The study of the class L+2\mathcal{L}^{+\,2} of Hilbert space operators which are the product of two bounded positive operators first arose in physics in the early 60s.

 

On finite dimensional Hilbert spaces, it is not hard to see that an operator is in this class if and only if it is similar to a positive operator. We extend the exploration of L+2\mathcal{L}^{+\,2} to separable infinite dimensional Hilbert spaces, where the structure is much richer, connecting (but not equivalent to) quasi-similarity and quasi-affinity to a positive operator.

 

The (generalized) spectral properties of elements of L+2\mathcal{L}^{+\,2} are also outlined, as well as membership in L+2\mathcal{L}^{+\,2} among various special classes of operators, including algebraic and compact operators.