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  .
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 and also determine the bounded linear operator A as the generators of one-parameter semigroups. Resolvent operators are particularly useful in the analysis of Sturm-Liouville operators and several others operators both bounded and unbounded.
Let X be a Banach space, be a finite set, the C0-semigroup which is strongly continuous one parameter semigroup of bounded linear operator in X, ω-OCPn be ω-order-preserving partial contraction mapping (semigroup of linear operator) which is an example of C0-semigroup. Similarly, let be a matrix, be a bounded linear operator on X, a partial transformation semigroup, a resolvent set, be spectrum and A is a generator of C0-semigroup.
This paper will focus on results of stability and spectra analysis of ω-OCPn on Banach space as an example of a semigroup of linear called C0-semigroup, and thereby establish the relationship between a semigroup, its generator and the resolvent as in Figure 1.
In  , 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.  . Chill and Tomilov  established some resolvent approach to stability operator semigroup. Räbiger and Wolf in  deduced some spectral and asymptotic properties of dominated operator. For relevant work on non-linear and one-parameter semigroups, see (  and  ). The aim of this work is, therefore, to obtain stability and spectra analysis on a new subclass of semigroup of linear operator.
The following definitions are crucial to the proof of our main results.
Definition 2.1: (Stable Semigroup  )
A strongly continuous semigroup is called
1) Uniformly exponentially stable if there exists such that
2) Uniformly stable if
3) Strongly stable if
Figure 1. Diagrammatical representation of relationship between a semigroup, its generator and its resolvent  .
Definition 2.2: (C0-Semigroup  )
A C0-Semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space.
Definition 2.3: (ω-OCPn  )
A transformation is called ω-order-preserving partial contraction mapping if and at least one of its transformation must satisfy such that whenever and otherwise for .
Definition 2.4: (Core  )
Let A be a closed linear operator with domain and range in a Banach space X. A subspace D of is called a core if A is the closure of its restriction to D.
Definition 2.5: (Resolvent Set  )
We define the resolvent set of A denoted by set of all such that is one-to-one with range equal to X.
Definition 2.6: (Spectrum  )
The spectrum of A denoted by is defined as the complement of the resolvent set.
Definition 2.7: (Hyperbolic  )
A semigroup on a Banach space X is called hyperbolic if X can be written as direct sum of two -invariant, closed subspaces , such that the restricted semigroups on and on satisfy the following conditions:
1) The semigroup is uniformly exponentially stable on .
2) The operator are invertible on , and is uniformly exponentially stable on .
Some Basic Spectral Properties
1) To any linear operator A we associate its spectral bound defined by
2) Resolvent set: .
3) Spectrum: .
4) Resolvent: .
5) Resolvent equation: .
and let , then
and let , then
matrix , we have
for each such that where is a resolvent set on X.
Suppose we have
and let , then
Let be the space of all bounded and uniformly continuous function from to , endowed with the sup-norm and let be defined by
For each and each , one may easily verify that satisfies the example 1 and 2 above.
3. Main Results
In this section, results of stability and spectral properties on ω-OCPn in Banach space and on C0-semigroup are considered:
Suppose X is a Banach space. Then a linear operator is an infinitesimal generator of a strongly continuous semigroup on X is uniformly exponentially stable if and only if for all one has
for all and .
If the semigroup is exponentially stable, then, the integral above is satisfied.
In order to show the converse implication, it suffices to verify that
So, we define for , the operators by
Then by assumption, the set is bounded for each , hence by the uniform boundedness principle, there exists such that
On the other hand, there exist and such that
From the previous two inequalities, we obtain
Hence, there exists a constant such that
Considering this, we conclude that
Hence the proof is complete.
Suppose X is a Banach space and where is the infinitesimal generator for a strongly continuous semigroup , then the following assertions are equivalent.
1) is hyperbolic.
2) for all .
The proof of implication 1) Þ 2) starts from the observation that because of the direct sum decomposition.
By assumption, is uniformly exponentially stable; hence for , and therefore
By the same argument, we obtain that . Suppose
we conclude that for each ; hence .
To prove 2) Þ 1), we fix such that and we use the existence at a spectral projection P corresponding to the spectral set
Then the space X is the direct sum of the -invariant subspaces and , where and . Then the restriction of T(s) has spectrum
hence, spectral radius . It follows that the semigroup is uniformly exponentially stable on .
Similarly, the restriction of in has spectrum
hence is invertible on . Clearly this implies that is invertible for , while for we choose such that . Then
hence is invertible, since is bijective.
Moreover, for the spectral radius, we have , and again this implies uniformly exponentially stable for the semigroup . Hence the proof.
Suppose and . Let be a linear operator which satisfies:
a) A is densely defined and closed; and
b) and for each , we have
1) for each ,
2) for each ,
3) for each and,
4) is the infinitesimal generator of a uniformly continuous semigroup satisfying
for each . In addition for each and , we have
Let and . Then we have
and as a result
for each .
Since is dense in X and
and from (3.12), we deduce
To show 2). Let us remark that we have successively
So, if , by 1), we have
which complete the proof of 2) and 3).
To show that for each . Since and , then by theorem of uniformly continuous semigroup, it follows that its generates a uniformly semigroup .
In order to show that , let us remark that, by virtue of for each and b), we have
Since , , and commute each to another for each and , we have
Hence the proof is complete.
For , we have to be a linear operator satisfying both and
for each and and if are regular values, i.e. and , then there exist:
3) for each and .
To prove 1), let us observe that
and this complete the proof of 1).
To prove 2), we assume for and let us define by
it’s obvious that
We want to prove that
for each .
So by resolvent Equation (3.17), we have
Consequently which proves (3.20).
From (3.19) and (3.20), we deduced that, for each and , we have
Passing to the Sup for on the left hand side of the inequality above, we now get for each . We can now define
Since 2) readily follows from (3.19), and 3) from (3.21) by taking , we have
Hence the proof.