The Curto–Muhly–Xia conjecture on commuting subnormal operators

Let $\mathbf{T}=(T_{1},T_{2})$ be a pair of commuting subnormal operators on a Hilbert space $\mathcal{H}$. A commuting tuple is subnormal if it has a commuting normal extension, and it is hyponormal if its self-commutator matrix is positive semidefinite.

Curto–Muhly–Xia conjecture. The pair $\mathbf{T}$ is subnormal if and only if it is hyponormal.

The paper announces three conceptually different families of counterexamples to this conjecture, so the conjecture is refuted: for commuting subnormal pairs, hyponormality does not imply joint subnormality.