Density-Dependent Operators on Density-Projection Condensation Spaces: Ambient Extensions, Zero-Density Defects, and Stability

Abstract

Let $X={\mathbb{R}}^n$ be equipped with Lebesgue measure $\hat{I}»$, and let $\mathcal{R}$ be the space of normalized nonnegative densities in $L^1(X,\hat{I}»)$. Each $\ddot{I}\in \mathcal{R}$ induces the weighted Hilbert space $H_{\ddot{I}}=L^2(X,\ddot{I}d\hat{I}»)$. Through the alignment isometry $U_{\ddot{I}}[f{]}_{\ddot{I}}=\sqrt{\ddot{I}}f$, the space $H_{\ddot{I}}$ is identified with the closed observable subspace $V_{\ddot{I}}=\{u\in L^2(X,\hat{I}»):u=0\hat{I}»\text{-a.e. on }\{\ddot{I}=0\}\}$. The companion density-projection completion theorem identifies the metric completion of the aligned object space with $L^2(X,\hat{I}»)\times \mathcal{R}$. We study bounded operator families $A_{\ddot{I}}:H_{\ddot{I}}\to H_{\ddot{I}}$ and characterize all bounded ambient extensions of the aligned operator $A_{\ddot{I}}{}^{♯}=U_{\ddot{I}}A_{\ddot{I}}U_{\ddot{I}}{}^{-1}$. Their collection is an affine space modeled on $\mathcal{B}(Z_{\ddot{I}},L^2(X,\hat{I}»))$, where $Z_{\ddot{I}}=V_{\ddot{I}}{}^{\perp }$ is the zero-density defect subspace. The defect-annihilating extension attains the minimum possible operator norm, and we classify self-adjoint, positive, and orthogonal-projection extensions. We prove that the induced map $(u,\ddot{I})\mapsto ({\tilde{A}}_{\ddot{I}}u,\ddot{I})$ is continuous exactly when the ambient family is strongly continuous, and that global Lipschitz continuity forces density independence. We also characterize convergence of support projections, show that $L^1$-convergence alone does not control support-dependent operators, and construct stable multiplication and density-weighted Hilbert--Schmidt families. The latter are $1/2$-Hölder continuous in operator norm with respect to the $L^1$-distance between densities.

22 pages, no figures