In this work, Fredholm joint spectrum of noncommuting operator family which generates nilpotent Lie algebra is defined and investigated some properties of it. The joint spectrum ofais defined to be the set &sgr;F(a) of those &lgr; ∈Cnfor which the complexKos(a− &lgr;,X) whose co‐homology spaces are not of finite dimension. It is proved that &sgr;F(a) is a compact set inCn. Polynomial spectral mapping theorem with respect to polynomials in noncommuting variables is proved such thatP(&sgr;F(a)) ⊂ &sgr;F(P(a)). © 2004 American Institute of Physics