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Homogeneous Herglotz class versus homogeneous Herglotz-Agler class Ball, Joseph
Description
The Herglotz class is defined as the class of holomorphic functions mapping the $d$-fold Cartesian product of the right half-plane with itself into the right half-plane. The Herglotz-Agler 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 positive-definite real part). Integral representation formulas and transfer-function realization formulas for the Herglotz class are rather involved as soon as $d > 2$ while such formulas are more like the single-variable case for the Herglotz-Agler class. A routine observation is that the Herglotz-Agler class is a sub-class 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 Herglotz-Agler 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. Kaliuzhnyi-Verbovetskyi. A parallel picture exists for the homogeneous Herglotz and homogeneous Herglotz-Agler classes: there is a better structural understanding of the homogeneous Herglotz-Agler class than for the general homogeneous Herglotz class, and it is an easy observation that the homogeneous Herglotz-Agler 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 Herglotz-Agler 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 Herglotz-Agler class
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| Creator | |
| Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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| Date Issued |
2019-10-08T14:05
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| Description |
The Herglotz class is defined as the class of holomorphic functions mapping the $d$-fold Cartesian product
of the right half-plane with itself into the right half-plane. The Herglotz-Agler 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 positive-definite real part). Integral representation
formulas and transfer-function realization formulas for the Herglotz class are rather involved as soon as $d > 2$ while
such formulas are more like the single-variable case for the Herglotz-Agler class. A routine observation is that
the Herglotz-Agler class is a sub-class 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 Herglotz-Agler 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.
Kaliuzhnyi-Verbovetskyi. A parallel picture exists for the homogeneous Herglotz and homogeneous Herglotz-Agler classes:
there is a better structural
understanding of the homogeneous Herglotz-Agler class than for the general homogeneous Herglotz class, and it
is an easy observation that the homogeneous Herglotz-Agler 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 Herglotz-Agler class are actually the same We make
some observations on this question and discuss a possible approach to resolving this question.
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| Extent |
35.0 minutes
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| Subject | |
| Type | |
| File Format |
video/mp4
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| Language |
eng
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| Notes |
Author affiliation: Virginia Tech
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| Series | |
| Date Available |
2020-04-06
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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| DOI |
10.14288/1.0389749
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| URI | |
| Affiliation | |
| Peer Review Status |
Unreviewed
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| Scholarly Level |
Faculty
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| Rights URI | |
| Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International