Fredholm equations on a Hilbert space of analytic functions

Abstract:

It is shown that the Hardy class ${\mathfrak {H}^2}$ for the upper half-plane is equal to the set of functions $f[r\exp (i\phi )]$ which are analytic in the open half-plane and square-integrable with respect to r for $0 < \phi < \pi$. A function f is in ${\mathfrak {H}^2}$ if and only if its Mellin transform with respect to r is a constant times $f(t)\exp (\phi t - i\phi /2)$, where f must belong to a certain ${\mathfrak {L}^2}$-space. This result enables f in ${\mathfrak {H}^2}$ to be constructed from its boundary values on the positive real axis. A study is made of a class $\mathfrak {N}$ consisting of integral operators K on ${\mathfrak {H}^2}$ having kernels $K(r,r’,\phi )$ which are square-integrable with respect to r and $r’$. It is found that $\mathfrak {N}$ is a Hilbert space and is a proper subset of the Schmidt class. The class $\mathfrak {N}$ is not an ideal in the algebra $\mathfrak {B}$ of all bounded operators on ${\mathfrak {H}^2}$, but there is a Banach algebra $\mathfrak {A}$ which is dense in $\mathfrak {B}$ and contains $\mathfrak {N}$ as an ideal. An operator A in $\mathfrak {A}$ is associated with a family of operators $A(\phi )$ on ${\mathfrak {L}^2}[0,\infty )$. As a result, a Fredholm equation on ${\mathfrak {H}^2}$ with a kernel K in $\mathfrak {N}$ is associated with a family of equations on ${\mathfrak {L}^2}[0,\infty )$ with kernels $K(\phi )$. The solution of the equation with kernel $K(\phi )$ has an analytic continuation which solves the equation on ${\mathfrak {H}^2}$ with kernel K, and all solutions in ${\mathfrak {H}^2}$ can be obtained in this way. Arguments based on the Mellin transform show that the kernels $K(r,r’,\phi )$ of operators in $\mathfrak {N}$ form a Hardy class ${\mathfrak {H}^2}(2)$ of functions of two variables, one complex and one real. A generalization leads to Hardy classes ${\mathfrak {H}^2}(n)$ of functions of n variables. On ${\mathfrak {H}^2}(n)$, there is a class of operators $\mathfrak {N}(n)$ whose kernels form a class ${\mathfrak {H}^2}(2n)$. This formalism was developed with a view to the n-body problem in quantum mechanics. It is explained that the results on ${\mathfrak {H}^2}(n - 1)$ are instrumental in evaluating quantities which occur in the theory of n-particle scattering.