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Nevanlinna function

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A complex analysis function
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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 H {\displaystyle \,{\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 + D z + ∫ R ( 1 λ − z − λ 1 + λ 2 ) d ⁡ μ ( λ ) , z ∈ H , {\displaystyle 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, H {\displaystyle {\mathcal {H}}} is the upper half-plane, and μ is a Borel measure on ℝ satisfying the growth condition

∫ R d ⁡ μ ( λ ) 1 + λ 2 < ∞ . {\displaystyle \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 = ℜ ( N ( i ) )  and  D = lim y → ∞ N ( i y ) i y {\displaystyle 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):

μ ( ( λ 1 , λ 2 ] ) = lim δ → 0 lim ε → 0 1 π ∫ λ 1 + δ λ 2 + δ ℑ ( N ( λ + i ε ) ) d ⁡ λ . {\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 .}

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 {\displaystyle z} can be replaced by z − a {\displaystyle z-a} for any real number a {\displaystyle a} .)

  • z p  with  0 ≤ p ≤ 1 {\displaystyle z^{p}{\text{ with }}0\leq p\leq 1}
  • − z p  with  − 1 ≤ p ≤ 0 {\displaystyle -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    with    − 1 ≤ p ≤ 1 {\displaystyle i(z/i)^{p}~{\text{ with }}~-1\leq p\leq 1} .
  • A sheet of ln ⁡ ( z ) {\displaystyle \ln(z)} such as the one with f ( 1 ) = 0 {\displaystyle f(1)=0} .
  • tan ⁡ ( z ) {\displaystyle \tan(z)} (an example that is surjective but not injective).
z ↦ a z + b c z + d {\displaystyle z\mapsto {\frac {az+b}{cz+d}}}
is a Nevanlinna function if (sufficient but not necessary) a ¯ d − b c ¯ {\displaystyle {\overline {a}}d-b{\overline {c}}} is a positive real number and ℑ ( b ¯ d ) = ℑ ( a ¯ c ) = 0 {\displaystyle \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 {\displaystyle {\frac {iz+i-2}{z+1+i}}}
  • 1 + i + z {\displaystyle 1+i+z} and i + e i z {\displaystyle 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 {\displaystyle f} is an arbitrary vector, then the function
⟨ ( S − z ) − 1 f , f ⟩ {\displaystyle \langle (S-z)^{-1}f,f\rangle }
is a Nevanlinna function.
  • If M ( z ) {\displaystyle M(z)} and N ( z ) {\displaystyle N(z)} are both Nevanlinna functions, then the composition M ( N ( z ) ) {\displaystyle 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

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

General

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