Local spectra and individual stability of uniformly bounded 𝐶₀-semigroups (original) (raw)

Asymptotic Behaviour ofC0-Semigroups with Bounded Local Resolvents

Mathematische Nachrichten, 2000

Let {T (t)} t≥0 be a C 0 -semigroup on a Banach space X with generator A, and let H ∞ T be the space of all x ∈ X such that the local resolvent λ → R(λ, A)x has a bounded holomorphic extension to the right half-plane. For the class of integrable functions φ on [0, ∞) whose Fourier transforms are integrable, we construct a functional calculus φ → T φ , as operators on H ∞ T . We show that each orbit T (·)T φ x is bounded and uniformly continuous, and T (t)T φ x → 0 weakly as t → ∞, and we give a new proof that T (t)R(µ, A)x = O(t). We also show that T (t)T φ x → 0 when T is sun-reflexive, and that T (t)R(µ, A)x = O(ln t) when T is a positive semigroup on a normal ordered space X and x is a positive vector in H ∞ T .

Asymptotic behavior of discrete and continuous semigroups on Hilbert spaces

2008

Let A : [0, ∞) → [0, ∞) be a nondecreasing function with A(t) > 0 for all t > 0, H be a complex Hilbert space and let T be a bounded linear operator acting on H. Among our results is the fact that T is power stable (i.e. its spectral radius is less than 1) if 1 X n=0 A(|h T n x, xi| ) < ∞ for all x ∈ H with ||x|| ≤ 1. In the continuous case we prove that a strongly continuous uniformly bounded semigroup of operators acting on a Hilbert space H is spectrally stable (i.e. the spectrum of its infinitesimal generator lies in the open left half plane) if and only if for each x ∈ H and each µ ∈ R one has: sup s¸0 þ þ Z s 0 e iiµt h T (t)x, xi dt þ þ < ∞.

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.

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].

A folk theorem in the spectral theory of C 0 -semigroups

Pacific Journal of Mathematics, 1984

If A is the infinitesimal generator of a C 0-semigroup T(t\ a classical theorem of Hille and Phillips relates the point spectrum of A and that of F(£) for £ > 0. Specifically, if μ is in the point spectrum of Γ(£) and μ Φ 0, then there exists <x 0 in the point spectrum of A with exp(ξα 0) = μ and the null space of μ-T(ξ) is the closed linear span of the null spaces of a n-A for a n = a 0 4-2πinξ~] and n ranging over the integers. In this note we shall extend the Hille-Phillips theorem by proving that the null space of (μ-T(ξ)) k is the closed linear span of the null spaces of (a n-A) k as n ranges over the integers. Such a result is useful in relating the order of poles of the resolvent of A and the order of poles of the resolvent of Γ(£), and as an example we shall give an application to the theory of positive (in the sense of cone-preserving) linear operators.

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.

Weakly and almost weakly stable C0-semigroups

International Journal of Dynamical Systems and Differential Equations, 2007

In this article we survey results concerning asymptotic properties of C0-semigroups on Banach spaces with respect to the weak operator topology. The property "no eigenvalues of the generator on the imaginary axis" is equivalent to weak stability for most time values; a phenomenon called "almost weak stability". Further, sufficient conditions implying the actual weak stability of a C0-semigroup are also given. The individual stability versions are considered as well, using both boundedness of the local resolvent and integrability conditions for the resolvent operator. By a number of examples we explain weak and almost weak stability and illustrate the fundamental difference between them. Where adequate, connections to strong stability are observed, while many historical and bibliographical remarks position the material in the literature. We conclude the paper with some open questions and comments.

A spectral mapping theorem for perturbed strongly continuous semigroups

Mathematische Annalen, 1991

We consider a strongly continuous semigroup (T(t)) t 0 with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup (S(t)) t 0 generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t)) t 0 . The results are applied to a cell equation with age structure and spatial distribution.

Tauberian theorems and stability of one-parameter semigroups

american mathematical society, 1988

The main result is the following stability theorem: Let T = (T(f))(>0 be a bounded Co-semigroup on a reflexive space X. Denote by A the generator of T and by a(A) the spectrum of A. If c(A) n ¿R is countable and no eigenvalue of A lies on the imaginary axis, then limt^oo T(t)x = 0 for all x e X.