Ellipticity of operators on a manifold with edges can be treated in the framework of a calculus of 2\times 2-block matrix operators with trace and potential operators on the edges. The picture is similar to the pseudodifferential analysis of boundary value problems. The extra conditions satisfy an analogue of the Shapiro-Lopatinskij condition, provided a topological obstruction for the elliptic edge-degenerate operator in the upper left corner vanishes; this is an analogue of a condition of Atiyah and Bott in boundary value problems. In general, however, we need global projection data, similarly to global boundary conditions, known for Dirac operators or other geometric operators. The present paper develops a new calculus with global projection data for operators on manifolds with edges. In particular, we show the Fredholm property in a suitable scale of spaces and construct parametrices within the calculus.
Edge operators with conditions of Toeplitz type
SEILER, JOERG
2006-01-01
Abstract
Ellipticity of operators on a manifold with edges can be treated in the framework of a calculus of 2\times 2-block matrix operators with trace and potential operators on the edges. The picture is similar to the pseudodifferential analysis of boundary value problems. The extra conditions satisfy an analogue of the Shapiro-Lopatinskij condition, provided a topological obstruction for the elliptic edge-degenerate operator in the upper left corner vanishes; this is an analogue of a condition of Atiyah and Bott in boundary value problems. In general, however, we need global projection data, similarly to global boundary conditions, known for Dirac operators or other geometric operators. The present paper develops a new calculus with global projection data for operators on manifolds with edges. In particular, we show the Fredholm property in a suitable scale of spaces and construct parametrices within the calculus.
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