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Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann \(\zeta\)-function. (English) Zbl 0816.30026
Summary: It is proved that a subspace of a holomorphic Hilbert space is completely determined by their distances to the reproducing kernels. A simple rule is established to localize common zeros of a subspace of the Hardy space of the unit disc. As an illustration we show a series of discs of the complex plane free of zeros of the Riemann \(\zeta\)-function.

MSC:
30H10 Hardy spaces
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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References:
[1] A. BEURLING, A closure problem related to the Riemann zeta-function, Proc. Nat. Acad. USA, 41, n° 5 (1995), 312-314. · Zbl 0065.30303
[2] P. DUREN, H. SHAPIRO and A. SHIELDS, Singular measures and domains not of Smirnov type, Duke Math. Journal, 33 (1966), 247-254. · Zbl 0174.37501
[3] H. HEDENMALM, A factorization theorem for square area-integrable analytic functions, J. reine angew. Math., 422 (1991), 45-68. · Zbl 0734.30040
[4] H. HEDENMALM, An invariant subspace of the Bergman space having the codimension two property, J. reine angew. Math., 443 (1993), 1-9. · Zbl 0783.30040
[5] K. HOFFMAN, Banach spaces of analytic functions, Prentice Hall, Englewood Cliffs, NJ, 1962. · Zbl 0117.34001
[6] B. KORENBLUM, Closed ideals of the ring an, Functional Analysis and its applications, 6, n° (1972), 38-52. · Zbl 0262.46053
[7] B. KORENBLUM, A Beurling type theorem, Acta Math., 138 (1977), 265-293. · Zbl 0354.30024
[8] T.L. KRIETE and M. ROSENBLUM, A phragmén-Lindelöf theorem with applications to M(u, v) functions, Pacific J. Math., 43 (1972), 175-188. · Zbl 0251.30019
[9] N.K. NIKOLSKI, Treatise on the shift operator, Springer-Verlag, Heidelberg, 1986.
[10] N.K. NIKOLSKI, Invitation aux techniques des espaces de Hardy, EDM, Université Bordeaux-1, 1992, p. 1-69.
[11] N.K. NIKOLSKI, Two problems on spectral synthesis, Lect. Notes Math., Springer-Verlag, 1043 (1984), 378-381.
[12] B. NYMAN, On some groups and semi-groups of translations, Thesis, Uppsala, 1950. · Zbl 0037.35401
[13] N. SHIROKOV, Analytic functions smooth up to the boundary, Lect. Notes Math., v. 1312, Springer-Verlag, 1988. · Zbl 0656.30029
[14] V.I. VASYUNIN, On a biorthogonal function system related to the Riemann hypothesis, Algebra i Analiz (St. Petersburg Math. J.), to appear. · Zbl 0851.11051
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