@article{13177,
  abstract     = {In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds cannot be established.},
  author       = {Hua, Bobo and Keller, Matthias and Schwarz, Michael and Wirth, Melchior},
  issn         = {1088-6826},
  journal      = {Proceedings of the American Mathematical Society},
  number       = {8},
  pages        = {3401--3414},
  publisher    = {American Mathematical Society},
  title        = {{Sobolev-type inequalities and eigenvalue growth on graphs with finite measure}},
  doi          = {10.1090/proc/14361},
  volume       = {151},
  year         = {2023},
}

@article{13319,
  abstract     = {We prove that the generator of the L2 implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule, extending earlier results by Cipriani and Sauvageot for tracially symmetric semigroups and the second-named author for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markov operators to symmetric Markov operators and is essential to obtain the required inner product on the Hilbert bimodule.},
  author       = {Vernooij, Matthijs and Wirth, Melchior},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  pages        = {381--416},
  publisher    = {Springer Nature},
  title        = {{Derivations and KMS-symmetric quantum Markov semigroups}},
  doi          = {10.1007/s00220-023-04795-6},
  volume       = {403},
  year         = {2023},
}

@article{12087,
  abstract     = {Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet–Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz–Schur multipliers over group algebras and generalized depolarizing semigroups.},
  author       = {Wirth, Melchior and Zhang, Haonan},
  issn         = {1424-0637},
  journal      = {Annales Henri Poincare},
  pages        = {717--750},
  publisher    = {Springer Nature},
  title        = {{Curvature-dimension conditions for symmetric quantum Markov semigroups}},
  doi          = {10.1007/s00023-022-01220-x},
  volume       = {24},
  year         = {2023},
}

@article{12104,
  abstract     = {We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.},
  author       = {Dello Schiavo, Lorenzo and Wirth, Melchior},
  issn         = {1424-3202},
  journal      = {Journal of Evolution Equations},
  number       = {1},
  publisher    = {Springer Nature},
  title        = {{Ergodic decompositions of Dirichlet forms under order isomorphisms}},
  doi          = {10.1007/s00028-022-00859-7},
  volume       = {23},
  year         = {2023},
}

@article{11330,
  abstract     = {In this article we study the noncommutative transport distance introduced by Carlen and Maas and its entropic regularization defined by Becker and Li. We prove a duality formula that can be understood as a quantum version of the dual Benamou–Brenier formulation of the Wasserstein distance in terms of subsolutions of a Hamilton–Jacobi–Bellmann equation.},
  author       = {Wirth, Melchior},
  issn         = {15729613},
  journal      = {Journal of Statistical Physics},
  number       = {2},
  publisher    = {Springer Nature},
  title        = {{A dual formula for the noncommutative transport distance}},
  doi          = {10.1007/s10955-022-02911-9},
  volume       = {187},
  year         = {2022},
}

@article{11916,
  abstract     = {A domain is called Kac regular for a quadratic form on L2 if every functions vanishing almost everywhere outside the domain can be approximated in form norm by functions with compact support in the domain. It is shown that this notion is stable under domination of quadratic forms. As applications measure perturbations of quasi-regular Dirichlet forms, Cheeger energies on metric measure spaces and Schrödinger operators on manifolds are studied. Along the way a characterization of the Sobolev space with Dirichlet boundary conditions on domains in infinitesimally Riemannian metric measure spaces is obtained.},
  author       = {Wirth, Melchior},
  issn         = {2538-225X},
  journal      = {Advances in Operator Theory},
  keywords     = {Algebra and Number Theory, Analysis},
  number       = {3},
  publisher    = {Springer Nature},
  title        = {{Kac regularity and domination of quadratic forms}},
  doi          = {10.1007/s43036-022-00199-w},
  volume       = {7},
  year         = {2022},
}

@article{9627,
  abstract     = {We compute the deficiency spaces of operators of the form 𝐻𝐴⊗̂ 𝐼+𝐼⊗̂ 𝐻𝐵, for symmetric 𝐻𝐴 and self-adjoint 𝐻𝐵. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already in Ibort et al. [Boundary dynamics driven entanglement, J. Phys. A: Math. Theor. 47(38) (2014) 385301], but only proven under the restriction of 𝐻𝐵 having discrete, non-degenerate spectrum.},
  author       = {Lenz, Daniel and Weinmann, Timon and Wirth, Melchior},
  issn         = {1464-3839},
  journal      = {Proceedings of the Edinburgh Mathematical Society},
  number       = {3},
  pages        = {443--447},
  publisher    = {Cambridge University Press},
  title        = {{Self-adjoint extensions of bipartite Hamiltonians}},
  doi          = {10.1017/S0013091521000080},
  volume       = {64},
  year         = {2021},
}

@article{9973,
  abstract     = {In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.},
  author       = {Wirth, Melchior and Zhang, Haonan},
  issn         = {1432-0916},
  journal      = {Communications in Mathematical Physics},
  keywords     = {Mathematical Physics, Statistical and Nonlinear Physics},
  pages        = {761–791},
  publisher    = {Springer Nature},
  title        = {{Complete gradient estimates of quantum Markov semigroups}},
  doi          = {10.1007/s00220-021-04199-4},
  volume       = {387},
  year         = {2021},
}