# Notes on the Propagators of Evolution Equations

- Yu Lin
^{1}, - Ti-Jun Xiao
^{2}Email author and - Jin Liang
^{3}

**2010**:795484

**DOI: **10.1155/2010/795484

© Yu Lin et al. 2010

**Received: **8 January 2010

**Accepted: **1 February 2010

**Published: **16 February 2010

## Abstract

We consider the propagator of an evolution equation, which is a semigroup of linear operators. Questions related to its operator norm function and its behavior at the *critical point* for norm continuity or compactness or differentiability are studied.

## 1. Introduction

As it is well known, each well-posed Cauchy problem for first-order evolution equation in Banach spaces

gives rise to a well-defined propagator, which is a semigroup of linear operators, and the theory of semigroups of linear operators on Banach spaces has developed quite rapidly since the discovery of the generation theorem by Hille and Yosida in 1948. By now, it is a rich theory with substantial applications to many fields (cf., e.g., [1–6]).

In this paper, we pay attention to some basic problems on the semigroups of linear operators and reveal some essential properties of theirs.

Let be a Banach space.

*A one-parameter family*

*of bounded linear operators on*

*is called a strongly continuous semigroup (or simply*

*-semigroup) if it satisfies the following conditions:*

- (i)
, with ( being the identity operator on X) ,

- (ii)
for ,

- (iii)
the map is continuous on for every .

The *infinitesimal generator*
of
is defined as

For a comprehensive theory of -semigroups we refer to [2].

## 2. Properties of the Function

- (I)
, for ;

- (II)
for .

Furthermore, we can infer from the strong continuity of that

- (III)
is lower-semicontinuous, that is,

(2.1)In fact,

(2.2)holds for all with . Thus, taking the supremum for all with on the left-hand side leads to (2.1).

We ask the following question

*For every function*
*satisfying*
*,*
*, and*
*, does there exist a*
*semigroup*
*on some Banach space*
*such that*
*for all*
?

We show that this is not true even if is a finite-dimensional space.

Theorem 2.1.

Then satisfies (I), (II), and (III), and there exists no semigroup on such that for all .

Proof.

First, we show that satisfies (I), (II), and (III). (I) is clearly satisfied.

To show (III) and (II), we write

hence satisfies (III).

For (II), suppose , and consider the following four cases.

Case 1 ( and ).

Case 2 ( and ).

that is, .

Case 3 ( , but and ).

Case 4 ( ).

Next, we prove that there does not exist any semigroup on such that . Suppose for some semigroup on and let be its infinitesimal generator.

First we note from (2.3) that

which is a contradiction to (2.16).

Open Problem 1.

Is it possible that there exists an with and a semigroup on such that for all ?

## 3. The Critical Point of Norm-Continuous (Compact, Differentiable) Semigroups

The following definitions are basic [1–6].

Definition 3.1.

*A*
*-semigroup*
*is called norm-continuous for*
*if*
*is continuous in the uniform operator topology for*
*.*

Definition 3.2.

*A*
*-semigroup*
*is called compact for*
*if*
*is a compact operator for*
*.*

Definition 3.3.

*A*
*-semigroup*
*is called differentiable for*
*if for every*
*,*
*is differentiable for*
*.*

It is known that if a -semigroup is norm continuous (compact, differentiable) at , then it remains so for all . For instance, the following holds.

Proposition 3.4.

If the map is right differentiable at , then it is also differentiable for .

Therefore, if we write

and suppose
(
,
), then
(
,
) takes the form of
for a nonnegative real number
. In other words, if
(
,
), then
is norm continuous (compact, differentiable) on the interval
but not at any point in
. We call
the *critical point* of the norm continuity (compactness, differentiability) of operator semigroup
.

A natural question is the following

*Suppose that*
*is the critical point of the norm continuity*
*compactness, differentiability*
*of the operator semigroup*
*. Is*
*also norm continuous (compact, differentiable) at*
? Of course, concerning norm continuity or differentiability at
we only mean right continuity or right differentiability.

We show that the answer is "yes" in some cases and "no" for other cases.

Example 3.5.

*Let*

*and*

Therefore, in this case we have . Since , we see that is compact and is differentiable at from the right.

Example 3.6.

*Let*

In view of that is unbounded, we know that is not norm continuous at .

For differentiability, we note that is differentiable at if and only if for each . From

it follows that when , for every . On the other hand, when and is any nonzero constant sequence, . Therefore the critical point for differentiability is . But is not differentiable at .

## Declarations

### Acknowledgments

The authors acknowledge the support from the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory of Modern Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).

## Authors’ Affiliations

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## Copyright

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