Page 216 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 216
RECENT DEVELOPMENTS ON PRESERVERS (MS-38)
Isometries of Wasserstein spaces
Dániel Virosztek, daniel.virosztek@ist.ac.at
Institute of Science and Technology Austria, Austria
Coauthors: György Pál Gehér, Tamás Titkos
I will report on our systematic study of isometries of classical Wasserstein spaces with various
underlying spaces. The starting point was the case of the discrete underlying space, where we
found a rich family of non-surjective isometries. However, we proved isometric rigidity on
the contrary, which means that every surjective isometry is governed by a permutation of the
underlying space [G.-T.-V., J. Math. Anal. Appl. 480 (2019), 123435].
The next step was the description of the isometries of Wp(R) for p = 2. Here, we proved
isometric rigidity and classified the non-surjective isometries for p > 1, as well. The study of
Wp([0, 1]) led to the discovery of a mass-splitting isometry for p = 1, which turned out to be
also a key step in giving an affirmative answer to Kloeckner’s questions from 2010 concerning
the existence of exotic and mass-splitting isometries on quadratic Wasserstein spaces [G.-T.-V.,
Trans. Amer. Math. Soc. 373 (2020), 5855-5883].
The most recent work of ours concerns Wasserstein-Hilbert spaces. We extended Kloeck-
ner’s result on the quadratic case to the infinite-dimensional setting and proved isometric rigidity
for the non-quadratic cases. The main tool we introduced to study the non-quadratic cases is the
Wasserstein potential of measures. As a byproduct of our results, we showed that Wp(X) is iso-
metrically rigid for every Polish space X and parameter 0 < p < 1 [G.-T.-V., arXiv:2102.02037
(2021)].
Positivity preservers forbidden to operate on diagonal blocks
Prateek Kumar Vishwakarma, prateekv@iisc.ac.in
Indian Institute of Science, Bangalore, India
The question of which functions acting entrywise preserve positive semidefiniteness has a long
history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely
monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on
matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942,
1959] shows the converse: there are no other such functions.
Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc.
2015] classified the entrywise positivity preservers in all dimensions, which act only on the
off-diagonal entries. These two results are at “opposite ends", and in both cases the preservers
have to be absolutely monotonic.
We complete the classification of positivity preservers that act entrywise except on specified
“diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a
more general framework.) This yields the first examples of dimension-free entrywise positivity
preservers – with certain forbidden principal blocks – that are not absolutely monotonic.
214
Isometries of Wasserstein spaces
Dániel Virosztek, daniel.virosztek@ist.ac.at
Institute of Science and Technology Austria, Austria
Coauthors: György Pál Gehér, Tamás Titkos
I will report on our systematic study of isometries of classical Wasserstein spaces with various
underlying spaces. The starting point was the case of the discrete underlying space, where we
found a rich family of non-surjective isometries. However, we proved isometric rigidity on
the contrary, which means that every surjective isometry is governed by a permutation of the
underlying space [G.-T.-V., J. Math. Anal. Appl. 480 (2019), 123435].
The next step was the description of the isometries of Wp(R) for p = 2. Here, we proved
isometric rigidity and classified the non-surjective isometries for p > 1, as well. The study of
Wp([0, 1]) led to the discovery of a mass-splitting isometry for p = 1, which turned out to be
also a key step in giving an affirmative answer to Kloeckner’s questions from 2010 concerning
the existence of exotic and mass-splitting isometries on quadratic Wasserstein spaces [G.-T.-V.,
Trans. Amer. Math. Soc. 373 (2020), 5855-5883].
The most recent work of ours concerns Wasserstein-Hilbert spaces. We extended Kloeck-
ner’s result on the quadratic case to the infinite-dimensional setting and proved isometric rigidity
for the non-quadratic cases. The main tool we introduced to study the non-quadratic cases is the
Wasserstein potential of measures. As a byproduct of our results, we showed that Wp(X) is iso-
metrically rigid for every Polish space X and parameter 0 < p < 1 [G.-T.-V., arXiv:2102.02037
(2021)].
Positivity preservers forbidden to operate on diagonal blocks
Prateek Kumar Vishwakarma, prateekv@iisc.ac.in
Indian Institute of Science, Bangalore, India
The question of which functions acting entrywise preserve positive semidefiniteness has a long
history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely
monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on
matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942,
1959] shows the converse: there are no other such functions.
Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc.
2015] classified the entrywise positivity preservers in all dimensions, which act only on the
off-diagonal entries. These two results are at “opposite ends", and in both cases the preservers
have to be absolutely monotonic.
We complete the classification of positivity preservers that act entrywise except on specified
“diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a
more general framework.) This yields the first examples of dimension-free entrywise positivity
preservers – with certain forbidden principal blocks – that are not absolutely monotonic.
214
