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 $\lambda$, and let $\mathcal{R}$ be the space of normalized nonnegative densities in $L^1(X,\lambda)$. Each $\rho\in\mathcal{R}$ induces the weighted Hilbert space $H_\rho=L^2(X,\rho\,d\lambda)$. Through the alignment isometry $U_\rho[f]_\rho=\sqrt{\rho}f$, the space $H_\rho$ is identified with the closed observable subspace $V_\rho=\{u\in L^2(X,\lambda):u=0\ \lambda\text{-a.e. on }\{\rho=0\}\}$. The companion density-projection completion theorem identifies the metric completion of the aligned object space with $L^2(X,\lambda)\times\mathcal{R}$. We study bounded operator families $A_\rho:H_\rho\to H_\rho$ and characterize all bounded ambient extensions of the aligned operator $A_\rho^\sharp=U_\rho A_\rho U_\rho^{-1}$. Their collection is an affine space modeled on $\mathcal{B}(Z_\rho,L^2(X,\lambda))$, where $Z_\rho=V_\rho^\perp$ 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 $(u,\rho)\mapsto(\widetilde A_\rho u,\rho)$ 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 $1/2$-H\"older continuous in operator norm with respect to the $L^1$-distance between densities.