Analytic continuation and stability of operator semigroups (original) (raw)

On the characterization of eventually norm continuous semigroups in Hilbert spaces

Archiv der Mathematik, 1994

It is an old problem to characterize eventually norm continuous semigroups through growth conditions for their resolvent. Besides the Hille-Yosida theorem for strongly continuous semigroups, known characterizations exist for nilpotent (see [4, w differentiable, eventually differentiable and analytic semigroups (see Chapt. 2.4, 2.5

Operators and the norm continuity problem for semigroups

Journal of Functional Analysis, 2009

We present a new method for constructing C 0 -semigroups for which properties of the resolvent of the generator and continuity properties in operator topology are controlled simultaneously. It allows us to show that a) there exists a C 0 -semigroup continuous in operator topology for no t ∈ [0, 1] such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; b) there exists a C 0 -semigroup continuous in operator topology for no t ∈ R + such that the resolvent of its generator has a decay along vertical lines arbitrarily close to a logarithmic one. These examples rule out any possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent decay.

Norm continuity and related notions for semigroups on Banach spaces

Archiv der Mathematik, 1996

We find some conditions on a c 0 -semigroup on a Banach space and its resolvent connected with the norm continuity of the semigroup. We use them to get characterizations of norm continuous, eventually norm continuous and eventually compact semigroups on Hilbert spaces in terms of the growth of the resolvent of their generator. * 1980 Mathematics Subject Classification (1985. 47D05.

Stability of C 0-semigroups and geometry of Banach spaces

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

We obtain new stability conditions for C 0 -semigroups on Banach spaces having nontrivial Fourier type. On Hilbert spaces these conditions are sharp. For C 0semigroups on general Banach spaces, we prove a new individual stability criterion. We also show that stronger versions of the range stability condition from [4]arenot necessary for stability. This answers an open problem from [4].

Stability estimates for semigroups on Banach spaces

Discrete and Continuous Dynamical Systems, 2013

For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant M satisfying the inequality T (t) M e ωt , for all t 0, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjöstrand for the Hilbert space case to estimate M in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.

Operator Valued Fourier Multipliers and Stability of Strongly Continuous Semigroups

Integral Equations and Operator Theory, 2005

Stability for strongly continuous semigroups on Banach spaces is described in terms of L p-Fourier multiplier properties of the resolvent of the generator. A discrete version of this theory is developed, that lead to a description of the spectra of the semigroups in terms of discrete L p-Fourier multipliers. Applications to stability of linear control systems on Banach spaces are given.

On convergence and asymptotic behaviour of semigroups of operators

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020

The classical and modern theorems on convergence, approximation and asymptotic stability of semigroups of operators are presented, and their applications to recent biological models are discussed. This article is part of the theme issue ‘Semigroup applications everywhere’.

Some Result of Stability and Spectra Properties on Semigroup of Linear Operator

Advances in Pure Mathematics

This paper consists of some properties of a new subclass of semigroup of linear operator. The stability and spectra analysis of ω-order preserving partial contraction mapping (ω-OCP n) are obtained. The results show that operators on the proposed ω-OCP n are densely defined and closed. Several existing results in the literature are contained in this work.

Weak Stability for Orbits of C 0-semigroups on Banach Spaces

Functional Analysis and Evolution Equations, 2007

A result of Huang and van Neerven [11] establishes weak individual stability for orbits of C0-semigroups under boundedness assumptions on the local resolvent of the generator. We present an elementary proof for this using only the inverse Fourier-transform representation of the orbits of the semigroup in terms of the local resolvent.