Abstract: 

A positive operator $A$ and a closed subspace $\cal S$ of a
Hilbert space $\cal H$ are called {\it compatible} if there
exists a projector $Q$ onto $\cal S$ such that $AQ=Q^*A$.
Compatibility is shown to depend on the existence of
certain decompositions of $\cal H$ and the ranges of $A$ and
$A^{1/2}$. It also depends on a certain angle between $A({\cal S})$
and the orthogonal of $\cal S$.

1. Departamento de Matemática
   FI-UBA, and IAM-CONICET
   Buenos Aires
   Argentina
   
2. Instituto de Ciencias, UNGS
   San Miguel, Argentina
   
3. Departamento de Matemática
   FCE-UNLP
   La Plata, Argentina