 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 BIRS Workshop Lecture Videos /
 Homogeneous Herglotz class versus homogeneous HerglotzAgler...
Open Collections
BIRS Workshop Lecture Videos
BIRS Workshop Lecture Videos
Homogeneous Herglotz class versus homogeneous HerglotzAgler class Ball, Joseph
Description
The Herglotz class is defined as the class of holomorphic functions mapping the $d$fold Cartesian product of the right halfplane with itself into the right halfplane. The HerglotzAgler class is defined as the space of such holomorphic functions such that (under any reasonable functional calculus) maps a $d$tuple of commuting accretive operators to an accretive operator (accretive operator here meaning having positivedefinite real part). Integral representation formulas and transferfunction realization formulas for the Herglotz class are rather involved as soon as $d > 2$ while such formulas are more like the singlevariable case for the HerglotzAgler class. A routine observation is that the HerglotzAgler class is a subclass of the Herglotz class with equality for $d=1,2$ and strict inclusion for $d>2$. Also of interest are the homogeneous Herglotz class and the homogeneous HerglotzAgler class where here homogeneous means "homogeneous of degree 1": $f(\lambda z_1, \dots, \lambda z_d) = \lambda f(z_1, \dots, z_d)$ for all complex scalars $\lambda$. This class has motivation from electrical circuit theory and has been studied at length by V. Bessmertnyi and D. KaliuzhnyiVerbovetskyi. A parallel picture exists for the homogeneous Herglotz and homogeneous HerglotzAgler classes: there is a better structural understanding of the homogeneous HerglotzAgler class than for the general homogeneous Herglotz class, and it is an easy observation that the homogeneous HerglotzAgler class is a subclass of the homogeneous Herglotz class. However there remains an open question: for what values of $d$ (the number of variables) is it the case that the homogeneous Herglotz class and the homogeneous HerglotzAgler class are actually the same We make some observations on this question and discuss a possible approach to resolving this question.
Item Metadata
Title 
Homogeneous Herglotz class versus homogeneous HerglotzAgler class

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20191008T14:05

Description 
The Herglotz class is defined as the class of holomorphic functions mapping the $d$fold Cartesian product
of the right halfplane with itself into the right halfplane. The HerglotzAgler class is defined as the space of such holomorphic
functions such that (under any reasonable functional calculus) maps a $d$tuple of commuting accretive operators
to an accretive operator (accretive operator here meaning having positivedefinite real part). Integral representation
formulas and transferfunction realization formulas for the Herglotz class are rather involved as soon as $d > 2$ while
such formulas are more like the singlevariable case for the HerglotzAgler class. A routine observation is that
the HerglotzAgler class is a subclass of the Herglotz class with equality for $d=1,2$ and strict inclusion for
$d>2$. Also of interest are the homogeneous Herglotz class and the homogeneous HerglotzAgler class
where here homogeneous means "homogeneous of degree 1":
$f(\lambda z_1, \dots, \lambda z_d) = \lambda f(z_1, \dots, z_d)$ for all complex scalars $\lambda$. This class
has motivation from electrical circuit theory and has been studied at length by V. Bessmertnyi and D.
KaliuzhnyiVerbovetskyi. A parallel picture exists for the homogeneous Herglotz and homogeneous HerglotzAgler classes:
there is a better structural
understanding of the homogeneous HerglotzAgler class than for the general homogeneous Herglotz class, and it
is an easy observation that the homogeneous HerglotzAgler class is a subclass of the homogeneous Herglotz class.
However there remains an open question: for what values of $d$ (the number of variables) is it the case that
the homogeneous Herglotz class and the homogeneous HerglotzAgler class are actually the same We make
some observations on this question and discuss a possible approach to resolving this question.

Extent 
35.0 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: Virginia Tech

Series  
Date Available 
20200406

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0389749

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Faculty

Rights URI  
Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International