Boundedness Types of Perturbations on the Growth of Semigroups ()
1. Introduction
Let
be the sequence of generators of a
-semigroup
on a Banach space X. Let
be a given function. That will say that the semigroup
is
-bounded, if
holds for some
and each
. If
for some
, then the semigroup is polynomially bounded and is bounded when
. It is known that there exist
and
such that
for all
. When the growth bounds of the semigroup
, are positive, the semigroup is exponentially bounded. We are concerned with the types of boundedness of
and its perturbation when
is exponentially bounded, i.e., for all
,
.
Many results are shown in this article. The first one states that if the Cesáro average of
and that of its adjoint are
-bounded, then so does the semigroup.
Next, give a Hille-Yosida type characterization of generators of
-bounded
-semigroups. Furthermore, Theorem 3.2 gives a sufficient condition that improves the results of (Shiand Feng [1] ) and (Eisner [2] ). Notice that a closer result including integrability conditions of each powers of the resolvent was given in (Batty et al. [3], Theorem 6.6) under more assumptions on
.
Finally, we consider a sequence perturbation
and establish the previous results for the perturbed semigroup
generated by
.
Throughout this article
stands for a closed densely defined linear sequence operators on Xwith domain
and spectre
. The pseudo-spectral bounded of sequence
is defined by
such that
whenever
, where
denotes the spectral bounded of sequence
given by
, with the convention
if
.
2. Boundedness Types of a Semigroup Interms of Cesáro-Boundedness of the Semigroup and Its Adjoint
It is shown in (Zwart [4] ) that if
is a
-semigroup on a Banach space X and
is its adjoint, then for each
,
,
and,
,
(1)
The following theorem is a consequence of this inequality and recovers (Zwart [4], Theorem 2.1, 2.2), (Van Casteren and Jan [5], Theorem 3.1, (iii ⇔ iv)), (Casteren and Jan [6], Proposition 3.1) and (Guo and Zwart [7], Theorem 8.2).
Theorem 2.1. Let
be a
-semigroup on a Banach space X,
with
. Let
and
be measurable positive functions. If for each
, and
(2)
and
(3)
hold, then
(4)
We indicate that one cannot omit the Condition (3). (Van Casteren and Jan [5], Example (2)), gives a polynomially bounded group,
of linear sequence operators acting on
, while its Cesáro average is bounded.
3. Boundedness Types of a Semigroup in Terms of the Resolvent
In order to deal with the converse when
is
-bounded, consider the set Λ of all continuous functions
such that
(5)
Note that if the functions
, and
, then
see (Boukdir [8] ).
Many classical functions are contained in Λ: The bounded functions
for which there exists
such that
, the polynomial functions,
for
. Indeed,
for each
and
. Also, the function
with
. In certainty, since
for each
then
.
Theorem 3.1. Let
be the sequence of operator on a Banach space X. Let
be a continuous function with
. Suppose the following assertions.
i)
is closed, densely defined, and for every
one has
and
(6)
for each
,
and some constant
.
ii)
generates a
-bounded
-semigroup
.
Then, (i) ⇒ (ii). Conversely, if in addition
satisfies (5), then (ii) ⇒ (i). Proof. (i) ⇒ (ii). Is deduced from the Hille-Yosida Theorem and the exponential formula,
for each
, with
and
.
(ii) ⇒ (i). Since
be the sequence of generates a
-semigroup, then it is closed and densely defined. The
-boundedness of
and (5) imply that there exist
such that
(7)
Consequently,
and hence
for all
. Let
and
. Then
Theorem 3.2. Suppose
be a closed and densely defined a sequence of operators in a Banach space X with
. For a continuous function
with
, we study the following assertions.
a) For all
, and
,
And
(8)
b) For each
, and
(9)
c)
generates a
-bounded
-semigroup
on X, for which
for each
(10)
Then (a) ⇒ (b) ⇒ (c). In this case, the semigroup
is given by
and for
,
, (11)
for each
and
. Furthermore, if X is a Hilbert space and
satisfies (5), then (c) implies (a) with
, instead of
, for some
.
Proof. (a) ⇒ (b) It is obtained by applying the Cauchy-Schwarz inequality.
(b) ⇒ (c). Deduce from (Gomilko [9] ) and ((p. 505) from (Chill & Tomilov [10] ) that the assumption (b) implies that the sequence of operator
generates
a
-semigroup
and for all
,
, and
for all
,
and
(12)
then the result is deduced by choosing
.
Conversely. Let
. As in (7) the Parseval identity yields
and the identical reasoning for the dual case.
Remark 3.3. 1) The condition
cannot be omitted in the above Theorem 3.2. If not, the semigroup may not be strongly continuous at the origin.
2) It is not enough that the condition (9) be satisfied by some
.
An example due to (Selim Grigorevich Krein [11] ) see also (Kaiser & Weis [12] ), exhibits that there exists a closed, densely defined sequence of operators
acting on a Hilbert space X such that the resolvent exists and uniformly bounded on
and
,
but the sequence
of generators of semigroup is not strongly continuous at the origin.
3) If
in (9) we recover the result of (Eisner [2] ) when
, and (Gomilko [9] ) with
.
4) If
for some
in (9), obtain (Laubenfels et al. [13] Corollary (3.5)), exactly, the semigroup satisfies
for all
. (13)
In order to give a generalization of (Eisner and Zwart [14], Theorem 2.1) supposes that the resolvent of the sequence of generator
is
-integrabe, i.e.
for all
, (14)
and
for all
, (15)
Note. We can deduce that:
for all
.
Proof. From (10) and (13).
Theorem 3.4. Let
be the sequence of generator of a
-semigroup
on a Banach space X such that
and its resolvent is
-integrable for
. Let
be a continuous function. If there exist
such that
(i)
for all
;
(ii)
for all
, (16)
then
(17)
for some
and
.
Proof. As it is shown in Eisner (2007), from (16) deduce that
and for all
there exists
such that for each
and
and
The Cauchy Schwarz inequality yields
. (18)
By the inverse formula we get
For
large enough, one can choose
.
Note. We can deduce that:
i)
for all
.
ii)
for all
.
Proof. i) From (10) and (17).
ii) From (13) and (17).
Corollary 3.5. Let
be the sequence of generator of a C0-semigroup Tj on a Banach space X such that
and the resolvent is
-integrable, for some
. If
(19)
and there exist constant
such that
, for all
for all
. (20)
Then the semigroup
is uniformly stable.
Proof. It is enough to choose
Remark 3.6. 1) By
, (19) and (20) are equivalent to
for each
and
. Hence with the
-integrability of the resolvent the uniform stability of the semigroup follows from (Eisner [15], Theorem 2.15).
2) Note that the Conditions (20) are satisfied by positivity-preserving semigroups, acting in
for some
and
, see (Davies [16], Lemma 9).
4. Boundedness Types of the Perturbed Semigroups
We exhibit that the conditions on
which ensure the
-boundedness of the semigroup
are sufficient to obtain the same property for the perturbed
of a semigroup sequence of generators
.
Let
be the sequence of generator of a
-semigroup
with
. Peekingan other closed sequence operator
such that
, and let
its dual.
Suppose that
for all
, and
for all
,
. (21)
Let
. Since
, and by
and the decomposition
, deduce that
. Furthermore the inverse
satisfies
(22)
where
. From
we obtain that
for each
, and similar arguments exhibit that
. (23)
Note that if (21) holds then the resolvent of
is
-integrable when that of
is.
Proposition 4.1. Let
be a closed and densely defined a sequence operator on a Banachspace X with
. Let
be a closed sequence of operator such that
and satisfy
and
. Let
be a continuous function for which (8) holds, then
generates a
-bounded
-semigroup.
Proof. Since
then
is dense. By the assumption
,
and
we deduce that
and the Conditions (9) for
are deduced from the Cauchy-Schwarz inequality, (22) and (23).
The following proposition organizes a connection between the φj-boundedness of the semigroup
and that of its perturbed S. Furthermore this result gives a generalization of (Kaiser and Weis [12], Theorem 3.1) and (Batty and Charles [17], Theorem 1).
Proposition 4.2. Let
be a sequences of generator of a
-semigroup
on a Hilbert spaceX. Let
be a closed sequence of operator satisfying
and for which the hypothesis
and
hold. Let
be a continuous function satisfying (5).
If
is
-bounded, then
generates a
-bounded
-semigroup.
Proof. from (22), (23) and Theorem 3.2.
Assuming
and
, we will give sufficient conditions on
confirming that both
and S have the same boundedness types for large
.
Proposition 4.3. Let
be a
-semigroup generated by the sequence of operator
for which
and having
-integrable resolvent for some
. Suppose that
and
hold for some closed sequence of operator
. Let
be a continuous function satisfying (16).
Then the semigroup S generated by
is strongly continuous sequence on
and satisfy (17).
Proof. a direct consequence of Theorem 3.4.
5. Conclusion
As discussed above the type of the boundedness of a semigroup
in terms of increment of its Cesàro-average and that of its adjoint
is
-bounded, then the semigroup is bounded (see Theorem 2.1). Also, we introduced a Hille-Yosida type characterization of generators of
-bounded
-semigroups (see Theorem 3.1). We presented some effect of a perturbation sequence operator
by sequence operator
, that satisfies some assumptions specified (see Proposition 4.1 and Proposition 4.2).
Acknowledgements
We would like to thank our colleagues for interesting discussions and helpful ideas.
Author Contributions
The authors approve and read the article.