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I was wondering whether there exists any kind of literature on the the powers of the discrete Laplacian, in particular the the discrete bi-Laplacian, possibly with weights on the edges.

In particular I wanted to ask the following question:

given some uniformly bounded, symmetric weights one can easily see that the one has the operator inequality

$$-\Delta_{w}\geq c(-\Delta)$$

where $-\Delta_{w}$ denotes the weighted discrete Laplacian and $-\Delta$ thee "free" one. $c$ is a positive constant given by the uniform bound on the weights. The inequality should be seen as an operator inequality in the sense of the quadratic Dirichlet form.

Obviously, $x^2$ is not an operator monotone function. But maybe there are some special properties of the discrete Laplacian, so that

$$(-\Delta_{w})^2\geq c(-\Delta)^2$$

could be true?

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