compact operator

Compact operators


For a linear operator between Banach spaces, compactness is a natural strengthening of continuity (boundedness). A linear operator is compact if it sends the bounded subsets to relatively compact subsets.

The quotient of the bounded operators by the compact operators is called the Calkin algebra.


Since every relatively compact subspace (in a Banach space, or indeed in any metric space) is bounded, every compact operator is bounded. Instead of checking compactness on all bounded subsets it is sufficient to check it for a ball of one fixed radius: an operator is compact iff it sends the ball of unit radius to a relatively compact set.

See also compact self-adjoint operator.

Completely continuous operators

In the setup of Hilbert spaces, instead of a compact operator, one sometimes says a completely continuous operator. However, in the full generality of Banach spaces, by a completely continuous operator one means slightly less: an operator that maps every weakly convergent? sequence to a norm convergent? sequence. For Hilbert spaces and, more generally, for reflexive Banach spaces, the two notions are equivalent.


category: analysis

Revised on October 17, 2016 09:13:36 by Urs Schreiber (