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.

**[1]**Einar Hille,*Analytic function theory. Vol. II*, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR**0201608****[2]**Kenneth Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR**0133008****[3]**Philip M. Morse and Herman Feshbach,*Methods of theoretical physics. 2 volumes*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR**0059774****[4]**Roger G. Newton,*Scattering theory of waves and particles*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0221823****[5]**Raymond E. A. C. Paley and Norbert Wiener,*Fourier transforms in the complex domain*, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR**1451142****[6]**Robert Schatten,*Norm ideals of completely continuous operators*, Ergebnisse der Mathematik und ihrer Grenzgebiete. N. F., Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR**0119112****[7]**L. I. Schiff,*Quantum mechanics*, 3rd ed., McGraw-Hill, New York, 1968.**[8]**F. Smithies,*Integral equations*, Cambridge Tracts in Mathematics and Mathematical Physics, no. 49, Cambridge University Press, New York, 1958. MR**0104991****[9]**E. C. Titchmarsh,*Introduction to the theory of Fourier integrals*, 2nd ed., Clarendon Press, Oxford, 1948.**[10]**Clasine van Winter,*Theory of finite systems of particles. I. The Green function*, Mat.-Fys. Skr. Danske Vid. Selsk.**2**(1964), no. 8, 60 pp. (1964). MR**0201168****[11]**Clasine van Winter,*Theory of finite systems of particles. II. Scattering theory*, Mat.-Fys. Skr. Danske Vid. Selsk.**2**(1965), no. 10, 94 pp. (1965). MR**0232584****[12]**-, ``The*n*-body problem on a Hilbert space of analytic functions,'' in R. P. Gilbert and R. G. Newton,*Analytic methods in mathematical physics*, Gordon and Breach, London, 1970, pp. 569-578.**[13]**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

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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