Let $M$ be a closed differentiable manifold. Let $E^{p}(M)$ be the vector space of $p$-forms on $M$ equipped with the $L^{2}$-inner product $(\alpha, \beta) = \int_{M}\alpha \wedge \star \beta$. The Laplace de Rahm operator is then defined by $\Delta = d^{*}d + dd^{*} : E^{p}(M) \rightarrow E^{p}(M)$. All these definitions are as in the ordinary Hodge theory. Is the operator $\Delta : E^{p}(M) \rightarrow E^{p}(M)$ bounded with respect to the $L^{2}$-inner product? I have not found any literature which treats these kind of questions. Is there any reference concerning functional analytical properties of this operator (spectrum etc....)?

Luigi

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