Quasinormal Operator - Quasinormal Invariant Subspaces

Quasinormal Invariant Subspaces

It is not known that, in general, whether a bounded operator A on a Hilbert space H has a nontrivial invariant subspace. However, when A is normal, an affirmative answer is given by the spectral theorem. Every normal operator A is obtained by integrating the identity function with respect to a spectral measure E = {E_B} on the spectrum of A, σ(A):

For any Borel set B ⊂ σ(A), the projection E_B commutes with A and therefore the range of E_B is an invariant subpsace of A.

The above can be extended directly to quasinormal operators. To say A commutes with A*A is to say that A commutes with (A*A)½. But this implies that A commutes with any projection E_B in the spectral measure of (A*A)½, which proves the invariant subspace claim. In fact, one can conclude something stronger. The range of E_B is actually a reducing subspace of A, i.e. its orthogonal complement is also invariant under A.