Relation between the boundedness and discreteness of conjugate operators

I have two very general questions about operators in quantum mechanics. Suppose $A$ and $B$ are self-adjoint operators associated to conjugate physical quantities (e.g. position/momentum), meaning that $[A,B]neq 0$.

If $A$ is bounded from one (or both) sides, can we conclude anything for the boundedness of $B$ ?

If $A$ has a discrete spectrum, can we conclude anything about the spectrum of $B$ ?