Nevanlinna function explained

l{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.

Integral representation

Every Nevanlinna function admits a representation

N(z) = C + D z + \int _{R} ( (1)/(λ- z) - (λ)/(1+λ ^{2}) ) \operatorname{d} \mu( λ ), z \in l{H},

where is a real constant, is a non-negative constant,

l{H} is the upper half-plane, and is a Borel measure on satisfying the growth condition \int _{R} (\operatorname{d)/(\mu( λ )}{ 1) + λ ^{2}} < inf ty.

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

C = \Re ( N(i) ) and D = \lim _{y} (N(i y))/(i y)

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

\mu ( ( λ _{1,} λ _{2} ] ) = \lim _{\delta →} \lim _{\varepsilon →} (1)/(\pi) \int ^{λ _{2+\delta}} _{λ _{1+\delta}} \Im ( N( λ + i \varepsilon) ) \operatorname{d} λ .

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 ^{pwith} 0 \le p \le 1 - z ^{pwith} -1 \le p \le 0

These are injective but when 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} ~ 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 (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: (i z + i -2)/(z +1+ i) 1 + i + z and i + \operatorname{e} ^{i} are examples which are entire functions. The second is neither injective nor surjective.

• If 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 ( N(z) ) is a Nevanlinna function as well.

Importance in operator theory

Nevanlinna functions appear in the study of Operator monotone functions.

References

General

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

Notes and References

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