Products of two normal operators

Abstract

We obtain some necessary or sufficient conditions for an operator on a complex separable Hilbert space to be expressible as a product of two normal operators. For example, it is shown that if T is such a product, then $$\dim \ker T \ge \dim \ (\ker T^*\ \cap \ \mathrm {ran}\, T^*)$$ . On the other hand, any operator T satisfying $$\dim \ker T^* \ge \dim \ker T$$ and $$\dim \ (\ker T\ \cap \ \overline{\mathrm {ran}\, T})\ge \dim \ (\ker T^*\ \cap \ \mathrm {ran}\, T^*$$ ) is a product of two normal operators. Such results complement our previous ones on the products of finitely many normal operators. We also obtain characterizations for products of two essentially normal operators.