Circular operators and their strong circularity

Circular operators have been studied extensively since the work of R. Gellar, who conjectured that every circular operator on a complex separable Hilbert space is strongly circular. In this short note, we show that circularity and strong circularity coincide for bounded operators that are finite or countably infinite direct sums of irreducible operators. This considerably narrows the search for potential counterexamples to Gellar’s conjecture. As an application, we prove that every circular operator in the Cowen-Douglas class is strongly circular. In addition, we obtain several general results on circular operators that reveal the significance of the hyper-range and the Cauchy dual.