  Let $\mathcal{H}$ be a (separable) Hilbert space and $\{e_k\}_{k\geq 1}$ a
fixed orthonormal basis of $\mathcal{H}$. Motivated by many papers on scaled
projections, angles of subspaces and oblique projections, we define and study
the notion of compatibility between a subspace and the abelian algebra of
diagonal operators in the given basis. This is used to refine previous work on
scaled projections, and to obtain a new characterization of Riesz frames.
