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Let $T$ be the Toeplitz operator on $\ell_p$ with symbol $\alpha(\lambda)=a/2\cdot \lambda-(a+1/2)+\lambda^{-1}$, where $a$ is complex. I want to solve the following

$$ Tx=y $$

for $x\in \ell_p$ and $y=(1,q,q^2,\ldots),|q|<1$. Therefore, I (Wiener-Hopf) Factorized the symbol:

$$\alpha(\lambda)=\alpha_{-}(\lambda)\alpha_{+}(\lambda)=(1/2-\lambda^{-1})(a\lambda-1).$$

So, $T^{-1}=T_{\alpha_+^{-1}}T_{\alpha_-^{-1}}$ (only if $a\not=1/\lambda$, not sure if this is true though. When is $T$ one-sided invertible?). Hence (again, not sure),

$$T^{-1}= 2a(S-2I)^{-1}(I-aS_{backw.})^{-1}$$

Now, I want tot compute $T^{-1}y$ to obtain $x$, but I got stuck here. Any hints? My second question is: How is the spectrum of $T$ defined, is there an easy way to compute $\sigma(T)$?

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