Fredholm equations on a Hilbert space of analytic functions

Author:
Clasine van Winter

Journal:
Trans. Amer. Math. Soc. **162** (1971), 103-139

MSC:
Primary 47B37; Secondary 81.47, 46E20

DOI:
https://doi.org/10.1090/S0002-9947-1971-0417849-8

MathSciNet review:
0417849

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Abstract: It is shown that the Hardy class for the upper half-plane is equal to the set of functions which are analytic in the open half-plane and square-integrable with respect to *r* for . A function *f* is in if and only if its Mellin transform with respect to *r* is a constant times , where *f* must belong to a certain -space. This result enables *f* in to be constructed from its boundary values on the positive real axis.

A study is made of a class consisting of integral operators *K* on having kernels which are square-integrable with respect to *r* and . It is found that is a Hilbert space and is a proper subset of the Schmidt class. The class is not an ideal in the algebra of all bounded operators on , but there is a Banach algebra which is dense in and contains as an ideal. An operator *A* in is associated with a family of operators on . As a result, a Fredholm equation on with a kernel *K* in is associated with a family of equations on with kernels . The solution of the equation with kernel has an analytic continuation which solves the equation on with kernel *K*, and all solutions in can be obtained in this way.

Arguments based on the Mellin transform show that the kernels of operators in form a Hardy class of functions of two variables, one complex and one real. A generalization leads to Hardy classes of functions of *n* variables. On , there is a class of operators whose kernels form a class .

This formalism was developed with a view to the *n*-body problem in quantum mechanics. It is explained that the results on are instrumental in evaluating quantities which occur in the theory of *n*-particle scattering.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0417849-8

Keywords:
Fredholm equation,
Hilbert space of analytic functions,
Hardy class,
Schmidt class,
integral operator,
operator algebra,
operator ideal,
Mellin transform,
Paley-Wiener theorem,
*n*-body problem in quantum mechanics,
scattering theory

Article copyright:
© Copyright 1971
American Mathematical Society