Spectral properties of the Laplace operator in \(L^ p(R)\) (Q919251)

Let \(1<p<\infty\) and let L denote the operator \(-d^ 2/dx^ 2\) acting on \(L^ p({\mathbb{R}})\), with its natural domain of definition. The aim of this paper is to show that, although L has a functional calculus based on the functions of finite total variation on \([0,\infty)\), L is not a scalar-type spectral operator unless \(p=2\). The idea of the proof is to show that, if L were scalar-type spectral, then so too would be its square root -id/dx, thereby contradicting (when \(p\neq 2)\) a known result.