In the previous chapter we treated Hankel operators defined by means of a Hankel matrix: we now turn our attention to a second kind of Hankel operator, the Hankel Integral Operator on L2(0, ∞). With the aid of the Laplace transform we are able to determine the action which such operators induce on the Hardy space H2(C+).

As with the Hankel operators of Chapter 3, the Hankel operators on H2(C+) can be regarded as being produced by applying an inversion, followed by a multiplication and finally an orthogonal projection. Using the isometric isomorphism between H2(C+) and H2 that was defined in Chapter 2, we then see that Hankel integral operators correspond to Hankel operators on H2.

This correspondence allows us to reap several corollaries, deducing versions of the Nehari, Kronecker and Hartman theorems from the corresponding results of Chapter 3.

The references most nearly related to the material of this chapter include Glover, Glover et al, and Power, although some of the calculations appear to have the status of folklore.

We begin with the Hankel integral operators on L2(0, ∞).

If h(x) ∈ L1(0, ∞) ∩ L2(0, ∞), then the Hankel Integral Operator Γh: L2(0, ∞) → L2(0, ∞) given by

is well-defined and bounded, with ‖Γh‖ ≤ ‖h‖I.

Proof Since h ∈ L2(0, ∞), it is clear from the Cauchy-Schwarz inequality that Λhu is defined pointwise. Now, if ν ∈ L2(0, ∞), we have

so that

letting z = x + y and using Fubini's theorem (see the Appendix) to justify rearranging the integral.