tailieunhanh - Báo cáo toán học: "A note on the space of pseudodifferential projections with the same principal symbol "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một lưu ý trên không gian của dự pseudodifferential với biểu tượng chính cùng. | J. OPERATOR THEORY 15 1986 207-216 Copyright by INCREST 1986 A NOTE ON THE SPACE OF PSEUDODIFFERENTIAL PROJECTIONS WITH THE SAME PRINCIPAL SYMBOL KRZYSZTOF WOJCIECHOWSKI 1. STATEMENT OF THE RESULT In this paper we study the topological structure of the following space ip e Ổ H p2 p dimRanP oo dimRan Id P Srp J 0 I and p - po e H here is the algebra of all bounded operators in a separable Hilbert space H and . H is the ideal of compact operators. Here is an important example. Let M be a closed smooth manifold E a complex vector bundle over M and let Pdiffp be the space of pseudodifferential projections p with the same principal symbol p - n E where n SM M is the natural projection from the cotangent sphere bundle we have fixed a Riemannian structure on M and a Hermitian structure on . In this casep is the projection onto some subbundle of n E actually the dimension of the fibre can change over connected components of SM . If p and Id p are not identically equal to 0 then the ranges of the projection Pand Id p are infinite dimensional. The reason at least for classical pseudodifferential operators is that if p 0 then p is of order 1 hence it is acompact operator. Theoperator is a projection so its range is fin ite dimensional. On the other hand if the range of p is finite dimensional then p is compact so the O-th order term in its symbol will vanish. Pseudodifferential projections described above are in many cases spectral projections of elliptic operators. The projections onto the Cauchy data spaces of elliptic differential operators are of this type too see for instance 1 2 3 4 8 . Now the space of projections in a separable Hilbert space with infinite dimensional range and kernel is contractible 7 . For the space the situation is different. Theorem . Let H be a separable Hilbert space po a projection with infinite dimensional range and kernel. Let s p be the space defined in . lPpa is a classi 208 KRZYSZTOF WOJCIECHOWSKI fying space for the .