TY - JOUR AU - Gehér, G.P. AU - Titkos, Tamás AU - Virosztek, Dániel TI - On the exotic isometry flow of the quadratic Wasserstein space over the real line JF - LINEAR ALGEBRA AND ITS APPLICATIONS J2 - LINEAR ALGEBRA APPL VL - Available online 6 March 2023 PY - 2024 SP - & SN - 0024-3795 DO - 10.1016/j.laa.2023.02.016 UR - https://m2.mtmt.hu/api/publication/33720234 ID - 33720234 N1 - Published online: 6 March 2023 Export Date: 28 March 2023 CODEN: LAAPA Correspondence Address: Virosztek, D.; Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, Hungary; email: virosztek.daniel@renyi.hu Funding details: Leverhulme Trust, ECF-2018-125 Funding details: European Research Council, ERC, 810115 Funding details: Magyar Tudományos Akadémia, MTA, LP2021-15/2021 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, K134944, PD128374 Funding text 1: Gy. P. Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF-2018-125), and also by the Hungarian National Research, Development and Innovation Office (Grant no. K134944).T. Titkos was supported by the Hungarian National Research, Development and Innovation Office – NKFIH (grant no. PD128374 and grant no. K134944) and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.D. Virosztek was supported by the Momentum program of the Hungarian Academy of Sciences under grant agreement no. LP2021-15/2021, and partially supported by the ERC Synergy Grant No. 810115. AB - Kloeckner discovered that the quadratic Wasserstein space over the real line (denoted by W2(R)) is quite peculiar, as its isometry group contains an exotic isometry flow. His result implies that it can happen that an isometry Φ fixes all Dirac measures, but still, Φ is not the identity of W2(R). This is the only known example of this surprising and counterintuitive phenomenon. Kloeckner also proved that the image of each finitely supported measure under these isometries (and thus under all isometry) is a finitely supported measure. Recently we showed that the exotic isometry flow can be represented as a unitary group on L2((0,1)). In this paper, we calculate the generator of this group, and we show that every exotic isometry (and thus every isometry) maps the set of all absolutely continuous measures belonging to W2(R) onto itself. © 2023 Elsevier Inc. LA - English DB - MTMT ER - TY - GEN AU - Gehér, György AU - Hruskova, Aranka AU - Titkos, Tamás AU - Virosztek, Dániel TI - Isometric rigidity of Wasserstein spaces over Euclidean spheres PY - 2023 UR - https://m2.mtmt.hu/api/publication/34131047 ID - 34131047 LA - English DB - MTMT ER - TY - JOUR AU - Gehér, György AU - Titkos, Tamás AU - Virosztek, Dániel TI - On isometries of Wasserstein spaces JF - RIMS KOKYUROKU BESSATSU J2 - RIMS KOKYUROKU BESSATSU VL - B93 PY - 2023 SP - 239 EP - 250 PG - 12 SN - 1881-6193 UR - https://m2.mtmt.hu/api/publication/33678624 ID - 33678624 LA - English DB - MTMT ER - TY - JOUR AU - Gehér, György AU - Titkos, Tamás AU - Virosztek, Dániel TI - Isometric rigidity of Wasserstein tori and spheres JF - MATHEMATIKA J2 - MATHEMATIKA VL - 69 PY - 2023 IS - 1 SP - 20 EP - 32 PG - 13 SN - 0025-5793 DO - 10.1112/mtk.12174 UR - https://m2.mtmt.hu/api/publication/33578758 ID - 33578758 N1 - Cited By :1 Export Date: 20 January 2023 Correspondence Address: Virosztek, D.; Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, Hungary; email: virosztek.daniel@renyi.hu Funding details: Leverhulme Trust, ECF‐2018‐125 Funding details: Magyar Tudományos Akadémia, MTA, K124152, KH129601, LP2021‐15/2021 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, K134944, PD128374 Funding text 1: Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF‐2018‐125), and also by the Hungarian National Research, Development and Innovation Office (Grant Number: K134944); Titkos was supported by the Hungarian National Research, Development and Innovation Office ‐ NKFIH (Grant Numbers: PD128374 and K134944) and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences; Virosztek was supported by the Momentum program of the Hungarian Academy of Sciences under Grant Agreement Number: LP2021‐15/2021, and partially supported by the Hungarian National Research, Development and Innovation Office ‐ NKFIH (Grant Numbers: K124152 and KH129601). AB - We prove isometric rigidity for p-Wasserstein spaces over finite-dimensional tori and spheres for all p. We present a unified approach to proving rigidity that relies on the robust method of recovering measures from their Wasserstein potentials. © 2022 The Authors. The publishing rights in this article are licensed to University College London under an exclusive licence. Mathematika is published by the London Mathematical Society on behalf of University College London. LA - English DB - MTMT ER - TY - JOUR AU - Gehér, György AU - Pitrik, József AU - Titkos, Tamás AU - Virosztek, Dániel TI - Quantum Wasserstein isometries on the qubit state space JF - JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS J2 - J MATH ANAL APPL VL - 522 PY - 2023 IS - 2 PG - 17 SN - 0022-247X DO - 10.1016/j.jmaa.2022.126955 UR - https://m2.mtmt.hu/api/publication/33578756 ID - 33578756 N1 - Export Date: 08 March 2024 AB - We describe Wasserstein isometries of the quantum bit state space with respect to distinguished cost operators. We derive a Wigner-type result for the cost operator involving all the Pauli matrices: in this case, the isometry group consists of unitary or anti-unitary conjugations. In the Bloch sphere model this means that the isometry group coincides with the classical symmetry group O(3). On the other hand, for the cost generated by the qubit ‘‘clock” and ‘‘shift” operators, we discovered non-surjective and non-injective isometries as well, beyond the regular ones. This phenomenon mirrors certain surprising properties of the quantum Wasserstein distance. © 2022 Elsevier Inc. LA - English DB - MTMT ER - TY - JOUR AU - Titkos, Tamás TI - A Tingley-sejtés JF - ÉRINTŐ : ELEKTRONIKUS MATEMATIKAI LAPOK J2 - ÉRINTŐ PY - 2022 IS - 26 SN - 2559-9275 UR - https://m2.mtmt.hu/api/publication/33578940 ID - 33578940 LA - Hungarian DB - MTMT ER - TY - JOUR AU - Kiss, Gergely AU - Titkos, Tamás TI - ISOMETRIC RIGIDITY OF WASSERSTEIN SPACES: THE GRAPH METRIC CASE JF - PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY J2 - P AM MATH SOC VL - 150 PY - 2022 IS - 9 SP - 4083 EP - 4097 PG - 15 SN - 0002-9939 DO - 10.1090/proc/15977 UR - https://m2.mtmt.hu/api/publication/33195355 ID - 33195355 N1 - Cited By :1 Export Date: 27 October 2022 Funding details: Magyar Tudományos Akadémia, MTA Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, K124749, K134944, LP2021-15/2021, PD128374 Funding text 1: Received by the editors September 29, 2021, and, in revised form, November 29, 2021. 2020 Mathematics Subject Classification. Primary 54E40, 46E27; Secondary 54E70, 05C12. Key words and phrases. Wasserstein space, graph metric space, isometry, isometric rigidity. The first author was supported by Premium Postdoctoral Fellowship of the Hungarian Academy of Sciences and by the Hungarian National Research, Development and Innovation Office - NKFIH (grant no. K124749). The second author was supported by the Hungarian National Research, Development and Innovation Office - NKFIH (grant no. PD128374 and grant no. K134944), by the János Bolyai Research Scholarship and the Momentum Program No. LP2021-15/2021 of the Hungarian Academy of Sciences, and by the ÚNKP-20-5-BGE-1 New National Excellence Program of the Ministry of Innovation and Technology. The second author is the corresponding author. AB - The aim of this paper is to prove that the p-Wasserstein space Wp(X) is isometrically rigid for all p ≥ 1 whenever X is a countable graph metric space. As a consequence, we obtain that for every countable group H and any p ≥ 1 there exists a p-Wasserstein space whose isometry group is isomorphic to H. © 2022 American Mathematical Society LA - English DB - MTMT ER - TY - JOUR AU - Gehér, György AU - Titkos, Tamás AU - Virosztek, Dániel TI - The isometry group of Wasserstein spaces: the Hilbertian case JF - JOURNAL OF THE LONDON MATHEMATICAL SOCIETY J2 - J LOND MATH SOC VL - 106 PY - 2022 IS - 4 SP - 3836 EP - 3894 PG - 59 SN - 0024-6107 DO - 10.1112/jlms.12676 UR - https://m2.mtmt.hu/api/publication/31855431 ID - 31855431 N1 - Department of Mathematics and Statistics, University of Reading, Reading, United Kingdom Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary BBS University of Applied Sciences, Budapest, Hungary Institute of Science and Technology Austria, Klosterneuburg, Austria Cited By :2 Export Date: 20 January 2023 Correspondence Address: Gehér, G.P.; Department of Mathematics and Statistics, Whiteknights, P.O. Box 220, United Kingdom; email: gehergyuri@gmail.com Funding details: Institute of Science and Technology Austria, ISTA Funding details: Leverhulme Trust, ECF‐2018‐125 Funding details: Magyar Tudományos Akadémia, MTA Funding details: Horizon 2020, 846294, K124152, KH129601, LP2021‐15/2021 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, K115383, K134944, PD128374 Funding text 1: This paper is based on discussions made during research visits at the Institute of Science and Technology (IST) Austria, Klosterneuburg. We are grateful to the Erdős group for the warm hospitality. We are also grateful to Lajos Molnár for his comments on an earlier version of the manuscript and to László Erdős for his suggestions on the structure and highlights of this paper. We thank the anonymous referee for his/her valuable comments on the manuscript. Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF-2018-125), and also by the Hungarian National Research, Development and Innovation Office - NKFIH (grant no. K115383 and K134944). Titkos was supported by the Hungarian National Research, Development and Innovation Office - NKFIH (grant no. PD128374, grant no. K115383 and K134944), by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP-20-5-BGE-1 New National Excellence Program of the Ministry of Innovation and Technology. Virosztek was supported by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 846294, by the Momentum program of the Hungarian Academy of Sciences under grant agreement no. LP2021-15/2021, and partially supported by the Hungarian National Research, Development and Innovation Office - NKFIH (grants no. K124152 and no. KH129601). Funding text 2: This paper is based on discussions made during research visits at the Institute of Science and Technology (IST) Austria, Klosterneuburg. We are grateful to the Erdős group for the warm hospitality. We are also grateful to Lajos Molnár for his comments on an earlier version of the manuscript and to László Erdős for his suggestions on the structure and highlights of this paper. We thank the anonymous referee for his/her valuable comments on the manuscript. Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF‐2018‐125), and also by the Hungarian National Research, Development and Innovation Office ‐ NKFIH (grant no. K115383 and K134944). Titkos was supported by the Hungarian National Research, Development and Innovation Office ‐ NKFIH (grant no. PD128374, grant no. K115383 and K134944), by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the ÚNKP‐20‐5‐BGE‐1 New National Excellence Program of the Ministry of Innovation and Technology. Virosztek was supported by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska‐Curie Grant Agreement No. 846294, by the Momentum program of the Hungarian Academy of Sciences under grant agreement no. LP2021‐15/2021, and partially supported by the Hungarian National Research, Development and Innovation Office ‐ NKFIH (grants no. K124152 and no. KH129601). AB - Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space (Formula presented.), we describe the isometry group (Formula presented.) for all parameters (Formula presented.) and for all separable real Hilbert spaces (Formula presented.). In particular, we show that (Formula presented.) is isometrically rigid for all Polish space (Formula presented.) whenever (Formula presented.). This is a consequence of our more general result: we prove that (Formula presented.) is isometrically rigid if (Formula presented.) 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 (Formula presented.), by solving Kloeckner's problem affirmatively on the existence of mass-splitting isometries. As a byproduct of our methods, we also obtain the isometric rigidity of (Formula presented.) for all complete and separable ultrametric spaces (Formula presented.) and parameters (Formula presented.). © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence. LA - English DB - MTMT ER - TY - JOUR AU - Tarcsay, Zsigmond AU - Titkos, Tamás TI - Operators on anti-dual pairs: Generalized Krein-von Neumann extension JF - MATHEMATISCHE NACHRICHTEN J2 - MATH NACHR VL - 294 PY - 2021 IS - 9 SP - 1821 EP - 1838 PG - 18 SN - 0025-584X DO - 10.1002/mana.201800431 UR - https://m2.mtmt.hu/api/publication/32163878 ID - 32163878 N1 - Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/c., Budapest, H-1117, Hungary Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, Budapest, H-1053, Hungary BBS University of Applied Sciences, Alkotmány u. 9, Budapest, H-1054, Hungary Export Date: 4 July 2022 Correspondence Address: Tarcsay, Z.; Department of Applied Analysis and Computational Mathematics, Pázmány Péter sétány 1/c., Hungary; email: tarcsay@cs.elte.hu AB - The main aim of this paper is to generalize the classical concept of a positive operator, and to develop a general extension theory, which overcomes not only the lack of a Hilbert space structure, but also the lack of a normable topology. The concept of anti-duality carries an adequate structure to define positivity in a natural way, and is still general enough to cover numerous important areas where the Hilbert space theory cannot be applied. Our running example - illustrating the applicability of the general setting to spaces bearing poor geometrical features - comes from noncommutative integration theory. Namely, representable extension of linear functionals of involutive algebras will be governed by their induced operators. The main theorem, to which the vast majority of the results is built, gives a complete and constructive characterization of those operators that admit a continuous positive extension to the whole space. Various properties such as commutation, or minimality and maximality of special extensions will be studied in detail. LA - English DB - MTMT ER - TY - JOUR AU - Tarcsay, Zsigmond AU - Titkos, Tamás TI - Operators on anti-dual pairs: Generalized Schur complement JF - LINEAR ALGEBRA AND ITS APPLICATIONS J2 - LINEAR ALGEBRA APPL VL - 614 PY - 2021 SP - 125 EP - 143 PG - 19 SN - 0024-3795 DO - 10.1016/j.laa.2020.02.031 UR - https://m2.mtmt.hu/api/publication/31203615 ID - 31203615 N1 - Közlésre elfogadva: 25-Feb-2020 Online megjelenés: 28-February-2020 AB - The goal of this paper is to develop the theory of Schur complementation in the context of operators acting on anti-dual pairs. As a byproduct, we obtain a natural generalization of the parallel sum and parallel difference, as well as the Lebesgue-type decomposition. To demonstrate how this operator approach works in application, we derive the corresponding results for operators acting on rigged Hilbert spaces, and for representable functionals of ⁎-algebras. LA - English DB - MTMT ER -