Oblique projections on metric spaces

It is known that complementary oblique projections $\hat{P}_0 + \hat{P}1 = I$onaHilbertspace$\mathscr{H}$havethesamestandardoperatornorm$|\hat{P}0| = |\hat{P}1|$ 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 $\mathcal{H}={\mathcal{H}}_0\oplus {\mathcal{H}}_1$, and of those ${\Gamma }_0,{\Gamma }_1$ induced on their algebraic duals, obey the relations \begin{align} \frac{\det+ L_1}{\det+ $\Gamma$0} = \frac{\det+ L_0}{\det+ $\Gamma$1} = {\det}+ G. \nonumber \end{align} Furthermore, we break this result down to eigenvalues, proving a “supersymmetry” of the two operators $\sqrt{{\Gamma }_0{L}_0}$ and $\sqrt{L_1{\Gamma }_1}$. We connect the former result to a well-known duality property of the weighted-spanning-tree polynomials in graph theory.