Compressions of linearly independent selfadjoint operators

The following question is considered: What is the smallest number γ(k) with the property that for every family X ₁,⋯,X _k of k selfadjoint and linearly independent operators on a real or complex Hilbert space H there exists a subspace H ₀⊂H of dimension γ(k) such that the compressions of X ₁,⋯,X _k to H ₀ are still linearly independent? Upper and lower bounds for γ(k) are established for any k, and the exact value is found for k=2,3. It is also shown that the set of all γ(k)-dimensional subspaces H ₀ with the desired property is open and dense in the respective Grassmannian. The k=3 case is used to prove that the ratio numerical range W(A/B) of a pair of operators on a Hilbert space either has a non-empty interior, or lies in a line or a circle.