Let X,Y be C-Hilbert spaces . A linear mapping f
defined on a dense subspace D(f) of X and taking values in Y is called an operator with index of X into Y if the following conditions are satisfied
-The graph G(f) of f is closed subspace of XxY .
- dim(Kerf) and codim(Imf) are finite .
In this case , we denote
-Dim(Kerf) - codim(Imf)=i(f)
and call it the index of f .
Now , let f : X-->Y be an operator with index .
a) Let g:Y-->Z be an operator with index . Show that , g.f is too and
i(g.f)=i(g)+i(f) .
b) Let f' is a extension of f , such that
=D(f)\bigoplus M)
, in which M is n-dimensional space . Show that , f' is an operator with index and i(f')=i(f)+n .
Bài viết đã được chỉnh sửa nội dung bởi pizza: 25-10-2005 - 19:48
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