Nevanlinna function

In mathematics, in the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to a real constant,[1] but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions.

Integral representation

Every Nevanlinna function N admits a representation

where C is a real constant, D is a non-negative constant, is the upper half-plane, and μ is a Borel measure on satisfying the growth condition

Conversely, every function of this form turns out to be a Nevanlinna function. The constants in this representation are related to the function N via

and the Borel measure μ can be recovered from N by employing the Stieltjes inversion formula (related to the inversion formula for the Stieltjes transformation):

A very similar representation of functions is also called the Poisson representation.[2]

Examples

Some elementary examples of Nevanlinna functions follow (with appropriately chosen branch cuts in the first three). ( can be replaced by for any real number .)

These are injective but when p does not equal 1 or −1 they are not surjective and can be rotated to some extent around the origin, such as .
  • A sheet of such as the one with .
  • (an example that is surjective but not injective).
is a Nevanlinna function if (sufficient but not necessary) is a positive real number and . This is equivalent to the set of such transformations that map the real axis to itself. One may then add any constant in the upper half-plane, and move the pole into the lower half-plane, giving new values for the parameters. Example:
  • and are examples which are entire functions. The second is neither injective nor surjective.
  • If S is a self-adjoint operator in a Hilbert space and is an arbitrary vector, then the function
is a Nevanlinna function.
  • If and are both Nevanlinna functions, then the composition is a Nevanlinna function as well.

Importance in operator theory

Nevanlinna functions appear in the study of Operator monotone functions.

References

  1. A real number is not considered to be in the upper half-plane.
  2. See for example Section 4, "Poisson representation" in Louis de Branges (1968). Hilbert Spaces of Entire Functions. Prentice-Hall. ASIN B0006BUXNM. De Branges gives a form for functions whose real part is non-negative in the upper half-plane.

General

  • Vadim Adamyan, ed. (2009). Modern analysis and applications. p. 27. ISBN 3-7643-9918-X.
  • Naum Ilyich Akhiezer and I. M. Glazman (1993). Theory of linear operators in Hilbert space. ISBN 0-486-67748-6.
  • Marvin Rosenblum and James Rovnyak (1994). Topics in Hardy Classes and Univalent Functions. ISBN 3-7643-5111-X.
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