$( \Leftarrow ) $ Assume that $tr(T) = tr ( T^2) = \cdots = tr(T^n)=0 $ and that T is not nilpotent.
Let $ m_i$ denote the multiplicity of the eigenvalue and let $\lambda_1, \cdots , \lambda_r $ denote all the distinct nonzero eigenvalues. using an above result and the assumption we have that
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In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by Radó (1925) and named after Heinz Prüfer.
== Construction ==
The Prüfer manifold can be constructed as follows (Spivak 1979, appendix A). Take an uncountable number of copies Xa of the plane, one for each real number a, and take a copy H of the upper half plane (of pairs (x, y) with y > 0). Then glue each plane Xa to the upper half plane H by identifying (x,y)∈Xa for y > 0 with the point (a + yx, y) in H. The resulting quotient space is the Prüfer...
