The aim of this paper is to give a characterization in Hilbert spaces of the generators of $C_0$-semigroups associated with closed, sectorial forms in terms of the convergence of a generalized Trotter's product formula. In the course of the proof of the main result we also present a similarity result which can be of independent interest: for any unbounded generator $A$ of a $C_0$-semigroup $e^{tA}$ it is possible to introduce an equivalent scalar product on the space, such that $e^{tA}$ becomes non-quasi-contractive with respect to the new scalar product.
The Lie-Trotter-Kato product formula has been recently extended into Hadamard spaces by [Stojkovic, Adv. Calc. Var., 2012]. The aim of our short note is to give a simpler proof relying upon weak convergence instead of an ultrapower technique.
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.
In this paper we prove a Lie-Trotter product formula for Markov semigroups in spaces of measures. We relate our results to "classical" results for strongly continuous linear semigroups on Banach spaces or Lipschitz semigroups in metric spaces and show that our approach is an extension of existing results. As Markov semigroups on measures are usually neither strongly continuous nor bounded linear operators for the relevant norms, we prove the convergence of the Lie-Trotter product formula assuming that the semigroups are locally equicontinuous and tight. A crucial tool we use in the proof is a …
In this paper we collect results concerning the {operator-norm} convergent {Trotter} product formula for two semigroups $\{\e^{- t A}\}_{t\geq 0}$, $\{\e^{- t B}\}_{t\geq 0}$, with densely defined generators $A$ and $B$ in a {Banach} space. Although the {strong} convergence in Banach space for contraction semigroups is known since the seminal paper by Trotter (1959), which after more than three decades was extended to convergence in the {operator-norm} topology in {Hilbert} spaces by Rogava (1993), the {operator-norm} convergence in a {Banach} space was established only in (2001).
Let $A$ and $B$ be positive semidefinite matrices. The limit of the expression $Z_p:=(A^{p/2}B^pA^{p/2})^{1/p}$ as $p$ tends to $0$ is given by the well known Lie-Trotter-Kato formula. A similar formula holds for the limit of $G_p:=(A^p\,\#\,B^p)^{2/p}$ as $p$ tends to $0$, where $X\,\#\,Y$ is the geometric mean of $X$ and $Y$. In this paper we study the complementary limit of $Z_p$ and $G_p$ as $p$ tends to $\infty$, with the ultimate goal of finding an explicit formula, which we call the anti Lie-Trotter formula. We show that the limit of $Z_p$ exists and find an explicit formula in a specia…
We constructed a unitary semigroup $(e^{tA})_{t \geq 0}$ on a Hilbert space and an orthogonal projection $P$ such that the limit $\lim_{n \to \infty} [ e^{\frac{t}{n}A}P ]^n$ does not exist strongly. A similar example with a positive contractive semigroup and positive contractive projection on $L_p$ is also constructed.
We extend Ando-Hiai's log-majorization for the weighted geometric mean of positive definite matrices into that for the Cartan barycenter in the general setting of probability measures on the Riemannian manifold of positive definite matrices equipped with trace metric. The main key is the settlement of the monotonicity problem of the Cartan barycenteric map on the space of probability measures with finite first moment for the stochastic order induced by the cone. We also derive a version of Lie-Trotter formula and related unitarily invariant norm inequalities for the Cartan barycenter as the ma…
It is shown that for a certain class of the Kato functions the Trotter-Kato product formulae converge in Dixmier ideal C 1,$\infty$ in topology, which is defined by the $\times$ 1,$\infty$-norm. Moreover, the rate of convergence in this topology inherits the error-bound estimate for the corresponding operator-norm convergence. 1 since [24], [14]. Note that a subtle point of this program is the question about the rate of convergence in the corresponding topology. Since the limit of the Trotter-Kato product formula is a strongly continuous semigroup, for the von Neumann-Schatten ideals this topo…
We consider the abstract Cauchy problem x'=Ax, x(0)=x_0\in D(A) for linear operators A on a Banach space X. We prove uniqueness of the (local) solution of this problem for a natural class of operators A. Moreover, we establish that the solution x(\cdot) can be represented as a limit of sequence F(t/n)^{n} as n\to\infty in the weak operator topology, where a function F:[0,\infty)\to L(X) satisfies F'(0)y=Ay, y\in D(A). As a consequence, we deduce necessary and sufficient conditions that a linear operator C is closable and its closure is a generator of C_0-semigroup. We also obtain some criteria…
We prove a Trotter product formula for gradient flows in metric spaces. This result is applied to establish convergence in the L^2-Wasserstein metric of the splitting method for some Fokker-Planck equations and porous medium type equations perturbed by a potential.
In this research project we presents the general properties, the spectral properties and the representation formulas for $C_0$-semigroups of linear operators in Banach spaces
A Trotter product formula is established for unitary quantum stochastic processes governed by quantum stochastic differential equations with constant bounded coefficients.
We develop new approach for studying the abstract Cauchy problem $\dot{x}=Ax$, $x(0)=x_0\in D(A)$ for linear operators $A$ defined on a locally convex space $X$. This approach was firstly introduced in the paper "Chernoff and Trotter type product formulas" to study the problem for Banach spaces. In this paper we not only generalize the results of the previous paper to more general topological spaces but also get new results for Banach spaces. In particular, we prove the "local" extension of Chernoff-Trotter-Kato type theorems. Applying this result, we prove Chernoff, Lie-Trotter and Trotter-Ka…
The paper is devoted to evolution equations of the form $\partial$ $\partial$t u(t) = --(A + B(t))u(t), t $\in$ I = [0, T ], on separable Hilbert spaces where A is a non-negative self-adjoint operator and B($\times$) is family of non-negative self-adjoint operators such that dom(A $α$) $\subseteq$ dom(B(t)) for some $α$ $\in$ [0, 1) and the map A --$α$ B($\times$)A --$α$ is H{ö}lder continuous with the H{ö}lder exponent $β$ $\in$ (0, 1). It is shown that the solution operator U(t, s) of the evolution equation can be approximated in the operator norm by a combination of semigroups generated by …
Let $A$ and $B$ be non-negative self-adjoint operators in a separable Hilbert space such that its form sum $C$ is densely defined. It is shown that the Trotter product formula holds for imaginary times in the $L^2$-norm, that is, one has % % \begin{displaymath} \lim_{n\to+\infty}\int^T_0 \|(e^{-itA/n}e^{-itB/n})^nh - e^{-itC}h\|^2dt = 0 \end{displaymath} % % for any element $h$ of the Hilbert space and any $T > 0$. The result remains true for the Trotter-Kato product formula % % \begin{displaymath} \lim_{n\to+\infty}\int^T_0 \|(f(itA/n)g(itB/n))^nh - e^{-itC}h\|^2dt = 0 \end{displaymath} % % w…
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 co…
A natural counterpart to the Lie-Trotter product formula for norm-continuous one-parameter semigroups is proved, for the class of quasicontractive quantum stochastic operator cocycles whose expectation semigroup is norm continuous. Compared to previous such results, the assumption of a strong form of independence of the constituent cocycles is overcome. The analysis is facilitated by the development of some quantum Ito algebra. It is also shown how the maximal Gaussian component of a quantum stochastic generator may be extracted - leading to a canonical decomposition of such genarators, and th…
In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space L^p(I,X), consisting of X-valued functions on the time-interval I. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in L^p(I,X). We show that the latter also allows to apply a full power of the operator-theoretical metho…
In this paper, we introduce formulations of the Trotter Kato theorem for approximation of bi continuous semigroups that provide a useful framework whenever convergence of numerical approximations to solutions of PDEs are studied with respect to an additional locally convex topology coarser than the norm topology to treat the lack of the strong continuity. Applicability of our results is demonstrated using a heat equation with infinite boundaries.
For any nonnegative self-adjoint operators A and B in a separable Hilbert space, we show that the Trotter-type formula $[(e^{i2tA/n}+e^{i2tB/n})/2]^n$ converges strongly in the closure of the intersection of the domains of A^{1/2} and B^{1/2}, along some subsequence and for almost every real number t. This result extends to the degenerate case and to Kato-functions following the method of Kato.
A new sufficient condition is given for the sum of linear m-accretive operator and accretive operator one in a Hilbert space to be m-accretive. As an application, an extended result to the operator-norm error bound estimate for the exponential Trotter-Kato product formula is given.
A basic result in semigroup theory states that every $C_0$-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula $(e^{\frac{t}{n}A}P)^n$ (where $P$ denotes a bounded projection), we prove that whenever the generator $A$ is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is {\it{not}} quasi-contractive.
Operator splitting for dissipative delay equations
Eötvös Loránd University（罗兰大学）；MTA-ELTE Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences（匈牙利科学院MTA-ELTE数值分析与大型网络研究组）；Bergische Universität Wuppertal（伍珀塔尔大学）
