1. Introduction

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

(1.1)

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.

Definition 1.1 (see [1–6]).

A one-parameter family of bounded linear operators on is called a strongly continuous semigroup (or simply -semigroup) if it satisfies the following conditions:

  1. (i)

    , with ( being the identity operator on X) ,

  2. (ii)

    for ,

  3. (iii)

    the map is continuous on for every .

The infinitesimal generator of is defined as

(1.2)

with domain

(1.3)

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

2. Properties of the Function  

Let be a -semigroup on and define for . Clearly, from Definition 1.1 we see that

  1. (I)

    , for ;

  2. (II)

    for .

Furthermore, we can infer from the strong continuity of that

  1. (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 functionsatisfying,, and, does there exist asemigroupon some Banach spacesuch thatfor all?

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

Theorem 2.1.

Let be an -dimensional Banach space with . Let

(2.3)

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

(2.4)

Then

(2.5)

hence satisfies (III).

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

Case 1 ( and ).

In this case

(2.6)

that is,

(2.7)

Case 2 ( and ).

Let

(2.8)

Then

(2.9)

and is a convex function on . So by Jensen's inequality, we have

(2.10)

that is,

(2.11)

Therefore

(2.12)

that is, .

Case 3 (, but and ).

It follows from Case 2 that

(2.13)

Case 4 ().

Again we have

(2.14)

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

(2.15)

for every , while

(2.16)

By the well-known Lyapunov theorem [2, Chapter I, Theorem ], all eigenvalues of (the infinitesimal generator of ) have negative parts for every . Letting be the eigenvalues of , we then have

(2.17)

and this implies that

(2.18)

It is known that there is an isomorphism of onto such that

(2.19)

where is the Jordan block corresponding to . Therefore

(2.20)

Set

(2.21)

where is the order of . Then is a th nilpotent matrix with for each . According to (2.20) and (2.18), we have

(2.22)

Observing

(2.23)

we see that

(2.24)

Thus,

(2.25)

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

(3.1)

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 thatis the critical point of the norm continuitycompactness, differentiabilityof the operator semigroup. Isalso 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

(3.2)

Then clearly for . Moreover, is not norm continuous (not compact, not differentiable) for any since

(3.3)

for sufficiently small , where

(3.4)

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

Example 3.6.

Let

(3.5)

with supremum norm. For any set

(3.6)

Then, is compact (hence norm-continuous) for since is the operator-norm limit of a sequence of finite-rank operators:

(3.7)

where

(3.8)

So the critical point for compactness and norm continuity is . However, the infinitesimal generator of is given by

(3.9)

with

(3.10)

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

(3.11)

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 .