Norm closure and extension of the symbolic calculus for the cone algebra
type: Item
instance of: scholarly article
title: Norm closure and extension of the symbolic calculus for the cone algebra
zbMATH Open document ID: 0843.46034
DOI: 10.1007/BF00773405
author: Frank Mantlik
published in: Annals of Global Analysis and Geometry
publication date: 1996-08-18T00:00:00Z
review text: The paper deals with the `cone algebra' \({\mathfrak C}= {\mathfrak C}^0(\mathbb{B}, {\mathfrak g}^\bullet_\gamma)\) of B.-W. Schulze. This is a *-algebra of zero-order pseudodifferential operators on a (stretched) manifold \(\mathbb{B}\) with conical singularities; each \(A\in {\mathfrak C}\) acts on a scale \(\{{\mathcal H}^{s, \gamma}(\mathbb{B})\}_{s\in \mathbb{R}}\) of weighted \(L_2\)-Sobolev spaces. The symbol of \(A\) contains a scalar part \(\sigma_\psi(A)\) and an operator function \(\sigma_M(A)\), the `Mellin symbol'. The author studies the closures \({\mathfrak A}_s\) and \({\mathfrak A}_\infty\) of \(\mathfrak C\) in \({\mathcal L}({\mathcal H}^{s, \gamma}(\mathbb{B}))\) and \(\bigcap_s {\mathcal L}({\mathcal H}^{s, \gamma}(\mathbb{B}))\), respectively. These sets turn out to be \(C^*\)-resp. Fréchet\(^*\)-algebras, equipped with natural extensions of \(_s\sigma_{{\mathfrak C}}\), \(_\infty\sigma_{{\mathfrak C}}\) of the symbol homomorphism \(\sigma_{{\mathfrak C}}= (\sigma_\psi, \sigma_M)\). The Fredholm property of \(A\in {\mathfrak A}_\times\) is characterized by the invertibility of \(_\times\sigma_{{\mathfrak C}}(A)\). A crucial step is to show that the symbol algebras \(_s\sigma_{{\mathfrak C}}({\mathfrak A}_s)\), \(_\infty\sigma_{{\mathfrak C}}({\mathcal A}_\infty)\) are again of type \(C^*\) resp. Fréchet\(^*\). As a central result one obtains spectral invariance of \({\mathfrak A}_\infty\) in \({\mathcal L}({\mathcal H}^{s, \gamma}(\mathbb{B}))\); one of the consequences is the existence of a holomorphic functional calculus for \({\mathfrak A}_\infty\). The methods employed are easily adaptable to other pseudodifferential algebras; previous approaches of H. O. Cordes and E. Schrohe for algebras with scalar symbol seem to be less flexible. In a forthcoming paper an axiomatic outline of the present approach will be given.
reviewed by: Frank Mantlik
Mathematics Subject Classification ID: 46H30
Mathematics Subject Classification ID: 46H35
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 839170
zbMATH Keywords: manifolds with singularities
zbMATH Keywords: cone algebra
zbMATH Keywords: Mellin symbol
zbMATH Keywords: *-algebra of zero-order pseudodifferential operators
zbMATH Keywords: \(L_ 2\)-Sobolev spaces
zbMATH Keywords: Fréchet\(^*\)-algebras
zbMATH Keywords: symbol homomorphism
zbMATH Keywords: Fredholm property
zbMATH Keywords: spectral invariance
zbMATH Keywords: holomorphic functional calculus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 839170