Compact Operator - Examples

Examples

  • For some fixed gC(R), define the linear operator T by
That the operator T is indeed compact follows from the Ascoli theorem.
  • More generally, if Ω is any domain in Rn and the integral kernel k : Ω × Ω → R is a Hilbert—Schmidt kernel, then the operator T on L2(Ω; R) defined by
is a compact operator.
  • By Riesz's lemma, the identity operator is a compact operator if and only if the space is finite dimensional.

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