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 X1,…,Xk of k selfadjoint and linearly independent operators on a real or complex Hilbert space H there exists a subspace H0⊂H of dimension γ(k) such that the compressions of X1,…,Xk to H0 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 H0 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.