Compact and Fredholm operators

Compact and finite-rank operators, Schauder's theorem, complete continuity, Hilbert–Schmidt operators, Riesz–Schauder theory, Fredholm alternative and index, closed-range theorems.

• Closed range theorem
• Complete continuity of compact operators
• Finite-rank approximation of compact operators
• Fredholm alternative
• Hilbert–Schmidt operators
• Homotopy invariance of the Fredholm index
• Riesz–Schauder theory
• Schauder's theorem on compact adjoints
• Spectrum of compact operators

TheoremThe kernel of a compact operator on a nonseparable Hilbert space is nonseparable#●ready to prove

Let \(H\) be a complex Hilbert space that is not separable and let \(A : H \to H\) be a compact operator. Then the kernel \(\ker A\) is not separable.

public theorem_wanted not_separableSpace_ker_of_isCompactOperator_of_not_separableSpace
    {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
    (A : H →L[ℂ] H)
    (hH : ¬ SeparableSpace H)
    (hA : IsCompactOperator A) :
    ¬ SeparableSpace (A.ker)

Sources

• MA-ProofBench (L. Pu et al., arXiv:2606.13782), Level II (Operator theory; Special classes of linear operators), problem ma_proofbench_l2_42.

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