1. Introduction
It is well known that many mathematical physics problems can be put into the perspective of infinite dimensional systems, which can be equivalently described by 
semigroups in proper function spaces. One important object to describe the long time dynamics of an infinite dimensional system is the global attractor, which is a connected and compact invariant set in some function space, and which attracts all bounded sets.
To show the existence of the global attractor, one normally needs to verify:
1) there exists an absorbing set, and
2) the semigroup is uniformly compact.
However, it is difficult or even impossible to verify the uniform compactness of the semigroup for many problems. In [1], the authors use the measure of noncompactness of a set to introduce a new concept of compactness called 
-limit compact, then they show that there exists a global attractor for a 
semigroup if and only if:
1) there is an absorbing set, and
2) the semigroup is 
-limit compact.
A well-known result (see [2-6]) is that a continuous semigroup has a global attractor if and only if:
1) it has a bounded absorbing set, and
2) it is asymptotically compact.
Furthermore, in [7], the author introduce the concept of asymptotically null and show that a lattice system has a global attractor if and only if:
1) it has a bounded absorbing set, and
2) it is asymptotically null.
Our main motivation of this paper is to prove that asymptotically compact Û 
-limit compact, and then we prove that the conditions in [1,7] are equivalent directly in
.
The concept of pullback random attractors for random dynamical systems, which is an extension of the attractors theory of deterministic systems, was introduced by the authors in [8-10]. We point out that our work in this paper also can be extended to pullback attractors.
2. Measure of Noncompactness and Its Properties
In this section, we recall the concept of measure of noncompactness and recapitulate its basic properties; see [11].
Definition 2.1 Let M be a metric space and A be a bounded subset of M. The measure of noncompactness 
of A is defined by
.
Lemma 2.1 Let M be a complete metric space, and 
be the measure of noncompactness of a set.
1) 
if and only if 
is compact;
2) If M is a Banach space, then
;
3) 
whenever
;
4)
;
5)
.
Proof. 1) (a) If 
is compact, then 
is precompact. 
is a complete metric space, thus for any
, there exists a finite subset 
of 
such that the balls of radii 
centered at 
form a finite covering of
. By Definition 2.1, 
admits a finite cover by sets of diameter
. The arbitrariness of 
implies that
.
(b) On the other hand, if
, then by Definition 2.1, we have that for any
, 
admits a finite cover by sets of diameter
. So for any
, 
always has a finite 
-net. Then 
is totally bounded. 
is complete, thus 
is precompact, and 
is compact.
2) If 
is a finite cover of
, and 
is a finite cover of
, then 
is a finite cover of
, thus
.
3) If
, then the finite cover of 
must be a finite cover of
, so
.
4) (a) The finite cover of 
must be a finite cover of both of 
and
. So we have 
and
. Thus
.
(b) For any
, we can find finite covers 
of 
and 
of 
with the diameter of 
and 
less than
. But 
is a cover of 
and the diameter of 
is less than
. Hence
. So
.
5) Since
, then
. For any
, 
has a finite cover by sets of diameter
. For any
, 
has a finite cover by sets of diameter
. From the arbitrariness of 
and Definition 2.1, we have
. Thus
. So
.
3. Main Results
In this section, firstly we recall some basic definitions in [1,7], then we show that the two necessary and sufficient conditions for the existence of global attractors for semigroups are equivalent directly.
Definition 3.1 Let 
be a complete metric space. A one parameter family 
of maps
, 
is called a 
semigroup if 1) 
is the identity map on M2) 
for all
3) the function 
is continuous at each point
.
Definition 3.2 Let 
be a 
semigroup in a complete metric space
. A subset 
of 
is called an absorbing set in
, if for any bounded subset 
of
, there exists some 
such that
, for all
.
Definition 3.3 A 
semigroup 
in a comple te metric space 
is called 
-limit compact, if for every bounded subset 
of 
and any
, there exists 
such that
Definition 3.4 A 
semigroup 
in a complete metric space 
is called asymptotically compact if, for every bounded subset
, for any 
and any
, 
has a convergent subsequence.
Let 
be a positive smooth function on R and
. Then define a weighted 
space as
with norm
.
Definition 3.5 
is said to be asymptotically null in 
if for any 
bounded in 
and
, the following holds
Theorem 3.1 Let 
be a 
semigroup in a complete metric space M, then we can have:
is 
-limit compact Û 
is asymptotically compact.
Proof. First, we prove the necessity.
It suffices to prove that for every bounded subset
, for any
, there exists
, such that
Assume otherwise, then there exists a bounded subset 
and
, such that for every 
we have
We take
, then
. Let 
and take
.
Let 
then
. By the definition of measure of noncompactness,
has no finite covering of balls of radii
.
Thus there exists 
and 
such that
Otherwise 
is the finite 
-net of
.
Next we take 
hence
. That is to say 
has no finite 
-net. Thus there exists 
and 
such that
Otherwise 
is the finite 
-net of
.
Repeat the previous procedure, then we have the sequence 
which satisfies
(1)
By the way of taking
, and
, we have
. Since 
and 
is a bounded subset of
, 
is asymptotically compact. Therefore 
has a convergent subsequence. This gives contradiction to (1).
Thus 
is 
-limit compact.
Next, we prove the sufficiency.
We need to prove that for every bounded subset
, for any 
and any
, 
has a convergent subsequence.
Since 
is 
-limit compact, then for the bounded subset 
above, for any
, there exists 
such that
For
, there exists
, such that 
when
. 
implies
Property (3) of the measure of noncompactness in Lemma 2.1 shows that
So
. Notice that 
contains only a finite number of elements (where 
is fixed such that 
as
).
Using properties in Lemma 2.1, we have
Thus
From the arbitrariness of
, it has
Hence 
is precompact. Thus 
has a convergent subsequence. Therefore 
is asymptotically compact. This completes the proof of Theorem.
Corollary 1 Let 
be a semigroup of continuous operators in
. Then 
has a bounded absorbing set and it is asymptotically null in 
has a bounded absorbing set and it is 
-limit compact.
Proof. By Corollary 3.4 in [7], we have 
is asymptotically compact in 
if and only if 
is asymptotically null in 
and 
is bounded in 
provided 
is bounded and
.
Using the Theorem 3.1 above , we have 
is 
-limit compact in 
if and only if 
is asymptotically null in 
and 
is bounded in 
provided 
is bounded and
. Thus the necessity of the corollary is obvious.
If 
has a bounded absorbing set, 
is bounded and
, then there exists 
such that 
is contained in the bounded absorbing set. 
is a finite set in
, so it is bounded. Thus 
is bounded. Now we can have the sufficiency immediately. This completes the proof of Corollary.
4. Acknowledgements
The author wishes to thank professor Yongluo Cao for his invaluable suggestions and encouragement.