Algebras of pseudodifferential operators with discontinuous oscillating symbols.
type: Item
instance of: scholarly article
title: Algebras of pseudodifferential operators with discontinuous oscillating symbols.
zbMATH Open document ID: 1218.47135
published in: Communications in Mathematical Analysis
publication date: 2011-03-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.cma/1298670007
review text: The authors study non-closed algebras \({\mathfrak{A}}_{\delta,\gamma}\) of pseudodifferential operators with slowly oscillating Lipschitz symbols in \(\Lambda^{SO}_{\delta,\gamma}({\mathbb R}\times{\mathbb R})\) with \(\delta,\gamma\in(1/2,1]\) as well as the minimal \(C^*\)-algebra \({\mathfrak{A}}\) containing all \({\mathfrak{A}}_{\delta,\gamma}\) on the Lebesgue space \(L^2({\mathbb R})\). Applying results on the boundedness and compactness of pseudodifferential operators \(A\in{\mathfrak{A}}\), a commutative algebra of their Fredholm symbols is described. A Fredholm criterion and an index formula for the operators \(A\in{\mathfrak{A}}\) are obtained. Then they study the Fredholmness for the more general \(C^*\)-algebra \({\mathfrak{B}}\) generated by the operators \(A\in{\mathfrak{A}}\) with multiplicatively slowly oscillating Lipschitz symbols and by the operators of multiplication \(aI\) and convolution operators \(W^0(b)\) with piecewise continuous functions \(a,b:\overline{{\mathbb R}}\to{\mathbb C}\). The algebra of Fredholm symbols for the operators \(A\in{\mathfrak{B}}\) is not commutative. In this more general setting, \(A\in{\mathfrak{B}}\), the authors establish again a Fredholm criterion.
reviewed by: Alexander Schnurr
Mathematics Subject Classification ID: 47L80
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 5858855
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: compactness
zbMATH Keywords: discontinuous oscillating symbol
zbMATH Keywords: oscillatory integral
zbMATH Keywords: slowly oscillating Lipschitz symbol
author: Yuri I. Karlovich
author: Vladimir Rabinovich
author: Nikolai Vasilevski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5858855