  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.
