Products of positive operators

On finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class ${\mathcal L}^{+2}$ of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in ${\mathcal L}^{+2}$ are developed, and membership in ${\mathcal L}^{+2}$ among special classes, including algebraic and compact operators, is examined.