Oblique projections on metric spaces

TL;DR

This paper extends the understanding of oblique projections from Hilbert spaces to metric spaces, revealing new relations between volume elements and eigenvalues, with implications for graph theory and operator analysis.

Contribution

It generalizes known properties of oblique projections to metric spaces, establishing new relations involving pseudodeterminants and eigenvalue symmetries.

Findings

Derived relations between volume elements of induced metrics.

Proved supersymmetry of certain operator square roots.

Connected results to duality in graph theory.

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.