Ijeas0312025

International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-12, December 2016

Results On Strongly Continuous Semigroup Radhi.Ali Zaboon, Arwa Nazar Mustafa, Rafah Alaa Abdulrazzaq  Abstract—Some results of strongly continuous semigroup

topologically transitive [1], and it's said to be a topologically

( -semigroup) of functional analysis of bounded linear operator defining on a separable Banach spaces like hypercyclic, topologically transitive, chaotic, mixing, weekly mixing and topologically ergrodic have been discussed. The generalization of hypercyclic, topologically transitive, chaotic, mixing, weakly mixing and topologically ergrodic for the direct sum of two and

ergodic if for every non-empty open subsets

/or hence to n -semigroup strongly continuous dynamic system of semigroups in infinite dimensional separable Banach space have been developed with proofs and discusses

periodic point of

of X,

the set is syndetic, that is

does not contains

arbitrarily long intervals[2]. A point

is called a

if there is some

such that

[3]. A -semigroup on X is called chaotic if it is hypercyclic and its set of periodic points is dense in X. Note that if spaces, then

Index Terms—semigroup, ergrodic, dynamic system.

are separable Banach the space is a Separable

I. INTRODUCTION Banach space, and if Let

be a separable infinite dimensional Banach space. A

one-parameter family of continuous (bounded) linear operators on X is a strongly continuous semigroup (

-semigroups)

if

,

-semigroup

for

every

.

of

and

there exists some

there

-semigroup exists

orb

by

II. SOME PROPERTIES THAT PRESERVED UNDER QUASICONJUAGACY:

of X, Definition(2.1) [2]: Let

[1], and it is said to be mixing if there exists some hand a

-semigroup on

[2].

A

, such that

such that the condition hold for all

is a defined

is said to be topologically

transitive if for any nonempty open subsets

are

-semigroups on X and Y respectively, then the direct sum

and ,

and

be

Y, respectively then

[2]. On the other

is said to be hypercyclic if some

and

whose is dense in X in this case we

-semigroups on

and

is called quasiconjugate to

if there exists a continuous map dense range such that

with . If

can be chosen to be a homeomorphism then

and

are called conjugate.

say that is called hypercyclic vector for this semigroup [2].

Note that the hypercyclicity and transitivity for a -semigroup are equivalent on

on a separable

Banach space [3]. It's clear that if

is a hypercyclic

operator for some

, then the

semigroup

is hypercylic. On the other hand if the semigroup hypercylic operator for all

is hypercylic then [1]. Also a

is said to be weakly mixing if

is

-semigroup is

Definition(2.2) [2]: A property P of

Radhi.Ali Zaboon, Department of mathematics, College of Science, AlMustansiriyah, University, Baghdad, IRAQ Arwa Nazar Mustafa, Department of mathematics, College of Science, Al- Mustansiriyah, University, Baghdad, IRAQ Rafah Alaa Abdulrazzaq, Department of mathematics, College of Science, Al- Mustansiriyah, University, Baghdad, IRAQ

-semigroup is said to be preserved

under (quasi) conjugacy if any (quasi)conjugate to a possesses property P.

51

-semigroup that is

-semigroup with property P also

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Results On Strongly Continuous Semigroup Proposition (2.3):

Proof: 1) Let

Topologically transitive for a under quasiconjugacy.

Proof: Let

-semigroup is preserved

and

are

X and Y, respectively such that to

is quasiconjugate

by

and

topologically transitive Let subsets of X, since

-semigroups on

is

be non-empty and open

is continuous and has dense range,

and

on X and Y, respectively quasiconjugate to

are

-semigroups

such that

is

by

and

is hypercyclic

-semigroup. Let

under

(since

If

have dense orbit is hypercyclic).

is a non-empty and open subset of X then

non-empty and open in Y since dense range

is

is continuous and has

are open and non-empty in Y, and since

,

is topologically transitive,

and since for

,thus

some

such ,

so

to

,

then , then

for

some

2)

Proposition (2.4): The property of having dense set of periodic points of -semigroup is preserved under quasiconjugacy .

-semigroups

by

has dense set of periodic points. Let U be a non-empty open subset of X since

be a

quasiconjugate to the

-semigroup on X that is

-semigroups

. Let

on Y by

are non-empty and open subsets of

is continuous and has dense range then are non-empty and open subsets of Y,

and since such

is mixing then there exists

that , then

is continuous and has

,

is open and non-empty subset of

Y, thus

is periodic point for such

that Now,

that is

thus

therefore

that is such

, quasiconjugate to

Let

X, since

, and

is

-semigroup.

-semigroup on X that is

on Y

dense range then

has dense orbit, therefore

hypercyclic

which mean that

be a

thus

that and

,

Proof: Let quasiconjugate to the

is quasiconjugate

and since is

quasiconjugate

to

so

, then

thus therefore

is

mixing 3) If ,therefore

and

are

-semigroup on X and

Y, respectively such that for some

, then has dense set of periodic point.

The following theorem was state in [2] without prove, we now proved it:

by

quasiconjugate to

, and

then

be a

quasiconjugate to the

is weakly mixing, -semigroup on X that is

-semigroups

by

Theorem (2.5): The following properties of a -semigroup are preserved under quasigonjugat: 1) Hypercyclicity 2) Mixing 3) Weakly mixing 4) Chaotic

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on Y

, since

is

weakly mixing then is topologically transitive and topologically transitive is preserved under quasiconjugate

then

transitive that is mean

is

topologically

is weakly mixing and

is topologically transitive.

4) From Propositions (2.3) and (2.4).

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International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-12, December 2016 Proposition(2.6):

i)

Topologically ergodic for a under quasiconjugacy.

Proof: Let quasiconjugate to the

be a

-semigroups

is hypercyclic then and

-semigroup on X that is

on Y by

, where

is topologically ergodic. Let

be non-empty

open subsets of X, since

If

-semigroup is preserved ii) If

is topologically

transitive then

iii) If

is chaotic then and

are chaotic.

iv)

is topologically ergodic and

is mixing if and only

if is synditic that is contains long interval

and

are mixing.

v) If

is weakly mixing

does not then

let

, for some

Since

are

topologically transitive.

are non-empty open subsets of Y

Now

and

is continuous and has dense range,

thus since

are hypercyclic.

such

and

are weakly

mixing.

that

vi)

,

is quasiconjugate to

, then

is topological ergodic

if and only if

and

are

topological ergodic. In particular, every for some

, thus

,

for some

topologically ergodic

)

weakly mixing.

is synditic that is

does not contains long

interval, therefore

Proof: i) Let

is topologically ergodic.

is hypercyclic, since

III. THE DIRECT SUM OF -SEMIGROUPS AND THE MAIN THEOREM

are quasiconjugate to , then by theorem (2.5)

Before we state the main theorem we consider the following: Let and Y, respectively to prove that the

be

are hypercyclic

-semigroups on X and ii) Let

and

transitive, since

and

is

topologically

and

are

are quasiconjugate to

-semigroup continuous

-semigroups is

, by has

dense

define

quasiconjugate

, then is range and

to

then

proposition (2.3)

by are

topologically transitive. iii) Let

is

chaotic,

since

are quasiconjugate to

, then

by

theorem

(2.5)

are chaotic. so

therefore

iv)

is

Let

quasiconjugate to

is mixing, since

and

By the same way we prove that

is quasiconjugate

are quasiconjugate to then

by

theorem

(2.5)

to and Main Theorem (3.1): Let and respectively, then

Let be

-semigroups on X and Y

are mixing and

any two non-empty open subsets

53

are mixing then for of X there

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Results On Strongly Continuous Semigroup exists

such

Let

that

and for any two non-empty open subsets

of Y

Now,

there exists such that

. Let , then we have Then does not contain long intervals.

Proposition (3.2) [4],[5]: There exists an operator T on a sequence space

since

and

where such that T and its adjoint direct sum of T and

thus

are hypercyclic and the

is not hypercyclic.

, Remark (3.3): The direct sum of two hypercyclic , and therefore

be hypercylic

is mixing.

v) Let and

are

to

quasiconjugate

, then by theorem (2.5) and

vi)

is

ergodic and since quasiconjugate

adjoint are

and

proposition (2.6) then

by

and

semigroup but the direct sum of

and

is not hypercyclic since if is hypercyclic

semigroup,

then every operator in

is

hypercyclic, thus is hypercyclic operator which is contradiction with proposition (3.2).

Consider the same example to see that the direct sum of two

and

are topological ergodic .

Let

and it's is not

hypercyclic from proposition (3.2)), thus

are

to

such that

are hypercyclic (but

topological

and

semigroup of operators on

, then the there exists

are weakly mixing .

Let

semigroup as the following example:

be the

is weakly mixing, since

semigroup need not

and

are

topologically transitive

semigroup need not be

topologically transitive separable Banach space).

topological ergodic

semigroup (since

is

thus for any two non-empty open subsets Proposition (3.4)[2]:

of X

Let

be a chaotic

-semigroup on X, then

is topologically ergodic. is synditic such that

Theorem (3.5):

does

not contains long interval. And for any two

The direct sum of two chaotic

non-empty open subsets

Proof : Let

on Y

-semigroup is chaotic.

and

are two chaotic

-semigroup on X and Y respectively, then by proposition (3.4)

and

are topological ergodic and

by the main theorem is synditic such that

does

ergodic, then by the same theorem (vi) hypercyclic.

not contains long interval.

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is topological is

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International Journal of Engineering and Applied Sciences (IJEAS) ISSN: 2394-3661, Volume-3, Issue-12, December 2016 Now to prove

Proposition (3.11): Let

has dense set of periodic

points. Consider that the sets of all points with

periodic

points

be a

and such that

provides a dense set of periodic points for

-semigroup then

, then is chaotic.

By the main theorem (3.1) and theorem (3.5), one can proof the generalization of the direct sum of chaotic as follow:

-semigroup

is topologically transitive is topologically transitive

-semigroup for each .

By the main theorem (3.1) , one can proof the generalization of the direct sum of weakly mixing

-semigroup

-semigroup as follow: Remark (3.6): Denote

by

the

Proposition (3.12): Let

-semigroup

be a such Proposition (3.7): Let By the main theorem (3.1) , one can proof the generalization of the direct sum of mixing follow:

is

-semigroup then only

if

is weakly mixing

semigroups( we have:

-semigroup) on X and Y, respectively then

mixing,

be a strongly continuous

-semigroup

is

(topologically

respectively)

transitive,

then

and

mixing, mixing, chaotic, topologically transitive, topologically ergodic, respectively)

Proposition(3.9): Let

2- The is

chaotic

is topological ergodic

is

and

(mixing,

respectively)

are

topologically

ergodic,

-semigroup.

And then proved it for n strongly continuous semigroup.

-semigroup as follow:

REFERENCES: [1] Banaslak, J., "An introduction to chaotic behavior of linear dynamical system with applications", 2nd edition, University of Natal, Durban, South Africa., 2000.

[2] Grosse. K.E., Peris., A. M., "Linear chaos", Springer, New York , 2011 [3] Jose, A., "Dynamical properties of skew-product operators", University of Valencia, Spain, 2014.

[4] Domingo, A., "Limits of hypercyclic and supercyclic operators", p 197-190Arezona state University, Arizona, 1990..

[5] Salas, H., " A hypercyclic operator whose adjoint is also hypercyclic", no.3,pp 765-770, American mathematical society, USA, 1991

Proposition (3.10): Let be a

-semigroup

is then

-semigroup

if and only if

-semigroup

-semigroup By the main theorem (3.1) , one can proof the generalization of the direct sum of hypercyclic

-semigroup

-semigroup.

chaotic (mixing, topologically ergodic, respectively)

-semigroup then

is topological ergodic

that

weakly

are hypercyclic (transitive, weakly

-semigroup as follow:

such

is chaotic

.

and

hypercyclic

is

mixing -semigroup.

By the main theorem (3.1) , one can proof the generalization of the direct sum of topological ergodic

if and only if

mixing

Let

1- If the

is mixing and

-semigroup then -semigroup then the direct sum -semigroup for each

-semigroup weakly

IV. CONCLUSIONS:

is

if

is

-semigroup as

Proposition(3.8): Let

-semigroup

that

is

hypercyclic hypercyclic

-semigroup for each .

By the main theorem (3.1) , one can proof the generalization of the direct sum of topologically transitive -semigroup as follow:

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