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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 0417849
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0417849-8
PII: S 0002-9947(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