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Math Help - Linear Operator

  1. #1
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    Linear Operator

    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!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    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.


    Quote Originally Posted by liquidpaper View Post
    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.
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  3. #3
    MHF Contributor
    Opalg's Avatar
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    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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