Spectrum of A Ring - Functional Analysis Perspective

Functional Analysis Perspective

The term "spectrum" comes from the use in operator theory. Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R=K, as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K (as a ring) equals the spectrum of T (as an operator).

Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module:

the 2×2 zero matrix has module

showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module

showing algebraic multiplicity 2 but geometric multiplicity 1.

In more detail:

• the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity;
• the primary decomposition of the module corresponds to the unreduced points of the variety;
• a diagonalizable (semisimple) operator corresponds to a reduced variety;
• a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the space);
• the first invariant factor of the module equals the minimal polynomial of the operator, and the last invariant factor equals the characteristic polynomial.