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 [math]\displaystyle{ \, \mathcal{H} \, }[/math] 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
- [math]\displaystyle{ N(z) = C + D z + \int_{\mathbb{R}} \bigg(\frac{1}{\lambda - z} - \frac{\lambda}{1 + \lambda^2} \bigg) \operatorname{d} \mu(\lambda), \quad z \in \mathcal{H}, }[/math]
where C is a real constant, D is a non-negative constant, [math]\displaystyle{ \mathcal{H} }[/math] is the upper half-plane, and μ is a Borel measure on ℝ satisfying the growth condition
- [math]\displaystyle{ \int_{\mathbb{R}} \frac{\operatorname{d} \mu(\lambda)}{1 + \lambda^2} \lt \infty. }[/math]
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
- [math]\displaystyle{ C = \Re \big( N(i) \big) \qquad \text{ and } \qquad D = \lim_{y \rightarrow \infty} \frac{N(i y)}{i y} }[/math]
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):
- [math]\displaystyle{ \mu \big( (\lambda_1, \lambda_2 ] \big) = \lim_{\delta\rightarrow 0} \lim_{\varepsilon\rightarrow 0} \frac{1}{\pi} \int_{\lambda_1+\delta}^{\lambda_2+\delta} \Im \big( N(\lambda + i \varepsilon) \big) \operatorname{d} \lambda. }[/math]
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). ([math]\displaystyle{ z }[/math] can be replaced by [math]\displaystyle{ z - a }[/math] for any real number [math]\displaystyle{ a }[/math].)
- [math]\displaystyle{ z^p\text{ with } 0 \le p \le 1 }[/math]
- [math]\displaystyle{ -z^p\text{ with } -1 \le p \le 0 }[/math]
- 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 [math]\displaystyle{ i(z/i)^p ~\text{ with }~-1\le p\le 1 }[/math].
- A sheet of [math]\displaystyle{ \ln(z) }[/math] such as the one with [math]\displaystyle{ f(1)=0 }[/math].
- [math]\displaystyle{ \tan(z) }[/math] (an example that is surjective but not injective).
- [math]\displaystyle{ z \mapsto \frac{az+b}{cz+d} }[/math]
- is a Nevanlinna function if (sufficient but not necessary) [math]\displaystyle{ \overline{a} d - b \overline{c} }[/math] is a positive real number and [math]\displaystyle{ \Im (\overline{b} d ) = \Im (\overline{a} c) = 0 }[/math]. 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: [math]\displaystyle{ \frac{i z + i - 2}{z + 1 + i} }[/math]
- [math]\displaystyle{ 1 + i + z }[/math] and [math]\displaystyle{ i + \operatorname{e}^{i z} }[/math] 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 [math]\displaystyle{ f }[/math] is an arbitrary vector, then the function
- [math]\displaystyle{ \langle (S-z)^{-1} f, f \rangle }[/math]
- is a Nevanlinna function.
- If [math]\displaystyle{ M(z) }[/math] and [math]\displaystyle{ N(z) }[/math] are both Nevanlinna functions, then the composition [math]\displaystyle{ M \big( N(z) \big) }[/math] is a Nevanlinna function as well.
Importance in operator theory
Nevanlinna functions appear in the study of Operator monotone functions.
References
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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