Eigenvalues in infinite dimensions

Abstract

We extend to infinite dimensions some classic results concerning the behavior of the eigenvalues under certain basic constructions, such as short exact sequences, tensor products and compositions.

An endomorphism of a vector space is called finitary if, for every vector, the subspace generated by its orbit is finite-dimensional. The characteristic roots of a finitary endomorphism are defined to be the roots of all minimal polynomials associated to each vector. The eigenvalues are instead defined in general after tensoring the endomorphism with an algebraic closure of the base field. The following theorems extend to infinite dimensions classic results known in finite dimensions:

Theorem 1If is a short exact sequence of endomorphisms, then ϕ is finitary if and only if so are ϕ^′ and ϕ^Dprime; Moreover.

Theorem 2 If are finilary endomorphisms, then so is and moreover .

Theorem 3 If are finitary commuting endomorphisms, then so is and moreover . We prove furthermore that the second part of Theorem 2 holds in general, without any condition on the endomorphisms, with the characteristic roots replaced by the eigenvalues. As an application of the above results, it is proved that the characteristic roots of an endomorphism of a connected graded algebra are contained in the multiplicative closure of the characteristic roots of the endomorphism induced on the indecomposable elements.

^*Current address: Department of Mathematics, MIT, Cambridge, MA 02139, USA.

^*Current address: Department of Mathematics, MIT, Cambridge, MA 02139, USA.

Notes

^*Current address: Department of Mathematics, MIT, Cambridge, MA 02139, USA.