Strict cosingularity versus strict singularity of adjoints

For a bounded operator T:X→Y with separable range space Y, is strict cosingularity of T equivalent to strict singularity of its adjoint T*?

Why it matters and what counts as progress

Why it matters: The equivalence connects quotient behavior of an operator to subspace behavior of its adjoint under a separability assumption.

What counts as progress: Prove the equivalence for all separable Y, extend it beyond separable range or construct a counterexample outside the valid scope.

How the problem developed

2012 · Open question [1]
Kevin Beanland Asked whether strict cosingularity of an operator between separable Banach spaces forces its adjoint to be strictly singular.
19 Jul 2026 · Theorem claim [2]
Antonio Acuaviva and Pablo Acuaviva Reported that for bounded operators T:X→Y with separable Y, T is strictly cosingular exactly when T* is strictly singular.