Oblique projections on metric spaces

Abstract

It is known that complementary oblique projections $\hat{P}_0 + \hat{P}_1 = I$ on a Hilbert space $\mathscr{H}$ have the same standard operator norm $\|\hat{P}_0\| = \|\hat{P}_1\|$ and the same singular values, but for the multiplicity of $0$ and $1$. We generalize these results to Hilbert spaces endowed with a positive-definite metric $G$ on top of the scalar product. Our main result is that the volume elements (pseudodeterminants $\det_+$) of the metrics $L_0,L_1$ induced by $G$ on the complementary oblique subspaces $\mathscr{H} = \mathscr{H}_0 \oplus \mathscr{H}_1$, and of those $\mathit{\Gamma}_0,\mathit{\Gamma}_1$ induced on their algebraic duals, obey the relations \begin{align} \frac{\det_+ L_1}{\det_+ \mathit{\Gamma}_0} = \frac{\det_+ L_0}{\det_+ \mathit{\Gamma}_1} = {\det}_+ G. \nonumber \end{align} Furthermore, we break this result down to eigenvalues, proving a "supersymmetry" of the two operators $\sqrt{\mathit{\Gamma}_0 L_0}$ and $\sqrt{L_1 \mathit{\Gamma}_1}$. We connect the former result to a well-known duality property of the weighted-spanning-tree polynomials in graph theory.