Zeros of nonpositive type of generalized Nevanlinna functions with one negative square

Henk de Snoo; Henrik Winkler; Michal Wojtylak

doi:10.1016/j.jmaa.2011.04.060

Zeros of nonpositive type of generalized Nevanlinna functions with one negative square

Henk de Snoo, Henrik Winkler, Michal Wojtylak^*

^*Corresponding author for this work

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Abstract

A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Qτ(z) = (Q(z) − τ )/(1 + τQ(z)), τ ∈ R ∪ {∞}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ) as a function of τ defines a path in the closed upper halfplane. Various properties of this path are studied in detail.

Original language	English
Pages (from-to)	399-417
Number of pages	19
Journal	Journal of Mathematical Analysis and Applications
Volume	382
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2011.04.060
Publication status	Published - 1-Oct-2011

Keywords

Pontryagin space
Generalized Nevanlinna function
Generalized pole of nonpositive type
Generalized zero of nonpositive type
Integral representation
Fractional linear transformation
SELF-ADJOINT EXTENSIONS
LINEAR RELATION
KREIN SPACE
DEFECT ONE
OPERATORS
IIX

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title = "Zeros of nonpositive type of generalized Nevanlinna functions with one negative square",

abstract = "A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Qτ(z) = (Q(z) − τ )/(1 + τQ(z)), τ ∈ R ∪ \{∞\}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ) as a function of τ defines a path in the closed upper halfplane. Various properties of this path are studied in detail.",

keywords = "Pontryagin space, Generalized Nevanlinna function, Generalized pole of nonpositive type, Generalized zero of nonpositive type, Integral representation, Fractional linear transformation, SELF-ADJOINT EXTENSIONS, LINEAR RELATION, KREIN SPACE, DEFECT ONE, OPERATORS, IIX",

author = "\{de Snoo\}, Henk and Henrik Winkler and Michal Wojtylak",

note = "Relation: http://www.rug.nl/fmns-research/bernoulli/index Rights: University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science",

year = "2011",

month = oct,

day = "1",

doi = "10.1016/j.jmaa.2011.04.060",

language = "English",

volume = "382",

pages = "399--417",

journal = "Journal of Mathematical Analysis and Applications",

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T1 - Zeros of nonpositive type of generalized Nevanlinna functions with one negative square

AU - de Snoo, Henk

AU - Winkler, Henrik

AU - Wojtylak, Michal

N1 - Relation: http://www.rug.nl/fmns-research/bernoulli/index Rights: University of Groningen, Johann Bernoulli Institute for Mathematics and Computer Science

PY - 2011/10/1

Y1 - 2011/10/1

N2 - A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Qτ(z) = (Q(z) − τ )/(1 + τQ(z)), τ ∈ R ∪ {∞}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ) as a function of τ defines a path in the closed upper halfplane. Various properties of this path are studied in detail.

AB - A generalized Nevanlinna function Q(z) with one negative square has precisely one generalized zero of nonpositive type in the closed extended upper halfplane. The fractional linear transformation defined by Qτ(z) = (Q(z) − τ )/(1 + τQ(z)), τ ∈ R ∪ {∞}, is a generalized Nevanlinna function with one negative square. Its generalized zero of nonpositive type α(τ) as a function of τ defines a path in the closed upper halfplane. Various properties of this path are studied in detail.

KW - Pontryagin space

KW - Generalized Nevanlinna function

KW - Generalized pole of nonpositive type

KW - Generalized zero of nonpositive type

KW - Integral representation

KW - Fractional linear transformation

KW - SELF-ADJOINT EXTENSIONS

KW - LINEAR RELATION

KW - KREIN SPACE

KW - DEFECT ONE

KW - OPERATORS

KW - IIX

U2 - 10.1016/j.jmaa.2011.04.060

DO - 10.1016/j.jmaa.2011.04.060

M3 - Article

SN - 0022-247X

VL - 382

SP - 399

EP - 417

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

Zeros of nonpositive type of generalized Nevanlinna functions with one negative square

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Keywords

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