Density of invertible operators

DefinitionScottish Book Problem 173#✓fully proved

Eidelheit's claim that for every real Banach space \(E\), the bounded linear operators \(E \to E\) that have a bounded linear two-sided inverse are dense, for the operator norm, in the space of all bounded linear operators on \(E\).

def ScottishBookProblem173Claim : Prop :=
  ∀ (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E],
    Dense {T : E →L[ℝ] E |
      ∃ S : E →L[ℝ] E,
        ContinuousLinearMap.comp T S = ContinuousLinearMap.id ℝ E ∧
          ContinuousLinearMap.comp S T = ContinuousLinearMap.id ℝ E}

Used by 1

Used byNegative answer to Scottish Book Problem 173

TheoremNegative answer to Scottish Book Problem 173#●ready to prove

Eidelheit's claim in Problem 173 is false: there is a real Banach space \(E\) in which the bounded linear operators \(E \to E\) that have a bounded linear two-sided inverse are not dense, for the operator norm, in the space of all bounded linear operators on \(E\).

For instance, in an infinite-dimensional Hilbert space the invertible operators are not dense, by the Fredholm-index obstruction: the unilateral shift cannot be approximated by invertible operators.

theorem_wanted eidelheit173_refuted : ¬ ScottishBookProblem173Claim

Sources

• M. Eidelheit, Problem 173 of the Scottish Book; see P. Halmos, A Hilbert Space Problem Book, Problem 109.

Uses 1

Statement usesScottish Book Problem 173

1 fully proved1 ready to prove

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