# On Some Approximation Theorems for Powerq-Bounded Operators on Locally Convex Vector Spaces

Authors: Ludovic Dan Lemle
Journal: The Scientific World Journal
 (2014)

---

## Abstract

This paper deals with the study of some operator inequalities involving the powerq-bounded operators along with the most known properties and results, in the
more general framework of locally convex vector spaces.


---

## Body

## 1. Introduction

LetX be a Hausdorff locally convex vector space over the complex field C. Bycalibration for the locally convex space X we understand a family P of seminorms generating the topology τ
P of X, in the sense that this topology is the coarsest with respect to the fact that all the seminorms in P are continuous. Such a family of seminorms was used by the author and Wu [1] and many others in different contexts (see [2–5]).It is well known that calibrationP is characterized by the property that the set
(1)
E
(
p
,
ϵ
)
=
{
x
∈
X
:
p
(
x
)
<
ϵ
}
,
ϵ
>
0
,
p
∈
P
is a neighborhood subbase at 0. Denote by (
X
,
P
) the locally convex space X endowed with calibration P.Recall that a locally convex algebra is an algebra with a locally convex topology in which the multiplication is separately continuous. Such an algebra is said to be locallym-convex (l.m.c.) if it has a neighborhood base U at 0 such that each U
∈
U is convex and balanced (i.e., λ
U
⊆
U for |
λ
|
≤
1) and satisfies the property U
2
⊆
U.Any algebra with identity will be called unital. It is well known that unital locallym-convex algebra A is characterized by the existence of calibration P such that each p
∈
P is submultiplicative (i.e., p
(
x
y
)
≤
p
(
x
)
p
(
y
), for all x
,
y
∈
A) and satisfies p
(
e
)
=
1, where e is the unit element.An elementa of locally convex algebra A is said to bebounded in
A if there exists α
∈
C such that the set {
(
α
x
)
n
}
n
≥
1 is bounded in A (see [6]). The set of all bounded elements in A will be denoted by A
0.LetC
∞
:
=
C
∪
{
∞
} be the Alexandroff one-point compactification of C. Following Waelbroeck [7, 8], we introduce the following.Definition 1.
We call resolvent set in the Waelbroeck sense of an elementx from a locally convex unital algebra (
X
,
P
) the set of all elements λ
0
∈
C
∞ for which there exists V
∈
V
λ
0 such that the following conditions hold:(a)
the elementλ
e
-
x is invertible in X, for any λ
∈
V
∖
{
∞
};
(b)
the set{
(
λ
e
-
x
)
-
1
:
λ
∈
V
∖
{
∞
}
} is bounded in (
X
,
P
).The resolvent set in Waelbroeck sense of an elementx will be denoted by ρ
W
(
x
). TheWaelbroeck spectrum of x will be defined as
(2)
σ
W
(
x
)
:
=
C
∞
∖
ρ
W
(
x
)
.


## 2.q-Bounded Operators

Following Michael [9] (see also [2, 10]), we introduce the following.Definition 2.
We say that a linear operatorT
:
X
→
X is q-bounded (quotient-bounded) with respect to P if for any p
∈
P there exists c
p
>
0 such that
(3)
p
(
T
x
)
≤
c
p
p
(
x
)
,
∀
x
∈
X
.Denote byQ
P
(
X
) the set which consists of all q-bounded operators with respect to calibration P.For a seminormp
∈
P, the application p
^
:
Q
P
(
X
)
→
R defined as
(4)
p
^
(
T
)
=
inf
⁡
{
r
>
0
:
p
(
T
x
)
≤
r
p
(
x
)
,
∀
x
∈
X
}
is also a seminorm. Note that
(5)
p
^
(
T
1
T
2
)
≤
p
^
(
T
1
)
p
^
(
T
2
)
,
T
1
,
T
2
∈
Q
P
(
X
)
,
p
∈
P
.
We denote by P
^ the family of seminorms {
p
^
:
p
∈
P
}. The space Q
P
(
X
) will be endowed with a topology τ
P
^ generated by P
^. Remark that [9, Proposition 2.4(j)] implies that under this topology Q
P
(
X
) becomes a Hausdorff locally m-convex topological algebra (in the sense of [9, Definition 2.1]).IfT
∈
Q
P
(
X
), the P-spectral radius, denoted by r
P
(
T
), is considered as the boundedness radius in the sense of Allan [6] (see also [11–13]),
(6)
r
P
(
T
)
=
inf
⁡
{
λ
>
0
:
the
sequence
(
(
λ
-
1
T
)
n
)
n
∈
N
h
h
h
h
h
is
bounded
in
Q
P
(
X
)
(
(
λ
-
1
T
)
n
)
n
∈
N
}
,
where, by common consent, inf
⁡
∅
:
=
+
∞.The set of all bounded elements inQ
P
(
X
) will be denoted by (
Q
P
(
X
)
)
0 (see [12]). It easily follows from [6, Proposition 2.14(ii)] that
(7)
(
Q
P
(
X
)
)
0
=
{
T
∈
Q
P
(
X
)
:
r
P
(
T
)
<
∞
}
.ForT
∈
(
Q
P
(
X
)
)
0 we denote by ρ
W
(
T
) theWaelbroeck resolvent set of
T and by σ
W
(
T
) theWaelbroeck spectrum of
T. The function
(8)
ρ
W
(
T
)
∋
λ
⟼
R
(
λ
,
T
)
∶
=
(
λ
I
-
T
)
-
1
∈
(
Q
P
(
X
)
)
0
is called theresolvent function of
T. It is well known that
(9)
R
(
λ
,
T
)
=
∑
n
=
0
∞
‍
T
n
λ
n
+
1
.In this paper we evaluate the behaviour of the power of aq-bounded operator from the algebra (
Q
P
(
X
)
)
0 by some type of approximations. The main results have been announced in [14].


## 3. The Main Results

We continue to employ the notations from the previous sections and we will introduce two types of operatorial approximations for operators from the algebra(
Q
P
(
X
)
)
0 which approximate a given operator T on a convergent power bounded series. The power boundedness problem for operators acting on Banach spaces was largely developed in various frameworks by many authors (see [15–17]).In the following, using the functional calculus from the(
Q
P
(
X
)
)
0 algebra (see [7, 8]), some important boundedness properties are obtained. Denote N
*
=
N
∖
{
0
}. First we have the following.Theorem 3.
IfT
∈
(
Q
P
(
X
)
)
0 satisfies
(10)
sup
⁡
p
^
∈
P
^
p
^
(
T
k
)
≤
C
,
for k
∈
N
*, then
(11)
sup
⁡
p
^
∈
P
^
p
^
[
R
(
λ
,
T
)
k
]
≤
C
(
|
λ
|
-
1
)
k
,
for k
∈
N
* and for all λ
∈
C with |
λ
|
>
1.Proof.
Assume thatsup
⁡
p
^
∈
P
^
p
^
(
T
k
)
≤
C for k
∈
N
*. Since
(12)
R
(
λ
,
T
)
=
∑
j
=
0
∞
‍
T
j
λ
j
+
1
,
for |
λ
|
>
1, then, by using the generalized binomial formula, we get
(13)
R
(
λ
,
T
)
k
=
λ
-
k
(
I
-
T
λ
)
-
k
=
1
λ
k
∑
j
=
0
∞
‍
(
j
+
k
-
1
j
)
T
j
λ
j
,
from where we deduce
(14)
p
^
[
R
(
λ
,
T
)
k
]
≤
C
|
λ
|
k
·
∑
j
=
0
∞
‍
(
j
+
k
-
1
j
)
(
1
|
λ
|
)
j
=
C
|
λ
|
k
·
1
(
1
-
1
/
|
λ
|
)
k
=
C
(
|
λ
|
-
1
)
k
,
for any k
∈
N
* and any p
^
∈
P
^. Therefore, the conclusion is verified.Conversely, we have the following.Theorem 4.
IfT
∈
(
Q
P
(
X
)
)
0 and
(15)
sup
⁡
p
^
∈
P
^
p
^
[
R
(
λ
,
T
)
]
≤
C
|
λ
|
-
1
,
for all λ
∈
C with |
λ
|
>
1, then
(16)
sup
⁡
p
^
∈
P
^
p
^
(
T
k
)
≤
C
e
(
k
+
1
)
,
for k
∈
N
*.Proof.
Let us suppose conditionp
^
[
R
(
λ
,
T
)
]
≤
C
/
(
|
λ
|
-
1
) is true for all p
^
∈
P
^, for any k
∈
N
* and |
λ
|
>
1. For k
∈
N
* fixed, by choosing the integration path Γ
:
|
λ
|
=
1
+
1
/
k, with the aid of the functional calculus from the algebra (
Q
P
(
X
)
)
0, we obtain
(
*
)
T
k
=
1
2
π
i
∫
Γ
λ
k
R
(
λ
,
T
)
d
λ
.
Thus, for all p
^
∈
P
^, we have
(17)
p
^
(
T
k
)
≤
1
2
π
∫
Γ
|
λ
|
k
p
^
(
R
(
λ
,
T
)
)
d
λ
≤
1
2
π
·
max
⁡
λ
∈
Γ
|
λ
|
k
·
max
⁡
λ
∈
Γ
C
|
λ
|
-
1
·
∫
Γ
d
λ
≤
1
2
π
·
(
1
+
1
k
)
k
·
C
k
·
2
π
(
1
+
1
k
)
≤
C
e
(
k
+
1
)
which implies the desired result.Moreover, we can formulate the following.Theorem 5.
IfT
∈
(
Q
P
(
X
)
)
0 and
(18)
sup
⁡
p
^
∈
P
^
p
^
[
R
(
λ
,
T
)
k
]
≤
C
(
|
λ
|
-
1
)
k
,
for k
∈
N
* and for all λ
∈
C with |
λ
|
>
1, then
(19)
sup
⁡
p
^
∈
P
^
p
^
(
T
k
)
≤
C
k
!
e
k
k
k
≤
C
2
π
(
k
+
1
)
,
k
∈
N
*
.Proof.
Integrating(
*
) by parts j
-
1 times, for j
>
2, we obtain
(20)
T
k
=
(
-
1
)
j
-
1
2
π
i
∫
Γ
(
j
+
1
)
!
λ
k
+
j
-
1
(
k
+
1
)
⋯
(
k
+
j
-
1
)
R
(
λ
,
T
)
j
d
λ
.
Now choosingΓ the circle of radius 1
+
j
/
k and by using the hypothesis, for j
→
∞, we get
(21)
sup
⁡
p
^
∈
P
^
p
^
(
T
k
)
≤
C
k
!
e
k
k
k
≤
C
2
π
(
k
+
1
)
.
The last inequality was obtained by using Stirling’s approximation.Now, forT
∈
(
Q
P
(
X
)
)
0 we introduce (see [18]) the following.Definition 6.
The Yosida approximationY
(
λ
,
T
) of T, for λ
∈
ρ
W
(
T
)
∩
C, is defined as
(22)
Y
(
λ
,
T
)
=
λ
T
R
(
λ
,
T
)
.Next theorem shows how an operator from the(
Q
P
(
X
)
)
0 algebra is related to its Yosida approximation.Theorem 7.
The Yosida approximationY
(
λ
,
T
) is analytic for λ
∈
ρ
W
(
T
)
∩
C and the series representation
(23)
Y
(
λ
,
T
)
=
∑
j
=
0
∞
‍
T
j
+
1
λ
j
converges for |
λ
|
>
r
P
(
T
). Moreover, (1)
Y
(
λ
,
T
)
=
λ
2
R
(
λ
,
T
)
-
λ
I;
(2)
if there existsp
^
∈
P
^ such that r
P
(
T
)
<
p
^
(
T
), then
(24)
p
^
(
Y
(
λ
,
T
)
-
T
)
≤
p
^
(
T
2
)
|
λ
|
-
p
^
(
T
)
,
for |
λ
|
>
p
^
(
T
);
(3)
σ
W
(
Y
(
λ
,
T
)
)
=
{
z
/
(
1
-
z
/
λ
)
,
z
∈
σ
W
(
T
)
}.Proof.
By evaluatingY
(
λ
,
T
) in terms of the resolvent R
(
λ
,
T
), for |
λ
|
>
r
P
(
T
) we obtain
(25)
Y
(
λ
,
T
)
=
λ
T
R
(
λ
,
T
)
=
λ
T
(
λ
I
-
T
)
-
1
=
λ
T
·
∑
j
=
0
∞
‍
T
j
λ
j
+
1
=
∑
j
=
0
∞
‍
T
j
+
1
λ
j
from where it follows that the assertion of the theorem is true. Moreover,
(26)
Y
(
λ
,
T
)
=
∑
j
=
0
∞
‍
T
j
+
1
λ
j
=
λ
I
+
T
+
T
2
λ
+
⋯
+
T
n
+
1
λ
n
+
⋯
-
λ
I
=
λ
2
∑
j
=
0
∞
‍
T
j
λ
j
+
1
-
λ
I
=
λ
2
R
(
λ
,
T
)
-
λ
I
,
so (1) is true.
To prove (2) one can observe that, from(27)
Y
(
λ
,
T
)
=
∑
j
=
0
∞
‍
T
j
+
1
λ
j
,
it follows that
(28)
Y
(
λ
,
T
)
-
T
=
∑
j
=
0
∞
‍
T
2
λ
(
T
j
λ
j
)
on a set for which |
λ
|
>
r
P
(
T
). Moreover,
(29)
p
^
(
Y
(
λ
,
T
)
-
T
)
≤
∑
j
=
0
∞
‍
p
^
(
T
2
λ
)
p
^
(
T
j
λ
j
)
≤
p
^
(
T
2
λ
)
·
∑
j
=
0
∞
‍
p
^
(
T
λ
)
j
=
p
^
(
T
2
λ
)
1
1
-
p
^
(
T
/
λ
)
=
p
^
(
T
2
)
|
λ
|
-
p
^
(
T
)
,
for |
λ
|
>
p
^
(
T
)
>
r
P
(
T
).
A simple reasoning shows thatR
(
λ
,
T
)
∈
(
Q
P
(
X
)
)
0; then it follows Y
(
λ
,
T
)
∈
(
Q
P
(
X
)
)
0.
From [19, Theorem 3.1.14], for |
λ
|
>
|
z
|, we have
(30)
σ
W
(
Y
(
λ
,
T
)
)
=
Y
(
λ
,
σ
W
(
T
)
)
,
for all z
∈
σ
W
(
T
), and
(31)
Y
(
λ
,
z
)
=
∑
j
=
0
∞
‍
z
j
+
1
λ
j
on |
λ
|
>
|
z
|, which could be written as Y
(
λ
,
z
)
=
z
/
(
1
-
z
/
λ
), for any z
∈
σ
W
(
T
), so (3) is proved.Below we state an equivalence between a power bounded operator from the(
Q
P
(
X
)
)
0 algebra and the power of its Yosida approximation.Theorem 8.
LetT
∈
(
Q
P
(
X
)
)
0 and Y
(
λ
,
T
) its Yosida approximation. Then the following assertions are equivalent:(i)
sup
⁡
p
^
∈
P
^
p
^
(
T
k
)
≤
c, for any k
∈
N
*;
(ii)
sup
⁡
p
^
∈
P
^
p
^
(
Y
(
λ
,
T
)
k
)
≤
c
/
(
1
-
1
/
|
λ
|
)
k, for any k
∈
N
* and for all λ
∈
C with |
λ
|
>
1.Proof.
Property (i) impliesr
P
(
T
)
≤
1 so that the argumentation given in the proof of Theorem 7 implies that any λ
∈
C with |
λ
|
>
1 belongs to the resolvent set of T. Hence, using the generalized binomial formula, we get
(32)
Y
(
λ
,
T
)
k
=
∑
j
=
0
∞
‍
(
k
+
j
-
1
j
)
T
j
+
k
λ
j
.
Now, by applying (i) again we obtain
(33)
p
^
(
Y
(
λ
,
T
)
k
)
≤
c
∑
j
=
0
∞
‍
(
k
+
j
-
1
j
)
(
1
|
λ
|
)
j
=
c
(
1
-
1
/
|
λ
|
)
k
for any p
^
∈
P
^, whence by passing to supremum, the inequality (ii) holds.
Conversely, (i) is a direct consequence of (ii).Forμ
∈
ρ
W
(
T
), consider now the followingMöbius transformation (see [20]):
(34)
ψ
λ
(
μ
)
=
{
(
λ
-
1
)
μ
λ
-
μ
,
if
λ
≠
∞
,
μ
,
if
λ
=
∞
.Definition 9.
The Möbius approximation ofT is defined as
(35)
A
(
λ
,
T
)
:
=
ψ
λ
(
T
)
.Proposition 10.
A
(
λ
,
T
) is holomorphic in λ
∈
ρ
W
(
T
)
∩
C and satisfies
(36)
A
(
λ
,
T
)
=
(
1
-
1
λ
)
Y
(
λ
,
T
)
,
λ
≠
0
.Proof.
Letλ
∈
ρ
W
(
T
)
∩
C
∖
{
0
}. By evaluating the right member of the above equality, we get successively
(37)
(
1
-
1
λ
)
Y
(
λ
,
T
)
=
(
1
-
1
λ
)
λ
T
R
(
λ
,
T
)
=
(
λ
-
1
)
T
R
(
λ
,
T
)
=
(
λ
-
1
)
T
λ
I
-
T
=
A
(
λ
,
T
)
for λ
≠
∞. If λ
=
∞, then from Definition 9 we have A
(
λ
,
T
)
=
T. On the other side (
1
-
1
/
λ
)
Y
(
λ
,
T
) converges to T, when λ
→
∞.A similar result as in Theorem8 is given below.Theorem 11.
LetT
∈
(
Q
P
(
X
)
)
0 and A
(
λ
,
T
) its approximation as above. Then the following assertions are equivalent:(i)
sup
⁡
p
^
∈
P
^
p
^
(
T
k
)
≤
C, for any k
∈
N
*;
(ii)
sup
⁡
p
^
∈
P
^
p
^
(
A
(
λ
,
T
)
k
)
≤
C, for any k
∈
N
* and for every λ
∈
C with |
λ
|
>
1.Proof.
From Theorem8, for T
∈
(
Q
P
(
X
)
)
0,
(38)
sup
⁡
p
^
∈
P
^
p
^
(
T
k
)
≤
C
is equivalent to
(39)
sup
⁡
p
^
∈
P
^
p
^
(
Y
(
λ
,
T
)
k
)
≤
C
(
1
-
1
/
|
λ
|
)
k
.
The conclusion follows taking into account that
(40)
A
(
λ
,
T
)
k
=
(
1
-
1
λ
)
k
·
Y
(
λ
,
T
)
k
,
for k
∈
N
*.


## 4. Application

ForL
>
0 let X
:
=
C
[
0
,
L
] be the space of continuous functions on [
0
,
L
] endowed with the norm |
u
|
L
:
=
max
⁡
[
0
,
L
]
|
u
(
t
)
|.ConsiderT
:
X
→
X, given by
(41)
T
u
(
t
)
=
∫
0
t
‍
u
(
s
)
d
s
.
Following [19], we see that the resolvent of T is given by
(42)
R
(
λ
,
T
)
u
(
t
)
=
1
λ
u
(
t
)
+
1
λ
2
∫
0
t
‍
e
(
t
-
s
)
/
λ
u
(
s
)
d
s
,
the Yosida approximation of T is
(43)
Y
(
λ
,
T
)
u
(
t
)
=
∫
0
t
‍
e
(
t
-
s
)
/
λ
u
(
s
)
d
s
,
and the Möbius approximation of T is
(44)
A
(
λ
,
T
)
u
(
t
)
=
(
1
-
1
λ
)
∫
0
t
‍
e
(
t
-
s
)
/
λ
u
(
s
)
d
s
.
Remark that, for all u
∈
C
[
0
,
L
], we have
(45)
|
T
u
|
L
=
max
⁡
t
∈
[
0
,
L
]
|
T
u
(
t
)
|
≤
max
⁡
t
∈
[
0
,
L
]
∫
0
t
‍
|
u
(
s
)
|
d
s
≤
max
⁡
t
∈
[
0
,
L
]
|
u
(
t
)
|
∫
0
L
‍
d
s
=
|
u
|
L
·
L
.
The above implies that T is a contraction for L
≤
1.IfL
>
1, then we can introduce for each ɛ
>
0 the following norm on C
[
0
,
L
]:
(46)
∥
u
∥
ɛ
:
=
max
⁡
t
∈
[
0
,
L
]
e
t
/
ɛ
|
u
(
t
)
|
,
u
∈
C
[
0
,
L
]
.
Then a simple computation gives that
(47)
∥
T
u
∥
ɛ
<
ɛ
∥
u
∥
ɛ
,
u
∈
C
[
0
,
L
]
.
On the other hand,
(48)
∥
u
∥
ɛ
≤
|
u
|
L
≤
e
L
/
ɛ
∥
u
∥
ɛ
.Remark that, by Theorem11, for all λ
>
1, we get
(49)
|
A
(
λ
,
T
)
|
L
=
(
λ
-
1
)
(
e
T
/
λ
-
1
)
≤
1
if and only if |
T
|
L
≤
1.It is clear that for estimating the powers ofT it seems to be better to use the Yosida approximation or Möbius approximation than the resolvent approximation.



---

*Source: 513162-2014-08-18.xml*