DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Compact linear inverse model between Hilbert spaces

Definition

Let $X, Y$ be real or complex Hilbert spaces over the same scalar field, and let $K : X \to Y$ be a compact linear operator. The associated compact linear inverse model has parameter space $X$ , data space $Y$ , forward map $K$ , and admissible truth class $X$ : for a true parameter $x^{†} \in X$ , the exact datum is $y = K x^{†} \in Y$ . Given an exact datum $y \in Y$ , solving this inverse problem means finding a parameter $x \in X$ such that $K x = y$ . No injectivity, surjectivity, or closed-range assumption is included in the model.

Definition references

General forward-map and solution-class interface

Real and complex Hilbert spaces

Compact linear operators between normed spaces

Depends on

Used by

A nonzero left-null datum passes the series test but is not in the range Counterexample
Noise underestimation defeats a truncated singular-value rule Counterexample
Centered conditional modulus of continuity Definition
Composite double regularization Definition
Error budget for computed Tikhonov inversion Definition
Gaussian sequence inverse model Definition
Noise-level-free parameter-choice rule Definition
Power source ball Definition
Spectral-index source balls for compact operators Definition
A compact diagonal map has a closure-only feature Example
A compact diagonal map with dense nonclosed range Example
An adjoint-range feature has an exact one-probe estimator Example
Conditional stability on a power-source class Example
Exponential singular decay, logarithmic balls, and spectral gaps Example
Polynomial singular decay with power and logarithmic source balls Example
Index source elements are minimum-norm solutions Lemma
Spectral interpolation for power source balls Lemma
Bakushinskii veto for discontinuous pseudoinverses Theorem
Closed range and boundedness of the Hilbert pseudoinverse Theorem
Conditional convergence of double-discrepancy composite inversion Theorem
Conditional modulus on a power source ball Theorem
Deterministic minimax lower bound on power source balls Theorem
Picard range and pseudoinverse-domain criterion Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christian Clason, Regularization of Inverse Problems: Lecture Notes, arXiv:2001.00617v2, §3.1 (standard reference, not scraped)
Otmar Scherzer and Leonidas Mindrinos, Inverse Problems: Lecture Notes, University of Vienna, 2015 (standard reference, not scraped)