Linear Operator

• May 25th 2011, 03:08 PM
liquidpaper
Linear Operator
Hello, I need some help with this exercise:

Let
T: $\displaystyle$L^1(R)$$--->\displaystyle L^1(R)$$

$\displaystyle$\left( {T\phi } \right)\left( t \right) = \phi \left( {t + 1} \right)\

1) Prove T es la linear operator bounded and calculate its norm.
2) Calculate ker(T) and ran(T) this is the range I believe.

I started with linear operators 2 days ago.. I dont know all the theory yet, thats why Iam asking for help.

Thanks!
• May 25th 2011, 08:23 PM
FernandoRevilla
A little help: using the substitution $\displaystyle u=t+1$ , prove that $\displaystyle T\phi\in L^1(\mathbb{R})$ that is $\displaystyle \int_{-\infty}^{+\infty}|\phi(t+1)|dt<+\infty$ so, $\displaystyle T$ is well defined . Now try to prove that $\displaystyle T$ is a linear map.

Quote:

Originally Posted by liquidpaper
I started with linear operators 2 days ago.. I dont know all the theory yet, thats why I`am asking for help.

You needn't all the theory, surely those two days are sufficient. :)
• May 25th 2011, 11:59 PM
Opalg
For the kernel and range, you might like to note that T has an inverse operator $\displaystyle (S\phi)(t) = \phi(t - 1)$.