Fredholm equations on a Hilbert space of analytic functions
Author:
Clasine van Winter
Journal:
Trans. Amer. Math. Soc. 162 (1971), 103139
MSC:
Primary 47B37; Secondary 81.47, 46E20
MathSciNet review:
0417849
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Abstract: It is shown that the Hardy class for the upper halfplane is equal to the set of functions which are analytic in the open halfplane and squareintegrable 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 squareintegrable 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 nbody problem in quantum mechanics. It is explained that the results on are instrumental in evaluating quantities which occur in the theory of nparticle scattering.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197104178498
PII:
S 00029947(1971)04178498
Keywords:
Fredholm equation,
Hilbert space of analytic functions,
Hardy class,
Schmidt class,
integral operator,
operator algebra,
operator ideal,
Mellin transform,
PaleyWiener theorem,
nbody problem in quantum mechanics,
scattering theory
Article copyright:
© Copyright 1971
American Mathematical Society
