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Thread: L_2 space for matrix-valued functions

  1. #1
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    Unhappy L_2 space for matrix-valued functions

    hi,

    here is the standard definition of L_2-space of matrix-valued functions: it's those measurable F that satisfy

    (i) $\displaystyle \int_0^1 ||F(x)||^2dx <\infty$

    (or equivalently, $\displaystyle \int_0^1 ||F(x)^*F(x)||dx <\infty$, or $\displaystyle \int_0^1 trace(F(x)^*F(x))dx <\infty$)

    (here $\displaystyle F^*$ is the adjoint of $\displaystyle F$)

    clearly this implies

    (ii) $\displaystyle \left\|\int_0^1 F(x)^* F(x)dx\right\| <\infty$

    what about the converse -- does (ii) necessarily implies (i), i.e. is (ii) enough for the function to be in L_2?

    for the scalar case (i)=(ii), but I suspect (and afraid) that this fails for matrices...

    ...help...
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  2. #2
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    okay, I think I got it, but I've been struggling with it for half of the day, so I might just be fancying -- please let me know if you see a flaw in the following proof.

    I claim (ii) implies (i).

    $\displaystyle
    \left\|\int_0^1 F(x)^* F(x)dx\right\| <\infty
    $, which means in particular that for any vector
    $\displaystyle \phi$,

    $\displaystyle
    \int_0^1 (F(x)^* F(x)\phi,\phi) dx =$
    $\displaystyle \int_0^1 ||F(x)\phi||^2 dx <\infty
    $

    Choose
    $\displaystyle \phi=e_{j}$. Then we obtain $\displaystyle \int_0^1 \sum_{k=1}^l |F_{kj}(x)|^2 dx <\infty$. Thus for any $\displaystyle k,j$, $\displaystyle \int_0^1 |F_{kj}(x)|^2 dx <\infty$, which implies $\displaystyle \int_0^1 trace(F(x)^*F(x))dx =\int_0^1 \sum_{j,k=1}^l |F_{kj}(x)|^2dx<\infty$, so $\displaystyle F$ is in $\displaystyle L_2$
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  3. #3
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    That proof looks good to me.
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