Characterization of similarities between two n-frames of projectors

An n-frame on a Banach space $\text{X}$ is E=(E₁,?, E_n) where the E_j's are bounded linear operators on $\text{X}$ such that E_j≠0,

$\text{∑}\text{j=1}\text{n}\text{E}{}_{\mathrm{j}}$

, and E_jE_k=δ_jkE_k (j, k=1,?, n). It is known that if two n-frames E and F are sufficiently close to each other, then they are similar, that is, F_j=TE_jT^-1 with T a bounded linear operator. Among the operators which realize the similarity of the two frames, there is the balanced transformation U(F, E)=(Σⁿ_j=1F_jE_j)(Σⁿ_j=1E_jF_jE_j)^{$\text{-1}\text{2}$}. One of our main results is a local characterization of the balanced transformation. Another operator which implements the similarity between E and F is the direct rotation R(F, E). It comes up in connection with the study of the set of all n-frames as a Banach manifold with an affine connection. Finally, it is shown that for quite a large set of pairs of 2-frames, the direct rotation has a global characterization.