Algebras of pseudodifferential operators with discontinuous oscillating symbols.
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.
scientific article; zbMATH DE number 5858855
scientific article