Compact Operator On Hilbert Space - Examples

Examples

  • Let H = L2. The multiplication operator M defined by

is a bounded self-adjoint operator on H that has no eigenvector and hence, by the spectral theorem, can not be compact.

  • Let K(x, y) be square integrable on 2 and define TK on H by

Then TK is compact on H; it is a Hilbert–Schmidt operator.

  • Suppose that the kernel K(x, y) satisfies the Hermiticity condition

Then TK is compact and self-adjoint on H; if {φn} is an orthonormal basis of eigenvectors, with eigenvalues {λn}, it can be proved that

where the sum of the series of functions is understood as L2 convergence for the Lebesgue measure on 2. Mercer's theorem gives conditions under which the series converges to K(x, y) pointwise, and uniformly on 2.

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