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On the angles between subspaces, the muckenhoupt condition, and projection from one co-invariant subspace onto another in the theory of character-automorphic hardy spaces on a multiply connected domain

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Abstract

It is known that in the case of the unit disk the invertibility of the orthogonal projection of one subspace of H2 which is co-invariant with respect to the shift operator onto another such subspace is connected with the Helson-Szegö theorem and the Muckenhoupt condition. In the present paper, we consider the same problem in character-automorphic Hardy spaces on a finitely connected planar domain. The problem is reduced to estimating the angles between certain subspaces of the weighted L2-space on the boundary of the domain. The answer is given in terms of the Muckenhoupt condition for certain weights. Bibliography: 29 titles.

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References

  1. S. D. Fisher,Function Theory on Planar Domains, Wiley (1983).

  2. N. K. Nikol'skii,Treatise on the Shift Operator, Springer-Verlag (1986).

  3. P. Koosis,Introduction to H p Spaces (London Math. Soc. Lecture Notes Series, 40), Cambridge University Press (1980).

  4. J. B. Garnett,Bounded Analytic Functions, Academic Press (1981).

  5. R. Hunt, B. Muckenhoupt, and R. Wheeden, “Weighted norm inequalities for conjugate functions and Hilbert transforms,”Trans. Am. Math. Soc.,176, 227–251 (1973).

    Google Scholar 

  6. D. Sarason, “Function theory on the unit circle,” Preprint (1978).

  7. B. Pavlov, “Basicity conditions of an exponential system and the Muckenhoupt condition,”Dokl. Akad. Nauk SSSR,20 (1979).

  8. S. Khrushchev (Chruschev), N. Nikol'skii (Nikolski), and B. Pavlov, “Unconditional bases of exponentials and reproducing kernels,”Lect. Notes Math.,864 (1981).

  9. S. I. Fedorov, “Harmonic analysis in a multiply connected domain. I, II,”Mat. Sb.,181 (1990).

  10. S. I. Fedorov, “On harmonic analysis in multiply connected domains and character-automorphic Hardy spaces,”Algebra Analiz,9, No. 2, 192–240 (1997).

    Google Scholar 

  11. S. I. Fedorov and V. L. Vinnikov, “On the Nevanlinna-Pick interpolation problem in multiply connected domains,”Zap. Nauchn. Semin. POMI (1998) (to appear).

  12. M. Voichick and L. Zalcman, “Inner and outer functions on Riemann surfaces,”Proc. Am. Math. Soc.,16, 1200–1204 (1965).

    Google Scholar 

  13. D. Sarason, “TheH p spaces of an annulus,”Mem. Am. Math. Soc.,127 (1965).

  14. M. Voichick, “Invariant subspaces on a Riemann surface,”Can. J. Math.,18, 399–403 (1966).

    Google Scholar 

  15. M. Voichick, “Ideals and invariant subspaces of analytic functions,”Trans. Am. Math. Soc.,111, 493–512 (1964).

    Google Scholar 

  16. M. Hasumi, “Hardy classes on infinitely connected Riemann surfaces,”Lecture Notes Math.,1027 (1983).

  17. B. S. Pavlov and S. I. Fedorov, “The group of shifts and harmonic analysis on a Riemann surface of genus one,”Algebra Analiz,1, No. 2, 132–168 (1989).

    Google Scholar 

  18. M. B. Abrahamse and R. G. Douglas, “A class of subnormal operators related to multiply connected domains,”Adv. Math.,19, 106–148 (1976).

    Google Scholar 

  19. H. Widom, “Extremal polynomials associated with a system of curves in the complex plane,”Adv. Math.,3, 127–232 (1969).

    Google Scholar 

  20. M. B. Abrahamse, “The Pick interpolation theorem for finitely connected domains,”Michigan Math. J.,26, 195–203 (1979).

    Google Scholar 

  21. M. Sodin and P. Yuditskii, “Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions,”KØbenhavns Univ. Matern. Inst. Notetrykkeri, Preprint, No. 20 (1994).

  22. S. Fedorov, “Weighted norm inequalities and the Muckenhoupt condition in a multiply connected domain,” Auckland Univ., Dept. Math. Report Series, No. 330 (1997).

  23. S. I. Fedorov, “The angle between subspaces of analytic and antianalytic functions in weightedL 2 space on a boundary of a multiply connected domain,” Auckland Univ., Dept. Math. Report Series, No. 368 (1997).

  24. S. I. Fedorov, “On a projection from one co-invariant subspace onto another in the character-automorphic Hardy space on a multiply connected domain” (to appear).

  25. K. F. Clancey, “Representing measures on multiply connected domains,”Illinois J. Math.,35, 286–311 (1991).

    Google Scholar 

  26. J. A. Ball and K. F. Clancey, “Reproducing kernels for Hardy spaces on multiply connected domains,”Integral Equations and Operator Theory,25, 35–57 (1996).

    Google Scholar 

  27. I. Kra,Automorphic Forms and Kleinian Groups, Math. Lecture Note Series (1972).

  28. H. M. Farkas and I. Kra,Riemann Surfaces, Springer-Verlag (1980).

  29. K. Hoffman,Banach Spaces of Analytic Functions, Prentice-Hall (1962).

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Dedicated to the 90th anniversary of G. M. Goluzin's birth

Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 161–193.

This research was supported by the Marsden Fund, grant 96-UOA-MIS-0098.

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Fedorov, S.I. On the angles between subspaces, the muckenhoupt condition, and projection from one co-invariant subspace onto another in the theory of character-automorphic hardy spaces on a multiply connected domain. J Math Sci 95, 2276–2294 (1999). https://doi.org/10.1007/BF02172472

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