Nevanlinna function

Template:Short description Template:More footnotes 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 $\mathscr{H}$ 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 Template:Mvar admits a representation

$N(z)=C+Dz+{\int }_{ℝ}(\frac{1}{\lambda -z}-\frac{\lambda }{1+{\lambda }^2})\mathrm{d}\mu (\lambda ),z\in \mathscr{H},$

where Template:Mvar is a real constant, Template:Mvar is a non-negative constant, $\mathscr{H}$ is the upper half-plane, and Template:Mvar is a Borel measure on [[real numbers|Template:Math]] satisfying the growth condition

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

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

$C=\mathrm{\Re }(N(i))\text{ and }D=\underset{y\to \mathrm{\infty }}{lim}\frac{N(iy)}{iy}$

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

$\mu (({\lambda }_1,{\lambda }_2])=\underset{\delta \to 0}{lim}\underset{\epsilon \to 0}{lim}\frac{1}{\pi }{\int }_{{\lambda }_1+\delta }^{{\lambda }_2+\delta }\mathrm{\Im }(N(\lambda +i\epsilon ))\mathrm{d}\lambda .$

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 $a$.)

• $z^p\text{ with }0\le p\le 1$

• $-z^p\text{ with }-1\le p\le 0$

These are injective but when Template:Mvar does not equal 1 or −1 they are not surjective and can be rotated to some extent around the origin, such as $i(z/i{)}^p\text{ with }-1\le p\le 1$.

• 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) $\bar{a}d-b\bar{c}$ is a positive real number and $\mathrm{\Im }(\bar{b}d)=\mathrm{\Im }(\bar{a}c)=0$. 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 $i+{\mathrm{e}}^{iz}$ are examples which are entire functions. The second is neither injective nor surjective.
• If Template:Mvar is a self-adjoint operator in a Hilbert space and $f$ is an arbitrary vector, then the function

$⟨(S-z{)}^{-1}f,f⟩$

is a Nevanlinna function.

• If $M(z)$ and $N(z)$ are both Nevanlinna functions, then the composition $M(N(z))$ is a Nevanlinna function as well.

Importance in operator theory

Nevanlinna functions appear in the study of Operator monotone functions.

References

General

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