  The aim of this work is to study the structure of bounded finite potent
endomorphisms on Hilbert spaces. In particular, for these operators, an answer
to the Invariant Subspace Problem is given and the main properties of its
adjoint operator are offered. Moreover, for every bounded finite potent
endomorphism we show that Tate's trace coincides with the Leray trace and with
the trace defined by R. Elliott for Riesz Trace Class operators.
