Page 190 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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OPERATOR SEMIGROUPS AND EVOLUTION EQUATIONS (MS-29)
Spectral aspects of eventually positive C0-semigroups
Sahiba Arora, sahiba.arora@mailbox.tu-dresden.de
Technical University Dresden, Germany
The theory of positive one-parameter semigroups is rich and has applications in various math-
ematical fields. A closely related notion that appeared recently is that of eventually positive
semigroups, i.e., semigroups that become (and stay) positive for sufficiently large times. A sys-
tematic study of this concept revealed that such semigroups exhibit spectral properties similar
to those already known for positive semigroups.
We start with a few examples of eventually positive semigroups and then highlight several
spectral theoretic behaviours of this notion. We follow this up with criteria for strong conver-
gence of an eventually positive semigroup. Further, we look at a characterization of conver-
gence of eventually positive semigroups in the operator norm which generalizes a result that
was previously only known for positive semigroups.
Towards the end, we look at some spectral and convergence implications of locally eventual
positive semigroups. In a loose sense, this means that the solution of the corresponding Cauchy
problem becomes positive in a part of the domain for large times. While examples of this
concept have been known for quite some time, a systematic study of it has only been recently
initiated.
References
[1] Rainer Nagel, editor. One-parameter semigroups of positive operators. Vol. 1184.
Springer, Cham, 1986.
[2] Sahiba Arora. Locally eventually positive operator semigroups. To appear in J. Oper. The-
ory. Preprint available online at https://arxiv.org/abs/2101.11386. (2021).
[3] Sahiba Arora and Jochen Glück. Spectrum and convergence of eventually positive operator
semigroups. Preprint. Available online at https://arxiv.org/abs/2011.04296v2. (2021).
[4] Daniel Daners, Jochen Glück, and James B. Kennedy. Eventually positive semigroups of
linear operators. J. Math. Anal. Appl. 433. 2(2016): 1561–1593.
[5] Filippo Gazzola and Hans-Christoph Grunau. Eventual local positivity for a biharmonic
heat equation in Rn. Discrete Contin. Dyn. Syst., Ser. S 1. 1(2008): 83–87.
Bounded functional calculi for unbounded operators
Charles Batty, charles.batty@sjc.ox.ac.uk
University of Oxford, United Kingdom
Several new functional calculi for unbounded operators have been discovered recently. In par-
ticular, many semigroup generators have a bounded functional calculus for a Banach algebra of
so-called analytic Besov functions. I will describe properties and significance of this calculus,
and then briefly mention further extensions of that calculus for generators of bounded semi-
groups on Hilbert spaces and for generators of bounded holomorphic semigroups on Banach
spaces.
This work has been in collaboration with Alexander Gomilko and Yuri Tomilov.
188
Spectral aspects of eventually positive C0-semigroups
Sahiba Arora, sahiba.arora@mailbox.tu-dresden.de
Technical University Dresden, Germany
The theory of positive one-parameter semigroups is rich and has applications in various math-
ematical fields. A closely related notion that appeared recently is that of eventually positive
semigroups, i.e., semigroups that become (and stay) positive for sufficiently large times. A sys-
tematic study of this concept revealed that such semigroups exhibit spectral properties similar
to those already known for positive semigroups.
We start with a few examples of eventually positive semigroups and then highlight several
spectral theoretic behaviours of this notion. We follow this up with criteria for strong conver-
gence of an eventually positive semigroup. Further, we look at a characterization of conver-
gence of eventually positive semigroups in the operator norm which generalizes a result that
was previously only known for positive semigroups.
Towards the end, we look at some spectral and convergence implications of locally eventual
positive semigroups. In a loose sense, this means that the solution of the corresponding Cauchy
problem becomes positive in a part of the domain for large times. While examples of this
concept have been known for quite some time, a systematic study of it has only been recently
initiated.
References
[1] Rainer Nagel, editor. One-parameter semigroups of positive operators. Vol. 1184.
Springer, Cham, 1986.
[2] Sahiba Arora. Locally eventually positive operator semigroups. To appear in J. Oper. The-
ory. Preprint available online at https://arxiv.org/abs/2101.11386. (2021).
[3] Sahiba Arora and Jochen Glück. Spectrum and convergence of eventually positive operator
semigroups. Preprint. Available online at https://arxiv.org/abs/2011.04296v2. (2021).
[4] Daniel Daners, Jochen Glück, and James B. Kennedy. Eventually positive semigroups of
linear operators. J. Math. Anal. Appl. 433. 2(2016): 1561–1593.
[5] Filippo Gazzola and Hans-Christoph Grunau. Eventual local positivity for a biharmonic
heat equation in Rn. Discrete Contin. Dyn. Syst., Ser. S 1. 1(2008): 83–87.
Bounded functional calculi for unbounded operators
Charles Batty, charles.batty@sjc.ox.ac.uk
University of Oxford, United Kingdom
Several new functional calculi for unbounded operators have been discovered recently. In par-
ticular, many semigroup generators have a bounded functional calculus for a Banach algebra of
so-called analytic Besov functions. I will describe properties and significance of this calculus,
and then briefly mention further extensions of that calculus for generators of bounded semi-
groups on Hilbert spaces and for generators of bounded holomorphic semigroups on Banach
spaces.
This work has been in collaboration with Alexander Gomilko and Yuri Tomilov.
188
