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.
Categories: math.FA math-ph math.MG math.MP