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Inverse of the adjoint of the shift operator

  1. Mar 14, 2014 #1
    Hi there,

    Let [itex]S[/itex] denote the shift operator on the Hardy space on the unit disc [itex]H^2[/itex], that is [itex](Sf)(z)=zf(z)[/itex].

    My question is to show the following identity

    [itex](1-\lambda S^*)^{-1}S^*f (z)=\frac{f(z)-f(\lambda)}{z-\lambda},[/itex]

    where [itex]\lambda,z\in\mathbb{D}[/itex]

    Thanks in advance
     
  2. jcsd
  3. Mar 14, 2014 #2

    micromass

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    First of all, can you figure out what ##S^*## does exactly? This can be made easy if you can figure out an orthonormal basis of ##H^2##.
     
  4. Mar 14, 2014 #3
    Yes, [itex]S^*=\frac{f(z)-f(0)}{z}[/itex]. But my problem is with the term [itex](1-\lambda S^*)^{-1}[/itex].
     
  5. Mar 14, 2014 #4

    micromass

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    Just put it on the other side. So you need to prove

    [tex]S^*f(z) = (1-\lambda S^*)\frac{f(z)-f(\lambda)}{z-\lambda}[/tex]
     
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