Products of projections and self-adjoint operators

Let H be a Hilbert space and L(H) be the algebra of all bounded linear operators from H to H. Our goal in this article is to study the set P⋅Lh of operators in L(H) that can be factorized as the product of an orthogonal projection and a self-adjoint operator. We describe P⋅Lh and present optimal factorizations, in different senses, for an operator in this set.