The Experts below are selected from a list of 27846 Experts worldwide ranked by ideXlab platform
Valentin A. Zagrebnov - One of the best experts on this subject based on the ideXlab platform.
-
Operator-Norm Convergence of the Trotter Product Formula on Hilbert and Banach Spaces: A Short Survey
Current Research in Nonlinear Analysis, 2018Co-Authors: Hagen Neidhardt, Artur Stephan, Valentin A. ZagrebnovAbstract:We give a review of results on the Operator-Norm convergence of the Trotter product formula on Hilbert and Banach spaces, which is focused on the problem of its convergence rates. Some recent results concerning evolution semigroups are presented in details.
-
Remarks on the Operator-Norm convergence of the Trotter product formula
arXiv: Mathematical Physics, 2017Co-Authors: Hagen Neidhardt, Artur Stephan, Valentin A. ZagrebnovAbstract:We revise the Operator-Norm convergence of the Trotter product formula for a pair {A,B} of generators of semigroups on a Banach space. Operator-Norm convergence holds true if the dominating Operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded Operator. Inspired by studies of evolution semigroups it is shown in the present paper that the Operator-Norm convergence generally fails even for bounded Operators B if A is not a holomorphic generator. Moreover, it is shown that Operator Norm convergence of the Trotter product formula can be arbitrary slow.
-
Operator-Norm Approximation of Semigroups by Quasi-sectorial Contractions
Journal of Functional Analysis, 2001Co-Authors: Vincent Cachia, Valentin A. ZagrebnovAbstract:Abstract We extend the Chernoff theory of approximation of contraction semigroups a la Trotter. We show that the Trotter–Neveu–Kato convergence theorem holds in Operator Norm for a family of uniformly m-sectorial generators in a Hilbert space. Then we obtain a Chernoff-type approximation theorem for quasi-sectorial contractions on a Hilbert space in the Operator Norm. We give necessary and sufficient conditions for the Operator-Norm convergence of Trotter-type product formulae.
-
Operator-Norm CONVERGENCE OF THE TROTTER PRODUCT FORMULA FOR HOLOMORPHIC SEMIGROUPS
2001Co-Authors: Vincent Cachia, Valentin A. ZagrebnovAbstract:We study the error estimates in Operator Norm for the Trotter product formula. It is shown that some recent results in this direction can be extended to non-self-adjoint generators and to the case of a Banach space for a class of holomorphic semigroups.
-
trotter kato product formula and Operator Norm convergence
Communications in Mathematical Physics, 1999Co-Authors: Hagen Neidhardt, Valentin A. ZagrebnovAbstract:We find necessary and sufficient conditions for the Operator-Norm convergence of the Trotter–Kato product formula. Using them we prove that this convergence takes place: (i) if the resolvent of one of the involved Operators is compact, either (ii) if one Operator is relatively compact with respect to another one, or (iii) if the product of resolvents of the involved Operators is compact.
V.a. Zagrebnov - One of the best experts on this subject based on the ideXlab platform.
-
Approximations of self-adjoint \(C_0\)-semigroups in the Operator-Norm topology
Annales Universitatis Mariae Curie-Sklodowska sectio A – Mathematica, 2020Co-Authors: V.a. ZagrebnovAbstract:The paper improves approximation theory based on the Trotter–Kato product formulae. For self-adjoint \(C_0\)-semigroups we develop a lifting of the strongly convergent Chernoff approximation (or product) formula to convergence in the Operator-Norm topology. This allows to obtain optimal estimate for the rate of Operator-Norm convergence of Trotter–Kato product formulae for Kato functions from the class \(K_2\).
-
Notes on the Chernoff Product Formula
2019Co-Authors: V.a. ZagrebnovAbstract:We revise the strong convergent Chernoff product formula and extend it, in a Hilbert space, to convergence in the Operator-Norm topology. Main results deal with the self-adjoint Chernoff product formula. The nonself-adjoint case concerns the quasi-sectorial contractions.
-
Trotter–Kato Product Formulae in Normed Ideals
2019Co-Authors: V.a. ZagrebnovAbstract:We show that for a certain class of Kato functions the Trotter–Kato product formulae converge in symmetrically-Normed ideals of compact Operators on a separable Hilbert space. The rate of convergence in topology of ideals inherits the error-bound estimate for the corresponding Operator-Norm convergence.
-
Comments on the Chernoff √ n-Lemma
2017Co-Authors: V.a. ZagrebnovAbstract:The Chernoff √ n-Lemma is revised. This concerns two aspects: an improvement of the Chernoff estimate in the strong Operator topol-ogy and an Operator-Norm estimate for quasi-sectorial contractions. Applications to the Lie-Trotter product formula approximation for semigroups is presented.
-
On convergence rate estimates for approximations of solution Operators of linear non-autonomous evolution equations
Nanosystems: Physics Chemistry Mathematics, 2017Co-Authors: Hagen Neidhardt, Artur Stephan, V.a. ZagrebnovAbstract:We improve some recent convergence rate estimates for approximations of solution Operators of linear non-autonomous evolution equations. The approximation results from the Trotter product formula which is proved to converge in the Operator-Norm and its convergence can be estimated. The result is applied to a diffusion perturbed by a time-dependent potential.
Vincent Cachia - One of the best experts on this subject based on the ideXlab platform.
-
Operator-Norm Approximation of Semigroups by Quasi-sectorial Contractions
Journal of Functional Analysis, 2001Co-Authors: Vincent Cachia, Valentin A. ZagrebnovAbstract:Abstract We extend the Chernoff theory of approximation of contraction semigroups a la Trotter. We show that the Trotter–Neveu–Kato convergence theorem holds in Operator Norm for a family of uniformly m-sectorial generators in a Hilbert space. Then we obtain a Chernoff-type approximation theorem for quasi-sectorial contractions on a Hilbert space in the Operator Norm. We give necessary and sufficient conditions for the Operator-Norm convergence of Trotter-type product formulae.
-
Operator-Norm CONVERGENCE OF THE TROTTER PRODUCT FORMULA FOR HOLOMORPHIC SEMIGROUPS
2001Co-Authors: Vincent Cachia, Valentin A. ZagrebnovAbstract:We study the error estimates in Operator Norm for the Trotter product formula. It is shown that some recent results in this direction can be extended to non-self-adjoint generators and to the case of a Banach space for a class of holomorphic semigroups.
-
Operator-Norm Convergence of the Trotter Product Formula for Sectorial Generators
Letters in Mathematical Physics, 1999Co-Authors: Vincent Cachia, Valentin A. ZagrebnovAbstract:The Operator-Norm convergence of the Trotter product formula is known for self-adjoint semigroups with compactness or smallness conditions on the generators involved in this formula. We generalize these two types of results to sectorial generators.
Hagen Neidhardt - One of the best experts on this subject based on the ideXlab platform.
-
Operator-Norm Convergence of the Trotter Product Formula on Hilbert and Banach Spaces: A Short Survey
Current Research in Nonlinear Analysis, 2018Co-Authors: Hagen Neidhardt, Artur Stephan, Valentin A. ZagrebnovAbstract:We give a review of results on the Operator-Norm convergence of the Trotter product formula on Hilbert and Banach spaces, which is focused on the problem of its convergence rates. Some recent results concerning evolution semigroups are presented in details.
-
On convergence rate estimates for approximations of solution Operators of linear non-autonomous evolution equations
Nanosystems: Physics Chemistry Mathematics, 2017Co-Authors: Hagen Neidhardt, Artur Stephan, V.a. ZagrebnovAbstract:We improve some recent convergence rate estimates for approximations of solution Operators of linear non-autonomous evolution equations. The approximation results from the Trotter product formula which is proved to converge in the Operator-Norm and its convergence can be estimated. The result is applied to a diffusion perturbed by a time-dependent potential.
-
Remarks on the Operator-Norm convergence of the Trotter product formula
arXiv: Mathematical Physics, 2017Co-Authors: Hagen Neidhardt, Artur Stephan, Valentin A. ZagrebnovAbstract:We revise the Operator-Norm convergence of the Trotter product formula for a pair {A,B} of generators of semigroups on a Banach space. Operator-Norm convergence holds true if the dominating Operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded Operator. Inspired by studies of evolution semigroups it is shown in the present paper that the Operator-Norm convergence generally fails even for bounded Operators B if A is not a holomorphic generator. Moreover, it is shown that Operator Norm convergence of the Trotter product formula can be arbitrary slow.
-
trotter kato product formula and Operator Norm convergence
Communications in Mathematical Physics, 1999Co-Authors: Hagen Neidhardt, Valentin A. ZagrebnovAbstract:We find necessary and sufficient conditions for the Operator-Norm convergence of the Trotter–Kato product formula. Using them we prove that this convergence takes place: (i) if the resolvent of one of the involved Operators is compact, either (ii) if one Operator is relatively compact with respect to another one, or (iii) if the product of resolvents of the involved Operators is compact.
Pierre Pudlo - One of the best experts on this subject based on the ideXlab platform.
-
Operator Norm Convergence of Spectral Clustering on Level Sets
Journal of Machine Learning Research, 2010Co-Authors: Bruno Pelletier, Pierre PudloAbstract:Following Hartigan (1975), a cluster is defined as a connected component of the t-level set of the underlying density, that is, the set of points for which the density is greater than t. A clustering algorithm which combines a density estimate with spectral clustering techniques is proposed. Our algorithm is composed of two steps. First, a nonparametric density estimate is used to extract the data points for which the estimated density takes a value greater than t. Next, the extracted points are clustered based on the eigenvectors of a graph Laplacian matrix. Under mild assumptions, we prove the almost sure convergence in Operator Norm of the empirical graph Laplacian Operator associated with the algorithm. Furthermore, we give the typical behavior of the representation of the data set into the feature space, which establishes the strong consistency of our proposed algorithm.
-
Operator Norm convergence of spectral clustering on level sets
arXiv: Machine Learning, 2010Co-Authors: Bruno Pelletier, Pierre PudloAbstract:Following Hartigan, a cluster is defined as a connected component of the t-level set of the underlying density, i.e., the set of points for which the density is greater than t. A clustering algorithm which combines a density estimate with spectral clustering techniques is proposed. Our algorithm is composed of two steps. First, a nonparametric density estimate is used to extract the data points for which the estimated density takes a value greater than t. Next, the extracted points are clustered based on the eigenvectors of a graph Laplacian matrix. Under mild assumptions, we prove the almost sure convergence in Operator Norm of the empirical graph Laplacian Operator associated with the algorithm. Furthermore, we give the typical behavior of the representation of the dataset into the feature space, which establishes the strong consistency of our proposed algorithm.
