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For $x\in \mathbb{R}^\mathbb{Z}$, let the discrete Laplacian be defined as \begin{align*} (\Delta x)_k = 2x_k-x_{k+1}-x_{k-1}. \end{align*}

I am looking for good references about its spectrum (or eigen-structure), compactness, and properties of the semigroup it generates.

The spaces to consider include the usual $l^p(\mathbb{Z})$, for my purpose, it is also interesting to work on weighted space $l^p_\rho(\mathbb{Z})$ in which, provided a summable weight $\rho$ with $\sum_k \rho_k <\infty$, the norm is given by $$\|x\|^p_{p,\rho}= \sum_{k\in\mathbb{Z}} |x_k|^p\rho_k.$$

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    $\begingroup$ There won't be any references that discuss this operator at length since it's very easy to analyze: just take Fourier transforms $Fx = \sum x_n e^{int}$ to represent $\Delta$ as multiplication by $2(1-\cos t)$ in $L^2(-\pi,\pi)$. This immediately answers all your questions (for example, the spectrum equals $[0,4]$, is purely ac of multiplicitly $2$). $\endgroup$
    – Christian Remling
    Commented Feb 2, 2020 at 19:06
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    $\begingroup$ Here I'm of course assuming that you defined the operator on $\ell^2(\mathbb Z)$. $\endgroup$
    – Christian Remling
    Commented Feb 2, 2020 at 19:07
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    $\begingroup$ If you are looking for earliest references then see Phillips and Wiener, "Nets and Dirichlet problem" (1923). $\endgroup$
    – Nemo
    Commented Feb 2, 2020 at 19:16
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    $\begingroup$ Could you be more precise about the space on which you want information about this operator. In particular, are you really interested about your questions in the whole topological vector space $\mathbf{R}^\mathbf{\mathbf{Z}}$? $\endgroup$
    – YCor
    Commented Feb 2, 2020 at 23:52

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