^n(x)$, also converge? It does if $f(0)=0$, but not in general.

 

[Calabi]

Hello guys and girls and sorry for the time of answer.

 

[Calabi]

: if TT is a linear operator satisfying ∥T∥<1\|T\|

Is it always true please?

 

[Samy_A]

Is it always true please?

What is true is that if $T$ is a linear operator on a Banach space satisfying $\|T\|<1$, then $I-T$ is invertible.
(A specific case of a more general property of unital Banach algebras.)

 

[Calabi]

Which property do you refere please?

 

[Calabi]

Forget what I said : in a Banch space the absolute convegrence make the convergence, so let be a Banach algebra : let be x with $$||x||< 1$$, as $$\Sigma ||x||^{n}$$ converge the $$\sigma x^{n}$$ converge,
and $$(x - Id)$$ is invertible his inverse is $$\Sigma x^{n}$$.

Since E is a banach space so all all the continious endomorphisme are a Banach algebra.

 

[Calabi]

But $$k < \frac{1}{||A||^{-1}}$$. We could have $$k \ge 1$$.

 

[Calabi]

It's not taht esay.

 

[Samy_A]

But $$k < \frac{1}{||A||^{-1}}$$. We could have $$k \ge 1$$.

If $T$ is a linear operator on a Banach space satisfying $\|T\|<1$, then $I-T$ is invertible.
You can deduce from this (see it as an exercise):
if A is an automorphism of a Banach space, and B is a linear operator on that space satisfying $\|A-B\|<\frac{1}{\|A^{−1}\|}$, then B is invertible too.

It's not taht esay.

It would be easier if you stated clearly what you want to prove. For now we are just trying to guess what you really mean. Strictly speaking your question as stated has been answered.

,

 

[Calabi]

I think we want to proove A + f is an automorphisme.

 

[Calabi]

As it at been said we have to proove f linear and bijectif.

 

[Samy_A]

Two different answers to the question "what do you want to prove".

I think we want to proove A + f is an automorphisme.

Then we are back to where we were two days ago. A + f will be an automorphism if and only if f is linear (given the properties of A and f in the OP).

As it at been said we have to proove f linear and bijectif.

No idea where this comes from. There is no way one can prove f is linear and/or bijective using only the properties of f given in the OP.

 

[S.G. Janssens]

Could a moderator please clean this thread up? It has become quite unreadable, notwithstanding the amount of patience that Samy_A has displayed. We require a clear problem statement from the OP, written in a single post and in clear English. Only then can others help.

 

[Dale]

Thread closed for moderation.

 

[Dale]

@Calabi after discussion with the mentors we recommend that you open up a new thread in the advanced homework section. Please use the template, that will greatly benefit the discussion and add some structure and clarity.

 

[Calabi]

Hello sorry I'm late, I accept and I'm sorry for 2 reason : first I was not very active on the conversation.
Then this exercicse is extract from a paper and the teacher made many mistake in the paper.