Suppose X is a Banach space, a finite set, the C0-semigroup that is strongly continuous one-parameter semigroup of bounded linear operator in X. Let ω-OCPn be ω-order-preserving partial contraction mapping (semigroup of linear operator) which is an example of C0-semigroup. Furthermore, let be a matrix, a bounded linear operator on X, a partial transformation semigroup, a resolvent set, a duality mapping on X and A is a generator of C0-semigroup. Taking the importance of the dissipative operator in a semigroup of linear operators into cognizance, dissipative properties characterized the generator of a semigroup of linear operator which does not require the explicit knowledge of the resolvent.
This paper will focus on results of dissipative operator on ω-OCPn on Banach space as an example of a semigroup of linear operator called C0-semigroup.
Yosida  proved some results on differentiability and representation of one-parameter semigroup of linear operators. Miyadera  , generated some strongly continuous semigroups of operators. Feller  , also obtained an unbounded semigroup of bounded linear operators. Balakrishnan  introduced fractional powers of closed operators and semigroups generated by them. Lumer and Phillips  , established dissipative operators in a Banach space and Hille & Philips  emphasized the theory required in the inclusion of an elaborate introduction to modern functional analysis with special emphasis on functional theory in Banach spaces and algebras. Batty  obtained asymptotic behaviour of semigroup of operator in Banach space. More relevant work and results on dissipative properties of ω-Order preserving partial contraction mapping in semigroup of linear operator could be seen in Engel and Nagel  , Vrabie  , Laradji and Umar  , Rauf and Akinyele  and Rauf et al.  .
Definition 2.1 (C0-Semigroup) 
C0-Semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space.
Definition 2.2 (ω-OCPn) 
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 2.3 (Subspace Semigroup) 
A subspace semigroup is the part of A in Y which is the operator defined by with domain .
Definition 2.4 (Duality set)
Let X be a Banach space, for every , a nonempty set defined by is called the duality set.
Definition 2.5 (Dissipative) 
A linear operator is dissipative if each , there exists such that .
2.1. Properties of Dissipative Operator
For dissipative operator , the following properties hold:
a) is injective for all and
for all y in the range .
b) is surjective for some if and only if it is surjective for each . In that case, we have , where is the resolvent of the generator A.
c) A is closed if and only if the range is closed for some .
d) If , that is if A is densely defined, then A is closable. its closure A is again dissipative and satisfies for all .
and let , then
and let , then
In any matrix , and for each such that where is a resolvent set on X.
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 , it is easily verified that satisfies Examples 1 and 2 above.
Let and consider the operator with domain . It is a closed operator whose domain is not dense. However, it is dissipative, since its resolvent can be computed explicitly as
for , . Moreover, for all . Therefore is dissipative.
2.2. Theorem (Hille-Yoshida  )
A linear operator is the infinitesimal generator for a C0-semigroup of contraction if and only if
1) A is densely defined and closed,
2) and for each
2.3. Theorem (Lumer-Phillips  )
Let X be a real, or complex Banach space with norm , and let us recall that the duality mapping is defined by
for each . In view of Hahn-Banach theorem, it follows that, for each , is nonempty.
2.4. Theorem (Hahn-Banach Theorem  )
Let V be a real vector space. Suppose is mapping satisfying the following conditions:
2) for all and real of ; and
3) for every .
Assume, furthermore that for each , either both and are or that both are finite.
3. Main Results
In this section, dissipative results on ω-OCPn as a semigroup of linear operator were established and the research results(Theorems) were given and proved appropriately:
Let where is a dissipative operator on a Banach space X such that is surjective for some . Then
1) the part A, of A in the subspace is densely defined and generates a constrain semigroup in , and
2) considering X to be a reflexive, A is densely defined and generates a contraction semigroup.
We recall from Definition 2.3 that
Since exists for , this implies that , hence
we need to show that is dense in .
Take and set . Then and
since . Therefore the operators converge pointwise on
to the identity. Since for all , we obtain the convergence of for all . If for each in , the density of in is shown which proved (i).
To prove (ii), we need to obtain the density of .
Let and define . The element , also belongs to . Moreover, by the proof of (i) the operators converges towards the identity pointwise on . It follows that
Since X is reflexive and is bounded, there exists a subsequence, still denoted by , that converges weakly to some . Since , implies that .
On the other hand, the elements converges weakly to z, so the weak closedness of A implies that and which proved (ii).
The linear operator is a dissipative if and only if for each and , where , then we have
Suppose A is dissipative, then, for each and , there exists such that . Therefore
and this completes the proof. Next, let and .
Let and let us observe that, by virtue of (3.3), .
So, in this case, we clearly have Therefore, by assuming that . As a consequence, , and thus
lies on the unit ball, i.e. . We have hence
and . Now, let us recall that the closed unit ball in
is weakly-star compact. Thus, the net has at least one weak-star cluster point with
From (3.4), it follows that and . Since , it follows that . Hence and and this completes the proof.
Let be infinitesimal generator of a C0-semigroup of contraction and . Suppose is endowed with the graph-norm defined by for . Then operator defined by
is the infinitesimal generator of a C0-semigroup of contractions on .
Let and and let us consider the equation Since A generates a C0-semigroup of contraction  , it follows that this equation has a unique solution .
Since , we conclude that and thus .
Thus . On the other hand, we have
which shows that satisfies condition (ii) in Theorem 2.2. Moreover, it follows that is closed in .
Indeed, as , it is closed, and consequently enjoys the same property which proves that is closed.
Now, let , , and let . Clearly , and in addition Thus, is dense in by virtue of Theorem 2.2, generates a C0-semigroup of contraction on . Hence the proof.
In this paper, it has been established that ω-OCPn possesses the properties of dissipative operators as a semigroup of linear operator, and obtaining some dissipative results on ω-OCPn.
 Yosida, K. (1948) On the Differentiability and Representation of One-Parameter Semigroups of Linear Operators. Journal of the Mathematical Society of Japan, 1, 15-21.
 Rauf, K. and Akinyele, A.Y. (2019) Properties of ω-Order-Preserving Partial Contraction Mapping and Its Relation to C0-Semigroup. International Journal of Mathematics and Computer Science, 14, 61-68.
 Rauf, K., Akinyele, A.Y., Etuk, M.O., Zubair, R.O. and Aasa, M.A. (2019) Some Results of Stability and Spectra Properties on Semigroupn of Linear Operator. Advances of Pure Mathematics, 9, 43-51.