Frames from normalized orbits of invertible normal operators

Can a bounded invertible normal operator T on ℓ²(N) have a vector g whose normalized orbit {Tᵏg/‖Tᵏg‖:k≥0} is a frame?

Why it matters and what counts as progress

Why it matters: It asks whether normalized orbit frames can exist for normal operators while keeping the narrower self-adjoint result distinct.

What counts as progress: An explicit bounded invertible normal operator with a normalized orbit frame, or a classification of normal operators permitting such a frame.

How the problem developed

2017 · Partial result [1]
Akram Aldroubi, Carlos Cabrelli, Ilya Krishtal and Ursula Molter Proved that normalized iterates of a self-adjoint operator cannot form a frame, motivating a broader conjecture for bounded normal operators.
18 Jun 2026 · Reported counterexample [2]
Ilya A. Krishtal and Götz E. Pfander Constructed a bounded diagonal normal operator and vector whose normalized orbit is a frame; the authors report using ChatGPT in the construction and manuscript preparation.