Calkin algebra - Knowlpedia

Let $H$ be an infinite-dimensional complex Hilbert space. The Calkin algebra is the quotient

\mathcal Q(H):=\mathcal B(H)/\mathcal K(H),

where $\mathcal B(H)$ is the bounded-operator algebra and $\mathcal K(H)$ is its compact-operator ideal.

Essential operator information

Two bounded operators define the same element of $\mathcal Q(H)$ exactly when their difference is compact.

The quotient map $q:\mathcal B(H)\to\mathcal Q(H)$ forgets compact perturbations. An operator $T$ is Fredholm exactly when $q(T)$ is invertible; this is the Calkin-algebra form of Atkinson's theorem. Thus the Calkin algebra records the operator's essential behavior and discards finite-dimensional errors.

Scope and conventions

The quotient is a $C^*$ -algebra because $\mathcal K(H)$ is a closed two-sided ideal. The infinite-dimensional hypothesis matters: if $H$ is finite-dimensional, every operator is compact and the quotient is the zero algebra. The notation $\mathcal Q(H)$ depends on $H$ , although Calkin algebras of Hilbert spaces with the same infinite dimension are canonically isomorphic up to the chosen Hilbert-space identification.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990. Springer DOI record. Relevant: Chapter XII, Fredholm operators and the Calkin algebra.
R. G. Douglas, Banach Algebra Techniques in Operator Theory, 2nd ed., Springer, 1998. Springer DOI record. Relevant: Chapter 3, Fredholm operators and the Calkin algebra.