Minimal geodesics of pencils of pairs of projections
Abstract
The set $\mathcal {P}_{T,\lambda }$ of pairs $(P, Q)$ of orthogonal projections with pencil $T=\lambda P+Q$ at any given $\lambda \in \mathbb {R}\setminus \{-1,0\}$ is shown to be an analytic Banach homogeneous space. In generic position, $\mathcal {P}_{T,\lambda }$ with a natural connection and the quotient Finsler metric of the operator norm becomes a classical Riemannian space in which any two pairs are joined by a minimal geodesic. Moreover, given a pair $(P, Q)\in \mathcal {P}_{T,\lambda }$, pairs in an open dense subset of $\mathcal {P}_{T,\lambda }$ can be joined to $(P, Q)$ by a unique minimal geodesic. In general, two pairs $(P_{0}, Q_{0}),(P, Q)$ in $\mathcal {P}_{T,\lambda }$ can be joined by a minimal geodesic in $\mathcal {P}_{T,\lambda }$ of length $\leq {\pi}/{2}$ if and only if $$ \begin {cases} \dim [R(P|_{N(T-I)})\cap N(P_{0}|_{N(T-I)})] \\ \quad \ =\dim [N(P|_{N(T-I)})\cap R(P_{0}|_{N(T-I)})], \quad \lambda =1,\\ \dim N(T-\lambda I)=\dim N(T-I), \quad \lambda \in \mathbb {R}\setminus \{0, -1, 1\}. \end {cases} $$
