1.Introduction
In paper [1] ,the author studies the problem whether is the generator of a bounded C0-semigroup if A generates a bounded C0-semigroup.We know that α-times resolvent operator family is generalization of C0-semigroup and C0-semigroup is 1-times resolvent operator family.So,in this paper,we will show that when the operator
generates a bounded α-times resolvent operator family,under certain condition,
is also the generator of a bounded α-times resolvent operator family.The representation of such bounded α-times resolvent operator family will be obtained,too.Furthermore,we will consider the problem whether
owns this property.
Let us first recall the definitions of α-times resolvent operator family.Let A be a closed densely defined linear operator on a Banach space X and.
is a Mittag-Leffler function.
Definition 1.1 [2] A family is called an α-times resolvent operator family for A if the following conditions are satisfied:
1) is strongly continuous for
and
;
2) and
for
and
;
3) For,
satisfies
where,
.
If where
,we write as
(or shortly
).Then we give the definitions of analytic α-times resolvent operator family.
Definition 1.2 [2] An α-times resolvent family is called analytic if
admits an analytic extension to a sector
for some
,where
.An analytic solution operator is said to be of analyticity type
if for each
and
,there is
such that
.
Then we give a Lemma which will be used later.
Lemma 1.1 [2] .Then
if and only if
and there is a strongly continuous operator-valued function
satisfying
,and such that
2.Main Theorem and Conclusion
Theorem 2.1.On a Hilbert space H,the following statements are equivalent:
(1),
;
(2) A is a closed,densely defined operator,,and for
,
;
(3) A is a closed,densely defined operator,for,
and
is invertible for some
.
Proof.(2) Þ (3) For,
,then we have for
,
(1)
hence we know is invertible from the proposition 1.5 of chapter 3 in book [3] .While,from equation (1),we can also have for
,
,then
.
(3) Þ (2) Since,
,then for
,
.
is invertible for some
imply that
is invertible for any
.Together with A is closed and densely defined,we have
,hence
and
.
(1) Þ (2) From lemma 1.3 of [4] ,we know that A is a closed,densely defined operator.And we can get the other conclusion from theorem 2.8 of [2] .
(2) Þ (1).Firstly,set.For every
,
is a bounded operator and can commute with one another.It follows from Theorem 2.5 of [2] that
generates an α-times resolvent family
which is also uniformly continuous and exponential bounded.
For,
.There exists a
,such that
,that is
.Then
Since and
,then we have that
.It means that for
,
.Consequently,
,and
.
From Lemma II,3.4(ii) of [5] ,we have that converges to A pointwise on
.If we can get the following properties,we will have
.
(a) (*) exists for
;
(b) is an α-times resolvent family which is generated by A;
(c).
(a) For is bounded,we can only need to prove (*) on
.For
,
where ((2.52) and (2.53) of [2] ).Together with
,we can get that
.Thus for
,
By Lemma II,3.4(ii) of [5] ,is a Cauchy sequence for each
.Therefore
converges uniformly on each interval
.
(b) I.For,
is the uniformly continuous functions and so is continuous itself.For each
,
is uniformly bounded on every interval
and
,then so is
.By Lemma I,5.2 of [5] ,
is strongly continuous and
.
II.For,
,
and
.Together with that A is an closed operator,we have that
.That is
.
We have,
and
converge to A and
pointwise,respectively.So,we have
.
III.We know that
And for,
converges uniformly on the interval
,then
For all the above,we can obtain that is an α-times resolvent family which is generated by A.
(c) For each,
and
converges to
pointwise,so
,too.That is
.
To sum up the above (a),(b) and (c),we can conclude that.
Theorem 2.2.,
Û A is a closed,densely defined operator,
,and for
,
.
Proof.In the proof of the previous theorem,we have only used the properties of Hilbert space in the acquisition of and we can get
without the properties of Hilbert space.
In fact,on a Banach space,for each,
can generate a C0-semigroup
.Moreover,
From the subordination principle,we have,where
.So
We can obtain that this theorem is tenable from the proof of the previous theorem.
Theorem 2.3.On a Hilbert space H,if,
and
exists as a closed,densely defined operator,then
.
Proof.From the above Theorem 2.1,we have,and for
,
.And
exists as a closed,densely defined operator,so it is easy to show that
is bounded invertible for some
,from (8) of [1] .Further more,for
,then
,thus there exists an
,such that
.Then
.By Theorem 2.1,we can obtain that
.
Theorem 2.4.If,
,
is the α-times resolvent family generated by it and
.And if
exists as a closed,densely defined operator,then
generates an α-times resolvent family
,which is given by
where is the first order Bessel function [6] .Moreover,there exists an
,such that
.
Proof.Since,then
.Together with the assumption that
is a closed,densely defined operator,we have that
.Because of the property of Bessel function
and for large t,
,then
Thus,the integral is well defined.Set .Obviously,
is strongly continuous and
.From the above discussion,we can get that
.For
,
Consequently,we can obtain a conclusion that generates an α-times resolvent family
from Lemma 2.1 and
,
.
Theorem 2.5.A satisfies the assumption of Theorem 2.4,for,
generates a bounded analytic α-times resolvent family
.If
,then
where
and
is oriented counterclockwise,where
and.
Proof.,so
and
.For
,
.It follows from the Remark 2.8(a) of [7] that
generates a bounded analytic α-times resolvent family
.If
,we set
Since,then
From [8] ,we have that there exists an,such that
.Next we estimate
,it follows from (2.4) of [8] that
The same estimate holds for the integral on.
The same estimate holds for the integral on.
To sum up,we can conclude that there exists an,such that
.So
.Then we should show that
is strongly continuous at 0.It following from the dominated convergence Theorem and Fubini Theorem that
For,it follows from Fubini Theorem that
From all the above,we can obtain a conclusion that if,
generates a bounded analytic α-times resolvent family
,
3.Conclusion
In this paper,we considered when the operator generates a bounded α-times resolvent operator family,under certain condition,
as well as
is also the generator of a bounded α-times resolvent operator family.Through the study of the problem whether
is the generator of a bounded α-times resolvent operator family if A generates a bounded α-times resolvent operator family,we can know the generator A more clearly.Furthermore,this work can improve the study of the inverse problem.
Acknowledgements
The author was supported by Scientific Research Starting Foundation of Chengdu University,No.2081915055.
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