Conditional stability on a power-source class

Given: Countable Choice, a compact infinite-rank operator $K:X\to Y$ between real or complex Hilbert spaces over one scalar field, a positive singular system $(s_j,v_j,u_j{)}_{j\ge 1}$, $\rho >0$, the class $M_{\rho }=X_{1,\rho }(K)$, and $\delta \ge 0$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): Countable Choice is assumed by the compact singular-value decomposition and Hilbert-space pseudoinverse definitions.

Compact linear inverse model between Hilbert spaces: The operator is compact and linear between real or complex Hilbert parameter/data spaces over one field.

Singular value decomposition for compact operators: The positive singular directions are orthonormal, with $v_j\in N(K{)}^{\perp }$, $\mathrm{\mid }K\mathrm{\mid }{v}_j=s_j{v}_j$, and $K{v}_j=s_j{u}_j$ for each positive singular value $s_j>0$.

Hilbert-space Moore–Penrose inverse: $D(K^{†})=R(K)\oplus R(K{)}^{\perp }$, and $K^{†}$ returns the minimum-norm preimage of the range component.

Power source ball: $M_{\rho }=X_{1,\rho }(K)=\{\mathrm{\mid }K\mathrm{\mid }w:w\in X,\mathrm{\parallel }w{\mathrm{\parallel }}_X\le \rho \}$.

Centered conditional modulus of continuity: ${\epsilon }_K(M_{\rho },\delta )$ is the supremum of $\mathrm{\parallel }x{\mathrm{\parallel }}_X$ over $x\in M_{\rho }$ with $\mathrm{\parallel }Kx{\mathrm{\parallel }}_Y\le \delta$.

Conditional modulus on a power source ball: For $\nu =1$, ${\epsilon }_K(X_{1,\rho }(K),\delta )\le {\delta }^{1\mathrm{/}2}{\rho }^{1\mathrm{/}2}$; equality holds at ${\delta }_j=\rho s_j^2$ with witness $x_j=\rho s_j{v}_j$.

Bakushinskii veto for discontinuous pseudoinverses: For discontinuous $K^{†}$, universal worst-case convergence over every exact datum and every admissible noise ball would imply continuity of $K^{†}$.