Weak and Strong Approximation of Semigroups on Hilbert Spaces

  title={Weak and Strong Approximation of Semigroups on Hilbert Spaces},
  author={R. Chill and A. F. M. Elst},
  journal={Integral Equations and Operator Theory},
  • R. Chill, A. T. Elst
  • Published 31 August 2016
  • Mathematics
  • Integral Equations and Operator Theory
For a sequence of uniformly bounded, degenerate semigroups on a Hilbert space, we compare various types of convergences to a limit semigroup. Among others, we show that convergence of the semigroups, or of the resolvents of the generators, in the weak operator topology, in the strong operator topology or in certain integral norms are equivalent under certain natural assumptions which are frequently met in applications. 
On homogenization of the first initial-boundary value problem for periodic hyperbolic systems
ABSTRACT Let be a bounded domain of class . In , we consider a self-adjoint matrix strongly elliptic second-order differential operator , , with the Dirichlet boundary condition. The coefficients of
Feynman-Kac formula for perturbations of order $\leq 1$ and noncommutative geometry
Let Q be a differential operator of order ď 1 on a complex metric vector bundle E Ñ M with metric connection ∇ over a possibly noncompact Riemannian manifold M . Under very mild regularity
On homogenization of periodic hyperbolic systems in $L_2(\mathbb{R}^d;\mathbb{C}^n)$. Variations on the theme of the Trotter-Kato theorem
In L2(R d;Cn), we consider a matrix elliptic second order differential operator Bε > 0. Coefficients of the operator Bε are periodic with respect to some lattice in Rd and depend on x/ε. We study the
Variations on the theme of the Trotter-Kato theorem for homogenization of periodic hyperbolic systems
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $B_\varepsilon >0$. Coefficients of the operator $B_\varepsilon$ are periodic with respect to


A note on approximation of semigroups of contractions on Hilbert spaces
We show that every contractive C0-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C0-groups in the weak operator topology uniformly on compact subsets of
By a continuous degenerate semigroup we mean a strongly continuous mapping T : R+ ! L(X) having the semigroup property. Thus, T (0) is a projection which may be different from the identity. The main
In the Trotter-Kato approximation theorem for $C_0$-semigroups on Banach spaces, we replace the strong by the weak operator topology and discuss the validity of the relevant implications.
Convergence of sectorial operators on varying Hilbert space
Convergence of operators acting on a given Hilbert space is an old and well studied topic in operator theory. The idea of introducing a related notion for operators acting on arying spaces is
A canonical decomposition for quadratic forms with applications to monotone convergence theorems
We prove monotone convergence theorems for quadratic forms on a Hilbert space which improve existing results. The main tool is a canonical decomposition for any positive quadratic form h = hr + hs
Dirichlet problems on varying domains
Abstract The aim of the paper is to characterise sequences of domains for which solutions to an elliptic equation with Dirichlet boundary conditions converge to a solution of the corresponding
Persistence of bounded solutions of parabolic equations under domain perturbation
The aim of this paper is to study the behavior of bounded solutions of parabolic equations on the whole real line under perturbation of the underlying domain. We give the convergence of bounded
Perturbation of semi-linear evolution equations under weak assumptions at initial time
Abstract We prove perturbation results for abstract semi-linear evolution equations in a Banach space. The main feature is that only very weak assumptions are needed at initial time. This allows to
Vector-valued laplace transforms and cauchy problems
Linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms. The central tool is an “integrated version” of Widder’s theorem (characterizing Laplace
Characterization for the Kuratowski Limits of a Sequence of Sobolev Spaces
The purpose of this paper is to study the strong lower and the weak upper limits in the sense of Kuratowski of a sequence of Sobolev spaces {W1, p0(Ωn)}n∈Nand compare them to a fixed spaceW1, p0(Ω).