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,^[1] but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions.

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.^[2]

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.

Nevanlinna function

Examples

Importance in operator theory

References

General

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