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CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22
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Atkinson in Calkin algebra language

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let X be an infinite-dimensional complex Banach space and let T∈B(X). Then the following are equivalent:

  1. T is Fredholm;
  2. the coset T+K(X) is invertible in the Calkin algebra C(X)=B(X)/K(X) (Calkin algebra);
  3. there is a bounded S∈B(X) with ST−IX and TS−IX both compact (Compact linear operator) — a bounded two-sided parametrix modulo compact operators.

Facts & Assumptions

Given: An assumed Axiom of Choice, an infinite-dimensional complex Banach space X, a bounded operator T∈B(X), and the Calkin algebra C(X) with unit 1=IX+K(X).

[L1]

T is Fredholm if and only if there is a bounded linear S with ST−IX and TS−IX compact (Atkinson).

[L2]

In the quotient algebra C(X) one has T+K invertible if and only if there is S+K with (T+K)(S+K)=1 and (S+K)(T+K)=1; these equations are exactly TS−IX∈K(X) and ST−IX∈K(X) (Calkin algebra).

[L3]

The Calkin algebra is built under Countable Choice, which is available here because the standing hypothesis is the stronger Axiom of Choice (The Axiom of Choice), whose standard consequences include Countable Choice (The Axiom of Countable Choice (ACω)).

Proof

technique · direct
1.1

Equivalence of 2 and 3: the equation (T+K)(S+K)=IX+K of [L2] holds exactly when TS−IX∈K(X), and the other product equation holds exactly when ST−IX∈K(X); so T+K is invertible precisely when T admits a bounded two-sided parametrix modulo compact operators.

L2L3
2.1

Equivalence of 1 and 3 is [L1]; combining it with [step 1.1] gives that 1, 2 and 3 are equivalent.

step 1.1L1∎

Remarks

  • No new Fredholm theory is hidden here. The corollary is a restatement of the Atkinson theorem in the quotient algebra: the only content beyond Atkinson is that quotient invertibility and the existence of a two-sided parametrix modulo compact operators are the same condition, which is the definition of the quotient multiplication.

  • Both products are required. One-sided quotient invertibility would only give one of the two compactness conditions; the theorem and the definition both ask for two-sided invertibility, and the remark here records that the order of the products is preserved: ST and TS appear in the two conditions separately.

Depends on

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