@article{12214,
  abstract     = {Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. },
  author       = {Gehér, György Pál and Titkos, Tamás and Virosztek, Daniel},
  issn         = {1469-7750},
  journal      = {Journal of the London Mathematical Society},
  keywords     = {General Mathematics},
  number       = {4},
  pages        = {3865--3894},
  publisher    = {Wiley},
  title        = {{The isometry group of Wasserstein spaces: The Hilbertian case}},
  doi          = {10.1112/jlms.12676},
  volume       = {106},
  year         = {2022},
}

@article{8373,
  abstract     = {It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim.},
  author       = {Pitrik, József and Virosztek, Daniel},
  issn         = {0024-3795},
  journal      = {Linear Algebra and its Applications},
  keywords     = {Kubo-Ando mean, weighted multivariate mean, barycenter},
  pages        = {203--217},
  publisher    = {Elsevier},
  title        = {{A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means}},
  doi          = {10.1016/j.laa.2020.09.007},
  volume       = {609},
  year         = {2021},
}

@article{9036,
  abstract     = {In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.},
  author       = {Virosztek, Daniel},
  issn         = {0001-8708},
  journal      = {Advances in Mathematics},
  keywords     = {General Mathematics},
  number       = {3},
  publisher    = {Elsevier},
  title        = {{The metric property of the quantum Jensen-Shannon divergence}},
  doi          = {10.1016/j.aim.2021.107595},
  volume       = {380},
  year         = {2021},
}

@article{7389,
  abstract     = {Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space
W_p(R) for all p \in [1,\infty) \setminus {2}. We show that W_2(R) is also exceptional regarding the
parameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying
space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact
interval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if
p is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1]))
cannot be embedded into Isom(W_1(R)).},
  author       = {Geher, Gyorgy Pal and Titkos, Tamas and Virosztek, Daniel},
  issn         = {10886850},
  journal      = {Transactions of the American Mathematical Society},
  keywords     = {Wasserstein space, isometric embeddings, isometric rigidity, exotic isometry flow},
  number       = {8},
  pages        = {5855--5883},
  publisher    = {American Mathematical Society},
  title        = {{Isometric study of Wasserstein spaces - the real line}},
  doi          = {10.1090/tran/8113},
  volume       = {373},
  year         = {2020},
}

@article{7618,
  abstract     = {This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. (Lett Math Phys 109:1777–1804, 2019) with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences that are of the form ϕ(A,B)=Tr((1−c)A+cB−AσB), where σ is an arbitrary Kubo–Ando mean, and c∈(0,1) is the weight of σ. We note that these divergences belong to the family of maximal quantum f-divergences, and hence are jointly convex, and satisfy the data processing inequality. We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate 1/2-power mean, that was claimed in Bhatia et al. (2019), is true in the case of commuting operators, but it is not correct in the general case. },
  author       = {Pitrik, Jozsef and Virosztek, Daniel},
  issn         = {1573-0530},
  journal      = {Letters in Mathematical Physics},
  number       = {8},
  pages        = {2039--2052},
  publisher    = {Springer Nature},
  title        = {{Quantum Hellinger distances revisited}},
  doi          = {10.1007/s11005-020-01282-0},
  volume       = {110},
  year         = {2020},
}

@article{6843,
  abstract     = {The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space Wp(X), where S is a countable discrete metric space and 0<p<∞ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of X×(0,1]-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of Wp(X) splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that Wp(X) is isometrically rigid for all 0<p<∞.},
  author       = {Gehér, György Pál and Titkos, Tamás and Virosztek, Daniel},
  issn         = {10960813},
  journal      = {Journal of Mathematical Analysis and Applications},
  number       = {2},
  publisher    = {Elsevier},
  title        = {{On isometric embeddings of Wasserstein spaces – the discrete case}},
  doi          = {10.1016/j.jmaa.2019.123435},
  volume       = {480},
  year         = {2019},
}

@inproceedings{7035,
  abstract     = {The aim of this short note is to expound one particular issue that was discussed during the talk [10] given at the symposium ”Researches on isometries as preserver problems and related topics” at Kyoto RIMS. That is,  the role of Dirac masses by  describing  the  isometry group of various metric spaces  of probability  measures.   This  article  is  of  survey  character,  and  it  does  not  contain  any  essentially  new results.From an isometric point of view, in some cases, metric spaces of measures are similar to C(K)-type function  spaces.   Similarity  means  here  that  their  isometries  are  driven  by  some  nice  transformations of  the  underlying  space.   Of  course,  it  depends  on  the  particular  choice  of  the  metric  how  nice  these transformations should be.  Sometimes, as we will see, being a homeomorphism is enough to generate an isometry.  But sometimes we need more:  the transformation must preserve the underlying distance as well.  Statements claiming that isometries in questions are necessarily induced by homeomorphisms are called Banach-Stone-type results, while results asserting that the underlying transformation is necessarily an isometry are termed as isometric rigidity results.As  Dirac  masses  can  be  considered  as  building  bricks  of  the  set  of  all  Borel  measures,  a  natural question arises:Is it enough to understand how an isometry acts on the set of Dirac masses?  Does this action extend uniquely to all measures?In what follows, we will thoroughly investigate this question.},
  author       = {Geher, Gyorgy Pal and Titkos, Tamas and Virosztek, Daniel},
  booktitle    = {Kyoto RIMS Kôkyûroku},
  location     = {Kyoto, Japan},
  pages        = {34--41},
  publisher    = {Research Institute for Mathematical Sciences, Kyoto University},
  title        = {{Dirac masses and isometric rigidity}},
  volume       = {2125},
  year         = {2019},
}

@article{405,
  abstract     = {We investigate the quantum Jensen divergences from the viewpoint of joint convexity. It turns out that the set of the functions which generate jointly convex quantum Jensen divergences on positive matrices coincides with the Matrix Entropy Class which has been introduced by Chen and Tropp quite recently.},
  author       = {Virosztek, Daniel},
  journal      = {Linear Algebra and Its Applications},
  pages        = {67--78},
  publisher    = {Elsevier},
  title        = {{Jointly convex quantum Jensen divergences}},
  doi          = {10.1016/j.laa.2018.03.002},
  volume       = {576},
  year         = {2019},
}

@article{284,
  abstract     = {Borel probability measures living on metric spaces are fundamental
mathematical objects. There are several meaningful distance functions that make the collection of the probability measures living on a certain space a metric space. We are interested in the description of the structure of the isometries of such metric spaces. We overview some of the recent results of the topic and we also provide some new ones concerning the Wasserstein distance. More specifically, we consider the space of all Borel probability measures on the unit sphere of a Euclidean space endowed with the Wasserstein metric W_p for arbitrary p &gt;= 1, and we show that the action of a Wasserstein isometry on the set of Dirac measures is induced by an isometry of the underlying unit sphere.},
  author       = {Virosztek, Daniel},
  issn         = {2064-8316},
  journal      = {Acta Scientiarum Mathematicarum},
  number       = {1-2},
  pages        = {65 -- 80},
  publisher    = {Springer Nature},
  title        = {{Maps on probability measures preserving certain distances - a survey and some new results}},
  doi          = {10.14232/actasm-018-753-y},
  volume       = {84},
  year         = {2018},
}