Let H be a Hilbert space and A, B two non-commuting bounded linear operators. Let Com(A,B) be the set of bounded linear operators C which commute both with A and B.

Question 1 : What is known about Com(A,B) ?

And in the quantum mechanical context :

QUestion 2 : What is known about Com(A,B) when H is a complex Hilbert space, and all the operators are Hermitian ?