Existence of nonnormal sum-normal tuples

Determine whether there exists a nonzero quasinilpotent sum-normal d-tuple, or more generally, a nonnormal sum-normal d-tuple, on a complex Hilbert space.

Background

The paper studies commuting d-tuples whose aggregate self-commutator satisfies [\mathbf{T}^*,\mathbf{T}] \ge 0, calling such tuples sum-hyponormal and calling them sum-normal when the aggregate self-commutator is zero. One of the introductory questions asks whether every sum-normal d-tuple must be normal.

The paper proves several positive results under additional hypotheses, including finite dimensionality, compactness-related conditions, commuting imaginary parts, and certain double-commutator assumptions. However, it does not resolve the general existence question for nonnormal sum-normal tuples. The authors specifically note that even the narrower case of a nonzero quasinilpotent sum-normal tuple remains unresolved. They further observe that the approximation of compact quasinilpotent operators by nilpotent operators, together with their nilpotent-tuple theorem, suggests that nonzero compact examples may not exist, but this suggestion is not established in general.