Spectral properties of the Laplace operator in \(L^ p(R)\)
type: Item
instance of: scholarly article
title: Spectral properties of the Laplace operator in \(L^ p(R)\)
zbMATH Open document ID: 0706.47028
author: Werner J. Ricker
published in: Osaka Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47B40
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 4159472
zbMATH Keywords: functional calculus
zbMATH Keywords: scalar-type spectral operator
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4159472