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Zeros of the derivative of a rational function and coinvariant subspaces for the shift operator on the Bergman space. (English. Russian original) Zbl 1069.30008

J. Math. Sci., New York 120, No. 5, 1657-1661 (2004); translation from Zap. Nauchn. Semin. POMI 282, 26-33 (2001).
Summary: If all \(n\) \((n>1)\) zeros of a rational function \(r\) with simple poles are in a half-plane, then the derivative of \(r\) has at least one zero in the same half-plane. This result is used to prove that the number of zeros of a linear combination of \(n\) Bergman kernels in the unit disk may range from 0 to \(2n-3\).

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
47B38 Linear operators on function spaces (general)
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