The following theorem of Fuglede is a classical result in the theory of normal operators. In the proof, one can appreciate how classical analysis are used in solving problems in operator theory through functional calculus.
Let be a complex Hilbert space and be the C*-algebra of all bounded operators on .
Theorem (Fuglede). Let be such that N is normal and NX = XN. Then we have N*X = XN*.
Proof. Firstly we have for any complex number z, by approximating the exponentials with polynomials and using the fact that X commutes with every polynomial of N. Now define
and we obtain a (-valued) entire function. Since N and N* commutes, we have
Note that is anti-hermitian (an operator A is said to be anti-hermitian if A* = -A), so for some self-adjoint . It follows that and thus f is bounded. By Liouville’s Theorem, f is a constant function. Therefore for any z,
Put z = 0, we get the result.
The following is a generalization of Fuglede’s Theorem. A way of proof is to imitate the above proof by considering a different bounded entire function. But we can directly apply the Fuglede’s Theorem and use a “2 x 2 matrix trick”.
Corollary (Putnam). Let be such that M and N are normal and MX = XN. Then we have M*X = XN*.
Proof. Observe that is normal and
i.e. and commute. By Fuglede’s Theorem, and commute. On the one hand,
On the other hand,
We are done.