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

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.


Introduction
The theory of stability is important since stability plays a central role in the structural theory of operators such as semigroup of linear operator, contraction semigroup, invariant subspace theory and to mention but few. The theory of stability is rich in which concerns the methods and ideas, and this shall be one of the main points of this paper. The recent advances deeply interact with modern topics from complex function theory, harmonic analysis, the geometry of Banach spaces, and spectra theory [1].
Another main focus of this paper is spectra analysis of a semigroup of linear operator, in which we use the resolvent to describe the relationship between the spectrum of A and of the semigroup operator ator in X, ω-OCP n be ω-order-preserving partial contraction mapping (semigroup of linear operator) which is an example of C 0 -semigroup. Similarly, let ( ) Mm  be a matrix, ( ) L X be a bounded linear operator on X, n P a partial transformation semigroup, ( ) A ρ a resolvent set, ( ) A σ be spectrum and A is a generator of C 0 -semigroup. This paper will focus on results of stability and spectra analysis of ω-OCP n on Banach space as an example of a semigroup of linear called C 0 -semigroup, and thereby establish the relationship between a semigroup, its generator and the resolvent as in Figure 1.
In [2], Batty obtained some spectral conditions for stability of one-parameter semigroup and also revealed some asymptotic behaviour of semigroup of operator, see also, Batty et al. [3]. Chill and Tomilov [4] established some resolvent approach to stability operator semigroup. Räbiger and Wolf in [5] deduced some spectral and asymptotic properties of dominated operator. For relevant work on non-linear and one-parameter semigroups, see ( [6] and [7]). The aim of this work is, therefore, to obtain stability and spectra analysis on a new subclass of semigroup of linear operator.

Preliminaries
The following definitions are crucial to the proof of our main results.
on u X satisfy the following conditions: is uniformly exponentially stable on s X .
2) The operator is uniformly exponentially stable on u X .

Some Basic Spectral Properties
1) To any linear operator A we associate its spectral bound defined by e e e I e e is a resolvent set on X.
Suppose we have For each f X ∈ and each , t s + ∈  , one may easily verify that satisfies the example 1 and 2 above.

Main Results
In this section, results of stability and spectral properties on ω-OCP n in Banach space and on C 0 -semigroup are considered:

Proof
If the semigroup is exponentially stable, then, the integral above is satisfied.
In order to show the converse implication, it suffices to verify that Considering this, we conclude that

Proof
The proof of implication 1) ⇒ 2) starts from the observation that we conclude that 1 and we use the existence at a spectral projection P corresponding to the spectral set Then the space X is the direct sum Similarly, the restriction hence is invertible on u X . Clearly this implies that and as a result which complete the proof of 2) and 3).
To show that In order to show that for each x X ∈ and 0 λ > .

Proof
To prove 1), let us observe that and this complete the proof of 1).
To prove 2), we assume for 0 µ > and let us define