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Let $\mathcal{H}$ be a Hilbert space and let {$e_j$}$_{j\in \mathbb{Z}}$ be an orthonormal basis for $\mathcal{H}$. Define a linear operator $T$ on $\mathcal{H}$ by $T(e_0) = 0$ and $T(e_j) = e_{j+1}$ for $j \neq$ 0. Define another linear operator on $\mathcal{H}$ by $A(e_0) = e_1$ and $A(e_j) = 0$ for $j \neq 0$. For $z \in \mathbb{C}$, define $T_z = T +zA$.

I'm being asked to find $\sigma(T_z)$ and to say what happens when $z \to 0$ (which I imagine is obvious if I can find $\sigma(T_z)$).

I've tried finding $\sigma(T_z)$ for certain values of $z$ with little success. I know that $\sigma(A)$ = {$0$} since the spectral radius of $A$ is $0$ ($A^n$ for $n>1$ is just going to be the trivial operator) and so $\sigma(zA)$ = {$0$}; I'm not sure if this will be useful. Correct me if I'm wrong, but $||T_z||$ = $1$ $\lor$ $||z||$ so $\sigma(T_z) \subset \overline{B}_{1 \lor ||z||}(0)$. I believe that the point spectrum of $T_z$ will be empty for $z \neq 0$ which doesn't help me either.

For context, the previous part of this problem asked for $\sigma(T)$.

Any pointers in the right direction would be great.

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1 Answer

up vote 1 down vote accepted

Hint: if $z \ne 0$, $T_z$ is closely related to the ordinary right shift operator.

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Would you mind elaborating on your hint? I can see that $T_z$ is related to the ordinary right shift operator, but only in a superficial way. –  Grasshopper Oct 3 '11 at 1:26
    
For example, I believe that the point spectrum of $T_z$ will be empty for $z \neq 0$ as in the case of the ordinary right shift operator, but this doesn't seem helfpul. –  Grasshopper Oct 3 '11 at 1:54
    
Not just superficially. They are similar. –  Robert Israel Oct 3 '11 at 5:16
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