For an infinite-dimensional complex Hilbert space, the Calkin algebra is the unital quotient of the bounded operators by the closed two-sided ideal of compact operators. Its spectrum records invertibility up to compact errors. An operator is Fredholm exactly when its image in this quotient is invertible.
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Calkin algebra refers to a specific type of algebraic structure in the realm of functional analysis, particularly associated with bounded linear operators on a Hilbert space. It is essentially the quotient algebra of bounded linear operators acting on a Hilbert space when identified modulo the ideal of compact operators.