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Let $T: H\to H$ be a bounded operator on Hilbert space $H$. $T(e_n) = a_n e_{n+1}$ where $\{e_n\}$ is orthonormal basis and $\{a_n\}$ is bounded sequence.

  1. What is the polar decomposition of $T$?
  2. For what sequences $T$ is Fredholm?
  3. For what sequences $T$ compact?
  • 1
    What have you tried? Where did you get stuck? Do you have any thoughts on the problem?2012-12-17
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    I know T = |T| U but cant seem to compute U ( U partial isometry )2012-12-17
  • 0
    Write out the 'matrix' for $T$. The partial isometry should be fairly clear.2012-12-17
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    cant seem to get this hint2012-12-17
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    $\begin{bmatrix}0 & 0 & \cdots \\ \text{sgn}\, a_1 & 0 & \cdots \\ 0 & \text{sgn}\, a_2 & \cdots \\ \vdots & &\end{bmatrix}$ (or something like $\frac{a_n}{|a_n|}$ if complex).2012-12-17
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    Oops, the above is the partial isometry part.2012-12-17

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