Linear Operator

**liquidpaper** · May 25th 2011, 03:08 PM

Hello, I need some help with this exercise:

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

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

1) Prove T es la linear operator bounded and calculate it`s 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, that`s why I`am asking for help.

Thanks!

**FernandoRevilla** · May 25th 2011, 08:23 PM

A little help: using the substitution $u=t+1$ , prove that $T\phi\in L^1(\mathbb{R})$ that is $\int_{-\infty}^{+\infty}|\phi(t+1)|dt<+\infty$ so, $T$ is well defined . Now try to prove that $T$ is a linear map.

Originally Posted by liquidpaper

I started with linear operators 2 days ago.. I dont know all the theory yet, that`s why I`am asking for help.

You needn't all the theory, surely those two days are sufficient.

**Opalg** · May 25th 2011, 11:59 PM

For the kernel and range, you might like to note that T has an inverse operator $(S\phi)(t) = \phi(t - 1)$ .

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