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 $\,{\mathcal {H}}\,$ and has a non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or a real constant, 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

$N(z)=C+Dz+\int _{\mathbb {R} }{\bigg (}{\frac {1}{\lambda -z}}-{\frac {\lambda }{1+\lambda ^{2}}}{\bigg )}\operatorname {d} \mu (\lambda ),\quad z\in {\mathcal {H}},$

where C is a real constant, D is a non-negative constant, ${\mathcal {H}}$ is the upper half-plane, and μ is a Borel measure on ℝ satisfying the growth condition

$\int _{\mathbb {R} }{\frac {\operatorname {d} \mu (\lambda )}{1+\lambda ^{2}}}<\infty .$

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

$C=\Re {\big (}N(i){\big )}\qquad {\text{ and }}\qquad D=\lim _{y\rightarrow \infty }{\frac {N(iy)}{iy}}$

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):

$\mu {\big (}(\lambda _{1},\lambda _{2}]{\big )}=\lim _{\delta \rightarrow 0}\lim _{\varepsilon \rightarrow 0}{\frac {1}{\pi }}\int _{\lambda _{1}+\delta }^{\lambda _{2}+\delta }\Im {\big (}N(\lambda +i\varepsilon ){\big )}\operatorname {d} \lambda .$

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

Examples

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

• $z^{p}{\text{ with }}0\leq p\leq 1$

• $-z^{p}{\text{ with }}-1\leq p\leq 0$

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 $i(z/i)^{p}~{\text{ with }}~-1\leq p\leq 1$.

• A sheet of $\ln(z)$ such as the one with $f(1)=0$.

• $\tan(z)$ (an example that is surjective but not injective).

• A Möbius transformation

$z\mapsto {\frac {az+b}{cz+d}}$

is a Nevanlinna function if (sufficient but not necessary) ${\overline {a}}d-b{\overline {c}}$ is a positive real number and $\Im ({\overline {b}}d)=\Im ({\overline {a}}c)=0$. 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: ${\frac {iz+i-2}{z+1+i}}$

• $1+i+z$ and $i+\operatorname {e} ^{iz}$ 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 $f$ is an arbitrary vector, then the function

$\langle (S-z)^{-1}f,f\rangle$

is a Nevanlinna function.

• If $M(z)$ and $N(z)$ are both Nevanlinna functions, then the composition $M{\big (}N(z){\big )}$ 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 978-3-7643-9918-4.
• Naum Ilyich Akhiezer and I. M. Glazman (1993). Theory of linear operators in Hilbert space. Dover Publications. ISBN 0-486-67748-6.
• Marvin Rosenblum and James Rovnyak (1994). Topics in Hardy Classes and Univalent Functions. Springer. ISBN 3-7643-5111-X.