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.